Next-generation Cosmic Microwave Background experiments such as the Simons Observatory, CMB-S4 and PICO aim to measure gravitational lensing of the Cosmic Microwave Background an order of magnitude better than current experiments. The lensing signal will be highly correlated with measurements of galaxy clustering from next-generation galaxy surveys such as LSST. This will help us understand whether cosmic inflation was driven by a single field or by multiple fields. It will also allow us to accurately measure the growth of structure as a function of time, which is a powerful probe of dark energy and the sum of neutrino masses. I will discuss the prospects for this, as well as recent progress on the theoretical modeling of galaxy clustering, which is key to realize the full potential of these anticipated datasets.

1. Future Cosmology with
CMB Lensing and Galaxy Clustering
Marcel Schmittfull
UW-Madison, 2/20/2020

2. I. Future cosmology with CMB lensing and galaxy clustering
II. Galaxy clustering: Theory & Analysis (~15 min.)
Outline
Figure 1. 2-d slices of the overdensity h(x) of simulated 1010.8
1011.8
h 1
M halos (top), compared wit
model (center), and the linear Standard Eulerian bias model (bottom). Each panel is 500 h 1
Mpc wide and 11
and each density is smoothed with a R = 2 h 1
Mpc Gaussian, WR(k) = exp[ (kR)2
/2]. The colorbar indica
this smoothed overdensity h(x).

3. Cosmic
Inflation
Dark Energy
PDG,D&JBaumann
Q
uantum
fluctuations
C
osm
ic
M
icrow
ave
Background
radiation
G
alaxies

4. Nature of each building block is unknown
Inflation
How did our Universe begin?
What drives the inflationary expansion?
Dark energy
What drives the current accelerated expansion?
Is it a cosmological constant? Is General Relativity valid?
Dark matter
What particle(s) is it made of?
Relativistic degrees of freedom
What is the mass (hierarchy) of neutrinos?
Are there additional light relic particles?

5. Nature of each building block is unknown
Inflation
How did our Universe begin?
What drives the inflationary expansion?
Dark energy
What drives the current accelerated expansion?
Is it a cosmological constant? Is General Relativity valid?
Dark matter
What particle(s) is it made of?
Relativistic degrees of freedom
What is the mass (hierarchy) of neutrinos?
Are there additional light relic particles?

6. Constraining the energy scale of inflation
Energy scale at which inflation takes place is completely unknown
and can range across 10 orders of magnitude
Highest-energy models (>1016 GeV) produce gravitational waves
GWamplituder
2.5 Implications of an improved upper limit on r 25
×
0.955 0.960 0.965 0.970 0.975 0.980 0.985 0.990 0.995 1.00
3 10-4
0.001
0.003
0.01
0.03
0.1
ns
m φ
2 2
μ φ3
47< N < 57*
V (1- (φ/M) )
CMB-S4
r
4
0
V tanh (φ/M)2
0
N = 50*
N = 57*
R2
Higgs
M=20 MP
M=12 MP
M=10 MP
M=2 M
P
BK14/Planck
47< N < 57*
φ
10/3 2/3
47< N < 57*μ
Figure 10. Forecast of CMB-S4 constraints in the ns–r plane for a fiducial model with r = 0.01.
Constraints on r are derived from the expected CMB-S4 sensitivity to the B-mode power spectrum as
described in Section 2.3. Constraints on ns are derived from expected CMB-S4 sensitivity to temperature
Slope of scalar fluctuations ns
CMB-S4 science book
⇣ r
0.01
⌘1/4
'
V 1/4
1016 GeV<latexit sha1_base64="LnGrsqCczCv0O2/KExVSvH0zTjk=">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</latexit>

7. Constraining inflation
Lower-energy models (<1016 GeV) produce no observable
primordial gravitational waves
The only way to probe this class of models is primordial non-
Gaussianity: Fluctuations not normally distributed.
Complements GW searches
e.g. Meerburg et al. (2019)

8. Q
uantum
fluctuations
C
osm
ic
M
icrow
ave
Background
radiation
G
alaxies
G
aussian
G
aussian
Peaks
ofG
aussian
field
Single field inflation

9. Q
uantum
fluctuations
C
osm
ic
M
icrow
ave
Background
radiation
G
alaxies
notG
aussian
notG
aussian
Peaks
ofnon-
G
aussian
field
Multi-field inflation

