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Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers

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Marcel Schmittfull

The lensing convergence measurable with future CMB experiments will be highly correlated with the clustering of galaxies that will be observed by deep imaging surveys such as LSST. I will discuss prospects for using that cross-correlation signal to constrain inflation models (fnl) and the sum of neutrino masses. A key limitation of such large-scale structure analyses is that dark matter halos and the galaxies within them only form at peaks of the dark matter density distribution, which is challenging to model. I will present recent work on this, where we found that a new basis of operators that account for large bulk flows successfully describes simulated halos at the field level over a surprisingly wide range of scales. Talk given by Marcel Schmittfull at IPMU, University of Tokyo, Oct 31st 2019

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Prospects for CMB lensing—galaxy clustering cross-correlations and modeling biased tracers Marcel Schmittfull
 
 IPMU, October 2019
Cosmic Microwave Background (CMB) Right after Big Bang light scatters frequently —> opaque As Universe expands, turns transparent See surface where light last scattered — 13.6996 bn yrs ago This is the CMB
Cosmic Microwave Background (CMB) Perr<latexit sha1_base64="HbsFOTY+FC07lz5Qau+blG/LzoY=">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</latexit> Hot and cold blobs: Picture of the Universe 13.6996 bn years ago
Gravitational lensing of the CMB 4 Figure 1.2: Illustration of gravitational lensing of CMB photons by large-scale structure (adapted from [10]). gravitationally deflected by galaxies and dark matter CMB
Unlensed$CMB$ 10°$ CMB$Temp.$ 4$T(ˆn)lensed = T(ˆn + d(ˆn))unlensedˆn + d(ˆn))unlensed Slide credit: Blake Sherwin CMB before lensing
Lensed$CMB$ 10°$ CMB$Temp.$ 5$T(ˆn)lensed = T(ˆn + d(ˆn))unlensed CMB after lensing Slide credit: Blake Sherwin
Statistics of the CMB What are the statistics of the CMB before and after lensing?
Statistics of the CMB before lensing Unlensed Lensed Normally distributed as far as we can tell
Statistics of the CMB before lensing Unlensed Lensed Variance
Statistics of the CMB before lensing Unlensed Lensed Variance Power spectrum
Unlensed Lensed Local power spectrum Local power spectrum Local power spectrum is the same in each patch Local power spectrum Statistics of the CMB before lensing
Unlensed Lensed Local power spectrum Local power is magnified or de-magnified Local power spectrum Local power spectrum Local power spectrum Statistics of the CMB after lensing
Unlensed Lensed 0 dz Mmin dM × |˜yi(M, z)| 2 |˜yj(M, z)| 2 , (8) where gν is the spectral function of the SZ effect, V (z) is the comoving volume of the universe integrated to a redshift of zmax = 4, M is the virial mass such that [log10(Mmin), log10(Mmax)] = [11, 16], dn/dM is the FIG. 1: The impact of varying the lensing scaling parameter on the lensed CMB temperature power spectrum, for AL = [0,2,5,10]. Smidt+ (2010) Global power spectrum unlensed lensed Peaks of global power are smeared out Statistics of the CMB after lensing
Unlensed Lensed Local power spectrum Local power spectrum Local power spectrum Local power spectrum Rather than averaging the modulation, measure it as a signal 
 —> magnification map Statistics of the CMB after lensing
Measured lensing magnification Planck Collaboration: Planck 2018 lensing Fig. 1. Mollweide projection in Galactic coordinates of the lensing-deflection reconstruction map from our baseline minimum-p
Cross-correlate with galaxy catalogs Galaxy catalog is sensitive to tails of the distribution of matter (galaxies form at peaks)
 Lensing magnification probes entire distribution (light is deflected by any mass)
 Some models of the Big Bang enhance peaks, so can test them Cross-correlations particularly well suited: Can determine ratio of two perfectly correlated random variables with infinite precision Planck Collaboration: Planck 2018 lensing Fig. 1. Mollweide projection in Galactic coordinates of the lensing-deflection reconstruction map from our baseline minimum- variance (MV) analysis. We show the Wiener-filtered displacement-like scalar field with multipoles ˆ↵MV LM = p L(L + 1)ˆMV LM , corre- sponding to the gradient mode (or E mode) of the lensing deflection angle. Modes with L < 8 have been filtered out. Our baseline lensing reconstruction map is shown in Fig. 1. In Sect. 2 we explain how this was obtained, and the changes compared to our analysis in PL2015. We also describe the new optimal filtering approach used for our best polarization anal- ysis. In Sect. 3 we present our main results, including power- spectrum estimates, cosmological parameter constraints, and a joint estimation of the lensing potential using the CIB. We end the section by using the estimates of the lensing map to delens the CMB, reducing the B-mode polarization power and sharpen- ing the acoustic peaks. In Sect. 4 we describe in detail a number of null and consistency tests, explaining the motivation for our data cuts and the limits of our understanding of the data. We also discuss possible contaminating signals, and assess whether they are potentially important for our results. In Sect. 5 we briefly de- scribe the various data products that are made available to the community, and we end with conclusions in Sect. 6. A series of appendices describe some technical details of the calculation of various biases that are subtracted, and derive the error model for the Monte Carlo estimates. 2. Data and methodology This final Planck lensing analysis is based on the 2018 Planck HFI maps as described in detail in Planck Collaboration III (2018). Our baseline analysis uses the SMICA foreground- cleaned CMB map described in Planck Collaboration IV (2018), and includes both temperature and polarization information. We use the Planck Full Focal Plane (FFP10) simulations, described in detail in Planck Collaboration III (2018), to remove a num- ber of bias terms and correctly normalize the lensing power- spectrum estimates. Our analysis methodology is based on the previous Planck analyses, as described in PL2013 and PL2015. After a summary of the methodology, Sect. 2.1 also lists the changes and improvements with respect to PL2015. Some de- tails of the covariance matrix are discussed in Sect. 2.2, and de- tails of the filtering in Sect. 2.3. The main set of codes applying the quadratic estimators will be made public as part of the CMB lensing toolbox LensIt.2 2.1. Lensing reconstruction The five main steps of the lensing reconstruction are as follows. 1. Filtering of the CMB maps. The observed sky maps are cut by a Galactic mask and have noise, so filtering is applied to remove the mask and approximately optimally weight for the noise. The lensing quadratic estimators use as input optimal Wiener-filtered X = T, E, and B CMB multipoles, as well as inverse-variance-weighted CMB maps. The latter maps can be obtained easily from the Wiener-filtered multipoles by divid- ing by the fiducial CMB power spectra Cfid ` before projecting onto maps. We write the observed temperature T and polariza- tion (written as the spin ±2 combination of Stokes parameters ±2P ⌘ Q ± iU) pixelized data as 0 BBBBBBBBBBBB@ Tdat 2Pdat 2Pdat 1 CCCCCCCCCCCCA = BY 0 BBBBBBBBBBBB@ T E B 1 CCCCCCCCCCCCA + noise, (1) 2 https://github.com/carronj/LensIt 3
Sample variance cancellation Galaxy number density: CMB magnification: Big Bang models are characterized by parameter Planck Collaboration: Planck 2018 lensing Fig. 1. Mollweide projection in Galactic coordinates of the lensing-deflection reconstruction map from our baseline minimum- variance (MV) analysis. We show the Wiener-filtered displacement-like scalar field with multipoles ˆ↵MV LM = p L(L + 1)ˆMV LM , corre- sponding to the gradient mode (or E mode) of the lensing deflection angle. Modes with L < 8 have been filtered out. Our baseline lensing reconstruction map is shown in Fig. 1. In Sect. 2 we explain how this was obtained, and the changes compared to our analysis in PL2015. We also describe the new optimal filtering approach used for our best polarization anal- ysis. In Sect. 3 we present our main results, including power- spectrum estimates, cosmological parameter constraints, and a joint estimation of the lensing potential using the CIB. We end the section by using the estimates of the lensing map to delens the CMB, reducing the B-mode polarization power and sharpen- ing the acoustic peaks. In Sect. 4 we describe in detail a number of null and consistency tests, explaining the motivation for our data cuts and the limits of our understanding of the data. We also discuss possible contaminating signals, and assess whether they are potentially important for our results. In Sect. 5 we briefly de- scribe the various data products that are made available to the community, and we end with conclusions in Sect. 6. A series of appendices describe some technical details of the calculation of various biases that are subtracted, and derive the error model for the Monte Carlo estimates. 2. Data and methodology This final Planck lensing analysis is based on the 2018 Planck HFI maps as described in detail in Planck Collaboration III (2018). Our baseline analysis uses the SMICA foreground- cleaned CMB map described in Planck Collaboration IV (2018), and includes both temperature and polarization information. We use the Planck Full Focal Plane (FFP10) simulations, described in detail in Planck Collaboration III (2018), to remove a num- ber of bias terms and correctly normalize the lensing power- spectrum estimates. Our analysis methodology is based on the previous Planck analyses, as described in PL2013 and PL2015. After a summary of the methodology, Sect. 2.1 also lists the changes and improvements with respect to PL2015. Some de- tails of the covariance matrix are discussed in Sect. 2.2, and de- tails of the filtering in Sect. 2.3. The main set of codes applying the quadratic estimators will be made public as part of the CMB lensing toolbox LensIt.2 2.1. Lensing reconstruction The five main steps of the lensing reconstruction are as follows. 1. Filtering of the CMB maps. The observed sky maps are cut by a Galactic mask and have noise, so filtering is applied to remove the mask and approximately optimally weight for the noise. The lensing quadratic estimators use as input optimal Wiener-filtered X = T, E, and B CMB multipoles, as well as inverse-variance-weighted CMB maps. The latter maps can be obtained easily from the Wiener-filtered multipoles by divid- ing by the fiducial CMB power spectra Cfid ` before projecting onto maps. We write the observed temperature T and polariza- tion (written as the spin ±2 combination of Stokes parameters ±2P ⌘ Q ± iU) pixelized data as 0 BBBBBBBBBBBB@ Tdat 2Pdat 2Pdat 1 CCCCCCCCCCCCA = BY 0 BBBBBBBBBBBB@ T E B 1 CCCCCCCCCCCCA + noise, (1) 2 https://github.com/carronj/LensIt 3  ⇠ N(0, 2 )<latexit sha1_base64="fh2RNmAwU1/7vj3GTJMoGF9vnfw=">AAACInicbVDLSgMxFM3UV62vqks3wSJUkTJTBV24KLhxJRXsAzpjuZOmbWgyMyQZoQzzD36E3+BW1+7EleDGPzF9LGzrgcDhnHO5N8ePOFPatr+szNLyyupadj23sbm1vZPf3aurMJaE1kjIQ9n0QVHOAlrTTHPajCQF4XPa8AfXI7/xSKViYXCvhxH1BPQC1mUEtJHa+RN3AFEE2FVMYFeA7hPgyW1atE+N1BPQngQeysftfMEu2WPgReJMSQFNUW3nf9xOSGJBA004KNVy7Eh7CUjNCKdpzo0VjYAMoEdbhgYgqPKS8Z9SfGSUDu6G0rxA47H6dyIBodRQ+CY5ulrNeyPxP68V6+6ll7AgijUNyGRRN+ZYh3hUEO4wSYnmQ0OASGZuxaQPEog2Nc5s8UVqOnHmG1gk9XLJOSuV784LlatpO1l0gA5RETnoAlXQDaqiGiLoCb2gV/RmPVvv1of1OYlmrOnMPpqB9f0LdyykHw==</latexit> g = (1 + fNL)<latexit sha1_base64="qSZyT7RHnm+puQb5ztG1Up+wy9s=">AAACEnicbVDLSsNAFJ3UV62vqDvdDBahIpSkCrpQKLhxIVLBPqAJYTKdtENnkjAzEUoI+BF+g1tduxO3/oBL/8Rpm4WtHrhwOOde7r3HjxmVyrK+jMLC4tLySnG1tLa+sbllbu+0ZJQITJo4YpHo+EgSRkPSVFQx0okFQdxnpO0Pr8Z++4EISaPwXo1i4nLUD2lAMVJa8sy9PryEFfs48FJHcHh7kx1BZ4jiGEHPLFtVawL4l9g5KYMcDc/8dnoRTjgJFWZIyq5txcpNkVAUM5KVnESSGOEh6pOupiHiRLrp5IcMHmqlB4NI6AoVnKi/J1LEpRxxX3dypAZy3huL/3ndRAXnbkrDOFEkxNNFQcKgiuA4ENijgmDFRpogLKi+FeIBEggrHdvMFp9nOhN7PoG/pFWr2ifV2t1puX6Rp1ME++AAVIANzkAdXIMGaAIMHsEzeAGvxpPxZrwbH9PWgpHP7IIZGJ8/vOGcLw==</latexit> fNL<latexit sha1_base64="J3TCtcachvp5KbIs7TN/zj+HhGo=">AAACAHicbVA9SwNBEJ2LXzF+RS1tFoNgFe6ioIVFwMZCJIL5wCSEvc1esmR379jdE8KRxt9gq7Wd2PpPLP0n7iVXmMQHA4/3ZpiZ50ecaeO6305uZXVtfSO/Wdja3tndK+4fNHQYK0LrJOShavlYU84krRtmOG1FimLhc9r0R9ep33yiSrNQPphxRLsCDyQLGMHGSo9BL+koge5uJ71iyS27U6Bl4mWkBBlqveJPpx+SWFBpCMdatz03Mt0EK8MIp5NCJ9Y0wmSEB7RtqcSC6m4yvXiCTqzSR0GobEmDpurfiQQLrcfCt50Cm6Fe9FLxP68dm+CymzAZxYZKMlsUxByZEKXvoz5TlBg+tgQTxeytiAyxwsTYkOa2+CLNxFtMYJk0KmXvrFy5Py9Vr7J08nAEx3AKHlxAFW6gBnUgIOEFXuHNeXbenQ/nc9aac7KZQ5iD8/UL/Y2XJA==</latexit>
Sample variance cancellation Galaxy number density: CMB magnification: Big Bang models are characterized by parameter Planck Collaboration: Planck 2018 lensing Fig. 1. Mollweide projection in Galactic coordinates of the lensing-deflection reconstruction map from our baseline minimum- variance (MV) analysis. We show the Wiener-filtered displacement-like scalar field with multipoles ˆ↵MV LM = p L(L + 1)ˆMV LM , corre- sponding to the gradient mode (or E mode) of the lensing deflection angle. Modes with L < 8 have been filtered out. Our baseline lensing reconstruction map is shown in Fig. 1. In Sect. 2 we explain how this was obtained, and the changes compared to our analysis in PL2015. We also describe the new optimal filtering approach used for our best polarization anal- ysis. In Sect. 3 we present our main results, including power- spectrum estimates, cosmological parameter constraints, and a joint estimation of the lensing potential using the CIB. We end the section by using the estimates of the lensing map to delens the CMB, reducing the B-mode polarization power and sharpen- ing the acoustic peaks. In Sect. 4 we describe in detail a number of null and consistency tests, explaining the motivation for our data cuts and the limits of our understanding of the data. We also discuss possible contaminating signals, and assess whether they are potentially important for our results. In Sect. 5 we briefly de- scribe the various data products that are made available to the community, and we end with conclusions in Sect. 6. A series of appendices describe some technical details of the calculation of various biases that are subtracted, and derive the error model for the Monte Carlo estimates. 2. Data and methodology This final Planck lensing analysis is based on the 2018 Planck HFI maps as described in detail in Planck Collaboration III (2018). Our baseline analysis uses the SMICA foreground- cleaned CMB map described in Planck Collaboration IV (2018), and includes both temperature and polarization information. We use the Planck Full Focal Plane (FFP10) simulations, described in detail in Planck Collaboration III (2018), to remove a num- ber of bias terms and correctly normalize the lensing power- spectrum estimates. Our analysis methodology is based on the previous Planck analyses, as described in PL2013 and PL2015. After a summary of the methodology, Sect. 2.1 also lists the changes and improvements with respect to PL2015. Some de- tails of the covariance matrix are discussed in Sect. 2.2, and de- tails of the filtering in Sect. 2.3. The main set of codes applying the quadratic estimators will be made public as part of the CMB lensing toolbox LensIt.2 2.1. Lensing reconstruction The five main steps of the lensing reconstruction are as follows. 1. Filtering of the CMB maps. The observed sky maps are cut by a Galactic mask and have noise, so filtering is applied to remove the mask and approximately optimally weight for the noise. The lensing quadratic estimators use as input optimal Wiener-filtered X = T, E, and B CMB multipoles, as well as inverse-variance-weighted CMB maps. The latter maps can be obtained easily from the Wiener-filtered multipoles by divid- ing by the fiducial CMB power spectra Cfid ` before projecting onto maps. We write the observed temperature T and polariza- tion (written as the spin ±2 combination of Stokes parameters ±2P ⌘ Q ± iU) pixelized data as 0 BBBBBBBBBBBB@ Tdat 2Pdat 2Pdat 1 CCCCCCCCCCCCA = BY 0 BBBBBBBBBBBB@ T E B 1 CCCCCCCCCCCCA + noise, (1) 2 https://github.com/carronj/LensIt 3  ⇠ N(0, 2 )<latexit sha1_base64="fh2RNmAwU1/7vj3GTJMoGF9vnfw=">AAACInicbVDLSgMxFM3UV62vqks3wSJUkTJTBV24KLhxJRXsAzpjuZOmbWgyMyQZoQzzD36E3+BW1+7EleDGPzF9LGzrgcDhnHO5N8ePOFPatr+szNLyyupadj23sbm1vZPf3aurMJaE1kjIQ9n0QVHOAlrTTHPajCQF4XPa8AfXI7/xSKViYXCvhxH1BPQC1mUEtJHa+RN3AFEE2FVMYFeA7hPgyW1atE+N1BPQngQeysftfMEu2WPgReJMSQFNUW3nf9xOSGJBA004KNVy7Eh7CUjNCKdpzo0VjYAMoEdbhgYgqPKS8Z9SfGSUDu6G0rxA47H6dyIBodRQ+CY5ulrNeyPxP68V6+6ll7AgijUNyGRRN+ZYh3hUEO4wSYnmQ0OASGZuxaQPEog2Nc5s8UVqOnHmG1gk9XLJOSuV784LlatpO1l0gA5RETnoAlXQDaqiGiLoCb2gV/RmPVvv1of1OYlmrOnMPpqB9f0LdyykHw==</latexit> If we measure only : If we measure and : g<latexit sha1_base64="lhiJBeb4nMmQQxTZcqh1G/By96k=">AAAB93icbVA9SwNBEJ3zM8avqKXNYRCswl0UtLAI2FgmYD4gOcLeZi5Zsrt37O4J4cgvsNXaTmz9OZb+EzfJFSbxwcDjvRlm5oUJZ9p43rezsbm1vbNb2CvuHxweHZdOTls6ThXFJo15rDoh0ciZxKZhhmMnUUhEyLEdjh9mfvsZlWaxfDKTBANBhpJFjBJjpcawXyp7FW8Od534OSlDjnq/9NMbxDQVKA3lROuu7yUmyIgyjHKcFnupxoTQMRli11JJBOogmx86dS+tMnCjWNmSxp2rfycyIrSeiNB2CmJGetWbif953dREd0HGZJIalHSxKEq5a2J39rU7YAqp4RNLCFXM3urSEVGEGpvN0pZQTG0m/moC66RVrfjXlWrjply7z9MpwDlcwBX4cAs1eIQ6NIECwgu8wpszcd6dD+dz0brh5DNnsATn6xdwBpN/</latexit> g<latexit sha1_base64="lhiJBeb4nMmQQxTZcqh1G/By96k=">AAAB93icbVA9SwNBEJ3zM8avqKXNYRCswl0UtLAI2FgmYD4gOcLeZi5Zsrt37O4J4cgvsNXaTmz9OZb+EzfJFSbxwcDjvRlm5oUJZ9p43rezsbm1vbNb2CvuHxweHZdOTls6ThXFJo15rDoh0ciZxKZhhmMnUUhEyLEdjh9mfvsZlWaxfDKTBANBhpJFjBJjpcawXyp7FW8Od534OSlDjnq/9NMbxDQVKA3lROuu7yUmyIgyjHKcFnupxoTQMRli11JJBOogmx86dS+tMnCjWNmSxp2rfycyIrSeiNB2CmJGetWbif953dREd0HGZJIalHSxKEq5a2J39rU7YAqp4RNLCFXM3urSEVGEGpvN0pZQTG0m/moC66RVrfjXlWrjply7z9MpwDlcwBX4cAs1eIQ6NIECwgu8wpszcd6dD+dz0brh5DNnsATn6xdwBpN/</latexit> <latexit sha1_base64="9bJNSyJCqgQXzYZuEwEOn7kVJFo=">AAAB/HicbVA9SwNBEN2LXzF+RS1tFoNgFe6ioIVFwMYygvmA5Ahzm71kze7esrsnhBB/g63WdmLrf7H0n7hJrjCJDwYe780wMy9SnBnr+99ebm19Y3Mrv13Y2d3bPygeHjVMkmpC6yThiW5FYChnktYts5y2lKYgIk6b0fB26jefqDYskQ92pGgooC9ZzAhYJzU6Q1AKusWSX/ZnwKskyEgJZah1iz+dXkJSQaUlHIxpB76y4Ri0ZYTTSaGTGqqADKFP245KENSE49m1E3zmlB6OE+1KWjxT/06MQRgzEpHrFGAHZtmbiv957dTG1+GYSZVaKsl8UZxybBM8fR33mKbE8pEjQDRzt2IyAA3EuoAWtkRi4jIJlhNYJY1KObgoV+4vS9WbLJ08OkGn6BwF6ApV0R2qoToi6BG9oFf05j17796H9zlvzXnZzDFagPf1C1gmlbM=</latexit> var( ˆfNL) ⇠ 2 <latexit sha1_base64="q7aORK2uPQFWV9/VujOi1K0JE98=">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</latexit> ˆfNL = g  1 ) var( ˆfNL) = 0 <latexit sha1_base64="acTMnBQP2Js1IiQWFlfVXRtc34c=">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</latexit> g = (1 + fNL)<latexit sha1_base64="qSZyT7RHnm+puQb5ztG1Up+wy9s=">AAACEnicbVDLSsNAFJ3UV62vqDvdDBahIpSkCrpQKLhxIVLBPqAJYTKdtENnkjAzEUoI+BF+g1tduxO3/oBL/8Rpm4WtHrhwOOde7r3HjxmVyrK+jMLC4tLySnG1tLa+sbllbu+0ZJQITJo4YpHo+EgSRkPSVFQx0okFQdxnpO0Pr8Z++4EISaPwXo1i4nLUD2lAMVJa8sy9PryEFfs48FJHcHh7kx1BZ4jiGEHPLFtVawL4l9g5KYMcDc/8dnoRTjgJFWZIyq5txcpNkVAUM5KVnESSGOEh6pOupiHiRLrp5IcMHmqlB4NI6AoVnKi/J1LEpRxxX3dypAZy3huL/3ndRAXnbkrDOFEkxNNFQcKgiuA4ENijgmDFRpogLKi+FeIBEggrHdvMFp9nOhN7PoG/pFWr2ifV2t1puX6Rp1ME++AAVIANzkAdXIMGaAIMHsEzeAGvxpPxZrwbH9PWgpHP7IIZGJ8/vOGcLw==</latexit> fNL<latexit sha1_base64="J3TCtcachvp5KbIs7TN/zj+HhGo=">AAACAHicbVA9SwNBEJ2LXzF+RS1tFoNgFe6ioIVFwMZCJIL5wCSEvc1esmR379jdE8KRxt9gq7Wd2PpPLP0n7iVXmMQHA4/3ZpiZ50ecaeO6305uZXVtfSO/Wdja3tndK+4fNHQYK0LrJOShavlYU84krRtmOG1FimLhc9r0R9ep33yiSrNQPphxRLsCDyQLGMHGSo9BL+koge5uJ71iyS27U6Bl4mWkBBlqveJPpx+SWFBpCMdatz03Mt0EK8MIp5NCJ9Y0wmSEB7RtqcSC6m4yvXiCTqzSR0GobEmDpurfiQQLrcfCt50Cm6Fe9FLxP68dm+CymzAZxYZKMlsUxByZEKXvoz5TlBg+tgQTxeytiAyxwsTYkOa2+CLNxFtMYJk0KmXvrFy5Py9Vr7J08nAEx3AKHlxAFW6gBnUgIOEFXuHNeXbenQ/nc9aac7KZQ5iD8/UL/Y2XJA==</latexit>
3Sample variance cancellation Planck Collaboration: Planck 2018 lensing Fig. 1. Mollweide projection in Galactic coordinates of the lensing-deflection reconstruction map from our baseline minimum- variance (MV) analysis. We show the Wiener-filtered displacement-like scalar field with multipoles ˆ↵MV LM = p L(L + 1)ˆMV LM , corre- sponding to the gradient mode (or E mode) of the lensing deflection angle. Modes with L < 8 have been filtered out. Our baseline lensing reconstruction map is shown in Fig. 1. In Sect. 2 we explain how this was obtained, and the changes compared to our analysis in PL2015. We also describe the new optimal filtering approach used for our best polarization anal- ysis. In Sect. 3 we present our main results, including power- spectrum estimates, cosmological parameter constraints, and a joint estimation of the lensing potential using the CIB. We end the section by using the estimates of the lensing map to delens the CMB, reducing the B-mode polarization power and sharpen- ing the acoustic peaks. In Sect. 4 we describe in detail a number of null and consistency tests, explaining the motivation for our data cuts and the limits of our understanding of the data. We also discuss possible contaminating signals, and assess whether they are potentially important for our results. In Sect. 5 we briefly de- scribe the various data products that are made available to the community, and we end with conclusions in Sect. 6. A series of appendices describe some technical details of the calculation of various biases that are subtracted, and derive the error model for the Monte Carlo estimates. 2. Data and methodology This final Planck lensing analysis is based on the 2018 Planck HFI maps as described in detail in Planck Collaboration III (2018). Our baseline analysis uses the SMICA foreground- cleaned CMB map described in Planck Collaboration IV (2018), and includes both temperature and polarization information. We use the Planck Full Focal Plane (FFP10) simulations, described in detail in Planck Collaboration III (2018), to remove a num- ber of bias terms and correctly normalize the lensing power- spectrum estimates. Our analysis methodology is based on the previous Planck analyses, as described in PL2013 and PL2015. After a summary of the methodology, Sect. 2.1 also lists the changes and improvements with respect to PL2015. Some de- tails of the covariance matrix are discussed in Sect. 2.2, and de- tails of the filtering in Sect. 2.3. The main set of codes applying the quadratic estimators will be made public as part of the CMB lensing toolbox LensIt.2 2.1. Lensing reconstruction The five main steps of the lensing reconstruction are as follows. 1. Filtering of the CMB maps. The observed sky maps are cut by a Galactic mask and have noise, so filtering is applied to remove the mask and approximately optimally weight for the noise. The lensing quadratic estimators use as input optimal Wiener-filtered X = T, E, and B CMB multipoles, as well as inverse-variance-weighted CMB maps. The latter maps can be obtained easily from the Wiener-filtered multipoles by divid- ing by the fiducial CMB power spectra Cfid ` before projecting onto maps. We write the observed temperature T and polariza- tion (written as the spin ±2 combination of Stokes parameters ±2P ⌘ Q ± iU) pixelized data as 0 BBBBBBBBBBBB@ Tdat 2Pdat 2Pdat 1 CCCCCCCCCCCCA = BY 0 BBBBBBBBBBBB@ T E B 1 CCCCCCCCCCCCA + noise, (1) 2 https://github.com/carronj/LensIt 3 Std. Big BangStd. Big Bang Enhanced tails (Angular scale)-1 Dalal+ (2008), Seljak (2009), McDonald & Seljak (2009), MS & Seljak (2018)
LSST Forecast for future experiments
Forecast for future experiments Compute Fisher information matrix assuming Gaussian likelihood (curvature of the likelihood near maximum = Hessian) Inverse gives lower bound on parameter error bars shot mber er- alize nal- t in bias mple rove and esti- and de- rove e in VII. oss- e 14 III. and wer one-dimensional data vector that starts with all spectra at `min, continues with all spectra at `min + 1, etc: d = d`min , d`min+1, . . . , d`max . (15) At each `, d` = C11 ` , C12 ` , . . . , CNN ` (16) contains N(N + 1)/2 spectra Cij ` with j i. Assum- ing Eq. (14), the covariance cov(d, d) is then a block- diagonal matrix with `max `min + 1 blocks of size N(N + 1)/2 ⇥ N(N + 1)/2, which is easily inverted if the number of fields is N . 100. The Fisher matrix at the power spectrum level is then Fab = `maxX `=`min @d` @✓a [cov(d`, d`)] 1 @d` @✓b . (17) We evaluate this without binning in `.2 for Fisher forecasts with Gaussian covariances, this may be more challenging for actual data analyses. In that case one may want to compress the observations before forming power spectra (see Appendix D). Although many of the cross-spectra have zero signal in the Limber approximation because they Fab = ⌧ @2 ln L(d|✓) @✓a@✓b<latexit sha1_base64="ta6ogSoEc9qH8kFkQwG7srESlKI=">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</latexit>
If models work and systematics under control, what can we hope for? Optimistic setting
0 1 2 3 4 5 6 7 z 10 3 10 2 10 1 1 101 102dn/dz[arcmin2 ] 0 0.2 0.4 0.6 0.8 1.0 W W LSST Optimistic LSST Gold FIG. 1: The redshift distribution of the CMB lensing convergence (red curve, normalized to a unit maximum) and LSST galaxy sam- ples, both Optimistic (light gray) and Gold (dark gray). We as- sume 16 tomographic redshift bins in the range 0 < z < 7, cross- FIG. 2: masses w configura primordi LSST number density Gold sample CMB lensing kernel Optimistic sample At low z, use clustering redshifts (Gorecki+ 2014)
 At high z, add Lyman-break galaxies (dropouts; extrapolated from HSC observations) Total of 66 galaxies per arcmin2 (MS & Seljak 2018)
Lyman-break galaxies (LBGs) at z>3 Photons blueward of 912 Å ionize neutral hydrogen in young star-forming galaxies
 Lyman break at (1+z) 912 Å Blue band ‘dropout’ 0.5 million z=4-7 LBGs found by HSC/Goldrush in 100 deg2
 
 
 Expect ~100 million in LSST Good for CMB lensing Xcorrel Ellis 1998 Ono, Ouchi+ (2018) 1704.06004
di↵erent fsky, maybe? MS: We do not include photo-z errors. Hope this is not too bad because lensing kernel is very broad. Could matter for tracer cross tracer though. Maybe include photo z errors in forecasts? FIG. 3. Redshift distribution and tomographic redshift bins. For CMB lensing (solid black), we plot the dn/dz that would yield C if integrated over, with arbitrary normalization. It at high smaller the low also ha at low ` For a contrib satisfyi ples, wh the cor black li ples are coe cie (see Re combin the CM peak co motivat niques. but is s Dropout galaxies Tomographic redshift bins MS & Seljak (2018)
Power spectra: CMB-S4 & LSST 5 FIG. 4. Left panel: Angular auto-power spectra of CMB-S4 lensing  map (black) and LSST galaxy density g (colored). Solid ines show the signal power (not including lensing noise or shot noise), and shaded regions show 1 error bars assuming the Gaussian covariance (27), fsky = 0.5, minimum variance lensing noise expected for CMB-S4, and LSST number density shown n Fig. 2. Dashed lines show lensing reconstruction noise (black) and shot noise (colored). Right panel: Angular cross-spectra between CMB lensing and LSST galaxy density. MS & Seljak (2018) gg κκ κg noise CXX0 ` = Z z P X X0 (k = `/ (z), z) ⇥ WX (z)bX(z)WX0 (z)bX0 (z). CXX0 ` = 2 ⇡ Z 1 0 dk k X,`(k) X0,`(k)k3 P X X0 (k, z = 0), X,`(k) ⌘ Z 1 0 d ¯WX ( )j`(k ). Low ell: High ell:
SNR of auto-power spectra MS & Seljak (2018) 7 the combined relation coef- be as high as g very broad 100. The rea- ical distances a very low k 0 2 hMpc 1 ). power spec- wer spectrum projection in- lues of and scales smaller the contribu- re important. urces at z > 4 noise reduces variance can- re the tracers `max SNR of CXX 500 1000 2000 CMB 233 406 539 BOSS LRG z=0-0.9 140 187 230 SDSS r < 22 z=0-0.5 247 487 936 SDSS r < 22 z=0.5-0.8 247 487 936 DESI BGS z=0-0.5 230 417 665 DESI ELG z=0.6-0.8 158 210 256 DESI ELG z=0.8-1.7 150 194 225 DESI LRG z=0.6-1.2 184 267 349 DESI QSO z=0.6-1.9 44.8 48.8 50.8 LSST i < 27 (3yr) z=0-0.5 250 496 982 LSST i < 27 (3yr) z=0.5-1 250 496 979 LSST i < 27 (3yr) z=1-2 249 492 956 LSST i < 27 (3yr) z=2-3 245 469 830 LSST i < 27 (3yr) z=3-4 239 444 724 LSST i < 27 (3yr) z=4-7 224 387 555 TABLE I. Total signal-to-noise of auto-power spectra CXX `
SNR of kg cross-power spectra Gaussian covariance (14), fsky = 0.5, minimum variance lensing noise expected in Fig. 4. Dashed lines show lensing reconstruction noise (black) and shot nois between CMB lensing and LSST galaxy density. `max SNR of CCMBX 500 1000 2000 BOSS LRG z=0-0.9 77.3 117 159 SDSS r < 22 z=0-0.5 88.3 167 284 SDSS r < 22 z=0.5-0.8 88.3 167 284 DESI BGS z=0-0.5 50.1 93.5 144 DESI ELG z=0.6-0.8 50.7 73.5 97 DESI ELG z=0.8-1.7 103 148 185 DESI LRG z=0.6-1.2 86.7 133 182 DESI QSO z=0.6-1.9 74.9 94.5 108 LSST i < 27 (3yr) z=0-0.5 78.1 150 258 LSST i < 27 (3yr) z=0.5-1 112 202 338 LSST i < 27 (3yr) z=1-2 144 259 406 LSST i < 27 (3yr) z=2-3 121 219 324 LSST i < 27 (3yr) z=3-4 101 182 261 LSST i < 27 (3yr) z=4-7 94 167 229 TABLE II. Like Table I but for CMB-lensing–clustering cross- spectra Cg ` . For galaxy by more than z & 2). At ` but less than of Fig. 7 com ance of each ments or exp shows that t 0.1 per mo lower-redshif ample, the f 0.05 per mo The g cro nal than gg a quence of com than with th signal-to-nois sponding gg the cross-cor and each ind Fig. 5 above,MS & Seljak (2018)
Correlation of CMB lensing and galaxies FIG. 3. Correlation r = Cg (C Cgg ) 1/2 of CMB-S4 lensing A. Local primordia 1. Motivation to The large-scale structure of primordial density fluctuation universe. Measuring the stat scale structures using galaxy can thus constrain the physics fluctuations. A non-Gaussian tribution function (pdf) can o inflation models, involving for higher-derivative operators. local type of primordial non- the primordial potential is th sian field and its square, (x)+ has a non-Gaussian pdf. If w amplitude, fNL & 1, it will r of the inflationary expansion robust way [25, 26]. This is o tional means to rule out a who early-universe models. In practice the measurement threshold signal fNL = 1 sepa and multi-field models has a v ables. The best constraints, f Planck’s CMB temperature a ments [27]. does not improve our forecasts because we cros against CMB lensing, so that redshift accurac less important than number density. We assu of b(z) = 1.7 ¯D 1 (z) where ¯D(z = 0) = 1. For DESI, we use five redshift samples, wit densities from Table 2.3 in Ref. [47]: The lo BGS sample at 0  z  0.5 with 9.6 ⇥ 106 o bias b(z) = 1.34 ¯D 1 (z), the LRG sample at 0.6 with 3.9 ⇥ 106 objects and bias b(z) = 1.7 ¯D ELG sample at 0.6  z  0.8 with 3.5 ⇥ 1 and bias b(z) = 0.84 ¯D 1 (z), a second ELG 0.8  z  1.7 with 1.3 ⇥ 107 objects and the and a QSO sample at 0.6  z  1.9 with 1.4⇥1 and bias 1.2 ¯D 1 (z). In each case, the number refers to a survey area of 14, 000 deg2 . D. CMB lensing–LSS correlation coe The performance of the cross-correlation an pends on the cross-correlation coe cient r` = Cg ` q ˆC ` ˆCgg ` between the measured CMB lensing converge the observed galaxy density g, where the pow ˆC include lensing reconstruction noise and s CMB lensing and galaxy maps are up to 95% correlated MS & Seljak (2018)
MS & Seljak (2018); Forecast papers for Simons Observatory, CMB-S4, NASA PICO satellite (Largest angular scale)-1 —> Potential to rule out all Big Bang models driven by a single field Forecast for local non-Gaussianity 12 FIG. 9. MS: Date: 20 Aug 2017 Constraints on primordial non-Gaussianity amplitude fNL as a function of minimum wavenumber `min, for di↵erent LSS surveys (colors), with full sky overlap between observations Threshold needed to distinguish Big Bang models Forecastfor Different experiments `min ⇠<latexit sha1_base64="JpqnNPjmjQoSX2eihOhMzZtQsZ0=">AAACDXicbVDLSsNAFJ3UV62vqLhyM1gEVyWpgi5cFNy4rGAf0IQwmU7aoTOTMDMRSsg3+A1ude1O3PoNLv0TJ20WtvXAhcM593IPJ0wYVdpxvq3K2vrG5lZ1u7azu7d/YB8edVWcSkw6OGax7IdIEUYF6WiqGeknkiAeMtILJ3eF33siUtFYPOppQnyORoJGFCNtpMA+8QhjgceRHkuecSpyT1Ee2HWn4cwAV4lbkjoo0Q7sH28Y45QToTFDSg1cJ9F+hqSmmJG85qWKJAhP0IgMDBWIE+Vns/g5PDfKEEaxNCM0nKl/LzLElZry0GwWOdWyV4j/eYNURzd+RkWSaiLw/FGUMqhjWHQBh1QSrNnUEIQlNVkhHiOJsDaNLXwJeW46cZcbWCXdZsO9bDQfruqt27KdKjgFZ+ACuOAatMA9aIMOwCADL+AVvFnP1rv1YX3OVytWeXMMFmB9/QJGU5zP</latexit> With/without sample variance cancellation (fNL)<latexitsha1_base64="SjRPaUwNSZsk4HkulZr/qbH5CbU=">AAACCnicbVDLSsNAFJ34rPUV69LNYBHqpiRV0IWLghsXIhXsA5oQJtNJO3RmEmYmYgn9A7/Bra7diVt/wqV/4rTNwrYeuHA4517O5YQJo0o7zre1srq2vrFZ2Cpu7+zu7dsHpZaKU4lJE8cslp0QKcKoIE1NNSOdRBLEQ0ba4fB64rcfiVQ0Fg96lBCfo76gEcVIGymwS56ifY4qUZB5ksO72/FpYJedqjMFXCZuTsogRyOwf7xejFNOhMYMKdV1nUT7GZKaYkbGRS9VJEF4iPqka6hAnCg/m/4+hidG6cEolmaEhlP170WGuFIjHppNjvRALXoT8T+vm+ro0s+oSFJNBJ4FRSmDOoaTImCPSoI1GxmCsKTmV4gHSCKsTV1zKSEfm07cxQaWSatWdc+qtfvzcv0qb6cAjsAxqAAXXIA6uAEN0AQYPIEX8ArerGfr3fqwPmerK1Z+cwjmYH39AhnnmmM=</latexit>
Growth of structure as function of time Expansion history / geometry / dark energy Sum of neutrino masses Galaxy formation? Other science from cross-correlations
30 Byeonghee Yu (PhD student @ UC Berkeley) Example: Sum of neutrino masses
Yu, Knight et al. (arXiv:1809.02120) 0 1 2 3 4 5 6 7 z 0.98 0.99 1.0 1.01 8(z)/8(z),fiducial a 3f /5 Marginalize over CDM parameters and Bi; S4 + DESI added Fixing CDM parameters FIG. 4: 1 constraints on the matter amplitude 8 in 6 tomo- graphic redshift bins, z = 0 0.5, 0.5 1, 1 2, 2 3, 3 4, 4 7, 60 meV neutrinos Massless neutrinos Example: Sum of neutrino masses Varianceoffluctuations (Size of the Universe)-1 - 1 Fix other cosmological params Marginalize over other cosmological params
Uses z<7 redshift lever arm Measures neutrino mass without optical depth to CMB Independent & competitive to usual probes Yu, Knight, Sherwin+ (1809.02120) 3 0.2 1 2 3 4 5 6 7 zmax 20 30 40 50 100 ( P m)[meV] S4Lens + LSST (Optimistic) S4Lens + LSST (Optimistic) + Planck(l > 30) S4Lens + LSST (Optimistic) + S4(l > 30) Planck/S4(l > 30) + DESI S4Lens + Planck/S4(l > 30) + DESI S4Lens + LSST (Gold) + Planck/S4(l > 30) + DESI S4Lens + LSST (Optimistic) +Planck/S4(l > 30) + DESI FIG. 2: Forecasted 1 constraints on the sum of the neutrino 100 30 20 50 3 0.2 1 2 3 4 5 6 7 zmax 20 30 40 50 100 ( P m)[meV] S4Lens + LSST (Optimistic) S4Lens + LSST (Optimistic) + Planck(l > 30) S4Lens + LSST (Optimistic) + S4(l > 30) Planck/S4(l > 30) + DESI S4Lens + Planck/S4(l > 30) + DESI S4Lens + LSST (Gold) + Planck/S4(l > 30) + DESI S4Lens + LSST (Optimistic) +Planck/S4(l > 30) + DESI FIG. 2: Forecasted 1 constraints on the sum of the neutrino masses without optical depth information, for di↵erent experiment configurations: CMB-S4 lensing and LSST clustering (black) + primordial CMB data (green dotted for Planck and green solid for Example: Sum of neutrino masses
33 Sensitive to kmax, but not too much when all data combined ( P m⌫ ) [meV] (Gold/Optimistic) kmax Lens + LSST + Planck/S4 T&P + DESI 0.05 307 / 243 94 / 68 32 / 29 0.1 176 / 129 68 / 53 31 / 27 0.2 107 / 71 47 / 38 28 / 25 0.3 84 / 55 40 / 33 27 / 24 0.4 79 / 49 38 / 31 26 / 23 TABLE I: Forecasts of the neutrino mass constraints without op- tical depth information, for di↵erent LSST number densities and redshift distributions, kmax limits, and lensing reconstruction noise levels. For di↵erent combinations of data, constraints provided on the left assume the LSST Gold sample, and those on the right assume the LSST Optimistic sample. spectra is given by n Len kmax (⌧) = 0.3 25 ( P Lens + kmax (⌧) = 0.3 22 / TABLE II: Forecasts of t flat priors on the optica S4 lensing, S4 primary BAO information. Botto numbers on the left assu assume the Optimistic s Yu, Knight, Sherwin+ (1809.02120) Ignores scale-dependent bias due to neutrinos LoVerde 2014, LoVerde 2016, Muñoz & Dvorkin 2018, MS & Seljak 2018, Chiang, LoVerde & Villaescusa-Navarro 2019 Example: Sum of neutrino masses
Modeling all power spectra for nonlinear scales (high L) Photometric redshift errors Relationship between galaxies and dark matter (galaxy bias) Challenges for growth measurements
Other cool things to do with CMB lensing (1) Joint analysis of CMB and CMB magnification, with covariance (2) Estimate unlensed CMB
 
 
 
 (3) Biases of the magnification estimator Figure 1.2: Illustration of gravitational lensing of CMB photons by large-scale structure (adapted from [10]). reconstructing these lenses from the observed CMB we can obtain crucial infor- mation on e.g. dark energy or the geometry of the universe. Several experiments will provide high quality CMB data in the next few years, e.g. full-mission Planck data including polarization, ACT/ACTPol, Polarbear and SPT/SPTpol. The contribution of this thesis, presented in Chapter 5, is a thorough analysis of how the reconstructed lensing information can be combined with the primary CMB data to perform reliably a joint parameter estimation. Such joint analy- ses are important to break degeneracies that limit the information that can be extracted from the CMB fluctuations laid down at recombination. The joint anal- ysis is non-trivial because the part of the lensing information that is present in the primary CMB power spectrum as well as in the lensing reconstruction is po- tentially double-counted. We quantify the temperature-lensing cross-correlation analytically, finding two physical contributions and confirming the results with simulated lensed CMB maps. This cross-correlation has not been considered be- fore and it could have turned out to be anywhere between zero and unity. We 15 (a) Theoretical matter cosmic variance contribution (D9) (b) Simulations FIG. 5. (a) Theoretical matter cosmic variance contribution (D9) to the correlation of the unbinned power spectra of recon- structed lensing potential and lensed temperature. The covariance (D9) is converted to a correlation using the same conversion factor as in (38). (b) Measured correlation ˆcorrel( ˆC ˆin ˆin L , ˆC ˜T ˜T LT ˆCT T LT ) of the input lensing potential power and the di↵erence of noise-free lensed and unlensed temperature powers in 1000 simulations. FIG. 6. Left: The approximate contribution to the covariance between [L(L + 1)]2 ˆC ˆˆ L /(2⇡) and L0 (L0 + 1) ˆC ˜T ˜T L0 /(2⇡) from cosmic variance of the lenses, derived from Eq. (D9). Right: The rank-one approximation to the matrix on the left from retaining only the largest singular value.TODOO: discuss this in main text or appendix CT correlations in all calculations we try to eliminate correlations between the lensing potential and the unlensed temperature in the simulations by considering ˆcov( ˆC ˆˆ L , ˆC ˜T ˜T L0 )|CV( ) = ˆcov( ˆC ˆin ˆin L , ˆC ˜T ˜T L0 ˆCT T L0 ), (52) where ˆC ˆin ˆin is the empirical power of the input lensing potential and ˆCT T is the empirical power of the unlensed temperature. Subtracting the unlensed from the lensed empirical power spectrum also reduces the noise of the covariance estimate because it eliminates the scatter due to cosmic variance of the unlensed temperature. We estimate the covariance in simulations similarly to (48). As shown in Fig. 5b these measurements agree with the theoretical expectation from (D9). `0 <latexit sha1_base64="X1y07k9ZmLbjfpS1W4qyfTp9Ng0=">AAAB+3icbVA9SwNBEJ3zM8avqKXNYhCtwl0UtLAI2FhG8JJAcoS9zV6yZHfv2N0TwnG/wVZrO7H1x1j6T9wkV5jEBwOP92aYmRcmnGnjut/O2vrG5tZ2aae8u7d/cFg5Om7pOFWE+iTmseqEWFPOJPUNM5x2EkWxCDlth+P7qd9+pkqzWD6ZSUIDgYeSRYxgYyW/Rzm/6Feqbs2dAa0SryBVKNDsV356g5ikgkpDONa667mJCTKsDCOc5uVeqmmCyRgPaddSiQXVQTY7NkfnVhmgKFa2pEEz9e9EhoXWExHaToHNSC97U/E/r5ua6DbImExSQyWZL4pSjkyMpp+jAVOUGD6xBBPF7K2IjLDCxNh8FraEIreZeMsJrJJWveZd1eqP19XGXZFOCU7hDC7BgxtowAM0wQcCDF7gFd6c3Hl3PpzPeeuaU8ycwAKcr18nCZUA</latexit> `<latexitsha1_base64="bfqfNP2WZfAkR6k1rfkz29U6stI=">AAAB+nicbVA9SwNBEJ3zM8avqKXNYhCswl0UtLAI2FhGMB+QHGFvM0mW7O0du3tCOPMXbLW2E1v/jKX/xL3kCpP4YODx3gwz84JYcG1c99tZW9/Y3Nou7BR39/YPDktHx00dJYphg0UiUu2AahRcYsNwI7AdK6RhILAVjO8yv/WESvNIPppJjH5Ih5IPOKMmk7ooRK9UdivuDGSVeDkpQ456r/TT7UcsCVEaJqjWHc+NjZ9SZTgTOC12E40xZWM6xI6lkoao/XR265ScW6VPBpGyJQ2ZqX8nUhpqPQkD2xlSM9LLXib+53USM7jxUy7jxKBk80WDRBATkexx0ucKmRETSyhT3N5K2IgqyoyNZ2FLEE5tJt5yAqukWa14l5Xqw1W5dpunU4BTOIML8OAaanAPdWgAgxG8wCu8Oc/Ou/PhfM5b15x85gQW4Hz9AsMDlM8=</latexit> 15 (a) Theoretical matter cosmic variance contribution (D9) (b) Simulations FIG. 5. (a) Theoretical matter cosmic variance contribution (D9) to the correlation of the unbinned power spectra of recon- structed lensing potential and lensed temperature. The covariance (D9) is converted to a correlation using the same conversion factor as in (38). (b) Measured correlation ˆcorrel( ˆC ˆin ˆin L , ˆC ˜T ˜T LT ˆCT T LT ) of the input lensing potential power and the di↵erence of noise-free lensed and unlensed temperature powers in 1000 simulations. FIG. 6. Left: The approximate contribution to the covariance between [L(L + 1)]2 ˆC ˆˆ L /(2⇡) and L0 (L0 + 1) ˆC ˜T ˜T L0 /(2⇡) from cosmic variance of the lenses, derived from Eq. (D9). Right: The rank-one approximation to the matrix on the left from retaining only the largest singular value.TODOO: discuss this in main text or appendix CT correlations in all calculations we try to eliminate correlations between the lensing potential and the unlensed temperature in the simulations by considering ˆcov( ˆC ˆˆ L , ˆC ˜T ˜T L0 )|CV( ) = ˆcov( ˆC ˆin ˆin L , ˆC ˜T ˜T L0 ˆCT T L0 ), (52) where ˆC ˆin ˆin is the empirical power of the input lensing potential and ˆCT T is the empirical power of the unlensed temperature. Subtracting the unlensed from the lensed empirical power spectrum also reduces the noise of the covariance estimate because it eliminates the scatter due to cosmic variance of the unlensed temperature. We estimate the covariance in simulations similarly to (48). As shown in Fig. 5b these measurements agree with the theoretical expectation from (D9). `0 <latexit sha1_base64="X1y07k9ZmLbjfpS1W4qyfTp9Ng0=">AAAB+3icbVA9SwNBEJ3zM8avqKXNYhCtwl0UtLAI2FhG8JJAcoS9zV6yZHfv2N0TwnG/wVZrO7H1x1j6T9wkV5jEBwOP92aYmRcmnGnjut/O2vrG5tZ2aae8u7d/cFg5Om7pOFWE+iTmseqEWFPOJPUNM5x2EkWxCDlth+P7qd9+pkqzWD6ZSUIDgYeSRYxgYyW/Rzm/6Feqbs2dAa0SryBVKNDsV356g5ikgkpDONa667mJCTKsDCOc5uVeqmmCyRgPaddSiQXVQTY7NkfnVhmgKFa2pEEz9e9EhoXWExHaToHNSC97U/E/r5ua6DbImExSQyWZL4pSjkyMpp+jAVOUGD6xBBPF7K2IjLDCxNh8FraEIreZeMsJrJJWveZd1eqP19XGXZFOCU7hDC7BgxtowAM0wQcCDF7gFd6c3Hl3PpzPeeuaU8ycwAKcr18nCZUA</latexit> MS, Challinor et al. (2013) Peloton, MS, et al. (2017) Planck collab. (2013, 2018) e.g. Sherwin & MS (2015) Böhm, MS & Sherwin (2016) Beck, Fabbian & Errard (2018) Böhm, Sherwin et al. (2018) Planck Collaboration: Gravitational lensing by large-scale structures with Planck 0.12 0.08 0.04 0.00 0.04 K 40 48 56 64 72 80 H0 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 Fig. 15. Two views of the geometric degeneracy in curved ⇤CDM models which is partially broken by lensing. Left: the degeneracy in the ⌦m-⌦⇤ plane, with samples from Planck+WP+highL colour coded by the value of H0. The contours delimit the 68% and 95% confidence regions, showing the further improvement from including the lensing likelihood. Right: the degeneracy in the ⌦K-H0 plane, with samples colour coded by ⌦⇤. Spatially-flat models lie along the grey dashed lines. 6.2. Correlation with the ISW Effect As CMB photons travel to us from the last-scattering surface, the gravitational potentials they traverse may undergo a non- negligible amount of evolution. This produces a net redshift or blueshift of the photons concerned, as they fall into and then escape from the evolving potentials. The overall result is a con- tribution to the CMB temperature anisotropy known as the late- time ISW e↵ect, or the Rees-Sciama (R-S) e↵ect depending on whether the evolution of the potentials concerned is in the linear (ISW) or non-linear (R-S) regime of structure formation (Sachs & Wolfe 1967; Rees & Sciama 1968). In the epoch of dark en- ergy domination, which occurs after z ⇠ 0.5 for the concor- dance ⇤CDM cosmology, large-scale potentials tend to decay over time as space expands, resulting in a net blueshifting of the CMB photons which traverse these potentials. In the concordance ⇤CDM model, there is significant over- lap between the large-scale structure which sources the CMB lensing potential and the ISW e↵ect (greater than 90% at L < 100), although it should be kept in mind that we cannot observe the ISW component by itself, and so the e↵ective cor- relation with the total CMB temperature is much smaller, on the order of 20%. The correlation between the lensing potential and the ISW e↵ect results in a non-zero bispectrum or three-point function for the observed CMB fluctuations. This bispectrum is peaked for “squeezed” configurations, in which one short leg at low-` Following Lewis et al. (2011), we begin with an estimator for the cross-spectrum of the lensing potential and the ISW e↵ect as ˆCT L = f 1 sky 2L + 1 X M ˆTLM ˆ⇤ LM, (45) where ˆT`m = CTT ` ¯T`m is the Wiener-filtered temperature map and ˆ is given in Eq. (13). In Fig. 16 we plot the measured cross- spectra for our individual frequency reconstructions at 100, 143, and 217 GHz as well as the MV reconstruction. We also plot the mean and scatter expected in the fiducial ⇤CDM model. To compare quantitatively the overall level of the measured CT L correlation to the value in ⇤CDM, we estimate an overall amplitude for the cross-spectrum as ˆAT = NT LmaxX L=Lmin (2L + 1)CT ,fid. L ˆCT L /(CTT L NL ). (46) The overall normalization NT is determined from Monte-Carlo simulations. For our processing of the data, we find that it is well approximated (at the 5% level) by the analytical approximation NT ⇡ 2 6666664 LmaxX L=Lmin (2L + 1) ⇣ CT ,fid. L ⌘2 /(CTT L NL ) 3 7777775 1 . (47) Curvature of space Expansionrate Amountofdarkenergy Joint analysis
Science with galaxy catalogs (1) Bias model at the field level (2) Cosmological parameter analysis (3) Accounting for skewness (4) Getting initial from final conditions 13 Figure 1. 2-d slices of the overdensity h(x) of simulated 1010.8 1011.8 h 1 M halos (top), compared with the cubic bias model (center), and the linear Standard Eulerian bias model (bottom). Each panel is 500 h 1 Mpc wide and 110 h 1 Mpc high, and each density is smoothed with a R = 2 h 1 Mpc Gaussian, WR(k) = exp[ (kR)2 /2]. The colorbar indicates the values of this smoothed overdensity h(x). Figure 2. Same as Fig. 1 but for more massive and less abundant 1011.8 1012.8 h 1 M halos. simulations. The variance, skewness and kurtosis of the densities shown in the histograms are listed in Table II for the full simulated and modeled densities, and in Table III for the model error. The linear Standard Eulerian bias model tends to underpredict troughs and overpredict peaks of the halo density, as shown in Fig. 3. The model error is not Gaussian for any of the shown smoothing scales; in particular its kurtosis is larger than 1 for all smoothing scales. The cubic model provides a more accurate description of the halo density pdf, as shown in Fig. 4. This emphasizes the importance of using nonlinear bias terms even on rather large scales. Still, the cubic model tends to underpredict the peaks of the true halo density, especially on small scales. This agrees with Fig. 2 where the model also underpredicts
Science with galaxy catalogs (1) Bias model at the field level (2) Cosmological parameter analysis (3) Accounting for skewness (4) Getting initial from final conditions 13 Figure 1. 2-d slices of the overdensity h(x) of simulated 1010.8 1011.8 h 1 M halos (top), compared with the cubic bias model (center), and the linear Standard Eulerian bias model (bottom). Each panel is 500 h 1 Mpc wide and 110 h 1 Mpc high, and each density is smoothed with a R = 2 h 1 Mpc Gaussian, WR(k) = exp[ (kR)2 /2]. The colorbar indicates the values of this smoothed overdensity h(x). Figure 2. Same as Fig. 1 but for more massive and less abundant 1011.8 1012.8 h 1 M halos. simulations. The variance, skewness and kurtosis of the densities shown in the histograms are listed in Table II for the full simulated and modeled densities, and in Table III for the model error. The linear Standard Eulerian bias model tends to underpredict troughs and overpredict peaks of the halo density, as shown in Fig. 3. The model error is not Gaussian for any of the shown smoothing scales; in particular its kurtosis is larger than 1 for all smoothing scales. The cubic model provides a more accurate description of the halo density pdf, as shown in Fig. 4. This emphasizes the importance of using nonlinear bias terms even on rather large scales. Still, the cubic model tends to underpredict the peaks of the true halo density, especially on small scales. This agrees with Fig. 2 where the model also underpredicts
Overview We calculate halo density field in PT and compare to simulations 1. How well does perturbative bias expansion work? 2. How correlated is the halo density field with the initial conditions? 3. What are the properties of the noise? Joint work with Marko Simonović, Valentin Assassi & Matias Zaldarriaga
Overview Most of the analyses use n-point functions. Disadvantages: These questions have been extensively explored in the past Desjacques, Jeong, Schmidt: Large-Scale Galaxy Bias — Cosmic variance, compromise on resolution/size of the box — At high k hard to disentangle different sources of nonlinearities — Overfitting (smooth curves, many parameters) — Only a few lowest n-point functions explored in practice — Difficult to isolate and study the noise
Overview Advantages: Use fields rather than summary statistics — No cosmic variance, small boxes with high resolution are sufficient — High S/N at low k, no need to go to the very nonlinear regime — No overfitting, each Fourier mode (amplitude and phase) is fitted — “All” n-point functions measured simultaneously — Easier to isolate and study the noise Roth & Porciani (2011) Baldauf, Schaan, Zaldarriaga (2015) Lazeyras, Schmit (2017) Abidi, Baldauf (2018) McQuinn, D’Aloisio (2018) Taruya, Nishimichi, Jeong (2018)
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First predict dark matter (purple), then halos (yellow) Modeling halo number density
Simple peak model Whenever dark matter density > threshold, a halo forms halo formation in peaks Gaussian fluctuations on various scales (described by the power spectrum) Peak-Background Split • Schematic Picture: 3 2 1 0 x δc Large Scale "Background" Enhanced "Peaks" →→first sites of halo formation W. Hu first sites of snowfall → → Halo forms
Simple peak model Whenever dark matter density > threshold, a halo forms halo formation in peaks Gaussian fluctuations on various scales (described by the power spectrum) Peak-Background Split • Schematic Picture: 3 2 1 0 x δc Large Scale "Background" Enhanced "Peaks" →→first sites of halo formation W. Hu first sites of snowfall → → Halo forms halo formation in peaks Gaussian fluctuations on various scales (described by the power spectrum) Peak-Background Split • Schematic Picture: 3 2 1 0 x δc Large Scale "Background" Enhanced "Peaks" →→first sites of halo formation W. Hu first sites of snowfall → → Snow falls
Simple peak model Whenever dark matter density > threshold, a halo forms halo formation in peaks Gaussian fluctuations on various scales (described by the power spectrum) Peak-Background Split • Schematic Picture: 3 2 1 0 x δc Large Scale "Background" Enhanced "Peaks" →→first sites of halo formation W. Hu first sites of snowfall → → Halo forms halo formation in peaks Gaussian fluctuations on various scales (described by the power spectrum) Peak-Background Split • Schematic Picture: 3 2 1 0 x δc Large Scale "Background" Enhanced "Peaks" →→first sites of halo formation W. Hu first sites of snowfall → → Snow falls If we zoom out, halos trace the large scale background that modulates the small-scale fluctuations and increases P(>threshold).