10. Inflation
Single field
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fNL & 1<latexit sha1_base64="S+sPnBQuqcMTZ6LOSc6h53tv+oA=">AAACAXicbVDLSgMxFM3UV62vUTeCm2ARXJWZKujCRcGNC5EK9gGdYcikmTY0yQxJRihD3fgrblwo4ta/cOffmGlnoa0HAodzziX3njBhVGnH+bZKS8srq2vl9crG5tb2jr2711ZxKjFp4ZjFshsiRRgVpKWpZqSbSIJ4yEgnHF3lfueBSEVjca/HCfE5GggaUYy0kQL7IAo8jvRQ8uz2ZgK9gTZZDl0Y2FWn5kwBF4lbkCoo0AzsL68f45QToTFDSvVcJ9F+hqSmmJFJxUsVSRAeoQHpGSoQJ8rPphdM4LFR+jCKpXlCw6n6eyJDXKkxD00y31bNe7n4n9dLdXThZ1QkqSYCzz6KUgZ1DPM6YJ9KgjUbG4KwpGZXiIdIIqxNaRVTgjt/8iJp12vuaa1+d1ZtXBZ1lMEhOAInwAXnoAGuQRO0AAaP4Bm8gjfryXqx3q2PWbRkFTP74A+szx+phpZY</latexit>
Multi-field
Gaussian fluctuations Non-Gaussian fluctuations
Non-Gaussian fluctuations from inflation

11. Schematically, expectation value of a quantum field perturbation
with Lagrangian during inflation:
Free theory generates Gaussian fluctuations
Interacting theory or couplings between multiple
fields generate non-Gaussian fluctuations
Maldacena (2003), Chen, Huang, Kachru & Shiu (2006), Chen (2010), lecture notes by L. Senatore (1609.00716)
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L ⇠ '2
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L ⇠ '3
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h⌦| 'k1
· · · 'kn
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Z
D[ '] 'k1
· · · 'kn
ei
R
C
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[ Non-Gaussian fluctuations from inflation ]

12. Single field theorem
For any single field inflation model, where there is only one degree
of freedom during inflation
Detection of would rule out all single field inflation
models regardless of
- the form of the potential
- the form of kinetic terms
- the initial vacuum state
Maldacena (2003), Creminelli & Zaldarriaga (2004), Seery & Lidsey (2005)
fNL '
5
12
(1 ns) ' 0.02
<latexit sha1_base64="d0h1DGZc9aIowaeuyxNmASwUswM=">AAACHXicbVBLSwMxGMzWV62vqkcvwSLUg2W3VvTgoeDFg0gF+4BuWbJptg1NsmuSFcqyf8SLf8WLB0U8eBH/jelD0NaBkGHm+0hm/IhRpW37y8osLC4tr2RXc2vrG5tb+e2dhgpjiUkdhyyULR8pwqggdU01I61IEsR9Rpr+4GLkN++JVDQUt3oYkQ5HPUEDipE2kpevBJ7Lke5LnlxfpdBVlJM76AYS4eQkTZxyWnSOhKcOfyy7ZJe9fMFcY8B54kxJAUxR8/IfbjfEMSdCY4aUajt2pDsJkppiRtKcGysSITxAPdI2VCBOVCcZp0vhgVG6MAilOULDsfp7I0FcqSH3zeQoiZr1RuJ/XjvWwVknoSKKNRF48lAQM6hDOKoKdqkkWLOhIQhLav4KcR+ZZrQpNGdKcGYjz5NGueQcl8o3lUL1fFpHFuyBfVAEDjgFVXAJaqAOMHgAT+AFvFqP1rP1Zr1PRjPWdGcX/IH1+Q0U1aCS</latexit>
fNL 0.02<latexit sha1_base64="gKKNmHaEeH6/0+kFPU8jep0UATs=">AAAB/3icbVDNS8MwHE3n15xfVcGLl+AQPI12CnrwMPDiQWSC+4C1lDRLt7AkLUkqjLqD/4oXD4p49d/w5n9juvWgmw9CHu/9fuTlhQmjSjvOt1VaWl5ZXSuvVzY2t7Z37N29topTiUkLxyyW3RApwqggLU01I91EEsRDRjrh6Cr3Ow9EKhqLez1OiM/RQNCIYqSNFNgHUeBxpIeSZ7c3E+gNBtCpOfXArpprCrhI3IJUQYFmYH95/RinnAiNGVKq5zqJ9jMkNcWMTCpeqkiC8AgNSM9QgThRfjbNP4HHRunDKJbmCA2n6u+NDHGlxjw0k3lWNe/l4n9eL9XRhZ9RkaSaCDx7KEoZ1DHMy4B9KgnWbGwIwpKarBAPkURYm8oqpgR3/suLpF2vuae1+t1ZtXFZ1FEGh+AInAAXnIMGuAZN0AIYPIJn8ArerCfrxXq3PmajJavY2Qd/YH3+AEAWlOs=</latexit>