using fractional deviations from the mean and proportionality constant (related to how likely a halo forms when changing the threshold density) Simple peak model Tracing the large scale background means halo number density is proportional to matter density, if both smoothed on large scales:
 Write this as g(x) = b1 m(x)<latexit sha1_base64="XaHcr7zAchcoHCuF4bu18V4pL3I=">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</latexit> g(x) ⌘ ng(x) hngi 1 <latexit sha1_base64="zPWmnTIRWzX9q862OsLLcLaQHNU=">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</latexit> m(x) ⌘ ⇢m(x) h⇢mi 1 <latexit sha1_base64="tVCTXJMrljgnXrmyqQ9zwu1iRlI=">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</latexit> ng(x) / ⇢m(x)<latexit sha1_base64="99dFa20NV5ubQOIpaYS4NMUSMrw=">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</latexit> b1<latexit sha1_base64="vFL/SH1AeDkXA67O23grfojqbUQ=">AAAB+XicbVA9SwNBEJ31M8avqKXNYhCswl0UtLAI2FhGNB+QHGFvs5cs2d07dveEcOQn2GptJ7b+Gkv/iZvkCpP4YODx3gwz88JEcGM97xutrW9sbm0Xdoq7e/sHh6Wj46aJU01Zg8Yi1u2QGCa4Yg3LrWDtRDMiQ8Fa4ehu6reemTY8Vk92nLBAkoHiEafEOukx7Pm9UtmreDPgVeLnpAw56r3ST7cf01QyZakgxnR8L7FBRrTlVLBJsZsalhA6IgPWcVQRyUyQzU6d4HOn9HEUa1fK4pn6dyIj0pixDF2nJHZolr2p+J/XSW10E2RcJallis4XRanANsbTv3Gfa0atGDtCqObuVkyHRBNqXToLW0I5cZn4ywmskma14l9Wqg9X5dptnk4BTuEMLsCHa6jBPdShARQG8AKv8IYy9I4+0Oe8dQ3lMyewAPT1C5XxlB4=</latexit>
using fractional deviations from the mean and proportionality constant (related to how likely a halo forms when changing the threshold density) Simple peak model Tracing the large scale background means halo number density is proportional to matter density, if both smoothed on large scales:
 Write this as g(x) = b1 m(x)<latexit sha1_base64="XaHcr7zAchcoHCuF4bu18V4pL3I=">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</latexit> g(x) ⌘ ng(x) hngi 1 <latexit sha1_base64="zPWmnTIRWzX9q862OsLLcLaQHNU=">AAACT3icbZA/TyMxEMW9AQ7IcUeAksYiQuKKi3Y5JK6gQKK5EiRCkOIomvXOBgvbu2d7c0Sr/Wh8CErKEy3UdAgnbMG/kSw9vTfjsX9xLoV1YXgbNObmF74sLi03v658+77aWls/s1lhOHZ5JjNzHoNFKTR2nXASz3ODoGKJvfjyaJr3xmisyPSpm+Q4UDDSIhUcnLeGrR7T+I9nSoFOSja+qkqmwF3EaXlVVU2WoHQwHO345Adl+LcQY8pSA7zUtesHJOiRROodZmay+hkNW+2wE86KfhRRLdqkruNh6z9LMl4o1I5LsLYfhbkblGCc4P7GJiss5sAvYYR9LzUotINyBqCi295JaJoZf7SjM/f1RAnK2omKfef0d/Z9NjU/y/qFS38PSqHzwqHmL4vSQlKX0SlNmgiD3MmJF8CN8G+l/AI8H+eZv9kSq8ozid4T+CjOdjvRr87uyV778KCms0Q2yRbZIRHZJ4fkDzkmXcLJNbkj9+QhuAkeg6dG3doIarFB3lRj+Rl0UbZI</latexit> m(x) ⌘ ⇢m(x) h⇢mi 1 <latexit sha1_base64="tVCTXJMrljgnXrmyqQ9zwu1iRlI=">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</latexit> ng(x) / ⇢m(x)<latexit sha1_base64="99dFa20NV5ubQOIpaYS4NMUSMrw=">AAACMXicbVDLSsNAFJ34tr6iLt0MFqFuSqKCCi4ENy4r2FpoQphMJ+3gPMLMRC0h3+JH+A1udd2duvUnnLZZWOuBC4dz7uXee+KUUW08b+jMzS8sLi2vrFbW1jc2t9ztnZaWmcKkiSWTqh0jTRgVpGmoYaSdKoJ4zMhdfH818u8eiNJUilszSEnIUU/QhGJkrBS554Egj1hyjkQ3Dx6eijzgyPTjJH8qioqIejUrHsIgVTI1EgaqLyM+1iK36tW9MeAs8UtSBSUakfsZdCXOOBEGM6R1x/dSE+ZIGYoZKSpBpkmK8D3qkY6lAnGiw3z8YgEPrNKFiVS2hIFj9fdEjrjWAx7bztH9+q83Ev/zOplJzsKcijQzRODJoiRj0D47ygt2qSLYsIElCCtqb4W4jxTCxqY6tSXmhc3E/5vALGkd1f3j+tHNSfXyokxnBeyBfVADPjgFl+AaNEATYPAMXsEbeHdenKHz4XxNWueccmYXTMH5/gHLg6ub</latexit> b1<latexit sha1_base64="vFL/SH1AeDkXA67O23grfojqbUQ=">AAAB+XicbVA9SwNBEJ31M8avqKXNYhCswl0UtLAI2FhGNB+QHGFvs5cs2d07dveEcOQn2GptJ7b+Gkv/iZvkCpP4YODx3gwz88JEcGM97xutrW9sbm0Xdoq7e/sHh6Wj46aJU01Zg8Yi1u2QGCa4Yg3LrWDtRDMiQ8Fa4ehu6reemTY8Vk92nLBAkoHiEafEOukx7Pm9UtmreDPgVeLnpAw56r3ST7cf01QyZakgxnR8L7FBRrTlVLBJsZsalhA6IgPWcVQRyUyQzU6d4HOn9HEUa1fK4pn6dyIj0pixDF2nJHZolr2p+J/XSW10E2RcJallis4XRanANsbTv3Gfa0atGDtCqObuVkyHRBNqXToLW0I5cZn4ywmskma14l9Wqg9X5dptnk4BTuEMLsCHa6jBPdShARQG8AKv8IYy9I4+0Oe8dQ3lMyewAPT1C5XxlB4=</latexit> linear regression
Simulations log M[h 1 M ] ¯n [(h 1 Mpc) 3 ] ¯n is comparable to 10.8 11.8 4.3 ⇥ 10 2 LSST [80, 81], Billion Object Apparatus [82] 11.8 12.8 5.7 ⇥ 10 3 SPHEREx [83, 84] 12.8 13.8 5.6 ⇥ 10 4 BOSS CMASS [85], DESI [86, 87], Euclid [88–90] 13.8 15.2 2.6 ⇥ 10 5 Cluster catalogs Table I. Simulated halo populations at z = 0.6. . The shifted squared density ˜2 and shifted tidal field ˜G2 are computed similarly, using 2 1( e mass. Ran 5 MP-Gadget1 DM-only N-body sims with 15363 DM particles, 30723 mesh for PM forces, L=500 Mpc/h, ~4000 time steps to evolve z=99 to z=0.6 4 FoF halo mass bins 1Feng et al. https://github.com/bluetides-project/MP-Gadget [derived from P-Gadget] mptcle = 2.9 ⇥ 109 M /h<latexit sha1_base64="5gOaYc51Lo066Ov3uoAnov8D0Bc=">AAACH3icbVDLSgMxFM3UV62vqks3wSK6kDpTBS0oFNy4ESrYB3TqkEnTNnQyGZI7Qhn6CX6E3+BW1+7EbZf+ieljYVsPBA7n3JuTHD8KuAbbHlqppeWV1bX0emZjc2t7J7u7V9UyVpRVqAykqvtEs4CHrAIcAlaPFCPCD1jN792O/NozU5rL8BH6EWsK0gl5m1MCRvKyx8JLXCVwBDRgg5tCvugCF0xjx34quqf3nitbEs66XjZn5+0x8CJxpiSHpih72R+3JWksWGhuJlo3HDuCZkIU8FFSxo01iwjtkQ5rGBoSE9pMxh8a4COjtHBbKnNCwGP170ZChNZ94ZtJQaCr572R+J/XiKF91Ux4GMXAQjoJascBBolH7eAWV4xC0DeEUMXNWzHtEkUomA5nUnwxMJ048w0skmoh75znCw8XudL1tJ00OkCH6AQ56BKV0B0qowqi6AW9oXf0Yb1an9aX9T0ZTVnTnX00A2v4CxZtokU=</latexit>
Test of linear model Reasonable prediction on large scales Missing some structure on small scales MS, Simonović et al. (2019) d slices of the overdensity h(x) of simulated 1010.8 1011.8 h 1 M halos (top), compared with the cubic b ), and the linear Standard Eulerian bias model (bottom). Each panel is 500 h 1 Mpc wide and 110 h 1 Mpc hi sity is smoothed with a R = 2 h 1 Mpc Gaussian, WR(k) = exp[ (kR)2 /2]. The colorbar indicates the value d slices of the overdensity h(x) of simulated 1010.8 1011.8 h 1 M halos (top), compared with the cubic b ), and the linear Standard Eulerian bias model (bottom). Each panel is 500 h 1 Mpc wide and 110 h 1 Mpc hi sity is smoothed with a R = 2 h 1 Mpc Gaussian, WR(k) = exp[ (kR)2 /2]. The colorbar indicates the value d overdensity h(x). Simulation (= truth) Model b1 m(x)<latexit sha1_base64="qPIVubm7zi7B0APirS0IQLfxYzc=">AAACJXicbVBNS8NAEN34WetX1aOXYBH0YEmqoAcPBS8eK1gtNCFsNhNd3N2E3U1tCfkV/gh/g1c9exPBk/hP3LY5WPXBwOO9GWbmhSmjSjvOhzUzOze/sFhZqi6vrK6t1zY2r1SSSQIdkrBEdkOsgFEBHU01g24qAfOQwXV4dzbyr/sgFU3EpR6m4HN8I2hMCdZGCmoHnoB7knCORZR7/UGRexzr2zDOB0VRDQPXi4BpHPA9Y+4HtbrTcMaw/xK3JHVUoh3UvrwoIRkHoQnDSvVcJ9V+jqWmhEFR9TIFKSZ3+AZ6hgrMQfn5+K3C3jVKZMeJNCW0PVZ/TuSYKzXkoekc3ax+eyPxP6+X6fjEz6lIMw2CTBbFGbN1Yo8ysiMqgWg2NAQTSc2tNrnFEhNtkpzaEvLCZOL+TuAvuWo23MNG8+Ko3jot06mgbbSD9pCLjlELnaM26iCCHtATekYv1qP1ar1Z75PWGauc2UJTsD6/AT9/psc=</latexit> g<latexit sha1_base64="wmzm750Tk9wCdwjQ/kl+uGkG6Vo=">AAACHXicbVC7TsMwFHV4lvIqMLJEVJWYqqQgwcBQiYWxSPQhNVXlODetVduJbKe0ivIFfATfwAozG2JFjPwJ7mOgLUeydHTOffn4MaNKO863tba+sbm1ndvJ7+7tHxwWjo4bKkokgTqJWCRbPlbAqIC6pppBK5aAuc+g6Q9uJ35zCFLRSDzocQwdjnuChpRgbaRuoeQJeCQR51gEqTccZanHse77YTrKsrwXANO42+sWik7ZmcJeJe6cFNEctW7hxwsiknAQmjCsVNt1Yt1JsdSUMDCDEwUxJgPcg7ahAnNQnXT6ncwuGSWww0iaJ7Q9Vf92pJgrNea+qZzcqpa9ifif1050eN1JqYgTDYLMFoUJs3VkT7KxAyqBaDY2BBNJza026WOJiTYJLmzxeWYycZcTWCWNStm9KFfuL4vVm3k6OXSKztA5ctEVqqI7VEN1RNATekGv6M16tt6tD+tzVrpmzXtO0AKsr1+F/aPk</latexit>
Nonlinear model So far used linear model g(x) = b1 m(x)<latexit sha1_base64="XaHcr7zAchcoHCuF4bu18V4pL3I=">AAACOHicbVDLSgMxFM34rPVVdekmWATdlBkVdKFQcOOyglWhU4ZM5k4NJpkhyahlmN/xI/wGtwrudCVu/QLTOgvbeiBwOOdeTu4JU860cd03Z2p6ZnZuvrJQXVxaXlmtra1f6iRTFNo04Ym6DokGziS0DTMcrlMFRIQcrsLb04F/dQdKs0RemH4KXUF6ksWMEmOloNb0JdzTRAgio9y/eyhyXxBzE8b5Q1FU/Qi4IUFvxzq7+ASHgYdLLRfFUA1qdbfhDoEniVeSOirRCmoffpTQTIA0lBOtO56bmm5OlGGUg83MNKSE3pIedCyVRIDu5sNLC7xtlQjHibJPGjxU/27kRGjdF6GdHJyhx72B+J/XyUx81M2ZTDMDkv4GxRnHJsGD2nDEFFDD+5YQqpj9K6Y3RBFqbLkjKaEobCfeeAOT5HKv4e039s4P6s3jsp0K2kRbaAd56BA10RlqoTai6BE9oxf06jw5786n8/U7OuWUOxtoBM73D5EtreE=</latexit>
Nonlinear model So far used linear model Include all nonlinear terms allowed by symmetries (EFT) Fit parameters by minimizing mean-squared error (least-squares ‘polynomial’ regression) bi<latexit sha1_base64="stVaTDZz5GHd9HUTbI9z2gtgqYY=">AAACK3icbVDLSgMxFM34tr7aunQTLIKrMqOCIi4ENy4rWhU6pSSZTBvMY0gytSXMJ7jVr/BrXClu/Q/T2oWtHggczr03956DM86MDcP3YG5+YXFpeWW1tLa+sblVrlRvjco1oU2iuNL3GBnKmaRNyyyn95mmSGBO7/DDxah+16faMCVv7DCjbYG6kqWMIOula9xhnXItrIdjwL8kmpAamKDRqQTVOFEkF1RawpExrSjMbNshbRnhtCjFuaEZIg+oS1ueSiSoabvxrQXc80oCU6X9kxaO1d8TDgljhgL7ToFsz8zWRuJ/tVZu05O2YzLLLZXkZ1Gac2gVHBmHCdOUWD70BBHN/K2Q9JBGxPp4SrGkj0QJgWTi4v6gcC4ercCpGxTFlCWHhf8cCnMKseLJ2IYp+RSj2cz+ktuDenRYP7g6qp2fTfJcATtgF+yDCByDc3AJGqAJCOiCJ/AMXoLX4C34CD5/WueCycw2mELw9Q0dSKcW</latexit> MS, Simonović et al. (2019) g(x) = b1 m(x) + b2 2 m(x) + tidal term + b3 3 m(x) + · · ·<latexit sha1_base64="hIhHT6q70TfpTrAHD6PG9mgELPM=">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</latexit>
Bulk flows 5 s x of a halo at the initial position q are given by x = q + (q). The overdensity 1 + h(x) = Z d3 q (1 + h(q)) D(x q (q)) , (12) Fourier transform of this field in Eulerian space is d3 x (1 + h(x)) e ik·x = Z d3 q (1 + h(q)) e ik·(q+ (q)) . (13) d in the rest of the paper we restrict the range of momenta to k 6= 0, so that the zero er our formulas. The nonlinear displacement from Lagrangian to Eulerian position ve series = 1 + 2 + · · · . At first order, we have the well-known Zel’dovich 1(q) = Z k eik·q ik k2 1(k) . (14) n be written as 2(q) = 3 14 Z k eik·q ik k2 G2(k) . (15) of the nonlinear displacement field and expanding the exponent e ik· (q) in Eq. (13) Standard Eulerian bias expansion. This procedure also fixes the relation between linear displacement is large 5 rdinates x of a halo at the initial position q are given by x = q + (q). The overdensity en by 1 + h(x) = Z d3 q (1 + h(q)) D(x q (q)) , (12) . The Fourier transform of this field in Eulerian space is k) ⌘ Z d3 x (1 + h(x)) e ik·x = Z d3 q (1 + h(q)) e ik·(q+ (q)) . (13) ion and in the rest of the paper we restrict the range of momenta to k 6= 0, so that the zero not enter our formulas. The nonlinear displacement from Lagrangian to Eulerian position urbative series = 1 + 2 + · · · . At first order, we have the well-known Zel’dovich 1(q) = Z k eik·q ik k2 1(k) . (14) ment can be written as 2(q) = 3 14 Z k eik·q ik k2 G2(k) . (15) ription of the nonlinear displacement field and expanding the exponent e ik· (q) in Eq. (13) usual Standard Eulerian bias expansion. This procedure also fixes the relation between s and their Standard Eulerian counterparts. Of course, this is not a surprise, as we expect ee order by order in perturbation theory. 5 f a halo at the initial position q are given by x = q + (q). The overdensity h(x) = Z d3 q (1 + h(q)) D(x q (q)) , (12) er transform of this field in Eulerian space is x (1 + h(x)) e ik·x = Z d3 q (1 + h(q)) e ik·(q+ (q)) . (13) he rest of the paper we restrict the range of momenta to k 6= 0, so that the zero ur formulas. The nonlinear displacement from Lagrangian to Eulerian position ries = 1 + 2 + · · · . At first order, we have the well-known Zel’dovich 1(q) = Z k eik·q ik k2 1(k) . (14) written as 2(q) = 3 14 Z k eik·q ik k2 G2(k) . (15) e nonlinear displacement field and expanding the exponent e ik· (q) in Eq. (13) Lagrangian space Eulerian space halo 5 t the initial position q are given by x = q + (q). The overdensity d3 q (1 + h(q)) D(x q (q)) , (12) rm of this field in Eulerian space is )) e ik·x = Z d3 q (1 + h(q)) e ik·(q+ (q)) . (13) the paper we restrict the range of momenta to k 6= 0, so that the zero s. The nonlinear displacement from Lagrangian to Eulerian position 1 + 2 + · · · . At first order, we have the well-known Zel’dovich q) = Z k eik·q ik k2 1(k) . (14) Slide credit: Marko Simonović
Large bulk flows lead to large model error 
 because fields are incoherent Need a model that takes into account large bulk flows Bulk flows Good model of x ! x +<latexit sha1_base64="+3MORj4q/XkrwqAejYnO59kK720=">AAACAHicbVBNS8NAEJ34WetX1IMHL4tFEISSVEGhBwtePFawH9CEstlu2qWbTdjdaEvpxb/ixYMiXv0Z3vw3btsctPXBwOO9GWbmBQlnSjvOt7W0vLK6tp7byG9ube/s2nv7dRWnktAaiXksmwFWlDNBa5ppTpuJpDgKOG0E/ZuJ33igUrFY3OthQv0IdwULGcHaSG37cOCVPcm6PY2ljB+9MhqceVXF2nbBKTpToEXiZqQAGapt+8vrxCSNqNCEY6VarpNof4SlZoTTcd5LFU0w6eMubRkqcESVP5o+MEYnRumgMJamhEZT9ffECEdKDaPAdEZY99S8NxH/81qpDq/8ERNJqqkgs0VhypGO0SQN1GGSEs2HhmAimbkVkR6WmGiTWd6E4M6/vEjqpaJ7XizdXRQq11kcOTiCYzgFFy6hArdQhRoQGMMzvMKb9WS9WO/Wx6x1ycpmDuAPrM8fPGuWKA==</latexit> Displace by a shifttruth<latexit sha1_base64="dorPueCGyAIyn+LUjl5+vXhBTBQ=">AAAB+3icbVBNS8NAEN34WetXrEcvi0XwVJIq6M2CF48V7Ac0IWw2m3bpZhN2J2IJ/StePCji1T/izX/jts1BWx8MPN6bYWZemAmuwXG+rbX1jc2t7cpOdXdv/+DQPqp1dZoryjo0Fanqh0QzwSXrAAfB+pliJAkF64Xj25nfe2RK81Q+wCRjfkKGksecEjBSYNe8iAkgQeGpBIPKYTQN7LrTcObAq8QtSR2VaAf2lxelNE+YBCqI1gPXycAviAJOBZtWvVyzjNAxGbKBoZIkTPvF/PYpPjNKhONUmZKA5+rviYIkWk+S0HQmBEZ62ZuJ/3mDHOJrv+Ayy4FJulgU5wJDimdB4IgrRkFMDCFUcXMrpiOiCAUTV9WE4C6/vEq6zYZ70WjeX9ZbN2UcFXSCTtE5ctEVaqE71EYdRNETekav6M2aWi/Wu/WxaF2zyplj9AfW5w9uWJSv</latexit> truth<latexit sha1_base64="dorPueCGyAIyn+LUjl5+vXhBTBQ=">AAAB+3icbVBNS8NAEN34WetXrEcvi0XwVJIq6M2CF48V7Ac0IWw2m3bpZhN2J2IJ/StePCji1T/izX/jts1BWx8MPN6bYWZemAmuwXG+rbX1jc2t7cpOdXdv/+DQPqp1dZoryjo0Fanqh0QzwSXrAAfB+pliJAkF64Xj25nfe2RK81Q+wCRjfkKGksecEjBSYNe8iAkgQeGpBIPKYTQN7LrTcObAq8QtSR2VaAf2lxelNE+YBCqI1gPXycAviAJOBZtWvVyzjNAxGbKBoZIkTPvF/PYpPjNKhONUmZKA5+rviYIkWk+S0HQmBEZ62ZuJ/3mDHOJrv+Ayy4FJulgU5wJDimdB4IgrRkFMDCFUcXMrpiOiCAUTV9WE4C6/vEq6zYZ70WjeX9ZbN2UcFXSCTtE5ctEVaqE71EYdRNETekav6M2aWi/Wu/WxaF2zyplj9AfW5w9uWJSv</latexit> <latexit sha1_base64="0kQQ4bQnE6qcjjlOoHqoSjdsqzk=">AAAB63icbVBNSwMxEJ34WetX1aOXYBE8ld0q6M2CF48V7Ae0S8mm2TY0yS5JVihL/4IXD4p49Q9589+YbfegrQ8GHu/NMDMvTAQ31vO+0dr6xubWdmmnvLu3f3BYOTpumzjVlLVoLGLdDYlhgivWstwK1k00IzIUrBNO7nK/88S04bF6tNOEBZKMFI84JTaX+k3DB5WqV/PmwKvEL0gVCjQHla/+MKapZMpSQYzp+V5ig4xoy6lgs3I/NSwhdEJGrOeoIpKZIJvfOsPnThniKNaulMVz9fdERqQxUxm6Tkns2Cx7ufif10ttdBNkXCWpZYouFkWpwDbG+eN4yDWjVkwdIVRzdyumY6IJtS6esgvBX355lbTrNf+yVn+4qjZuizhKcApncAE+XEMD7qEJLaAwhmd4hTck0Qt6Rx+L1jVUzJzAH6DPH/PXjis=</latexit> ✏(x) ⌘ truth model<latexit sha1_base64="j8JWHHDkGfzWSFy/jB9sc9v3ags=">AAACOXicbVDLSgNBEJz1Gd9Rj14Gg6AHw64KevAQ8eJRwaiQDWF20jGD81hnesWw5Hv8CL/Bq+LRm3j1B5yNOWi0YKC6qpvuqSSVwmEYvgZj4xOTU9Olmdm5+YXFpfLyyoUzmeVQ50Yae5UwB1JoqKNACVepBaYSCZfJzXHhX96BdcLoc+yl0FTsWouO4Ay91CofxZA6IY3evN+iMdxm4o7SuA0SWStWDLtW5Wgz7Pbp9qiujK/7rXIlrIYD0L8kGpIKGeK0VX6L24ZnCjRyyZxrRGGKzZxZFFxCfzbOHKSM37BraHiqmQLXzAdf7dMNr7Rpx1j/NNKB+nMiZ8q5nkp8Z3GlG/UK8T+vkWHnoJkLnWYImn8v6mSSoqFFbrQtLHCUPU8Yt8LfSnmXWcbRp/trS6KKTKLRBP6Si51qtFvdOdur1A6H6ZTIGlknmyQi+6RGTsgpqRNOHsgTeSYvwWPwFrwHH9+tY8FwZpX8QvD5BQAFrsc=</latexit>
Including bulk flows in the model licity, in this equation and in the rest of the paper we restrict the range of momenta to k 6= 0, so that the ze r mean density do not enter our formulas. The nonlinear displacement from Lagrangian to Eulerian positi xpanded in a perturbative series = 1 + 2 + · · · . At first order, we have the well-known Zel’dovi mation [69] 1(q) = Z k eik·q ik k2 1(k) . (1 nd-order displacement can be written as 2(q) = 3 14 Z k eik·q ik k2 G2(k) . (1 e perturbative description of the nonlinear displacement field and expanding the exponent e ik· (q) in Eq. (1 sible to recover the usual Standard Eulerian bias expansion. This procedure also fixes the relation betwe an bias parameters and their Standard Eulerian counterparts. Of course, this is not a surprise, as we expe descriptions to agree order by order in perturbation theory. e other hand we do not want to expand the full nonlinear displacement. We are going to keep the large q) exponentiated and expand only the higher-order terms.3 In this way the largest part of the problema acements is not expanded in perturbation theory. With this in mind, we can rewrite Eq. (13) in the followi h(k) = Z d3 q ⇣ 1 + bL 1 1(q) + bL 2 ( 2 1(q) 2 1) + bL G2 G2(q) + · · · ik · 2(q) + · · · ⌘ e ik·(q+ 1(q)) , (1 e new contributions come from expanding the second (and higher) order displacement field in the exponen ortant to stress that at leading order this new term can be expressed through the second order operator (15)). Therefore, at second order in perturbation theory, expanding the nonlinear terms in the displaceme ) only shifts some of the standard Lagrangian bias parameters by a calculable constant. We will give mo Usual approximation in (C)LPT for example: Vlah, Castorina, White (2016) art with the description of biased tracers in Lagrangian space. The displacement field is then t linear contribution and smaller higher order corrections. The nonlinear corrections to are while the linear piece is kept in the exponent. In this way, the dominant part of the large treated exactly, and the resulting operators once written in Eulerian space are automatically est of this section we give the details of this construction. ity field at Lagrangian position q is modeled using a bias expansion in the linear Lagrangian- L h(q) = bL 1 1(q) + bL 2 ( 2 1(q) 2 1) + bL G2 G2(q) + · · · , (8) re Lagrangian bias parameters, 2 1 is the r.m.s. fluctuation of the linear density field 2 1 = ⌦ 2 1(q) ↵ = Z 1 0 dk 2⇡2 k2 P11(k) , (9) ) is defined as2 G2(q) ⌘  @i@j @2 1(q) 2 2 1(q) . (10) this operator in momentum space is given by G2(k) = Z p  (p · (k p))2 p2|k p|2 1 1(p) 1(k p) . (11) ng notation in which R p ⌘ R d3 p/(2⇡)3 . In the bias expansion (8) we kept only terms up to bation theory. We will continue to work at this order throughout this section, because it is ng notation and motivating the bias model that we are going to use to make comparisons with er order or higher derivative operators needed for the consistent one-loop calculation can be ded. We will come back to this in Section VIII. n Eq. (8) is in Lagrangian space. In order to go to Eulerian space, let us start from Eq. (8) and split into the dominant linear contribution and smaller higher order corrections. The nonlinear corrections treated perturbatively, while the linear piece is kept in the exponent. In this way, the dominant part of displacements can be treated exactly, and the resulting operators once written in Eulerian space are auto IR-resummed. In the rest of this section we give the details of this construction. The proto-halo density field at Lagrangian position q is modeled using a bias expansion in the linear La space density 1(q): L h(q) = bL 1 1(q) + bL 2 ( 2 1(q) 2 1) + bL G2 G2(q) + · · · , where bL 1, bL 2, bL G2 , . . . are Lagrangian bias parameters, 2 1 is the r.m.s. fluctuation of the linear density field 2 1 = ⌦ 2 1(q) ↵ = Z 1 0 dk 2⇡2 k2 P11(k) , and the operator G2(q) is defined as2 G2(q) ⌘  @i@j @2 1(q) 2 2 1(q) . The representation of this operator in momentum space is given by G2(k) = Z p  (p · (k p))2 p2|k p|2 1 1(p) 1(k p) . Notice that we are using notation in which R p ⌘ R d3 p/(2⇡)3 . In the bias expansion (8) we kept only te second order in perturbation theory. We will continue to work at this order throughout this section, bec su cient for introducing notation and motivating the bias model that we are going to use to make compar simulations. The higher order or higher derivative operators needed for the consistent one-loop calculati straightforwardly included. We will come back to this in Section VIII. The bias expansion in Eq. (8) is in Lagrangian space. In order to go to Eulerian space, let us start from E include the gravitational evolution. The gravitational evolution is encoded in the nonlinear displacement fi 5 Eulerian coordinates x of a halo at the initial position q are given by x = q + (q). The overdensity is way is given by 1 + h(x) = Z d3 q (1 + h(q)) D(x q (q)) , (12) Dirac delta. The Fourier transform of this field in Eulerian space is h(k) ⌘ Z d3 x (1 + h(x)) e ik·x = Z d3 q (1 + h(q)) e ik·(q+ (q)) . (13) n this equation and in the rest of the paper we restrict the range of momenta to k 6= 0, so that the zero density do not enter our formulas. The nonlinear displacement from Lagrangian to Eulerian position ed in a perturbative series = 1 + 2 + · · · . At first order, we have the well-known Zel’dovich [69] 1(q) = Z k eik·q ik k2 1(k) . (14) er displacement can be written as 2(q) = 3 14 Z k eik·q ik k2 G2(k) . (15) ik· (q)
Shifted operators Z(k) = ˜1(k) + 1 2 ˜G2(k) 1 3 ˜G3(k) + · · · , (20) or analogous to ˜G2 (see Appendix D). In other words, Z(k) can be absorbed in the bias ging the bias parameters. Of course, this is just a choice, and there is nothing wrong in he formulas. As we are going to see later, di↵erent choices may be more appropriate for us point out that in the formula (20) the displacements 1(q) are treated exactly. In other k· 1(q) is never expanded in 1(q). The only expansion parameter is the derivative of the 1(q), which is a small quantity.5 This is consistent with the way the shifted operators asis of shifted operators we can write the bias expansion of the halo density field in Eulerian order in perturbation theory, in the following way h(k) = b1 ˜1(k) + b2 ˜2(k) + bG2 ˜G2(k) + · · · . (21) this section. Notice that the new bias parameters bi di↵er from the original Lagrangian his di↵erence comes from expanding the nonlinear part of the displacement (Eq. (16)) and sity field in terms of shifted operators (Eq. (20)). We give the explicit relation of bi and bL i (21) has a similar structure as the usual Standard Eulerian bias expansion h(k) = bE 1 (k) + bE 2 2(k) + bE G2 G2(k) + · · · , (22) ce that all fields in this equation are nonlinear. Apart from the IR resummation of the large ant di↵erence compared to the expansion in terms of ˜O is that in Eq. (21) all operators are inear field 1. As we are going to see, for the purposes of describing the biased tracers on her important virtue of the expansion (21). + noise ik · 2(q) + · · · e , (16) me from expanding the second (and higher) order displacement field in the exponent. t leading order this new term can be expressed through the second order operator G2 cond order in perturbation theory, expanding the nonlinear terms in the displacement he standard Lagrangian bias parameters by a calculable constant. We will give more s in Section VIII. tivates us to write down the bias expansion in Eulerian space in terms of shifted he following way ˜O(k) ⌘ Z d3 q O(q) e ik·(q+ 1(q)) , (17) 2 1), G2, . . .}.4 We would like to stress again a few important advantages that this d operators are written in Eulerian space and therefore allow for easy comparisons tion of their importance. (b) The large displacement terms 1(q) are kept resummed, s with simulations on the level of realizations. Notice that this also implies that in ass filter, compared to the wavelength of a Fourier mode 1(k). For a given wavenumber k, the nto the long-wavelength and short-wavelength part: 1 = L 1 + S 1 , where L 1 = W(k) 1 and L 1 on the short modes is fixed by the Equivalence Principle. Therefore, strictly speaking, only L 1 in any perturbative calculation S 1 has to be expanded order by order in perturbation theory. The y keeping the full 1 in the exponent is always higher order in S 1 than terms we calculate. Also, all scales. In order to keep the formulas simple, we decide not to do the long-short splitting in our not just given by a translation of the position argument because they implicitly include the inverse n @xi/@qj due to the coordinate transformation. This is similar to the Zel’dovich density, which is gian space shifted by 1(q). Motivates bias expansion in “shifted” operators (incl. bulk flows)
Shifted operators Z(k) = ˜1(k) + 1 2 ˜G2(k) 1 3 ˜G3(k) + · · · , (20) or analogous to ˜G2 (see Appendix D). In other words, Z(k) can be absorbed in the bias ging the bias parameters. Of course, this is just a choice, and there is nothing wrong in he formulas. As we are going to see later, di↵erent choices may be more appropriate for us point out that in the formula (20) the displacements 1(q) are treated exactly. In other k· 1(q) is never expanded in 1(q). The only expansion parameter is the derivative of the 1(q), which is a small quantity.5 This is consistent with the way the shifted operators asis of shifted operators we can write the bias expansion of the halo density field in Eulerian order in perturbation theory, in the following way h(k) = b1 ˜1(k) + b2 ˜2(k) + bG2 ˜G2(k) + · · · . (21) this section. Notice that the new bias parameters bi di↵er from the original Lagrangian his di↵erence comes from expanding the nonlinear part of the displacement (Eq. (16)) and sity field in terms of shifted operators (Eq. (20)). We give the explicit relation of bi and bL i (21) has a similar structure as the usual Standard Eulerian bias expansion h(k) = bE 1 (k) + bE 2 2(k) + bE G2 G2(k) + · · · , (22) ce that all fields in this equation are nonlinear. Apart from the IR resummation of the large ant di↵erence compared to the expansion in terms of ˜O is that in Eq. (21) all operators are inear field 1. As we are going to see, for the purposes of describing the biased tracers on her important virtue of the expansion (21). + noise ik · 2(q) + · · · e , (16) me from expanding the second (and higher) order displacement field in the exponent. t leading order this new term can be expressed through the second order operator G2 cond order in perturbation theory, expanding the nonlinear terms in the displacement he standard Lagrangian bias parameters by a calculable constant. We will give more s in Section VIII. tivates us to write down the bias expansion in Eulerian space in terms of shifted he following way ˜O(k) ⌘ Z d3 q O(q) e ik·(q+ 1(q)) , (17) 2 1), G2, . . .}.4 We would like to stress again a few important advantages that this d operators are written in Eulerian space and therefore allow for easy comparisons tion of their importance. (b) The large displacement terms 1(q) are kept resummed, s with simulations on the level of realizations. Notice that this also implies that in ass filter, compared to the wavelength of a Fourier mode 1(k). For a given wavenumber k, the nto the long-wavelength and short-wavelength part: 1 = L 1 + S 1 , where L 1 = W(k) 1 and L 1 on the short modes is fixed by the Equivalence Principle. Therefore, strictly speaking, only L 1 in any perturbative calculation S 1 has to be expanded order by order in perturbation theory. The y keeping the full 1 in the exponent is always higher order in S 1 than terms we calculate. Also, all scales. In order to keep the formulas simple, we decide not to do the long-short splitting in our not just given by a translation of the position argument because they implicitly include the inverse n @xi/@qj due to the coordinate transformation. This is similar to the Zel’dovich density, which is gian space shifted by 1(q). Motivates bias expansion in “shifted” operators (incl. bulk flows) Result is in Eulerian space so easy to compare against simulations Has IR resummation, giving correct halo positions and BAO spread Model is perturbative, only linear fields used in the construction Power spectrum agrees with resummed 1-loop PT
Model on the grid Distribute 15363 particles on regular grid Assign artificial particle masses Displace by linear displacement Interpolate to Eulerian grid using CIC weighted by particle masses [Very similar to generating N-body initial conds./Zeldovich density] mi = O(qi)<latexit sha1_base64="OIMXXFzMNCtfp4mc7I6Q6k7Bu8E=">AAACF3icbZC7TsMwFIadcivlFmBkwKJCKkuVFCRYEJVY2CgSvUhtFTmu01q1k2A7SFWUkYfgGVhhZkOsjIy8CU6agbb8kqVP/zlH5/h3Q0alsqxvo7C0vLK6VlwvbWxube+Yu3stGUQCkyYOWCA6LpKEUZ80FVWMdEJBEHcZabvj67TefiRC0sC/V5OQ9Dka+tSjGCltOeYhdyi8hD2O1AgjFt8mlYxdL35IHHrimGWramWCi2DnUAa5Go750xsEOOLEV5ghKbu2Fap+jISimJGk1IskCREeoyHpavQRJ7IfZx9J4LF2BtALhH6+gpn7dyJGXMoJd3VneqScr6Xmf7VupLyLfkz9MFLEx9NFXsSgCmCaChxQQbBiEw0IC6pvhXiEBMJKZzezxeWJzsSeT2ARWrWqfVqt3Z2V61d5OkVwAI5ABdjgHNTBDWiAJsDgCbyAV/BmPBvvxofxOW0tGPnMPpiR8fULYv6f9w==</latexit> q<latexit sha1_base64="Ft5o3g7n3eKWwngUx/MdJMYDuiE=">AAACAHicbVDLSgMxFL3js9ZX1aWbYBFclZkq6M6CG5cV7APboWTSTBuaZMYkI5ShG7/Bra7diVv/xKV/YqadhW09EDiccy/35AQxZ9q47rezsrq2vrFZ2Cpu7+zu7ZcODps6ShShDRLxSLUDrClnkjYMM5y2Y0WxCDhtBaObzG89UaVZJO/NOKa+wAPJQkawsdJDV2AzDML0cdIrld2KOwVaJl5OypCj3iv9dPsRSQSVhnCsdcdzY+OnWBlGOJ0Uu4mmMSYjPKAdSyUWVPvpNPEEnVqlj8JI2ScNmqp/N1IstB6LwE5mCfWil4n/eZ3EhFd+ymScGCrJ7FCYcGQilH0f9ZmixPCxJZgoZrMiMsQKE2NLmrsSiKwTb7GBZdKsVrzzSvXuoly7ztspwDGcwBl4cAk1uIU6NICAhBd4hTfn2Xl3PpzP2eiKk+8cwRycr1/QxZes</latexit> q ! q + 1(q)<latexit sha1_base64="WM5LTA6NvvqFh353BeqMK/OW1Ew=">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</latexit> 1(q)<latexit sha1_base64="TJaAHFoYRyspIjBVZgsQyBThdy0=">AAACKXicbVDLSsNAFJ34rPVVdekmWIS6KUkVdGfBjcsK9gFNKJPppB06j3RmUikh3+FH+A1ude1O3Yk/4rSNYFsPDBzOuZdz5wQRJUo7zoe1srq2vrGZ28pv7+zu7RcODhtKxBLhOhJUyFYAFaaE47ommuJWJDFkAcXNYHAz8ZsjLBUR/F6PI+wz2OMkJAhqI3UKrsfxAxKMQd5NvNEwTTwGdT8Ik2Ga5n+5V1Mk7bglM3DWKRSdsjOFvUzcjBRBhlqn8OV1BYoZ5hpRqFTbdSLtJ1Bqgig2IbHCEUQD2MNtQzlkWPnJ9GupfWqUrh0KaR7X9lT9u5FAptSYBWZycqta9Cbif1471uGVnxAexRpzNAsKY2prYU96srtEYqTp2BCIJDG32qgPJUTatDmXErDUdOIuNrBMGpWye16u3F0Uq9dZOzlwDE5ACbjgElTBLaiBOkDgETyDF/BqPVlv1rv1ORtdsbKdIzAH6/sHurmolw==</latexit> O(q)<latexit sha1_base64="fNNlinCcTxszFIVZ7rBkeZk8/5Y=">AAACJXicbVDLSgMxFM3UV62vqks3g0WoC8tMFXRnwY07K9gHtEPJZDJtaB7TJFMpw3yFH+E3uNW1OxFciX9i+ljY1gOBwzn3cm6OH1GitON8WZmV1bX1jexmbmt7Z3cvv39QVyKWCNeQoEI2fagwJRzXNNEUNyOJIfMpbvj9m7HfGGKpiOAPehRhj8EuJyFBUBupkz9rc/yIBGOQB0l7OEiTNoO654fJIE1zE44gTe7SojFPO/mCU3ImsJeJOyMFMEO1k/9pBwLFDHONKFSq5TqR9hIoNUEUm4BY4QiiPuzilqEcMqy8ZPKt1D4xSmCHQprHtT1R/24kkCk1Yr6ZHN+pFr2x+J/XinV45SWER7HGHE2DwpjaWtjjjuyASIw0HRkCkSTmVhv1oIRImybnUnyWmk7cxQaWSb1ccs9L5fuLQuV61k4WHIFjUAQuuAQVcAuqoAYQeAIv4BW8Wc/Wu/VhfU5HM9Zs5xDMwfr+Ba4ppw4=</latexit> ˜O(x)<latexit sha1_base64="jp0htYvlhK/ioIFAQhH9fbs5T6c=">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</latexit>
d slices of the overdensity h(x) of simulated 1010.8 1011.8 h 1 M halos (top), compared with the cubic b ), and the linear Standard Eulerian bias model (bottom). Each panel is 500 h 1 Mpc wide and 110 h 1 Mpc hi sity is smoothed with a R = 2 h 1 Mpc Gaussian, WR(k) = exp[ (kR)2 /2]. The colorbar indicates the value d overdensity h(x). g<latexit sha1_base64="wmzm750Tk9wCdwjQ/kl+uGkG6Vo=">AAACHXicbVC7TsMwFHV4lvIqMLJEVJWYqqQgwcBQiYWxSPQhNVXlODetVduJbKe0ivIFfATfwAozG2JFjPwJ7mOgLUeydHTOffn4MaNKO863tba+sbm1ndvJ7+7tHxwWjo4bKkokgTqJWCRbPlbAqIC6pppBK5aAuc+g6Q9uJ35zCFLRSDzocQwdjnuChpRgbaRuoeQJeCQR51gEqTccZanHse77YTrKsrwXANO42+sWik7ZmcJeJe6cFNEctW7hxwsiknAQmjCsVNt1Yt1JsdSUMDCDEwUxJgPcg7ahAnNQnXT6ncwuGSWww0iaJ7Q9Vf92pJgrNea+qZzcqpa9ifif1050eN1JqYgTDYLMFoUJs3VkT7KxAyqBaDY2BBNJza026WOJiTYJLmzxeWYycZcTWCWNStm9KFfuL4vVm3k6OXSKztA5ctEVqqI7VEN1RNATekGv6M16tt6tD+tzVrpmzXtO0AKsr1+F/aPk</latexit> Test of nonlinear model MS, Simonović et al. (2019) b1 m(x)<latexit sha1_base64="qPIVubm7zi7B0APirS0IQLfxYzc=">AAACJXicbVBNS8NAEN34WetX1aOXYBH0YEmqoAcPBS8eK1gtNCFsNhNd3N2E3U1tCfkV/gh/g1c9exPBk/hP3LY5WPXBwOO9GWbmhSmjSjvOhzUzOze/sFhZqi6vrK6t1zY2r1SSSQIdkrBEdkOsgFEBHU01g24qAfOQwXV4dzbyr/sgFU3EpR6m4HN8I2hMCdZGCmoHnoB7knCORZR7/UGRexzr2zDOB0VRDQPXi4BpHPA9Y+4HtbrTcMaw/xK3JHVUoh3UvrwoIRkHoQnDSvVcJ9V+jqWmhEFR9TIFKSZ3+AZ6hgrMQfn5+K3C3jVKZMeJNCW0PVZ/TuSYKzXkoekc3ax+eyPxP6+X6fjEz6lIMw2CTBbFGbN1Yo8ysiMqgWg2NAQTSc2tNrnFEhNtkpzaEvLCZOL+TuAvuWo23MNG8+Ko3jot06mgbbSD9pCLjlELnaM26iCCHtATekYv1qP1ar1Z75PWGauc2UJTsD6/AT9/psc=</latexit> sim g (x)<latexit sha1_base64="dAEZhBUIThUpHuyxiN4tqPX1ous=">AAACMXicbVDLSsNAFJ3U97vq0k2wCHVTEhVUcCG4cVnBPqCJYTK5aQdnJmFmopaQb/Ej/Aa3uu5O3foTTh8LbT0wcDjnXs6dE6aMKu04A6s0N7+wuLS8srq2vrG5Vd7eaaokkwQaJGGJbIdYAaMCGppqBu1UAuYhg1Z4fzX0Ww8gFU3Ere6n4HPcFTSmBGsjBeVzT8AjSTjHIsq9h6ci9zjWvTDOn4pi1YuAaRx078aq5LmivCiqZvAwKFecmjOCPUvcCamgCepB+dOLEpJxEJowrFTHdVLt51hqShiYsExBisk97kLHUIE5KD8ffbGwD4wS2XEizRPaHqm/N3LMlerz0EwOL1XT3lD8z+tkOj7zcyrSTIMg46A4Y7ZO7GFfdkQlEM36hmAiqbnVJj0sMdGm1T8pIS9MJ+50A7OkeVRzj2tHNyeVy4tJO8toD+2jKnLRKbpE16iOGoigZ/SK3tC79WINrA/razxasiY7u+gPrO8fryasww==</latexit> d slices of the overdensity h(x) of simulated 1010.8 1011.8 h 1 M halos (top), compared with the cubic b ), and the linear Standard Eulerian bias model (bottom). Each panel is 500 h 1 Mpc wide and 110 h 1 Mpc hi sity is smoothed with a R = 2 h 1 Mpc Gaussian, WR(k) = exp[ (kR)2 /2]. The colorbar indicates the value Nonlinear model d slices of the overdensity h(x) of simulated 1010.8 1011.8 h 1 M halos (top), compared with the cubic b ), and the linear Standard Eulerian bias model (bottom). Each panel is 500 h 1 Mpc wide and 110 h 1 Mpc hi sity is smoothed with a R = 2 h 1 Mpc Gaussian, WR(k) = exp[ (kR)2 /2]. The colorbar indicates the value Much better agreement than previous model Simulation (= truth)
Measures of success Error power spectrum (= MSE per wavenumber k) Cross-correlation between model and truth rcc(k) = h truth(k) ⇤ model(k)i p Ptruth(k)Pmodel(k)<latexit sha1_base64="cRIq0DTnUaRL8LfD6bOtGkmKbM8=">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</latexit> For best-fit model, , so focus on herePerr = Ptruth(1 r2 cc)<latexit sha1_base64="tRkpT8H4AD9a4kRuQlWo+CYRCgo=">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</latexit> Perr(k) ⌘ h| truth(k) model(k)|2 i<latexit sha1_base64="okwU7UK76wFFi51s3hFUJl6za4k=">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</latexit> Perr<latexit sha1_base64="HbsFOTY+FC07lz5Qau+blG/LzoY=">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</latexit>
Error power spectrum (= MSE) Poisson noise 1/¯n Perr = h| truth h model h |2 i Simulated halos (truth) Quadr. bias (model) isson noise 1/¯n h model h |2 i s (model) Poisson noise 1/¯n Perr = h| truth h model h |2 i Simulated halos (truth) Quadr. bias (model) Fourier wavenumber k Powerspectrum Error is quite white, similar to Poisson shot noise (discrete sampling) MS, Simonović et al. (2019)
Tried many other nonlinear bias operators Give few times larger model error Fourier wavenumber k Figure 20. Left panel: Model error power spectrum for Standard Eulerian bias models, for nonlinear dark matter NL from simulations as the input for the Standard Eulerian bias mo large scales because it involves squaring NL, which is rather UV-sensitive. Alternatively, density as the input to the bias model (dark orange) is treating large bulk flows perturbaMS, Simonović et al. (2019) Errorpowerspectrum
Scale dependence of the error ±1% of Phh Expanding Z Quadratic bias Cubic bias Linear Std. Eul. bias Scale dependence of the noise important around the nonlinear scale Potentially dangerous because can bias cosmological parameters
kmax up to which noise is constant kmax when scale dependence of Perr is detectable with 1σ in V=10 h-3Gpc3 volume (or V=0.5 in brackets): kmax [hMpc 1 ] log M[h 1 M ] ¯n [(h 1 Mpc) 3 ] Lin. Std. Eul. Cubic 10.8 11.8 4.3 ⇥ 10 2 0.1 (0.14) 0.3 (0.37) 11.8 12.8 5.7 ⇥ 10 3 0.08 (0.1) 0.18 (0.24) 12.8 13.8 5.6 ⇥ 10 4 0.07 (0.1) 0.13 (0.18) 13.8 15.2 2.6 ⇥ 10 5 0.1 (0.14) 0.24 (0.32) ximum wavenumber kmax when a scale dependence of the model error can be detected with 1 in a 10 h a 0.5 h 3 Gpc3 volume, shown in brackets), for the linear Standard Eulerian and the cubic bias models. Th model is typically 2 to 3 higher than that of linear Standard Eulerian bias. This can improve measurem arameters that a↵ect the galaxy power spectrum at these scales, for example the sum of neutrino masse significant uncertainty due to the noise in our measurement of the model error, as shown in Fig. 11. of the survey volume. We also summarize this in Table IV for two survey volumes. ear Standard Eulerian bias we find the following. For the lowest halo mass bin, the scale depen ificant for kmax ' 0.1 hMpc 1 in a 10 h 3 Gpc3 volume or for kmax ' 0.14 hMpc 1 in a 0.5 h Nonlinear bias has 2-3x higher kmax 8-30x more Fourier modes
 4-5x smaller error bars (in principle; also have more params!)
Weighting halos by their mass Halo number density
 (how many halos per cell) Halo mass density
 (how much halo mass per cell) used so far more similar to dark matter
 smaller shot noise Seljak, Hamaus & Desjacques (2009)
 Hamaus, Seljak & Desjacques (2010, 2011, 2012) Cai, Bernstein & Sheth (2011)
Weighting halos by their mass Shot noise (squared model error) 17x lower for light halos, 2-7x lower for heavy halos With 60% halo mass scatter (green), still get factor few How well can we do observationally? 49 mass weighting on the mean-square model error divided by the Poisson expectation, h| obs h (k) obs obs truth ?
Mass weighting questions How well can halo masses be measured (e.g. BOSS, DESI)? What observable properties of galaxies can we use? What sims? New ideas to get halo masses? For shot noise limited applications, gain may be large What if mass estimates are biased? Use for BAO reconstruction? (Suffers from high shot noise)
Science with galaxy catalogs (1) Bias model at the field level (2) Cosmological parameter analysis (3) Accounting for skewness (4) Getting initial from final conditions 13 Figure 1. 2-d slices of the overdensity h(x) of simulated 1010.8 1011.8 h 1 M halos (top), compared with the cubic bias model (center), and the linear Standard Eulerian bias model (bottom). Each panel is 500 h 1 Mpc wide and 110 h 1 Mpc high, and each density is smoothed with a R = 2 h 1 Mpc Gaussian, WR(k) = exp[ (kR)2 /2]. The colorbar indicates the values of this smoothed overdensity h(x). Figure 2. Same as Fig. 1 but for more massive and less abundant 1011.8 1012.8 h 1 M halos. simulations. The variance, skewness and kurtosis of the densities shown in the histograms are listed in Table II for the full simulated and modeled densities, and in Table III for the model error. The linear Standard Eulerian bias model tends to underpredict troughs and overpredict peaks of the halo density, as shown in Fig. 3. The model error is not Gaussian for any of the shown smoothing scales; in particular its kurtosis is larger than 1 for all smoothing scales. The cubic model provides a more accurate description of the halo density pdf, as shown in Fig. 4. This emphasizes the importance of using nonlinear bias terms even on rather large scales. Still, the cubic model tends to underpredict the peaks of the true halo density, especially on small scales. This agrees with Fig. 2 where the model also underpredicts
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers
Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers

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Prospects for CMB lensing-galaxy clustering cross-correlations and modeling biased tracers

  • 1. Prospects for CMB lensing—galaxy clustering cross-correlations and modeling biased tracers Marcel Schmittfull
 
 IPMU, October 2019
  • 2. Cosmic Microwave Background (CMB) Right after Big Bang light scatters frequently —> opaque As Universe expands, turns transparent See surface where light last scattered — 13.6996 bn yrs ago This is the CMB
  • 3. Cosmic Microwave Background (CMB) Perr<latexit sha1_base64="HbsFOTY+FC07lz5Qau+blG/LzoY=">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</latexit> Hot and cold blobs: Picture of the Universe 13.6996 bn years ago
  • 4. Gravitational lensing of the CMB 4 Figure 1.2: Illustration of gravitational lensing of CMB photons by large-scale structure (adapted from [10]). gravitationally deflected by galaxies and dark matter CMB
  • 5. Unlensed$CMB$ 10°$ CMB$Temp.$ 4$T(ˆn)lensed = T(ˆn + d(ˆn))unlensedˆn + d(ˆn))unlensed Slide credit: Blake Sherwin CMB before lensing
  • 6. Lensed$CMB$ 10°$ CMB$Temp.$ 5$T(ˆn)lensed = T(ˆn + d(ˆn))unlensed CMB after lensing Slide credit: Blake Sherwin
  • 7. Statistics of the CMB What are the statistics of the CMB before and after lensing?
  • 8. Statistics of the CMB before lensing Unlensed Lensed Normally distributed as far as we can tell
  • 9. Statistics of the CMB before lensing Unlensed Lensed Variance
  • 10. Statistics of the CMB before lensing Unlensed Lensed Variance Power spectrum
  • 11. Unlensed Lensed Local power spectrum Local power spectrum Local power spectrum is the same in each patch Local power spectrum Statistics of the CMB before lensing
  • 12. Unlensed Lensed Local power spectrum Local power is magnified or de-magnified Local power spectrum Local power spectrum Local power spectrum Statistics of the CMB after lensing
  • 13. Unlensed Lensed 0 dz Mmin dM × |˜yi(M, z)| 2 |˜yj(M, z)| 2 , (8) where gν is the spectral function of the SZ effect, V (z) is the comoving volume of the universe integrated to a redshift of zmax = 4, M is the virial mass such that [log10(Mmin), log10(Mmax)] = [11, 16], dn/dM is the FIG. 1: The impact of varying the lensing scaling parameter on the lensed CMB temperature power spectrum, for AL = [0,2,5,10]. Smidt+ (2010) Global power spectrum unlensed lensed Peaks of global power are smeared out Statistics of the CMB after lensing
  • 14. Unlensed Lensed Local power spectrum Local power spectrum Local power spectrum Local power spectrum Rather than averaging the modulation, measure it as a signal 
 —> magnification map Statistics of the CMB after lensing
  • 15. Measured lensing magnification Planck Collaboration: Planck 2018 lensing Fig. 1. Mollweide projection in Galactic coordinates of the lensing-deflection reconstruction map from our baseline minimum-p
  • 16. Cross-correlate with galaxy catalogs Galaxy catalog is sensitive to tails of the distribution of matter (galaxies form at peaks)
 Lensing magnification probes entire distribution (light is deflected by any mass)
 Some models of the Big Bang enhance peaks, so can test them Cross-correlations particularly well suited: Can determine ratio of two perfectly correlated random variables with infinite precision Planck Collaboration: Planck 2018 lensing Fig. 1. Mollweide projection in Galactic coordinates of the lensing-deflection reconstruction map from our baseline minimum- variance (MV) analysis. We show the Wiener-filtered displacement-like scalar field with multipoles ˆ↵MV LM = p L(L + 1)ˆMV LM , corre- sponding to the gradient mode (or E mode) of the lensing deflection angle. Modes with L < 8 have been filtered out. Our baseline lensing reconstruction map is shown in Fig. 1. In Sect. 2 we explain how this was obtained, and the changes compared to our analysis in PL2015. We also describe the new optimal filtering approach used for our best polarization anal- ysis. In Sect. 3 we present our main results, including power- spectrum estimates, cosmological parameter constraints, and a joint estimation of the lensing potential using the CIB. We end the section by using the estimates of the lensing map to delens the CMB, reducing the B-mode polarization power and sharpen- ing the acoustic peaks. In Sect. 4 we describe in detail a number of null and consistency tests, explaining the motivation for our data cuts and the limits of our understanding of the data. We also discuss possible contaminating signals, and assess whether they are potentially important for our results. In Sect. 5 we briefly de- scribe the various data products that are made available to the community, and we end with conclusions in Sect. 6. A series of appendices describe some technical details of the calculation of various biases that are subtracted, and derive the error model for the Monte Carlo estimates. 2. Data and methodology This final Planck lensing analysis is based on the 2018 Planck HFI maps as described in detail in Planck Collaboration III (2018). Our baseline analysis uses the SMICA foreground- cleaned CMB map described in Planck Collaboration IV (2018), and includes both temperature and polarization information. We use the Planck Full Focal Plane (FFP10) simulations, described in detail in Planck Collaboration III (2018), to remove a num- ber of bias terms and correctly normalize the lensing power- spectrum estimates. Our analysis methodology is based on the previous Planck analyses, as described in PL2013 and PL2015. After a summary of the methodology, Sect. 2.1 also lists the changes and improvements with respect to PL2015. Some de- tails of the covariance matrix are discussed in Sect. 2.2, and de- tails of the filtering in Sect. 2.3. The main set of codes applying the quadratic estimators will be made public as part of the CMB lensing toolbox LensIt.2 2.1. Lensing reconstruction The five main steps of the lensing reconstruction are as follows. 1. Filtering of the CMB maps. The observed sky maps are cut by a Galactic mask and have noise, so filtering is applied to remove the mask and approximately optimally weight for the noise. The lensing quadratic estimators use as input optimal Wiener-filtered X = T, E, and B CMB multipoles, as well as inverse-variance-weighted CMB maps. The latter maps can be obtained easily from the Wiener-filtered multipoles by divid- ing by the fiducial CMB power spectra Cfid ` before projecting onto maps. We write the observed temperature T and polariza- tion (written as the spin ±2 combination of Stokes parameters ±2P ⌘ Q ± iU) pixelized data as 0 BBBBBBBBBBBB@ Tdat 2Pdat 2Pdat 1 CCCCCCCCCCCCA = BY 0 BBBBBBBBBBBB@ T E B 1 CCCCCCCCCCCCA + noise, (1) 2 https://github.com/carronj/LensIt 3
  • 17. Sample variance cancellation Galaxy number density: CMB magnification: Big Bang models are characterized by parameter Planck Collaboration: Planck 2018 lensing Fig. 1. Mollweide projection in Galactic coordinates of the lensing-deflection reconstruction map from our baseline minimum- variance (MV) analysis. We show the Wiener-filtered displacement-like scalar field with multipoles ˆ↵MV LM = p L(L + 1)ˆMV LM , corre- sponding to the gradient mode (or E mode) of the lensing deflection angle. Modes with L < 8 have been filtered out. Our baseline lensing reconstruction map is shown in Fig. 1. In Sect. 2 we explain how this was obtained, and the changes compared to our analysis in PL2015. We also describe the new optimal filtering approach used for our best polarization anal- ysis. In Sect. 3 we present our main results, including power- spectrum estimates, cosmological parameter constraints, and a joint estimation of the lensing potential using the CIB. We end the section by using the estimates of the lensing map to delens the CMB, reducing the B-mode polarization power and sharpen- ing the acoustic peaks. In Sect. 4 we describe in detail a number of null and consistency tests, explaining the motivation for our data cuts and the limits of our understanding of the data. We also discuss possible contaminating signals, and assess whether they are potentially important for our results. In Sect. 5 we briefly de- scribe the various data products that are made available to the community, and we end with conclusions in Sect. 6. A series of appendices describe some technical details of the calculation of various biases that are subtracted, and derive the error model for the Monte Carlo estimates. 2. Data and methodology This final Planck lensing analysis is based on the 2018 Planck HFI maps as described in detail in Planck Collaboration III (2018). Our baseline analysis uses the SMICA foreground- cleaned CMB map described in Planck Collaboration IV (2018), and includes both temperature and polarization information. We use the Planck Full Focal Plane (FFP10) simulations, described in detail in Planck Collaboration III (2018), to remove a num- ber of bias terms and correctly normalize the lensing power- spectrum estimates. Our analysis methodology is based on the previous Planck analyses, as described in PL2013 and PL2015. After a summary of the methodology, Sect. 2.1 also lists the changes and improvements with respect to PL2015. Some de- tails of the covariance matrix are discussed in Sect. 2.2, and de- tails of the filtering in Sect. 2.3. The main set of codes applying the quadratic estimators will be made public as part of the CMB lensing toolbox LensIt.2 2.1. Lensing reconstruction The five main steps of the lensing reconstruction are as follows. 1. Filtering of the CMB maps. The observed sky maps are cut by a Galactic mask and have noise, so filtering is applied to remove the mask and approximately optimally weight for the noise. The lensing quadratic estimators use as input optimal Wiener-filtered X = T, E, and B CMB multipoles, as well as inverse-variance-weighted CMB maps. The latter maps can be obtained easily from the Wiener-filtered multipoles by divid- ing by the fiducial CMB power spectra Cfid ` before projecting onto maps. We write the observed temperature T and polariza- tion (written as the spin ±2 combination of Stokes parameters ±2P ⌘ Q ± iU) pixelized data as 0 BBBBBBBBBBBB@ Tdat 2Pdat 2Pdat 1 CCCCCCCCCCCCA = BY 0 BBBBBBBBBBBB@ T E B 1 CCCCCCCCCCCCA + noise, (1) 2 https://github.com/carronj/LensIt 3  ⇠ N(0, 2 )<latexit sha1_base64="fh2RNmAwU1/7vj3GTJMoGF9vnfw=">AAACInicbVDLSgMxFM3UV62vqks3wSJUkTJTBV24KLhxJRXsAzpjuZOmbWgyMyQZoQzzD36E3+BW1+7EleDGPzF9LGzrgcDhnHO5N8ePOFPatr+szNLyyupadj23sbm1vZPf3aurMJaE1kjIQ9n0QVHOAlrTTHPajCQF4XPa8AfXI7/xSKViYXCvhxH1BPQC1mUEtJHa+RN3AFEE2FVMYFeA7hPgyW1atE+N1BPQngQeysftfMEu2WPgReJMSQFNUW3nf9xOSGJBA004KNVy7Eh7CUjNCKdpzo0VjYAMoEdbhgYgqPKS8Z9SfGSUDu6G0rxA47H6dyIBodRQ+CY5ulrNeyPxP68V6+6ll7AgijUNyGRRN+ZYh3hUEO4wSYnmQ0OASGZuxaQPEog2Nc5s8UVqOnHmG1gk9XLJOSuV784LlatpO1l0gA5RETnoAlXQDaqiGiLoCb2gV/RmPVvv1of1OYlmrOnMPpqB9f0LdyykHw==</latexit> g = (1 + fNL)<latexit sha1_base64="qSZyT7RHnm+puQb5ztG1Up+wy9s=">AAACEnicbVDLSsNAFJ3UV62vqDvdDBahIpSkCrpQKLhxIVLBPqAJYTKdtENnkjAzEUoI+BF+g1tduxO3/oBL/8Rpm4WtHrhwOOde7r3HjxmVyrK+jMLC4tLySnG1tLa+sbllbu+0ZJQITJo4YpHo+EgSRkPSVFQx0okFQdxnpO0Pr8Z++4EISaPwXo1i4nLUD2lAMVJa8sy9PryEFfs48FJHcHh7kx1BZ4jiGEHPLFtVawL4l9g5KYMcDc/8dnoRTjgJFWZIyq5txcpNkVAUM5KVnESSGOEh6pOupiHiRLrp5IcMHmqlB4NI6AoVnKi/J1LEpRxxX3dypAZy3huL/3ndRAXnbkrDOFEkxNNFQcKgiuA4ENijgmDFRpogLKi+FeIBEggrHdvMFp9nOhN7PoG/pFWr2ifV2t1puX6Rp1ME++AAVIANzkAdXIMGaAIMHsEzeAGvxpPxZrwbH9PWgpHP7IIZGJ8/vOGcLw==</latexit> fNL<latexit sha1_base64="J3TCtcachvp5KbIs7TN/zj+HhGo=">AAACAHicbVA9SwNBEJ2LXzF+RS1tFoNgFe6ioIVFwMZCJIL5wCSEvc1esmR379jdE8KRxt9gq7Wd2PpPLP0n7iVXmMQHA4/3ZpiZ50ecaeO6305uZXVtfSO/Wdja3tndK+4fNHQYK0LrJOShavlYU84krRtmOG1FimLhc9r0R9ep33yiSrNQPphxRLsCDyQLGMHGSo9BL+koge5uJ71iyS27U6Bl4mWkBBlqveJPpx+SWFBpCMdatz03Mt0EK8MIp5NCJ9Y0wmSEB7RtqcSC6m4yvXiCTqzSR0GobEmDpurfiQQLrcfCt50Cm6Fe9FLxP68dm+CymzAZxYZKMlsUxByZEKXvoz5TlBg+tgQTxeytiAyxwsTYkOa2+CLNxFtMYJk0KmXvrFy5Py9Vr7J08nAEx3AKHlxAFW6gBnUgIOEFXuHNeXbenQ/nc9aac7KZQ5iD8/UL/Y2XJA==</latexit>
  • 18. Sample variance cancellation Galaxy number density: CMB magnification: Big Bang models are characterized by parameter Planck Collaboration: Planck 2018 lensing Fig. 1. Mollweide projection in Galactic coordinates of the lensing-deflection reconstruction map from our baseline minimum- variance (MV) analysis. We show the Wiener-filtered displacement-like scalar field with multipoles ˆ↵MV LM = p L(L + 1)ˆMV LM , corre- sponding to the gradient mode (or E mode) of the lensing deflection angle. Modes with L < 8 have been filtered out. Our baseline lensing reconstruction map is shown in Fig. 1. In Sect. 2 we explain how this was obtained, and the changes compared to our analysis in PL2015. We also describe the new optimal filtering approach used for our best polarization anal- ysis. In Sect. 3 we present our main results, including power- spectrum estimates, cosmological parameter constraints, and a joint estimation of the lensing potential using the CIB. We end the section by using the estimates of the lensing map to delens the CMB, reducing the B-mode polarization power and sharpen- ing the acoustic peaks. In Sect. 4 we describe in detail a number of null and consistency tests, explaining the motivation for our data cuts and the limits of our understanding of the data. We also discuss possible contaminating signals, and assess whether they are potentially important for our results. In Sect. 5 we briefly de- scribe the various data products that are made available to the community, and we end with conclusions in Sect. 6. A series of appendices describe some technical details of the calculation of various biases that are subtracted, and derive the error model for the Monte Carlo estimates. 2. Data and methodology This final Planck lensing analysis is based on the 2018 Planck HFI maps as described in detail in Planck Collaboration III (2018). Our baseline analysis uses the SMICA foreground- cleaned CMB map described in Planck Collaboration IV (2018), and includes both temperature and polarization information. We use the Planck Full Focal Plane (FFP10) simulations, described in detail in Planck Collaboration III (2018), to remove a num- ber of bias terms and correctly normalize the lensing power- spectrum estimates. Our analysis methodology is based on the previous Planck analyses, as described in PL2013 and PL2015. After a summary of the methodology, Sect. 2.1 also lists the changes and improvements with respect to PL2015. Some de- tails of the covariance matrix are discussed in Sect. 2.2, and de- tails of the filtering in Sect. 2.3. The main set of codes applying the quadratic estimators will be made public as part of the CMB lensing toolbox LensIt.2 2.1. Lensing reconstruction The five main steps of the lensing reconstruction are as follows. 1. Filtering of the CMB maps. The observed sky maps are cut by a Galactic mask and have noise, so filtering is applied to remove the mask and approximately optimally weight for the noise. The lensing quadratic estimators use as input optimal Wiener-filtered X = T, E, and B CMB multipoles, as well as inverse-variance-weighted CMB maps. The latter maps can be obtained easily from the Wiener-filtered multipoles by divid- ing by the fiducial CMB power spectra Cfid ` before projecting onto maps. We write the observed temperature T and polariza- tion (written as the spin ±2 combination of Stokes parameters ±2P ⌘ Q ± iU) pixelized data as 0 BBBBBBBBBBBB@ Tdat 2Pdat 2Pdat 1 CCCCCCCCCCCCA = BY 0 BBBBBBBBBBBB@ T E B 1 CCCCCCCCCCCCA + noise, (1) 2 https://github.com/carronj/LensIt 3  ⇠ N(0, 2 )<latexit sha1_base64="fh2RNmAwU1/7vj3GTJMoGF9vnfw=">AAACInicbVDLSgMxFM3UV62vqks3wSJUkTJTBV24KLhxJRXsAzpjuZOmbWgyMyQZoQzzD36E3+BW1+7EleDGPzF9LGzrgcDhnHO5N8ePOFPatr+szNLyyupadj23sbm1vZPf3aurMJaE1kjIQ9n0QVHOAlrTTHPajCQF4XPa8AfXI7/xSKViYXCvhxH1BPQC1mUEtJHa+RN3AFEE2FVMYFeA7hPgyW1atE+N1BPQngQeysftfMEu2WPgReJMSQFNUW3nf9xOSGJBA004KNVy7Eh7CUjNCKdpzo0VjYAMoEdbhgYgqPKS8Z9SfGSUDu6G0rxA47H6dyIBodRQ+CY5ulrNeyPxP68V6+6ll7AgijUNyGRRN+ZYh3hUEO4wSYnmQ0OASGZuxaQPEog2Nc5s8UVqOnHmG1gk9XLJOSuV784LlatpO1l0gA5RETnoAlXQDaqiGiLoCb2gV/RmPVvv1of1OYlmrOnMPpqB9f0LdyykHw==</latexit> If we measure only : If we measure and : g<latexit sha1_base64="lhiJBeb4nMmQQxTZcqh1G/By96k=">AAAB93icbVA9SwNBEJ3zM8avqKXNYRCswl0UtLAI2FgmYD4gOcLeZi5Zsrt37O4J4cgvsNXaTmz9OZb+EzfJFSbxwcDjvRlm5oUJZ9p43rezsbm1vbNb2CvuHxweHZdOTls6ThXFJo15rDoh0ciZxKZhhmMnUUhEyLEdjh9mfvsZlWaxfDKTBANBhpJFjBJjpcawXyp7FW8Od534OSlDjnq/9NMbxDQVKA3lROuu7yUmyIgyjHKcFnupxoTQMRli11JJBOogmx86dS+tMnCjWNmSxp2rfycyIrSeiNB2CmJGetWbif953dREd0HGZJIalHSxKEq5a2J39rU7YAqp4RNLCFXM3urSEVGEGpvN0pZQTG0m/moC66RVrfjXlWrjply7z9MpwDlcwBX4cAs1eIQ6NIECwgu8wpszcd6dD+dz0brh5DNnsATn6xdwBpN/</latexit> g<latexit sha1_base64="lhiJBeb4nMmQQxTZcqh1G/By96k=">AAAB93icbVA9SwNBEJ3zM8avqKXNYRCswl0UtLAI2FgmYD4gOcLeZi5Zsrt37O4J4cgvsNXaTmz9OZb+EzfJFSbxwcDjvRlm5oUJZ9p43rezsbm1vbNb2CvuHxweHZdOTls6ThXFJo15rDoh0ciZxKZhhmMnUUhEyLEdjh9mfvsZlWaxfDKTBANBhpJFjBJjpcawXyp7FW8Od534OSlDjnq/9NMbxDQVKA3lROuu7yUmyIgyjHKcFnupxoTQMRli11JJBOogmx86dS+tMnCjWNmSxp2rfycyIrSeiNB2CmJGetWbif953dREd0HGZJIalHSxKEq5a2J39rU7YAqp4RNLCFXM3urSEVGEGpvN0pZQTG0m/moC66RVrfjXlWrjply7z9MpwDlcwBX4cAs1eIQ6NIECwgu8wpszcd6dD+dz0brh5DNnsATn6xdwBpN/</latexit> <latexit sha1_base64="9bJNSyJCqgQXzYZuEwEOn7kVJFo=">AAAB/HicbVA9SwNBEN2LXzF+RS1tFoNgFe6ioIVFwMYygvmA5Ahzm71kze7esrsnhBB/g63WdmLrf7H0n7hJrjCJDwYe780wMy9SnBnr+99ebm19Y3Mrv13Y2d3bPygeHjVMkmpC6yThiW5FYChnktYts5y2lKYgIk6b0fB26jefqDYskQ92pGgooC9ZzAhYJzU6Q1AKusWSX/ZnwKskyEgJZah1iz+dXkJSQaUlHIxpB76y4Ri0ZYTTSaGTGqqADKFP245KENSE49m1E3zmlB6OE+1KWjxT/06MQRgzEpHrFGAHZtmbiv957dTG1+GYSZVaKsl8UZxybBM8fR33mKbE8pEjQDRzt2IyAA3EuoAWtkRi4jIJlhNYJY1KObgoV+4vS9WbLJ08OkGn6BwF6ApV0R2qoToi6BG9oFf05j17796H9zlvzXnZzDFagPf1C1gmlbM=</latexit> var( ˆfNL) ⇠ 2 <latexit sha1_base64="q7aORK2uPQFWV9/VujOi1K0JE98=">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</latexit> ˆfNL = g  1 ) var( ˆfNL) = 0 <latexit sha1_base64="acTMnBQP2Js1IiQWFlfVXRtc34c=">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</latexit> g = (1 + fNL)<latexit sha1_base64="qSZyT7RHnm+puQb5ztG1Up+wy9s=">AAACEnicbVDLSsNAFJ3UV62vqDvdDBahIpSkCrpQKLhxIVLBPqAJYTKdtENnkjAzEUoI+BF+g1tduxO3/oBL/8Rpm4WtHrhwOOde7r3HjxmVyrK+jMLC4tLySnG1tLa+sbllbu+0ZJQITJo4YpHo+EgSRkPSVFQx0okFQdxnpO0Pr8Z++4EISaPwXo1i4nLUD2lAMVJa8sy9PryEFfs48FJHcHh7kx1BZ4jiGEHPLFtVawL4l9g5KYMcDc/8dnoRTjgJFWZIyq5txcpNkVAUM5KVnESSGOEh6pOupiHiRLrp5IcMHmqlB4NI6AoVnKi/J1LEpRxxX3dypAZy3huL/3ndRAXnbkrDOFEkxNNFQcKgiuA4ENijgmDFRpogLKi+FeIBEggrHdvMFp9nOhN7PoG/pFWr2ifV2t1puX6Rp1ME++AAVIANzkAdXIMGaAIMHsEzeAGvxpPxZrwbH9PWgpHP7IIZGJ8/vOGcLw==</latexit> fNL<latexit sha1_base64="J3TCtcachvp5KbIs7TN/zj+HhGo=">AAACAHicbVA9SwNBEJ2LXzF+RS1tFoNgFe6ioIVFwMZCJIL5wCSEvc1esmR379jdE8KRxt9gq7Wd2PpPLP0n7iVXmMQHA4/3ZpiZ50ecaeO6305uZXVtfSO/Wdja3tndK+4fNHQYK0LrJOShavlYU84krRtmOG1FimLhc9r0R9ep33yiSrNQPphxRLsCDyQLGMHGSo9BL+koge5uJ71iyS27U6Bl4mWkBBlqveJPpx+SWFBpCMdatz03Mt0EK8MIp5NCJ9Y0wmSEB7RtqcSC6m4yvXiCTqzSR0GobEmDpurfiQQLrcfCt50Cm6Fe9FLxP68dm+CymzAZxYZKMlsUxByZEKXvoz5TlBg+tgQTxeytiAyxwsTYkOa2+CLNxFtMYJk0KmXvrFy5Py9Vr7J08nAEx3AKHlxAFW6gBnUgIOEFXuHNeXbenQ/nc9aac7KZQ5iD8/UL/Y2XJA==</latexit>
  • 19. 3Sample variance cancellation Planck Collaboration: Planck 2018 lensing Fig. 1. Mollweide projection in Galactic coordinates of the lensing-deflection reconstruction map from our baseline minimum- variance (MV) analysis. We show the Wiener-filtered displacement-like scalar field with multipoles ˆ↵MV LM = p L(L + 1)ˆMV LM , corre- sponding to the gradient mode (or E mode) of the lensing deflection angle. Modes with L < 8 have been filtered out. Our baseline lensing reconstruction map is shown in Fig. 1. In Sect. 2 we explain how this was obtained, and the changes compared to our analysis in PL2015. We also describe the new optimal filtering approach used for our best polarization anal- ysis. In Sect. 3 we present our main results, including power- spectrum estimates, cosmological parameter constraints, and a joint estimation of the lensing potential using the CIB. We end the section by using the estimates of the lensing map to delens the CMB, reducing the B-mode polarization power and sharpen- ing the acoustic peaks. In Sect. 4 we describe in detail a number of null and consistency tests, explaining the motivation for our data cuts and the limits of our understanding of the data. We also discuss possible contaminating signals, and assess whether they are potentially important for our results. In Sect. 5 we briefly de- scribe the various data products that are made available to the community, and we end with conclusions in Sect. 6. A series of appendices describe some technical details of the calculation of various biases that are subtracted, and derive the error model for the Monte Carlo estimates. 2. Data and methodology This final Planck lensing analysis is based on the 2018 Planck HFI maps as described in detail in Planck Collaboration III (2018). Our baseline analysis uses the SMICA foreground- cleaned CMB map described in Planck Collaboration IV (2018), and includes both temperature and polarization information. We use the Planck Full Focal Plane (FFP10) simulations, described in detail in Planck Collaboration III (2018), to remove a num- ber of bias terms and correctly normalize the lensing power- spectrum estimates. Our analysis methodology is based on the previous Planck analyses, as described in PL2013 and PL2015. After a summary of the methodology, Sect. 2.1 also lists the changes and improvements with respect to PL2015. Some de- tails of the covariance matrix are discussed in Sect. 2.2, and de- tails of the filtering in Sect. 2.3. The main set of codes applying the quadratic estimators will be made public as part of the CMB lensing toolbox LensIt.2 2.1. Lensing reconstruction The five main steps of the lensing reconstruction are as follows. 1. Filtering of the CMB maps. The observed sky maps are cut by a Galactic mask and have noise, so filtering is applied to remove the mask and approximately optimally weight for the noise. The lensing quadratic estimators use as input optimal Wiener-filtered X = T, E, and B CMB multipoles, as well as inverse-variance-weighted CMB maps. The latter maps can be obtained easily from the Wiener-filtered multipoles by divid- ing by the fiducial CMB power spectra Cfid ` before projecting onto maps. We write the observed temperature T and polariza- tion (written as the spin ±2 combination of Stokes parameters ±2P ⌘ Q ± iU) pixelized data as 0 BBBBBBBBBBBB@ Tdat 2Pdat 2Pdat 1 CCCCCCCCCCCCA = BY 0 BBBBBBBBBBBB@ T E B 1 CCCCCCCCCCCCA + noise, (1) 2 https://github.com/carronj/LensIt 3 Std. Big BangStd. Big Bang Enhanced tails (Angular scale)-1 Dalal+ (2008), Seljak (2009), McDonald & Seljak (2009), MS & Seljak (2018)
  • 21. Forecast for future experiments Compute Fisher information matrix assuming Gaussian likelihood (curvature of the likelihood near maximum = Hessian) Inverse gives lower bound on parameter error bars shot mber er- alize nal- t in bias mple rove and esti- and de- rove e in VII. oss- e 14 III. and wer one-dimensional data vector that starts with all spectra at `min, continues with all spectra at `min + 1, etc: d = d`min , d`min+1, . . . , d`max . (15) At each `, d` = C11 ` , C12 ` , . . . , CNN ` (16) contains N(N + 1)/2 spectra Cij ` with j i. Assum- ing Eq. (14), the covariance cov(d, d) is then a block- diagonal matrix with `max `min + 1 blocks of size N(N + 1)/2 ⇥ N(N + 1)/2, which is easily inverted if the number of fields is N . 100. The Fisher matrix at the power spectrum level is then Fab = `maxX `=`min @d` @✓a [cov(d`, d`)] 1 @d` @✓b . (17) We evaluate this without binning in `.2 for Fisher forecasts with Gaussian covariances, this may be more challenging for actual data analyses. In that case one may want to compress the observations before forming power spectra (see Appendix D). Although many of the cross-spectra have zero signal in the Limber approximation because they Fab = ⌧ @2 ln L(d|✓) @✓a@✓b<latexit sha1_base64="ta6ogSoEc9qH8kFkQwG7srESlKI=">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</latexit>
  • 22. If models work and systematics under control, what can we hope for? Optimistic setting
  • 23. 0 1 2 3 4 5 6 7 z 10 3 10 2 10 1 1 101 102dn/dz[arcmin2 ] 0 0.2 0.4 0.6 0.8 1.0 W W LSST Optimistic LSST Gold FIG. 1: The redshift distribution of the CMB lensing convergence (red curve, normalized to a unit maximum) and LSST galaxy sam- ples, both Optimistic (light gray) and Gold (dark gray). We as- sume 16 tomographic redshift bins in the range 0 < z < 7, cross- FIG. 2: masses w configura primordi LSST number density Gold sample CMB lensing kernel Optimistic sample At low z, use clustering redshifts (Gorecki+ 2014)
 At high z, add Lyman-break galaxies (dropouts; extrapolated from HSC observations) Total of 66 galaxies per arcmin2 (MS & Seljak 2018)
  • 24. Lyman-break galaxies (LBGs) at z>3 Photons blueward of 912 Å ionize neutral hydrogen in young star-forming galaxies
 Lyman break at (1+z) 912 Å Blue band ‘dropout’ 0.5 million z=4-7 LBGs found by HSC/Goldrush in 100 deg2
 
 
 Expect ~100 million in LSST Good for CMB lensing Xcorrel Ellis 1998 Ono, Ouchi+ (2018) 1704.06004
  • 25. di↵erent fsky, maybe? MS: We do not include photo-z errors. Hope this is not too bad because lensing kernel is very broad. Could matter for tracer cross tracer though. Maybe include photo z errors in forecasts? FIG. 3. Redshift distribution and tomographic redshift bins. For CMB lensing (solid black), we plot the dn/dz that would yield C if integrated over, with arbitrary normalization. It at high smaller the low also ha at low ` For a contrib satisfyi ples, wh the cor black li ples are coe cie (see Re combin the CM peak co motivat niques. but is s Dropout galaxies Tomographic redshift bins MS & Seljak (2018)
  • 26. Power spectra: CMB-S4 & LSST 5 FIG. 4. Left panel: Angular auto-power spectra of CMB-S4 lensing  map (black) and LSST galaxy density g (colored). Solid ines show the signal power (not including lensing noise or shot noise), and shaded regions show 1 error bars assuming the Gaussian covariance (27), fsky = 0.5, minimum variance lensing noise expected for CMB-S4, and LSST number density shown n Fig. 2. Dashed lines show lensing reconstruction noise (black) and shot noise (colored). Right panel: Angular cross-spectra between CMB lensing and LSST galaxy density. MS & Seljak (2018) gg κκ κg noise CXX0 ` = Z z P X X0 (k = `/ (z), z) ⇥ WX (z)bX(z)WX0 (z)bX0 (z). CXX0 ` = 2 ⇡ Z 1 0 dk k X,`(k) X0,`(k)k3 P X X0 (k, z = 0), X,`(k) ⌘ Z 1 0 d ¯WX ( )j`(k ). Low ell: High ell:
  • 27. SNR of auto-power spectra MS & Seljak (2018) 7 the combined relation coef- be as high as g very broad 100. The rea- ical distances a very low k 0 2 hMpc 1 ). power spec- wer spectrum projection in- lues of and scales smaller the contribu- re important. urces at z > 4 noise reduces variance can- re the tracers `max SNR of CXX 500 1000 2000 CMB 233 406 539 BOSS LRG z=0-0.9 140 187 230 SDSS r < 22 z=0-0.5 247 487 936 SDSS r < 22 z=0.5-0.8 247 487 936 DESI BGS z=0-0.5 230 417 665 DESI ELG z=0.6-0.8 158 210 256 DESI ELG z=0.8-1.7 150 194 225 DESI LRG z=0.6-1.2 184 267 349 DESI QSO z=0.6-1.9 44.8 48.8 50.8 LSST i < 27 (3yr) z=0-0.5 250 496 982 LSST i < 27 (3yr) z=0.5-1 250 496 979 LSST i < 27 (3yr) z=1-2 249 492 956 LSST i < 27 (3yr) z=2-3 245 469 830 LSST i < 27 (3yr) z=3-4 239 444 724 LSST i < 27 (3yr) z=4-7 224 387 555 TABLE I. Total signal-to-noise of auto-power spectra CXX `
  • 28. SNR of kg cross-power spectra Gaussian covariance (14), fsky = 0.5, minimum variance lensing noise expected in Fig. 4. Dashed lines show lensing reconstruction noise (black) and shot nois between CMB lensing and LSST galaxy density. `max SNR of CCMBX 500 1000 2000 BOSS LRG z=0-0.9 77.3 117 159 SDSS r < 22 z=0-0.5 88.3 167 284 SDSS r < 22 z=0.5-0.8 88.3 167 284 DESI BGS z=0-0.5 50.1 93.5 144 DESI ELG z=0.6-0.8 50.7 73.5 97 DESI ELG z=0.8-1.7 103 148 185 DESI LRG z=0.6-1.2 86.7 133 182 DESI QSO z=0.6-1.9 74.9 94.5 108 LSST i < 27 (3yr) z=0-0.5 78.1 150 258 LSST i < 27 (3yr) z=0.5-1 112 202 338 LSST i < 27 (3yr) z=1-2 144 259 406 LSST i < 27 (3yr) z=2-3 121 219 324 LSST i < 27 (3yr) z=3-4 101 182 261 LSST i < 27 (3yr) z=4-7 94 167 229 TABLE II. Like Table I but for CMB-lensing–clustering cross- spectra Cg ` . For galaxy by more than z & 2). At ` but less than of Fig. 7 com ance of each ments or exp shows that t 0.1 per mo lower-redshif ample, the f 0.05 per mo The g cro nal than gg a quence of com than with th signal-to-nois sponding gg the cross-cor and each ind Fig. 5 above,MS & Seljak (2018)
  • 29. Correlation of CMB lensing and galaxies FIG. 3. Correlation r = Cg (C Cgg ) 1/2 of CMB-S4 lensing A. Local primordia 1. Motivation to The large-scale structure of primordial density fluctuation universe. Measuring the stat scale structures using galaxy can thus constrain the physics fluctuations. A non-Gaussian tribution function (pdf) can o inflation models, involving for higher-derivative operators. local type of primordial non- the primordial potential is th sian field and its square, (x)+ has a non-Gaussian pdf. If w amplitude, fNL & 1, it will r of the inflationary expansion robust way [25, 26]. This is o tional means to rule out a who early-universe models. In practice the measurement threshold signal fNL = 1 sepa and multi-field models has a v ables. The best constraints, f Planck’s CMB temperature a ments [27]. does not improve our forecasts because we cros against CMB lensing, so that redshift accurac less important than number density. We assu of b(z) = 1.7 ¯D 1 (z) where ¯D(z = 0) = 1. For DESI, we use five redshift samples, wit densities from Table 2.3 in Ref. [47]: The lo BGS sample at 0  z  0.5 with 9.6 ⇥ 106 o bias b(z) = 1.34 ¯D 1 (z), the LRG sample at 0.6 with 3.9 ⇥ 106 objects and bias b(z) = 1.7 ¯D ELG sample at 0.6  z  0.8 with 3.5 ⇥ 1 and bias b(z) = 0.84 ¯D 1 (z), a second ELG 0.8  z  1.7 with 1.3 ⇥ 107 objects and the and a QSO sample at 0.6  z  1.9 with 1.4⇥1 and bias 1.2 ¯D 1 (z). In each case, the number refers to a survey area of 14, 000 deg2 . D. CMB lensing–LSS correlation coe The performance of the cross-correlation an pends on the cross-correlation coe cient r` = Cg ` q ˆC ` ˆCgg ` between the measured CMB lensing converge the observed galaxy density g, where the pow ˆC include lensing reconstruction noise and s CMB lensing and galaxy maps are up to 95% correlated MS & Seljak (2018)
  • 30. MS & Seljak (2018); Forecast papers for Simons Observatory, CMB-S4, NASA PICO satellite (Largest angular scale)-1 —> Potential to rule out all Big Bang models driven by a single field Forecast for local non-Gaussianity 12 FIG. 9. MS: Date: 20 Aug 2017 Constraints on primordial non-Gaussianity amplitude fNL as a function of minimum wavenumber `min, for di↵erent LSS surveys (colors), with full sky overlap between observations Threshold needed to distinguish Big Bang models Forecastfor Different experiments `min ⇠<latexit sha1_base64="JpqnNPjmjQoSX2eihOhMzZtQsZ0=">AAACDXicbVDLSsNAFJ3UV62vqLhyM1gEVyWpgi5cFNy4rGAf0IQwmU7aoTOTMDMRSsg3+A1ude1O3PoNLv0TJ20WtvXAhcM593IPJ0wYVdpxvq3K2vrG5lZ1u7azu7d/YB8edVWcSkw6OGax7IdIEUYF6WiqGeknkiAeMtILJ3eF33siUtFYPOppQnyORoJGFCNtpMA+8QhjgceRHkuecSpyT1Ee2HWn4cwAV4lbkjoo0Q7sH28Y45QToTFDSg1cJ9F+hqSmmJG85qWKJAhP0IgMDBWIE+Vns/g5PDfKEEaxNCM0nKl/LzLElZry0GwWOdWyV4j/eYNURzd+RkWSaiLw/FGUMqhjWHQBh1QSrNnUEIQlNVkhHiOJsDaNLXwJeW46cZcbWCXdZsO9bDQfruqt27KdKjgFZ+ACuOAatMA9aIMOwCADL+AVvFnP1rv1YX3OVytWeXMMFmB9/QJGU5zP</latexit> With/without sample variance cancellation (fNL)<latexitsha1_base64="SjRPaUwNSZsk4HkulZr/qbH5CbU=">AAACCnicbVDLSsNAFJ34rPUV69LNYBHqpiRV0IWLghsXIhXsA5oQJtNJO3RmEmYmYgn9A7/Bra7diVt/wqV/4rTNwrYeuHA4517O5YQJo0o7zre1srq2vrFZ2Cpu7+zu7dsHpZaKU4lJE8cslp0QKcKoIE1NNSOdRBLEQ0ba4fB64rcfiVQ0Fg96lBCfo76gEcVIGymwS56ifY4qUZB5ksO72/FpYJedqjMFXCZuTsogRyOwf7xejFNOhMYMKdV1nUT7GZKaYkbGRS9VJEF4iPqka6hAnCg/m/4+hidG6cEolmaEhlP170WGuFIjHppNjvRALXoT8T+vm+ro0s+oSFJNBJ4FRSmDOoaTImCPSoI1GxmCsKTmV4gHSCKsTV1zKSEfm07cxQaWSatWdc+qtfvzcv0qb6cAjsAxqAAXXIA6uAEN0AQYPIEX8ArerGfr3fqwPmerK1Z+cwjmYH39AhnnmmM=</latexit>
  • 31. Growth of structure as function of time Expansion history / geometry / dark energy Sum of neutrino masses Galaxy formation? Other science from cross-correlations
  • 32. 30 Byeonghee Yu (PhD student @ UC Berkeley) Example: Sum of neutrino masses
  • 33. Yu, Knight et al. (arXiv:1809.02120) 0 1 2 3 4 5 6 7 z 0.98 0.99 1.0 1.01 8(z)/8(z),fiducial a 3f /5 Marginalize over CDM parameters and Bi; S4 + DESI added Fixing CDM parameters FIG. 4: 1 constraints on the matter amplitude 8 in 6 tomo- graphic redshift bins, z = 0 0.5, 0.5 1, 1 2, 2 3, 3 4, 4 7, 60 meV neutrinos Massless neutrinos Example: Sum of neutrino masses Varianceoffluctuations (Size of the Universe)-1 - 1 Fix other cosmological params Marginalize over other cosmological params
  • 34. Uses z<7 redshift lever arm Measures neutrino mass without optical depth to CMB Independent & competitive to usual probes Yu, Knight, Sherwin+ (1809.02120) 3 0.2 1 2 3 4 5 6 7 zmax 20 30 40 50 100 ( P m)[meV] S4Lens + LSST (Optimistic) S4Lens + LSST (Optimistic) + Planck(l > 30) S4Lens + LSST (Optimistic) + S4(l > 30) Planck/S4(l > 30) + DESI S4Lens + Planck/S4(l > 30) + DESI S4Lens + LSST (Gold) + Planck/S4(l > 30) + DESI S4Lens + LSST (Optimistic) +Planck/S4(l > 30) + DESI FIG. 2: Forecasted 1 constraints on the sum of the neutrino 100 30 20 50 3 0.2 1 2 3 4 5 6 7 zmax 20 30 40 50 100 ( P m)[meV] S4Lens + LSST (Optimistic) S4Lens + LSST (Optimistic) + Planck(l > 30) S4Lens + LSST (Optimistic) + S4(l > 30) Planck/S4(l > 30) + DESI S4Lens + Planck/S4(l > 30) + DESI S4Lens + LSST (Gold) + Planck/S4(l > 30) + DESI S4Lens + LSST (Optimistic) +Planck/S4(l > 30) + DESI FIG. 2: Forecasted 1 constraints on the sum of the neutrino masses without optical depth information, for di↵erent experiment configurations: CMB-S4 lensing and LSST clustering (black) + primordial CMB data (green dotted for Planck and green solid for Example: Sum of neutrino masses
  • 35. 33 Sensitive to kmax, but not too much when all data combined ( P m⌫ ) [meV] (Gold/Optimistic) kmax Lens + LSST + Planck/S4 T&P + DESI 0.05 307 / 243 94 / 68 32 / 29 0.1 176 / 129 68 / 53 31 / 27 0.2 107 / 71 47 / 38 28 / 25 0.3 84 / 55 40 / 33 27 / 24 0.4 79 / 49 38 / 31 26 / 23 TABLE I: Forecasts of the neutrino mass constraints without op- tical depth information, for di↵erent LSST number densities and redshift distributions, kmax limits, and lensing reconstruction noise levels. For di↵erent combinations of data, constraints provided on the left assume the LSST Gold sample, and those on the right assume the LSST Optimistic sample. spectra is given by n Len kmax (⌧) = 0.3 25 ( P Lens + kmax (⌧) = 0.3 22 / TABLE II: Forecasts of t flat priors on the optica S4 lensing, S4 primary BAO information. Botto numbers on the left assu assume the Optimistic s Yu, Knight, Sherwin+ (1809.02120) Ignores scale-dependent bias due to neutrinos LoVerde 2014, LoVerde 2016, Muñoz & Dvorkin 2018, MS & Seljak 2018, Chiang, LoVerde & Villaescusa-Navarro 2019 Example: Sum of neutrino masses
  • 36. Modeling all power spectra for nonlinear scales (high L) Photometric redshift errors Relationship between galaxies and dark matter (galaxy bias) Challenges for growth measurements
  • 37. Other cool things to do with CMB lensing (1) Joint analysis of CMB and CMB magnification, with covariance (2) Estimate unlensed CMB
 
 
 
 (3) Biases of the magnification estimator Figure 1.2: Illustration of gravitational lensing of CMB photons by large-scale structure (adapted from [10]). reconstructing these lenses from the observed CMB we can obtain crucial infor- mation on e.g. dark energy or the geometry of the universe. Several experiments will provide high quality CMB data in the next few years, e.g. full-mission Planck data including polarization, ACT/ACTPol, Polarbear and SPT/SPTpol. The contribution of this thesis, presented in Chapter 5, is a thorough analysis of how the reconstructed lensing information can be combined with the primary CMB data to perform reliably a joint parameter estimation. Such joint analy- ses are important to break degeneracies that limit the information that can be extracted from the CMB fluctuations laid down at recombination. The joint anal- ysis is non-trivial because the part of the lensing information that is present in the primary CMB power spectrum as well as in the lensing reconstruction is po- tentially double-counted. We quantify the temperature-lensing cross-correlation analytically, finding two physical contributions and confirming the results with simulated lensed CMB maps. This cross-correlation has not been considered be- fore and it could have turned out to be anywhere between zero and unity. We 15 (a) Theoretical matter cosmic variance contribution (D9) (b) Simulations FIG. 5. (a) Theoretical matter cosmic variance contribution (D9) to the correlation of the unbinned power spectra of recon- structed lensing potential and lensed temperature. The covariance (D9) is converted to a correlation using the same conversion factor as in (38). (b) Measured correlation ˆcorrel( ˆC ˆin ˆin L , ˆC ˜T ˜T LT ˆCT T LT ) of the input lensing potential power and the di↵erence of noise-free lensed and unlensed temperature powers in 1000 simulations. FIG. 6. Left: The approximate contribution to the covariance between [L(L + 1)]2 ˆC ˆˆ L /(2⇡) and L0 (L0 + 1) ˆC ˜T ˜T L0 /(2⇡) from cosmic variance of the lenses, derived from Eq. (D9). Right: The rank-one approximation to the matrix on the left from retaining only the largest singular value.TODOO: discuss this in main text or appendix CT correlations in all calculations we try to eliminate correlations between the lensing potential and the unlensed temperature in the simulations by considering ˆcov( ˆC ˆˆ L , ˆC ˜T ˜T L0 )|CV( ) = ˆcov( ˆC ˆin ˆin L , ˆC ˜T ˜T L0 ˆCT T L0 ), (52) where ˆC ˆin ˆin is the empirical power of the input lensing potential and ˆCT T is the empirical power of the unlensed temperature. Subtracting the unlensed from the lensed empirical power spectrum also reduces the noise of the covariance estimate because it eliminates the scatter due to cosmic variance of the unlensed temperature. We estimate the covariance in simulations similarly to (48). As shown in Fig. 5b these measurements agree with the theoretical expectation from (D9). `0 <latexit sha1_base64="X1y07k9ZmLbjfpS1W4qyfTp9Ng0=">AAAB+3icbVA9SwNBEJ3zM8avqKXNYhCtwl0UtLAI2FhG8JJAcoS9zV6yZHfv2N0TwnG/wVZrO7H1x1j6T9wkV5jEBwOP92aYmRcmnGnjut/O2vrG5tZ2aae8u7d/cFg5Om7pOFWE+iTmseqEWFPOJPUNM5x2EkWxCDlth+P7qd9+pkqzWD6ZSUIDgYeSRYxgYyW/Rzm/6Feqbs2dAa0SryBVKNDsV356g5ikgkpDONa667mJCTKsDCOc5uVeqmmCyRgPaddSiQXVQTY7NkfnVhmgKFa2pEEz9e9EhoXWExHaToHNSC97U/E/r5ua6DbImExSQyWZL4pSjkyMpp+jAVOUGD6xBBPF7K2IjLDCxNh8FraEIreZeMsJrJJWveZd1eqP19XGXZFOCU7hDC7BgxtowAM0wQcCDF7gFd6c3Hl3PpzPeeuaU8ycwAKcr18nCZUA</latexit> `<latexitsha1_base64="bfqfNP2WZfAkR6k1rfkz29U6stI=">AAAB+nicbVA9SwNBEJ3zM8avqKXNYhCswl0UtLAI2FhGMB+QHGFvM0mW7O0du3tCOPMXbLW2E1v/jKX/xL3kCpP4YODx3gwz84JYcG1c99tZW9/Y3Nou7BR39/YPDktHx00dJYphg0UiUu2AahRcYsNwI7AdK6RhILAVjO8yv/WESvNIPppJjH5Ih5IPOKMmk7ooRK9UdivuDGSVeDkpQ456r/TT7UcsCVEaJqjWHc+NjZ9SZTgTOC12E40xZWM6xI6lkoao/XR265ScW6VPBpGyJQ2ZqX8nUhpqPQkD2xlSM9LLXib+53USM7jxUy7jxKBk80WDRBATkexx0ucKmRETSyhT3N5K2IgqyoyNZ2FLEE5tJt5yAqukWa14l5Xqw1W5dpunU4BTOIML8OAaanAPdWgAgxG8wCu8Oc/Ou/PhfM5b15x85gQW4Hz9AsMDlM8=</latexit> 15 (a) Theoretical matter cosmic variance contribution (D9) (b) Simulations FIG. 5. (a) Theoretical matter cosmic variance contribution (D9) to the correlation of the unbinned power spectra of recon- structed lensing potential and lensed temperature. The covariance (D9) is converted to a correlation using the same conversion factor as in (38). (b) Measured correlation ˆcorrel( ˆC ˆin ˆin L , ˆC ˜T ˜T LT ˆCT T LT ) of the input lensing potential power and the di↵erence of noise-free lensed and unlensed temperature powers in 1000 simulations. FIG. 6. Left: The approximate contribution to the covariance between [L(L + 1)]2 ˆC ˆˆ L /(2⇡) and L0 (L0 + 1) ˆC ˜T ˜T L0 /(2⇡) from cosmic variance of the lenses, derived from Eq. (D9). Right: The rank-one approximation to the matrix on the left from retaining only the largest singular value.TODOO: discuss this in main text or appendix CT correlations in all calculations we try to eliminate correlations between the lensing potential and the unlensed temperature in the simulations by considering ˆcov( ˆC ˆˆ L , ˆC ˜T ˜T L0 )|CV( ) = ˆcov( ˆC ˆin ˆin L , ˆC ˜T ˜T L0 ˆCT T L0 ), (52) where ˆC ˆin ˆin is the empirical power of the input lensing potential and ˆCT T is the empirical power of the unlensed temperature. Subtracting the unlensed from the lensed empirical power spectrum also reduces the noise of the covariance estimate because it eliminates the scatter due to cosmic variance of the unlensed temperature. We estimate the covariance in simulations similarly to (48). As shown in Fig. 5b these measurements agree with the theoretical expectation from (D9). `0 <latexit sha1_base64="X1y07k9ZmLbjfpS1W4qyfTp9Ng0=">AAAB+3icbVA9SwNBEJ3zM8avqKXNYhCtwl0UtLAI2FhG8JJAcoS9zV6yZHfv2N0TwnG/wVZrO7H1x1j6T9wkV5jEBwOP92aYmRcmnGnjut/O2vrG5tZ2aae8u7d/cFg5Om7pOFWE+iTmseqEWFPOJPUNM5x2EkWxCDlth+P7qd9+pkqzWD6ZSUIDgYeSRYxgYyW/Rzm/6Feqbs2dAa0SryBVKNDsV356g5ikgkpDONa667mJCTKsDCOc5uVeqmmCyRgPaddSiQXVQTY7NkfnVhmgKFa2pEEz9e9EhoXWExHaToHNSC97U/E/r5ua6DbImExSQyWZL4pSjkyMpp+jAVOUGD6xBBPF7K2IjLDCxNh8FraEIreZeMsJrJJWveZd1eqP19XGXZFOCU7hDC7BgxtowAM0wQcCDF7gFd6c3Hl3PpzPeeuaU8ycwAKcr18nCZUA</latexit> MS, Challinor et al. (2013) Peloton, MS, et al. (2017) Planck collab. (2013, 2018) e.g. Sherwin & MS (2015) Böhm, MS & Sherwin (2016) Beck, Fabbian & Errard (2018) Böhm, Sherwin et al. (2018) Planck Collaboration: Gravitational lensing by large-scale structures with Planck 0.12 0.08 0.04 0.00 0.04 K 40 48 56 64 72 80 H0 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 Fig. 15. Two views of the geometric degeneracy in curved ⇤CDM models which is partially broken by lensing. Left: the degeneracy in the ⌦m-⌦⇤ plane, with samples from Planck+WP+highL colour coded by the value of H0. The contours delimit the 68% and 95% confidence regions, showing the further improvement from including the lensing likelihood. Right: the degeneracy in the ⌦K-H0 plane, with samples colour coded by ⌦⇤. Spatially-flat models lie along the grey dashed lines. 6.2. Correlation with the ISW Effect As CMB photons travel to us from the last-scattering surface, the gravitational potentials they traverse may undergo a non- negligible amount of evolution. This produces a net redshift or blueshift of the photons concerned, as they fall into and then escape from the evolving potentials. The overall result is a con- tribution to the CMB temperature anisotropy known as the late- time ISW e↵ect, or the Rees-Sciama (R-S) e↵ect depending on whether the evolution of the potentials concerned is in the linear (ISW) or non-linear (R-S) regime of structure formation (Sachs & Wolfe 1967; Rees & Sciama 1968). In the epoch of dark en- ergy domination, which occurs after z ⇠ 0.5 for the concor- dance ⇤CDM cosmology, large-scale potentials tend to decay over time as space expands, resulting in a net blueshifting of the CMB photons which traverse these potentials. In the concordance ⇤CDM model, there is significant over- lap between the large-scale structure which sources the CMB lensing potential and the ISW e↵ect (greater than 90% at L < 100), although it should be kept in mind that we cannot observe the ISW component by itself, and so the e↵ective cor- relation with the total CMB temperature is much smaller, on the order of 20%. The correlation between the lensing potential and the ISW e↵ect results in a non-zero bispectrum or three-point function for the observed CMB fluctuations. This bispectrum is peaked for “squeezed” configurations, in which one short leg at low-` Following Lewis et al. (2011), we begin with an estimator for the cross-spectrum of the lensing potential and the ISW e↵ect as ˆCT L = f 1 sky 2L + 1 X M ˆTLM ˆ⇤ LM, (45) where ˆT`m = CTT ` ¯T`m is the Wiener-filtered temperature map and ˆ is given in Eq. (13). In Fig. 16 we plot the measured cross- spectra for our individual frequency reconstructions at 100, 143, and 217 GHz as well as the MV reconstruction. We also plot the mean and scatter expected in the fiducial ⇤CDM model. To compare quantitatively the overall level of the measured CT L correlation to the value in ⇤CDM, we estimate an overall amplitude for the cross-spectrum as ˆAT = NT LmaxX L=Lmin (2L + 1)CT ,fid. L ˆCT L /(CTT L NL ). (46) The overall normalization NT is determined from Monte-Carlo simulations. For our processing of the data, we find that it is well approximated (at the 5% level) by the analytical approximation NT ⇡ 2 6666664 LmaxX L=Lmin (2L + 1) ⇣ CT ,fid. L ⌘2 /(CTT L NL ) 3 7777775 1 . (47) Curvature of space Expansionrate Amountofdarkenergy Joint analysis
  • 38. Science with galaxy catalogs (1) Bias model at the field level (2) Cosmological parameter analysis (3) Accounting for skewness (4) Getting initial from final conditions 13 Figure 1. 2-d slices of the overdensity h(x) of simulated 1010.8 1011.8 h 1 M halos (top), compared with the cubic bias model (center), and the linear Standard Eulerian bias model (bottom). Each panel is 500 h 1 Mpc wide and 110 h 1 Mpc high, and each density is smoothed with a R = 2 h 1 Mpc Gaussian, WR(k) = exp[ (kR)2 /2]. The colorbar indicates the values of this smoothed overdensity h(x). Figure 2. Same as Fig. 1 but for more massive and less abundant 1011.8 1012.8 h 1 M halos. simulations. The variance, skewness and kurtosis of the densities shown in the histograms are listed in Table II for the full simulated and modeled densities, and in Table III for the model error. The linear Standard Eulerian bias model tends to underpredict troughs and overpredict peaks of the halo density, as shown in Fig. 3. The model error is not Gaussian for any of the shown smoothing scales; in particular its kurtosis is larger than 1 for all smoothing scales. The cubic model provides a more accurate description of the halo density pdf, as shown in Fig. 4. This emphasizes the importance of using nonlinear bias terms even on rather large scales. Still, the cubic model tends to underpredict the peaks of the true halo density, especially on small scales. This agrees with Fig. 2 where the model also underpredicts
  • 39. Science with galaxy catalogs (1) Bias model at the field level (2) Cosmological parameter analysis (3) Accounting for skewness (4) Getting initial from final conditions 13 Figure 1. 2-d slices of the overdensity h(x) of simulated 1010.8 1011.8 h 1 M halos (top), compared with the cubic bias model (center), and the linear Standard Eulerian bias model (bottom). Each panel is 500 h 1 Mpc wide and 110 h 1 Mpc high, and each density is smoothed with a R = 2 h 1 Mpc Gaussian, WR(k) = exp[ (kR)2 /2]. The colorbar indicates the values of this smoothed overdensity h(x). Figure 2. Same as Fig. 1 but for more massive and less abundant 1011.8 1012.8 h 1 M halos. simulations. The variance, skewness and kurtosis of the densities shown in the histograms are listed in Table II for the full simulated and modeled densities, and in Table III for the model error. The linear Standard Eulerian bias model tends to underpredict troughs and overpredict peaks of the halo density, as shown in Fig. 3. The model error is not Gaussian for any of the shown smoothing scales; in particular its kurtosis is larger than 1 for all smoothing scales. The cubic model provides a more accurate description of the halo density pdf, as shown in Fig. 4. This emphasizes the importance of using nonlinear bias terms even on rather large scales. Still, the cubic model tends to underpredict the peaks of the true halo density, especially on small scales. This agrees with Fig. 2 where the model also underpredicts
  • 40. Overview We calculate halo density field in PT and compare to simulations 1. How well does perturbative bias expansion work? 2. How correlated is the halo density field with the initial conditions? 3. What are the properties of the noise? Joint work with Marko Simonović, Valentin Assassi & Matias Zaldarriaga
  • 41. Overview Most of the analyses use n-point functions. Disadvantages: These questions have been extensively explored in the past Desjacques, Jeong, Schmidt: Large-Scale Galaxy Bias — Cosmic variance, compromise on resolution/size of the box — At high k hard to disentangle different sources of nonlinearities — Overfitting (smooth curves, many parameters) — Only a few lowest n-point functions explored in practice — Difficult to isolate and study the noise
  • 42. Overview Advantages: Use fields rather than summary statistics — No cosmic variance, small boxes with high resolution are sufficient — High S/N at low k, no need to go to the very nonlinear regime — No overfitting, each Fourier mode (amplitude and phase) is fitted — “All” n-point functions measured simultaneously — Easier to isolate and study the noise Roth & Porciani (2011) Baldauf, Schaan, Zaldarriaga (2015) Lazeyras, Schmit (2017) Abidi, Baldauf (2018) McQuinn, D’Aloisio (2018) Taruya, Nishimichi, Jeong (2018)
  • 43. Overview Same initial conditions Sim ulation model<latexit sha1_base64="fJ3rOzfgxg04Png+ZlhcqA4B/Z4=">AAACCnicbVC7TsMwFHXKq5RXaEcWiwoJMVRJGehGJRgYi0QfUhNFjuO0Vu0ksh1EFfUP+AZWYGVDrHwAKyN/gtN2oC1HutK559yre3X8hFGpLOvbKKytb2xuFbdLO7t7+wfmYbkj41Rg0sYxi0XPR5IwGpG2ooqRXiII4j4jXX90lfvdeyIkjaM7NU6Iy9EgoiHFSGnJM8tOQJhCXuYIDnmsm4lnVq2aNQVcJfacVJuVxvVX5fWs5Zk/ThDjlJNIYYak7NtWotwMCUUxI5OSk0qSIDxCA9LXNEKcSDeb/j6BJ1oJYBgLXZGCU/XvRoa4lGPu60mO1FAue7n4n9dPVdhwMxolqSIRnh0KUwZVDPMgYEAFwYqNNUFYUP0rxEMkEFY6roUrPs8zsZcTWCWdes0+r9VvdTiXYIYiOALH4BTY4AI0wQ1ogTbA4AE8gWfwYjwab8a78TEbLRjznQpYgPH5C8f9naA=</latexit> truth<latexit sha1_base64="IB7CHTpqOeRx7pTZPfheU1Oiklg=">AAACCnicbVC7TgJBFJ3FF+JrhdJmIjExFmQXC+kk0cISE3kksCGzwwATZnY3M3eNZMMf+A22amtnbP0AW0v/xFmgEPAkNzk5596cm+NHgmtwnG8rs7a+sbmV3c7t7O7tH9iH+YYOY0VZnYYiVC2faCZ4wOrAQbBWpBiRvmBNf3SV+s17pjQPgzsYR8yTZBDwPqcEjNS1850eE0C6SUdJDCqG4aRrF52SMwVeJe6cFKuFyvVX4fWs1rV/Or2QxpIFQAXRuu06EXgJUcCpYJNcJ9YsInREBqxtaEAk014y/X2CT4zSw/1QmQkAT9W/FwmRWo+lbzYlgaFe9lLxP68dQ7/iJTyIYmABnQX1Y4EhxGkRuMcVoyDGhhCquPkV0yFRhIKpayHFl2kn7nIDq6RRLrnnpfKtKecSzZBFR+gYnSIXXaAqukE1VEcUPaAn9IxerEfrzXq3PmarGWt+U0ALsD5/AQRancY=</latexit> PT m odel
  • 44. First predict dark matter (purple), then halos (yellow) Modeling halo number density
  • 45. Simple peak model Whenever dark matter density > threshold, a halo forms halo formation in peaks Gaussian fluctuations on various scales (described by the power spectrum) Peak-Background Split • Schematic Picture: 3 2 1 0 x δc Large Scale "Background" Enhanced "Peaks" →→first sites of halo formation W. Hu first sites of snowfall → → Halo forms
  • 46. Simple peak model Whenever dark matter density > threshold, a halo forms halo formation in peaks Gaussian fluctuations on various scales (described by the power spectrum) Peak-Background Split • Schematic Picture: 3 2 1 0 x δc Large Scale "Background" Enhanced "Peaks" →→first sites of halo formation W. Hu first sites of snowfall → → Halo forms halo formation in peaks Gaussian fluctuations on various scales (described by the power spectrum) Peak-Background Split • Schematic Picture: 3 2 1 0 x δc Large Scale "Background" Enhanced "Peaks" →→first sites of halo formation W. Hu first sites of snowfall → → Snow falls
  • 47. Simple peak model Whenever dark matter density > threshold, a halo forms halo formation in peaks Gaussian fluctuations on various scales (described by the power spectrum) Peak-Background Split • Schematic Picture: 3 2 1 0 x δc Large Scale "Background" Enhanced "Peaks" →→first sites of halo formation W. Hu first sites of snowfall → → Halo forms halo formation in peaks Gaussian fluctuations on various scales (described by the power spectrum) Peak-Background Split • Schematic Picture: 3 2 1 0 x δc Large Scale "Background" Enhanced "Peaks" →→first sites of halo formation W. Hu first sites of snowfall → → Snow falls If we zoom out, halos trace the large scale background that modulates the small-scale fluctuations and increases P(>threshold).
  • 48. using fractional deviations from the mean and proportionality constant (related to how likely a halo forms when changing the threshold density) Simple peak model Tracing the large scale background means halo number density is proportional to matter density, if both smoothed on large scales:
 Write this as g(x) = b1 m(x)<latexit sha1_base64="XaHcr7zAchcoHCuF4bu18V4pL3I=">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</latexit> g(x) ⌘ ng(x) hngi 1 <latexit sha1_base64="zPWmnTIRWzX9q862OsLLcLaQHNU=">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</latexit> m(x) ⌘ ⇢m(x) h⇢mi 1 <latexit sha1_base64="tVCTXJMrljgnXrmyqQ9zwu1iRlI=">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</latexit> ng(x) / ⇢m(x)<latexit sha1_base64="99dFa20NV5ubQOIpaYS4NMUSMrw=">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</latexit> b1<latexit sha1_base64="vFL/SH1AeDkXA67O23grfojqbUQ=">AAAB+XicbVA9SwNBEJ31M8avqKXNYhCswl0UtLAI2FhGNB+QHGFvs5cs2d07dveEcOQn2GptJ7b+Gkv/iZvkCpP4YODx3gwz88JEcGM97xutrW9sbm0Xdoq7e/sHh6Wj46aJU01Zg8Yi1u2QGCa4Yg3LrWDtRDMiQ8Fa4ehu6reemTY8Vk92nLBAkoHiEafEOukx7Pm9UtmreDPgVeLnpAw56r3ST7cf01QyZakgxnR8L7FBRrTlVLBJsZsalhA6IgPWcVQRyUyQzU6d4HOn9HEUa1fK4pn6dyIj0pixDF2nJHZolr2p+J/XSW10E2RcJallis4XRanANsbTv3Gfa0atGDtCqObuVkyHRBNqXToLW0I5cZn4ywmskma14l9Wqg9X5dptnk4BTuEMLsCHa6jBPdShARQG8AKv8IYy9I4+0Oe8dQ3lMyewAPT1C5XxlB4=</latexit>
  • 49. using fractional deviations from the mean and proportionality constant (related to how likely a halo forms when changing the threshold density) Simple peak model Tracing the large scale background means halo number density is proportional to matter density, if both smoothed on large scales:
 Write this as g(x) = b1 m(x)<latexit sha1_base64="XaHcr7zAchcoHCuF4bu18V4pL3I=">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</latexit> g(x) ⌘ ng(x) hngi 1 <latexit sha1_base64="zPWmnTIRWzX9q862OsLLcLaQHNU=">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</latexit> m(x) ⌘ ⇢m(x) h⇢mi 1 <latexit sha1_base64="tVCTXJMrljgnXrmyqQ9zwu1iRlI=">AAACVXicbZBLTxsxFIWdgQJNaQl02Y1FVAkWjWZopXbRBVI3LEFqACkeRXc8dxILP6a2JxCN5tfxI6ouu2TbrivhJLMojytZOj7H19f+slIK5+P4VydaW3+xsbn1svtq+/Wbnd7u3rkzleU45EYae5mBQyk0Dr3wEi9Li6AyiRfZ1bdFfjFD64TR3/28xFTBRItCcPDBGvdSpvGaG6VA5zWb3TQ1U+CnWVHfNE2X5Sg9jNVBSA4pwx+VmFFWWOA1s1PTBqFHgp5IpCuT2eWu+ZCMe/14EC+LPhVJK/qkrdNx7zfLDa8Uas8lODdK4tKnNVgveLixyyqHJfArmOAoSA0KXVovMTT0fXByWhgblvZ06f7fUYNybq6ycHLxR/c4W5jPZaPKF1/SWuiy8qj5alBRSeoNXTClubDIvZwHAdyK8FbKpxAo+UD+wZRMNYFJ8pjAU3F+NEg+Do7OPvWPv7Z0tsg7sk8OSEI+k2NyQk7JkHByS+7IH/K387PzL1qPNlZHo07b85Y8qGjnHvnPuAU=</latexit> ng(x) / ⇢m(x)<latexit sha1_base64="99dFa20NV5ubQOIpaYS4NMUSMrw=">AAACMXicbVDLSsNAFJ34tr6iLt0MFqFuSqKCCi4ENy4r2FpoQphMJ+3gPMLMRC0h3+JH+A1udd2duvUnnLZZWOuBC4dz7uXee+KUUW08b+jMzS8sLi2vrFbW1jc2t9ztnZaWmcKkiSWTqh0jTRgVpGmoYaSdKoJ4zMhdfH818u8eiNJUilszSEnIUU/QhGJkrBS554Egj1hyjkQ3Dx6eijzgyPTjJH8qioqIejUrHsIgVTI1EgaqLyM+1iK36tW9MeAs8UtSBSUakfsZdCXOOBEGM6R1x/dSE+ZIGYoZKSpBpkmK8D3qkY6lAnGiw3z8YgEPrNKFiVS2hIFj9fdEjrjWAx7bztH9+q83Ev/zOplJzsKcijQzRODJoiRj0D47ygt2qSLYsIElCCtqb4W4jxTCxqY6tSXmhc3E/5vALGkd1f3j+tHNSfXyokxnBeyBfVADPjgFl+AaNEATYPAMXsEbeHdenKHz4XxNWueccmYXTMH5/gHLg6ub</latexit> b1<latexit sha1_base64="vFL/SH1AeDkXA67O23grfojqbUQ=">AAAB+XicbVA9SwNBEJ31M8avqKXNYhCswl0UtLAI2FhGNB+QHGFvs5cs2d07dveEcOQn2GptJ7b+Gkv/iZvkCpP4YODx3gwz88JEcGM97xutrW9sbm0Xdoq7e/sHh6Wj46aJU01Zg8Yi1u2QGCa4Yg3LrWDtRDMiQ8Fa4ehu6reemTY8Vk92nLBAkoHiEafEOukx7Pm9UtmreDPgVeLnpAw56r3ST7cf01QyZakgxnR8L7FBRrTlVLBJsZsalhA6IgPWcVQRyUyQzU6d4HOn9HEUa1fK4pn6dyIj0pixDF2nJHZolr2p+J/XSW10E2RcJallis4XRanANsbTv3Gfa0atGDtCqObuVkyHRBNqXToLW0I5cZn4ywmskma14l9Wqg9X5dptnk4BTuEMLsCHa6jBPdShARQG8AKv8IYy9I4+0Oe8dQ3lMyewAPT1C5XxlB4=</latexit> linear regression
  • 50. Simulations log M[h 1 M ] ¯n [(h 1 Mpc) 3 ] ¯n is comparable to 10.8 11.8 4.3 ⇥ 10 2 LSST [80, 81], Billion Object Apparatus [82] 11.8 12.8 5.7 ⇥ 10 3 SPHEREx [83, 84] 12.8 13.8 5.6 ⇥ 10 4 BOSS CMASS [85], DESI [86, 87], Euclid [88–90] 13.8 15.2 2.6 ⇥ 10 5 Cluster catalogs Table I. Simulated halo populations at z = 0.6. . The shifted squared density ˜2 and shifted tidal field ˜G2 are computed similarly, using 2 1( e mass. Ran 5 MP-Gadget1 DM-only N-body sims with 15363 DM particles, 30723 mesh for PM forces, L=500 Mpc/h, ~4000 time steps to evolve z=99 to z=0.6 4 FoF halo mass bins 1Feng et al. https://github.com/bluetides-project/MP-Gadget [derived from P-Gadget] mptcle = 2.9 ⇥ 109 M /h<latexit sha1_base64="5gOaYc51Lo066Ov3uoAnov8D0Bc=">AAACH3icbVDLSgMxFM3UV62vqks3wSK6kDpTBS0oFNy4ESrYB3TqkEnTNnQyGZI7Qhn6CX6E3+BW1+7EbZf+ieljYVsPBA7n3JuTHD8KuAbbHlqppeWV1bX0emZjc2t7J7u7V9UyVpRVqAykqvtEs4CHrAIcAlaPFCPCD1jN792O/NozU5rL8BH6EWsK0gl5m1MCRvKyx8JLXCVwBDRgg5tCvugCF0xjx34quqf3nitbEs66XjZn5+0x8CJxpiSHpih72R+3JWksWGhuJlo3HDuCZkIU8FFSxo01iwjtkQ5rGBoSE9pMxh8a4COjtHBbKnNCwGP170ZChNZ94ZtJQaCr572R+J/XiKF91Ux4GMXAQjoJascBBolH7eAWV4xC0DeEUMXNWzHtEkUomA5nUnwxMJ048w0skmoh75znCw8XudL1tJ00OkCH6AQ56BKV0B0qowqi6AW9oXf0Yb1an9aX9T0ZTVnTnX00A2v4CxZtokU=</latexit>
  • 51. Test of linear model Reasonable prediction on large scales Missing some structure on small scales MS, Simonović et al. (2019) d slices of the overdensity h(x) of simulated 1010.8 1011.8 h 1 M halos (top), compared with the cubic b ), and the linear Standard Eulerian bias model (bottom). Each panel is 500 h 1 Mpc wide and 110 h 1 Mpc hi sity is smoothed with a R = 2 h 1 Mpc Gaussian, WR(k) = exp[ (kR)2 /2]. The colorbar indicates the value d slices of the overdensity h(x) of simulated 1010.8 1011.8 h 1 M halos (top), compared with the cubic b ), and the linear Standard Eulerian bias model (bottom). Each panel is 500 h 1 Mpc wide and 110 h 1 Mpc hi sity is smoothed with a R = 2 h 1 Mpc Gaussian, WR(k) = exp[ (kR)2 /2]. The colorbar indicates the value d overdensity h(x). Simulation (= truth) Model b1 m(x)<latexit sha1_base64="qPIVubm7zi7B0APirS0IQLfxYzc=">AAACJXicbVBNS8NAEN34WetX1aOXYBH0YEmqoAcPBS8eK1gtNCFsNhNd3N2E3U1tCfkV/gh/g1c9exPBk/hP3LY5WPXBwOO9GWbmhSmjSjvOhzUzOze/sFhZqi6vrK6t1zY2r1SSSQIdkrBEdkOsgFEBHU01g24qAfOQwXV4dzbyr/sgFU3EpR6m4HN8I2hMCdZGCmoHnoB7knCORZR7/UGRexzr2zDOB0VRDQPXi4BpHPA9Y+4HtbrTcMaw/xK3JHVUoh3UvrwoIRkHoQnDSvVcJ9V+jqWmhEFR9TIFKSZ3+AZ6hgrMQfn5+K3C3jVKZMeJNCW0PVZ/TuSYKzXkoekc3ax+eyPxP6+X6fjEz6lIMw2CTBbFGbN1Yo8ysiMqgWg2NAQTSc2tNrnFEhNtkpzaEvLCZOL+TuAvuWo23MNG8+Ko3jot06mgbbSD9pCLjlELnaM26iCCHtATekYv1qP1ar1Z75PWGauc2UJTsD6/AT9/psc=</latexit> g<latexit sha1_base64="wmzm750Tk9wCdwjQ/kl+uGkG6Vo=">AAACHXicbVC7TsMwFHV4lvIqMLJEVJWYqqQgwcBQiYWxSPQhNVXlODetVduJbKe0ivIFfATfwAozG2JFjPwJ7mOgLUeydHTOffn4MaNKO863tba+sbm1ndvJ7+7tHxwWjo4bKkokgTqJWCRbPlbAqIC6pppBK5aAuc+g6Q9uJ35zCFLRSDzocQwdjnuChpRgbaRuoeQJeCQR51gEqTccZanHse77YTrKsrwXANO42+sWik7ZmcJeJe6cFNEctW7hxwsiknAQmjCsVNt1Yt1JsdSUMDCDEwUxJgPcg7ahAnNQnXT6ncwuGSWww0iaJ7Q9Vf92pJgrNea+qZzcqpa9ifif1050eN1JqYgTDYLMFoUJs3VkT7KxAyqBaDY2BBNJza026WOJiTYJLmzxeWYycZcTWCWNStm9KFfuL4vVm3k6OXSKztA5ctEVqqI7VEN1RNATekGv6M16tt6tD+tzVrpmzXtO0AKsr1+F/aPk</latexit>
  • 52. Nonlinear model So far used linear model g(x) = b1 m(x)<latexit sha1_base64="XaHcr7zAchcoHCuF4bu18V4pL3I=">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</latexit>
  • 53. Nonlinear model So far used linear model Include all nonlinear terms allowed by symmetries (EFT) Fit parameters by minimizing mean-squared error (least-squares ‘polynomial’ regression) bi<latexit sha1_base64="stVaTDZz5GHd9HUTbI9z2gtgqYY=">AAACK3icbVDLSgMxFM34tr7aunQTLIKrMqOCIi4ENy4rWhU6pSSZTBvMY0gytSXMJ7jVr/BrXClu/Q/T2oWtHggczr03956DM86MDcP3YG5+YXFpeWW1tLa+sblVrlRvjco1oU2iuNL3GBnKmaRNyyyn95mmSGBO7/DDxah+16faMCVv7DCjbYG6kqWMIOula9xhnXItrIdjwL8kmpAamKDRqQTVOFEkF1RawpExrSjMbNshbRnhtCjFuaEZIg+oS1ueSiSoabvxrQXc80oCU6X9kxaO1d8TDgljhgL7ToFsz8zWRuJ/tVZu05O2YzLLLZXkZ1Gac2gVHBmHCdOUWD70BBHN/K2Q9JBGxPp4SrGkj0QJgWTi4v6gcC4ercCpGxTFlCWHhf8cCnMKseLJ2IYp+RSj2cz+ktuDenRYP7g6qp2fTfJcATtgF+yDCByDc3AJGqAJCOiCJ/AMXoLX4C34CD5/WueCycw2mELw9Q0dSKcW</latexit> MS, Simonović et al. (2019) g(x) = b1 m(x) + b2 2 m(x) + tidal term + b3 3 m(x) + · · ·<latexit sha1_base64="hIhHT6q70TfpTrAHD6PG9mgELPM=">AAACqnicbVBdaxQxFM2MX3Wtuq3oiy/BRagoy8yuoPgBBV98rOK2lZ11SDKZbWg+huRO7RIG/Jv+AX+Hmemo3a0XAifn5Nzce2glhYMk+RnF167fuHlr6/bgzvbde/eHO7uHztSW8Rkz0thjShyXQvMZCJD8uLKcKCr5ET390OpHZ9w6YfQXWFV8ochSi1IwAoHKhz+ygksg+XIvUwROaOnPm2fvaZ72vLrMD57TfPJH+DZZl7qLVR5EQWT2FrhVTWuY/jVMNwysMODy4SgZJ13hqyDtwQj1dZDvRLtZYVituAYmiXPzNKlg4YkFwSRvBlnteEXYKVnyeYCaKO4WvouqwU8DU+DS2HA04I697PBEObdSNLxsZ3WbWkv+T5vXUL5eeKGrGrhmFx+VtcRgcJs7LoTlDOQqAMKsCLNidkIsYSGo0Enz78woRXThs7Pzxvt/UTVrK3mqQnOs3BtMjSy6NdwgpJhuZnYVHE7G6XQ8+fRytP+uz3MLPUZP0B5K0Su0jz6iAzRDDP2KtqOH0aP4Rfw5/hrPL57GUe95gNYqLn4DTrLRww==</latexit>
  • 54. Bulk flows 5 s x of a halo at the initial position q are given by x = q + (q). The overdensity 1 + h(x) = Z d3 q (1 + h(q)) D(x q (q)) , (12) Fourier transform of this field in Eulerian space is d3 x (1 + h(x)) e ik·x = Z d3 q (1 + h(q)) e ik·(q+ (q)) . (13) d in the rest of the paper we restrict the range of momenta to k 6= 0, so that the zero er our formulas. The nonlinear displacement from Lagrangian to Eulerian position ve series = 1 + 2 + · · · . At first order, we have the well-known Zel’dovich 1(q) = Z k eik·q ik k2 1(k) . (14) n be written as 2(q) = 3 14 Z k eik·q ik k2 G2(k) . (15) of the nonlinear displacement field and expanding the exponent e ik· (q) in Eq. (13) Standard Eulerian bias expansion. This procedure also fixes the relation between linear displacement is large 5 rdinates x of a halo at the initial position q are given by x = q + (q). The overdensity en by 1 + h(x) = Z d3 q (1 + h(q)) D(x q (q)) , (12) . The Fourier transform of this field in Eulerian space is k) ⌘ Z d3 x (1 + h(x)) e ik·x = Z d3 q (1 + h(q)) e ik·(q+ (q)) . (13) ion and in the rest of the paper we restrict the range of momenta to k 6= 0, so that the zero not enter our formulas. The nonlinear displacement from Lagrangian to Eulerian position urbative series = 1 + 2 + · · · . At first order, we have the well-known Zel’dovich 1(q) = Z k eik·q ik k2 1(k) . (14) ment can be written as 2(q) = 3 14 Z k eik·q ik k2 G2(k) . (15) ription of the nonlinear displacement field and expanding the exponent e ik· (q) in Eq. (13) usual Standard Eulerian bias expansion. This procedure also fixes the relation between s and their Standard Eulerian counterparts. Of course, this is not a surprise, as we expect ee order by order in perturbation theory. 5 f a halo at the initial position q are given by x = q + (q). The overdensity h(x) = Z d3 q (1 + h(q)) D(x q (q)) , (12) er transform of this field in Eulerian space is x (1 + h(x)) e ik·x = Z d3 q (1 + h(q)) e ik·(q+ (q)) . (13) he rest of the paper we restrict the range of momenta to k 6= 0, so that the zero ur formulas. The nonlinear displacement from Lagrangian to Eulerian position ries = 1 + 2 + · · · . At first order, we have the well-known Zel’dovich 1(q) = Z k eik·q ik k2 1(k) . (14) written as 2(q) = 3 14 Z k eik·q ik k2 G2(k) . (15) e nonlinear displacement field and expanding the exponent e ik· (q) in Eq. (13) Lagrangian space Eulerian space halo 5 t the initial position q are given by x = q + (q). The overdensity d3 q (1 + h(q)) D(x q (q)) , (12) rm of this field in Eulerian space is )) e ik·x = Z d3 q (1 + h(q)) e ik·(q+ (q)) . (13) the paper we restrict the range of momenta to k 6= 0, so that the zero s. The nonlinear displacement from Lagrangian to Eulerian position 1 + 2 + · · · . At first order, we have the well-known Zel’dovich q) = Z k eik·q ik k2 1(k) . (14) Slide credit: Marko Simonović
  • 55. Large bulk flows lead to large model error 
 because fields are incoherent Need a model that takes into account large bulk flows Bulk flows Good model of x ! x +<latexit sha1_base64="+3MORj4q/XkrwqAejYnO59kK720=">AAACAHicbVBNS8NAEJ34WetX1IMHL4tFEISSVEGhBwtePFawH9CEstlu2qWbTdjdaEvpxb/ixYMiXv0Z3vw3btsctPXBwOO9GWbmBQlnSjvOt7W0vLK6tp7byG9ube/s2nv7dRWnktAaiXksmwFWlDNBa5ppTpuJpDgKOG0E/ZuJ33igUrFY3OthQv0IdwULGcHaSG37cOCVPcm6PY2ljB+9MhqceVXF2nbBKTpToEXiZqQAGapt+8vrxCSNqNCEY6VarpNof4SlZoTTcd5LFU0w6eMubRkqcESVP5o+MEYnRumgMJamhEZT9ffECEdKDaPAdEZY99S8NxH/81qpDq/8ERNJqqkgs0VhypGO0SQN1GGSEs2HhmAimbkVkR6WmGiTWd6E4M6/vEjqpaJ7XizdXRQq11kcOTiCYzgFFy6hArdQhRoQGMMzvMKb9WS9WO/Wx6x1ycpmDuAPrM8fPGuWKA==</latexit> Displace by a shifttruth<latexit sha1_base64="dorPueCGyAIyn+LUjl5+vXhBTBQ=">AAAB+3icbVBNS8NAEN34WetXrEcvi0XwVJIq6M2CF48V7Ac0IWw2m3bpZhN2J2IJ/StePCji1T/izX/jts1BWx8MPN6bYWZemAmuwXG+rbX1jc2t7cpOdXdv/+DQPqp1dZoryjo0Fanqh0QzwSXrAAfB+pliJAkF64Xj25nfe2RK81Q+wCRjfkKGksecEjBSYNe8iAkgQeGpBIPKYTQN7LrTcObAq8QtSR2VaAf2lxelNE+YBCqI1gPXycAviAJOBZtWvVyzjNAxGbKBoZIkTPvF/PYpPjNKhONUmZKA5+rviYIkWk+S0HQmBEZ62ZuJ/3mDHOJrv+Ayy4FJulgU5wJDimdB4IgrRkFMDCFUcXMrpiOiCAUTV9WE4C6/vEq6zYZ70WjeX9ZbN2UcFXSCTtE5ctEVaqE71EYdRNETekav6M2aWi/Wu/WxaF2zyplj9AfW5w9uWJSv</latexit> truth<latexit sha1_base64="dorPueCGyAIyn+LUjl5+vXhBTBQ=">AAAB+3icbVBNS8NAEN34WetXrEcvi0XwVJIq6M2CF48V7Ac0IWw2m3bpZhN2J2IJ/StePCji1T/izX/jts1BWx8MPN6bYWZemAmuwXG+rbX1jc2t7cpOdXdv/+DQPqp1dZoryjo0Fanqh0QzwSXrAAfB+pliJAkF64Xj25nfe2RK81Q+wCRjfkKGksecEjBSYNe8iAkgQeGpBIPKYTQN7LrTcObAq8QtSR2VaAf2lxelNE+YBCqI1gPXycAviAJOBZtWvVyzjNAxGbKBoZIkTPvF/PYpPjNKhONUmZKA5+rviYIkWk+S0HQmBEZ62ZuJ/3mDHOJrv+Ayy4FJulgU5wJDimdB4IgrRkFMDCFUcXMrpiOiCAUTV9WE4C6/vEq6zYZ70WjeX9ZbN2UcFXSCTtE5ctEVaqE71EYdRNETekav6M2aWi/Wu/WxaF2zyplj9AfW5w9uWJSv</latexit> <latexit sha1_base64="0kQQ4bQnE6qcjjlOoHqoSjdsqzk=">AAAB63icbVBNSwMxEJ34WetX1aOXYBE8ld0q6M2CF48V7Ae0S8mm2TY0yS5JVihL/4IXD4p49Q9589+YbfegrQ8GHu/NMDMvTAQ31vO+0dr6xubWdmmnvLu3f3BYOTpumzjVlLVoLGLdDYlhgivWstwK1k00IzIUrBNO7nK/88S04bF6tNOEBZKMFI84JTaX+k3DB5WqV/PmwKvEL0gVCjQHla/+MKapZMpSQYzp+V5ig4xoy6lgs3I/NSwhdEJGrOeoIpKZIJvfOsPnThniKNaulMVz9fdERqQxUxm6Tkns2Cx7ufif10ttdBNkXCWpZYouFkWpwDbG+eN4yDWjVkwdIVRzdyumY6IJtS6esgvBX355lbTrNf+yVn+4qjZuizhKcApncAE+XEMD7qEJLaAwhmd4hTck0Qt6Rx+L1jVUzJzAH6DPH/PXjis=</latexit> ✏(x) ⌘ truth model<latexit sha1_base64="j8JWHHDkGfzWSFy/jB9sc9v3ags=">AAACOXicbVDLSgNBEJz1Gd9Rj14Gg6AHw64KevAQ8eJRwaiQDWF20jGD81hnesWw5Hv8CL/Bq+LRm3j1B5yNOWi0YKC6qpvuqSSVwmEYvgZj4xOTU9Olmdm5+YXFpfLyyoUzmeVQ50Yae5UwB1JoqKNACVepBaYSCZfJzXHhX96BdcLoc+yl0FTsWouO4Ay91CofxZA6IY3evN+iMdxm4o7SuA0SWStWDLtW5Wgz7Pbp9qiujK/7rXIlrIYD0L8kGpIKGeK0VX6L24ZnCjRyyZxrRGGKzZxZFFxCfzbOHKSM37BraHiqmQLXzAdf7dMNr7Rpx1j/NNKB+nMiZ8q5nkp8Z3GlG/UK8T+vkWHnoJkLnWYImn8v6mSSoqFFbrQtLHCUPU8Yt8LfSnmXWcbRp/trS6KKTKLRBP6Si51qtFvdOdur1A6H6ZTIGlknmyQi+6RGTsgpqRNOHsgTeSYvwWPwFrwHH9+tY8FwZpX8QvD5BQAFrsc=</latexit>
  • 56. Including bulk flows in the model licity, in this equation and in the rest of the paper we restrict the range of momenta to k 6= 0, so that the ze r mean density do not enter our formulas. The nonlinear displacement from Lagrangian to Eulerian positi xpanded in a perturbative series = 1 + 2 + · · · . At first order, we have the well-known Zel’dovi mation [69] 1(q) = Z k eik·q ik k2 1(k) . (1 nd-order displacement can be written as 2(q) = 3 14 Z k eik·q ik k2 G2(k) . (1 e perturbative description of the nonlinear displacement field and expanding the exponent e ik· (q) in Eq. (1 sible to recover the usual Standard Eulerian bias expansion. This procedure also fixes the relation betwe an bias parameters and their Standard Eulerian counterparts. Of course, this is not a surprise, as we expe descriptions to agree order by order in perturbation theory. e other hand we do not want to expand the full nonlinear displacement. We are going to keep the large q) exponentiated and expand only the higher-order terms.3 In this way the largest part of the problema acements is not expanded in perturbation theory. With this in mind, we can rewrite Eq. (13) in the followi h(k) = Z d3 q ⇣ 1 + bL 1 1(q) + bL 2 ( 2 1(q) 2 1) + bL G2 G2(q) + · · · ik · 2(q) + · · · ⌘ e ik·(q+ 1(q)) , (1 e new contributions come from expanding the second (and higher) order displacement field in the exponen ortant to stress that at leading order this new term can be expressed through the second order operator (15)). Therefore, at second order in perturbation theory, expanding the nonlinear terms in the displaceme ) only shifts some of the standard Lagrangian bias parameters by a calculable constant. We will give mo Usual approximation in (C)LPT for example: Vlah, Castorina, White (2016) art with the description of biased tracers in Lagrangian space. The displacement field is then t linear contribution and smaller higher order corrections. The nonlinear corrections to are while the linear piece is kept in the exponent. In this way, the dominant part of the large treated exactly, and the resulting operators once written in Eulerian space are automatically est of this section we give the details of this construction. ity field at Lagrangian position q is modeled using a bias expansion in the linear Lagrangian- L h(q) = bL 1 1(q) + bL 2 ( 2 1(q) 2 1) + bL G2 G2(q) + · · · , (8) re Lagrangian bias parameters, 2 1 is the r.m.s. fluctuation of the linear density field 2 1 = ⌦ 2 1(q) ↵ = Z 1 0 dk 2⇡2 k2 P11(k) , (9) ) is defined as2 G2(q) ⌘  @i@j @2 1(q) 2 2 1(q) . (10) this operator in momentum space is given by G2(k) = Z p  (p · (k p))2 p2|k p|2 1 1(p) 1(k p) . (11) ng notation in which R p ⌘ R d3 p/(2⇡)3 . In the bias expansion (8) we kept only terms up to bation theory. We will continue to work at this order throughout this section, because it is ng notation and motivating the bias model that we are going to use to make comparisons with er order or higher derivative operators needed for the consistent one-loop calculation can be ded. We will come back to this in Section VIII. n Eq. (8) is in Lagrangian space. In order to go to Eulerian space, let us start from Eq. (8) and split into the dominant linear contribution and smaller higher order corrections. The nonlinear corrections treated perturbatively, while the linear piece is kept in the exponent. In this way, the dominant part of displacements can be treated exactly, and the resulting operators once written in Eulerian space are auto IR-resummed. In the rest of this section we give the details of this construction. The proto-halo density field at Lagrangian position q is modeled using a bias expansion in the linear La space density 1(q): L h(q) = bL 1 1(q) + bL 2 ( 2 1(q) 2 1) + bL G2 G2(q) + · · · , where bL 1, bL 2, bL G2 , . . . are Lagrangian bias parameters, 2 1 is the r.m.s. fluctuation of the linear density field 2 1 = ⌦ 2 1(q) ↵ = Z 1 0 dk 2⇡2 k2 P11(k) , and the operator G2(q) is defined as2 G2(q) ⌘  @i@j @2 1(q) 2 2 1(q) . The representation of this operator in momentum space is given by G2(k) = Z p  (p · (k p))2 p2|k p|2 1 1(p) 1(k p) . Notice that we are using notation in which R p ⌘ R d3 p/(2⇡)3 . In the bias expansion (8) we kept only te second order in perturbation theory. We will continue to work at this order throughout this section, bec su cient for introducing notation and motivating the bias model that we are going to use to make compar simulations. The higher order or higher derivative operators needed for the consistent one-loop calculati straightforwardly included. We will come back to this in Section VIII. The bias expansion in Eq. (8) is in Lagrangian space. In order to go to Eulerian space, let us start from E include the gravitational evolution. The gravitational evolution is encoded in the nonlinear displacement fi 5 Eulerian coordinates x of a halo at the initial position q are given by x = q + (q). The overdensity is way is given by 1 + h(x) = Z d3 q (1 + h(q)) D(x q (q)) , (12) Dirac delta. The Fourier transform of this field in Eulerian space is h(k) ⌘ Z d3 x (1 + h(x)) e ik·x = Z d3 q (1 + h(q)) e ik·(q+ (q)) . (13) n this equation and in the rest of the paper we restrict the range of momenta to k 6= 0, so that the zero density do not enter our formulas. The nonlinear displacement from Lagrangian to Eulerian position ed in a perturbative series = 1 + 2 + · · · . At first order, we have the well-known Zel’dovich [69] 1(q) = Z k eik·q ik k2 1(k) . (14) er displacement can be written as 2(q) = 3 14 Z k eik·q ik k2 G2(k) . (15) ik· (q)
  • 57. Shifted operators Z(k) = ˜1(k) + 1 2 ˜G2(k) 1 3 ˜G3(k) + · · · , (20) or analogous to ˜G2 (see Appendix D). In other words, Z(k) can be absorbed in the bias ging the bias parameters. Of course, this is just a choice, and there is nothing wrong in he formulas. As we are going to see later, di↵erent choices may be more appropriate for us point out that in the formula (20) the displacements 1(q) are treated exactly. In other k· 1(q) is never expanded in 1(q). The only expansion parameter is the derivative of the 1(q), which is a small quantity.5 This is consistent with the way the shifted operators asis of shifted operators we can write the bias expansion of the halo density field in Eulerian order in perturbation theory, in the following way h(k) = b1 ˜1(k) + b2 ˜2(k) + bG2 ˜G2(k) + · · · . (21) this section. Notice that the new bias parameters bi di↵er from the original Lagrangian his di↵erence comes from expanding the nonlinear part of the displacement (Eq. (16)) and sity field in terms of shifted operators (Eq. (20)). We give the explicit relation of bi and bL i (21) has a similar structure as the usual Standard Eulerian bias expansion h(k) = bE 1 (k) + bE 2 2(k) + bE G2 G2(k) + · · · , (22) ce that all fields in this equation are nonlinear. Apart from the IR resummation of the large ant di↵erence compared to the expansion in terms of ˜O is that in Eq. (21) all operators are inear field 1. As we are going to see, for the purposes of describing the biased tracers on her important virtue of the expansion (21). + noise ik · 2(q) + · · · e , (16) me from expanding the second (and higher) order displacement field in the exponent. t leading order this new term can be expressed through the second order operator G2 cond order in perturbation theory, expanding the nonlinear terms in the displacement he standard Lagrangian bias parameters by a calculable constant. We will give more s in Section VIII. tivates us to write down the bias expansion in Eulerian space in terms of shifted he following way ˜O(k) ⌘ Z d3 q O(q) e ik·(q+ 1(q)) , (17) 2 1), G2, . . .}.4 We would like to stress again a few important advantages that this d operators are written in Eulerian space and therefore allow for easy comparisons tion of their importance. (b) The large displacement terms 1(q) are kept resummed, s with simulations on the level of realizations. Notice that this also implies that in ass filter, compared to the wavelength of a Fourier mode 1(k). For a given wavenumber k, the nto the long-wavelength and short-wavelength part: 1 = L 1 + S 1 , where L 1 = W(k) 1 and L 1 on the short modes is fixed by the Equivalence Principle. Therefore, strictly speaking, only L 1 in any perturbative calculation S 1 has to be expanded order by order in perturbation theory. The y keeping the full 1 in the exponent is always higher order in S 1 than terms we calculate. Also, all scales. In order to keep the formulas simple, we decide not to do the long-short splitting in our not just given by a translation of the position argument because they implicitly include the inverse n @xi/@qj due to the coordinate transformation. This is similar to the Zel’dovich density, which is gian space shifted by 1(q). Motivates bias expansion in “shifted” operators (incl. bulk flows)
  • 58. Shifted operators Z(k) = ˜1(k) + 1 2 ˜G2(k) 1 3 ˜G3(k) + · · · , (20) or analogous to ˜G2 (see Appendix D). In other words, Z(k) can be absorbed in the bias ging the bias parameters. Of course, this is just a choice, and there is nothing wrong in he formulas. As we are going to see later, di↵erent choices may be more appropriate for us point out that in the formula (20) the displacements 1(q) are treated exactly. In other k· 1(q) is never expanded in 1(q). The only expansion parameter is the derivative of the 1(q), which is a small quantity.5 This is consistent with the way the shifted operators asis of shifted operators we can write the bias expansion of the halo density field in Eulerian order in perturbation theory, in the following way h(k) = b1 ˜1(k) + b2 ˜2(k) + bG2 ˜G2(k) + · · · . (21) this section. Notice that the new bias parameters bi di↵er from the original Lagrangian his di↵erence comes from expanding the nonlinear part of the displacement (Eq. (16)) and sity field in terms of shifted operators (Eq. (20)). We give the explicit relation of bi and bL i (21) has a similar structure as the usual Standard Eulerian bias expansion h(k) = bE 1 (k) + bE 2 2(k) + bE G2 G2(k) + · · · , (22) ce that all fields in this equation are nonlinear. Apart from the IR resummation of the large ant di↵erence compared to the expansion in terms of ˜O is that in Eq. (21) all operators are inear field 1. As we are going to see, for the purposes of describing the biased tracers on her important virtue of the expansion (21). + noise ik · 2(q) + · · · e , (16) me from expanding the second (and higher) order displacement field in the exponent. t leading order this new term can be expressed through the second order operator G2 cond order in perturbation theory, expanding the nonlinear terms in the displacement he standard Lagrangian bias parameters by a calculable constant. We will give more s in Section VIII. tivates us to write down the bias expansion in Eulerian space in terms of shifted he following way ˜O(k) ⌘ Z d3 q O(q) e ik·(q+ 1(q)) , (17) 2 1), G2, . . .}.4 We would like to stress again a few important advantages that this d operators are written in Eulerian space and therefore allow for easy comparisons tion of their importance. (b) The large displacement terms 1(q) are kept resummed, s with simulations on the level of realizations. Notice that this also implies that in ass filter, compared to the wavelength of a Fourier mode 1(k). For a given wavenumber k, the nto the long-wavelength and short-wavelength part: 1 = L 1 + S 1 , where L 1 = W(k) 1 and L 1 on the short modes is fixed by the Equivalence Principle. Therefore, strictly speaking, only L 1 in any perturbative calculation S 1 has to be expanded order by order in perturbation theory. The y keeping the full 1 in the exponent is always higher order in S 1 than terms we calculate. Also, all scales. In order to keep the formulas simple, we decide not to do the long-short splitting in our not just given by a translation of the position argument because they implicitly include the inverse n @xi/@qj due to the coordinate transformation. This is similar to the Zel’dovich density, which is gian space shifted by 1(q). Motivates bias expansion in “shifted” operators (incl. bulk flows) Result is in Eulerian space so easy to compare against simulations Has IR resummation, giving correct halo positions and BAO spread Model is perturbative, only linear fields used in the construction Power spectrum agrees with resummed 1-loop PT
  • 59. Model on the grid Distribute 15363 particles on regular grid Assign artificial particle masses Displace by linear displacement Interpolate to Eulerian grid using CIC weighted by particle masses [Very similar to generating N-body initial conds./Zeldovich density] mi = O(qi)<latexit sha1_base64="OIMXXFzMNCtfp4mc7I6Q6k7Bu8E=">AAACF3icbZC7TsMwFIadcivlFmBkwKJCKkuVFCRYEJVY2CgSvUhtFTmu01q1k2A7SFWUkYfgGVhhZkOsjIy8CU6agbb8kqVP/zlH5/h3Q0alsqxvo7C0vLK6VlwvbWxube+Yu3stGUQCkyYOWCA6LpKEUZ80FVWMdEJBEHcZabvj67TefiRC0sC/V5OQ9Dka+tSjGCltOeYhdyi8hD2O1AgjFt8mlYxdL35IHHrimGWramWCi2DnUAa5Go750xsEOOLEV5ghKbu2Fap+jISimJGk1IskCREeoyHpavQRJ7IfZx9J4LF2BtALhH6+gpn7dyJGXMoJd3VneqScr6Xmf7VupLyLfkz9MFLEx9NFXsSgCmCaChxQQbBiEw0IC6pvhXiEBMJKZzezxeWJzsSeT2ARWrWqfVqt3Z2V61d5OkVwAI5ABdjgHNTBDWiAJsDgCbyAV/BmPBvvxofxOW0tGPnMPpiR8fULYv6f9w==</latexit> q<latexit sha1_base64="Ft5o3g7n3eKWwngUx/MdJMYDuiE=">AAACAHicbVDLSgMxFL3js9ZX1aWbYBFclZkq6M6CG5cV7APboWTSTBuaZMYkI5ShG7/Bra7diVv/xKV/YqadhW09EDiccy/35AQxZ9q47rezsrq2vrFZ2Cpu7+zu7ZcODps6ShShDRLxSLUDrClnkjYMM5y2Y0WxCDhtBaObzG89UaVZJO/NOKa+wAPJQkawsdJDV2AzDML0cdIrld2KOwVaJl5OypCj3iv9dPsRSQSVhnCsdcdzY+OnWBlGOJ0Uu4mmMSYjPKAdSyUWVPvpNPEEnVqlj8JI2ScNmqp/N1IstB6LwE5mCfWil4n/eZ3EhFd+ymScGCrJ7FCYcGQilH0f9ZmixPCxJZgoZrMiMsQKE2NLmrsSiKwTb7GBZdKsVrzzSvXuoly7ztspwDGcwBl4cAk1uIU6NICAhBd4hTfn2Xl3PpzP2eiKk+8cwRycr1/QxZes</latexit> q ! q + 1(q)<latexit sha1_base64="WM5LTA6NvvqFh353BeqMK/OW1Ew=">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</latexit> 1(q)<latexit sha1_base64="TJaAHFoYRyspIjBVZgsQyBThdy0=">AAACKXicbVDLSsNAFJ34rPVVdekmWIS6KUkVdGfBjcsK9gFNKJPppB06j3RmUikh3+FH+A1ude1O3Yk/4rSNYFsPDBzOuZdz5wQRJUo7zoe1srq2vrGZ28pv7+zu7RcODhtKxBLhOhJUyFYAFaaE47ommuJWJDFkAcXNYHAz8ZsjLBUR/F6PI+wz2OMkJAhqI3UKrsfxAxKMQd5NvNEwTTwGdT8Ik2Ga5n+5V1Mk7bglM3DWKRSdsjOFvUzcjBRBhlqn8OV1BYoZ5hpRqFTbdSLtJ1Bqgig2IbHCEUQD2MNtQzlkWPnJ9GupfWqUrh0KaR7X9lT9u5FAptSYBWZycqta9Cbif1471uGVnxAexRpzNAsKY2prYU96srtEYqTp2BCIJDG32qgPJUTatDmXErDUdOIuNrBMGpWye16u3F0Uq9dZOzlwDE5ACbjgElTBLaiBOkDgETyDF/BqPVlv1rv1ORtdsbKdIzAH6/sHurmolw==</latexit> O(q)<latexit sha1_base64="fNNlinCcTxszFIVZ7rBkeZk8/5Y=">AAACJXicbVDLSgMxFM3UV62vqks3g0WoC8tMFXRnwY07K9gHtEPJZDJtaB7TJFMpw3yFH+E3uNW1OxFciX9i+ljY1gOBwzn3cm6OH1GitON8WZmV1bX1jexmbmt7Z3cvv39QVyKWCNeQoEI2fagwJRzXNNEUNyOJIfMpbvj9m7HfGGKpiOAPehRhj8EuJyFBUBupkz9rc/yIBGOQB0l7OEiTNoO654fJIE1zE44gTe7SojFPO/mCU3ImsJeJOyMFMEO1k/9pBwLFDHONKFSq5TqR9hIoNUEUm4BY4QiiPuzilqEcMqy8ZPKt1D4xSmCHQprHtT1R/24kkCk1Yr6ZHN+pFr2x+J/XinV45SWER7HGHE2DwpjaWtjjjuyASIw0HRkCkSTmVhv1oIRImybnUnyWmk7cxQaWSb1ccs9L5fuLQuV61k4WHIFjUAQuuAQVcAuqoAYQeAIv4BW8Wc/Wu/VhfU5HM9Zs5xDMwfr+Ba4ppw4=</latexit> ˜O(x)<latexit sha1_base64="jp0htYvlhK/ioIFAQhH9fbs5T6c=">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</latexit>
  • 60. d slices of the overdensity h(x) of simulated 1010.8 1011.8 h 1 M halos (top), compared with the cubic b ), and the linear Standard Eulerian bias model (bottom). Each panel is 500 h 1 Mpc wide and 110 h 1 Mpc hi sity is smoothed with a R = 2 h 1 Mpc Gaussian, WR(k) = exp[ (kR)2 /2]. The colorbar indicates the value d overdensity h(x). g<latexit sha1_base64="wmzm750Tk9wCdwjQ/kl+uGkG6Vo=">AAACHXicbVC7TsMwFHV4lvIqMLJEVJWYqqQgwcBQiYWxSPQhNVXlODetVduJbKe0ivIFfATfwAozG2JFjPwJ7mOgLUeydHTOffn4MaNKO863tba+sbm1ndvJ7+7tHxwWjo4bKkokgTqJWCRbPlbAqIC6pppBK5aAuc+g6Q9uJ35zCFLRSDzocQwdjnuChpRgbaRuoeQJeCQR51gEqTccZanHse77YTrKsrwXANO42+sWik7ZmcJeJe6cFNEctW7hxwsiknAQmjCsVNt1Yt1JsdSUMDCDEwUxJgPcg7ahAnNQnXT6ncwuGSWww0iaJ7Q9Vf92pJgrNea+qZzcqpa9ifif1050eN1JqYgTDYLMFoUJs3VkT7KxAyqBaDY2BBNJza026WOJiTYJLmzxeWYycZcTWCWNStm9KFfuL4vVm3k6OXSKztA5ctEVqqI7VEN1RNATekGv6M16tt6tD+tzVrpmzXtO0AKsr1+F/aPk</latexit> Test of nonlinear model MS, Simonović et al. (2019) b1 m(x)<latexit sha1_base64="qPIVubm7zi7B0APirS0IQLfxYzc=">AAACJXicbVBNS8NAEN34WetX1aOXYBH0YEmqoAcPBS8eK1gtNCFsNhNd3N2E3U1tCfkV/gh/g1c9exPBk/hP3LY5WPXBwOO9GWbmhSmjSjvOhzUzOze/sFhZqi6vrK6t1zY2r1SSSQIdkrBEdkOsgFEBHU01g24qAfOQwXV4dzbyr/sgFU3EpR6m4HN8I2hMCdZGCmoHnoB7knCORZR7/UGRexzr2zDOB0VRDQPXi4BpHPA9Y+4HtbrTcMaw/xK3JHVUoh3UvrwoIRkHoQnDSvVcJ9V+jqWmhEFR9TIFKSZ3+AZ6hgrMQfn5+K3C3jVKZMeJNCW0PVZ/TuSYKzXkoekc3ax+eyPxP6+X6fjEz6lIMw2CTBbFGbN1Yo8ysiMqgWg2NAQTSc2tNrnFEhNtkpzaEvLCZOL+TuAvuWo23MNG8+Ko3jot06mgbbSD9pCLjlELnaM26iCCHtATekYv1qP1ar1Z75PWGauc2UJTsD6/AT9/psc=</latexit> sim g (x)<latexit sha1_base64="dAEZhBUIThUpHuyxiN4tqPX1ous=">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</latexit> d slices of the overdensity h(x) of simulated 1010.8 1011.8 h 1 M halos (top), compared with the cubic b ), and the linear Standard Eulerian bias model (bottom). Each panel is 500 h 1 Mpc wide and 110 h 1 Mpc hi sity is smoothed with a R = 2 h 1 Mpc Gaussian, WR(k) = exp[ (kR)2 /2]. The colorbar indicates the value Nonlinear model d slices of the overdensity h(x) of simulated 1010.8 1011.8 h 1 M halos (top), compared with the cubic b ), and the linear Standard Eulerian bias model (bottom). Each panel is 500 h 1 Mpc wide and 110 h 1 Mpc hi sity is smoothed with a R = 2 h 1 Mpc Gaussian, WR(k) = exp[ (kR)2 /2]. The colorbar indicates the value Much better agreement than previous model Simulation (= truth)
  • 61. Measures of success Error power spectrum (= MSE per wavenumber k) Cross-correlation between model and truth rcc(k) = h truth(k) ⇤ model(k)i p Ptruth(k)Pmodel(k)<latexit sha1_base64="cRIq0DTnUaRL8LfD6bOtGkmKbM8=">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</latexit> For best-fit model, , so focus on herePerr = Ptruth(1 r2 cc)<latexit sha1_base64="tRkpT8H4AD9a4kRuQlWo+CYRCgo=">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</latexit> Perr(k) ⌘ h| truth(k) model(k)|2 i<latexit sha1_base64="okwU7UK76wFFi51s3hFUJl6za4k=">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</latexit> Perr<latexit sha1_base64="HbsFOTY+FC07lz5Qau+blG/LzoY=">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</latexit>
  • 62. Error power spectrum (= MSE) Poisson noise 1/¯n Perr = h| truth h model h |2 i Simulated halos (truth) Quadr. bias (model) isson noise 1/¯n h model h |2 i s (model) Poisson noise 1/¯n Perr = h| truth h model h |2 i Simulated halos (truth) Quadr. bias (model) Fourier wavenumber k Powerspectrum Error is quite white, similar to Poisson shot noise (discrete sampling) MS, Simonović et al. (2019)
  • 63. Tried many other nonlinear bias operators Give few times larger model error Fourier wavenumber k Figure 20. Left panel: Model error power spectrum for Standard Eulerian bias models, for nonlinear dark matter NL from simulations as the input for the Standard Eulerian bias mo large scales because it involves squaring NL, which is rather UV-sensitive. Alternatively, density as the input to the bias model (dark orange) is treating large bulk flows perturbaMS, Simonović et al. (2019) Errorpowerspectrum
  • 64. Scale dependence of the error ±1% of Phh Expanding Z Quadratic bias Cubic bias Linear Std. Eul. bias Scale dependence of the noise important around the nonlinear scale Potentially dangerous because can bias cosmological parameters
  • 65. kmax up to which noise is constant kmax when scale dependence of Perr is detectable with 1σ in V=10 h-3Gpc3 volume (or V=0.5 in brackets): kmax [hMpc 1 ] log M[h 1 M ] ¯n [(h 1 Mpc) 3 ] Lin. Std. Eul. Cubic 10.8 11.8 4.3 ⇥ 10 2 0.1 (0.14) 0.3 (0.37) 11.8 12.8 5.7 ⇥ 10 3 0.08 (0.1) 0.18 (0.24) 12.8 13.8 5.6 ⇥ 10 4 0.07 (0.1) 0.13 (0.18) 13.8 15.2 2.6 ⇥ 10 5 0.1 (0.14) 0.24 (0.32) ximum wavenumber kmax when a scale dependence of the model error can be detected with 1 in a 10 h a 0.5 h 3 Gpc3 volume, shown in brackets), for the linear Standard Eulerian and the cubic bias models. Th model is typically 2 to 3 higher than that of linear Standard Eulerian bias. This can improve measurem arameters that a↵ect the galaxy power spectrum at these scales, for example the sum of neutrino masse significant uncertainty due to the noise in our measurement of the model error, as shown in Fig. 11. of the survey volume. We also summarize this in Table IV for two survey volumes. ear Standard Eulerian bias we find the following. For the lowest halo mass bin, the scale depen ificant for kmax ' 0.1 hMpc 1 in a 10 h 3 Gpc3 volume or for kmax ' 0.14 hMpc 1 in a 0.5 h Nonlinear bias has 2-3x higher kmax 8-30x more Fourier modes
 4-5x smaller error bars (in principle; also have more params!)
  • 66. Weighting halos by their mass Halo number density
 (how many halos per cell) Halo mass density
 (how much halo mass per cell) used so far more similar to dark matter
 smaller shot noise Seljak, Hamaus & Desjacques (2009)
 Hamaus, Seljak & Desjacques (2010, 2011, 2012) Cai, Bernstein & Sheth (2011)
  • 67. Weighting halos by their mass Shot noise (squared model error) 17x lower for light halos, 2-7x lower for heavy halos With 60% halo mass scatter (green), still get factor few How well can we do observationally? 49 mass weighting on the mean-square model error divided by the Poisson expectation, h| obs h (k) obs obs truth ?
  • 68. Mass weighting questions How well can halo masses be measured (e.g. BOSS, DESI)? What observable properties of galaxies can we use? What sims? New ideas to get halo masses? For shot noise limited applications, gain may be large What if mass estimates are biased? Use for BAO reconstruction? (Suffers from high shot noise)
  • 69. Science with galaxy catalogs (1) Bias model at the field level (2) Cosmological parameter analysis (3) Accounting for skewness (4) Getting initial from final conditions 13 Figure 1. 2-d slices of the overdensity h(x) of simulated 1010.8 1011.8 h 1 M halos (top), compared with the cubic bias model (center), and the linear Standard Eulerian bias model (bottom). Each panel is 500 h 1 Mpc wide and 110 h 1 Mpc high, and each density is smoothed with a R = 2 h 1 Mpc Gaussian, WR(k) = exp[ (kR)2 /2]. The colorbar indicates the values of this smoothed overdensity h(x). Figure 2. Same as Fig. 1 but for more massive and less abundant 1011.8 1012.8 h 1 M halos. simulations. The variance, skewness and kurtosis of the densities shown in the histograms are listed in Table II for the full simulated and modeled densities, and in Table III for the model error. The linear Standard Eulerian bias model tends to underpredict troughs and overpredict peaks of the halo density, as shown in Fig. 3. The model error is not Gaussian for any of the shown smoothing scales; in particular its kurtosis is larger than 1 for all smoothing scales. The cubic model provides a more accurate description of the halo density pdf, as shown in Fig. 4. This emphasizes the importance of using nonlinear bias terms even on rather large scales. Still, the cubic model tends to underpredict the peaks of the true halo density, especially on small scales. This agrees with Fig. 2 where the model also underpredicts