13. Single field theorem
For any single field inflation model, where there is only one degree
of freedom during inflation
Detection of would rule out all single field inflation
models regardless of
- the form of the potential
- the form of kinetic terms
- the initial vacuum state
Maldacena (2003), Creminelli & Zaldarriaga (2004), Seery & Lidsey (2005)
fNL '
5
12
(1 ns) ' 0.02
<latexit sha1_base64="d0h1DGZc9aIowaeuyxNmASwUswM=">AAACHXicbVBLSwMxGMzWV62vqkcvwSLUg2W3VvTgoeDFg0gF+4BuWbJptg1NsmuSFcqyf8SLf8WLB0U8eBH/jelD0NaBkGHm+0hm/IhRpW37y8osLC4tr2RXc2vrG5tb+e2dhgpjiUkdhyyULR8pwqggdU01I61IEsR9Rpr+4GLkN++JVDQUt3oYkQ5HPUEDipE2kpevBJ7Lke5LnlxfpdBVlJM76AYS4eQkTZxyWnSOhKcOfyy7ZJe9fMFcY8B54kxJAUxR8/IfbjfEMSdCY4aUajt2pDsJkppiRtKcGysSITxAPdI2VCBOVCcZp0vhgVG6MAilOULDsfp7I0FcqSH3zeQoiZr1RuJ/XjvWwVknoSKKNRF48lAQM6hDOKoKdqkkWLOhIQhLav4KcR+ZZrQpNGdKcGYjz5NGueQcl8o3lUL1fFpHFuyBfVAEDjgFVXAJaqAOMHgAT+AFvFqP1rP1Zr1PRjPWdGcX/IH1+Q0U1aCS</latexit>
fNL 0.02<latexit sha1_base64="gKKNmHaEeH6/0+kFPU8jep0UATs=">AAAB/3icbVDNS8MwHE3n15xfVcGLl+AQPI12CnrwMPDiQWSC+4C1lDRLt7AkLUkqjLqD/4oXD4p49d/w5n9juvWgmw9CHu/9fuTlhQmjSjvOt1VaWl5ZXSuvVzY2t7Z37N29topTiUkLxyyW3RApwqggLU01I91EEsRDRjrh6Cr3Ow9EKhqLez1OiM/RQNCIYqSNFNgHUeBxpIeSZ7c3E+gNBtCpOfXArpprCrhI3IJUQYFmYH95/RinnAiNGVKq5zqJ9jMkNcWMTCpeqkiC8AgNSM9QgThRfjbNP4HHRunDKJbmCA2n6u+NDHGlxjw0k3lWNe/l4n9eL9XRhZ9RkaSaCDx7KEoZ1DHMy4B9KgnWbGwIwpKarBAPkURYm8oqpgR3/suLpF2vuae1+t1ZtXFZ1FEGh+AInAAXnIMGuAZN0AIYPIJn8ArerCfrxXq3PmajJavY2Qd/YH3+AEAWlOs=</latexit>
[ Can also detect derivative operators and non-standard vacuum state using
shape of skewness or 3-point function ]

14. G
alaxies
WMAP satellite:
Planck satellite:
Both consistent with zero (2σ)
fNL = 37 ± 19.9 (1 )<latexit sha1_base64="wSUoQB1rZdZ9oTOmFg+16fboMnU=">AAACD3icbVDLSsNAFJ3UV62vqEs3g0WpICVphVpQKLhxIVLBPqApZTKdtENnkjAzEUroH7jxV9y4UMStW3f+jZM2C209cOFwzr3ce48bMiqVZX0bmaXlldW17HpuY3Nre8fc3WvKIBKYNHDAAtF2kSSM+qShqGKkHQqCuMtIyx1dJX7rgQhJA/9ejUPS5WjgU49ipLTUM4+9nsORGgoe395M4CUsV5yQQ7tarDqnsGA7kg44OumZeatoTQEXiZ2SPEhR75lfTj/AESe+wgxJ2bGtUHVjJBTFjExyTiRJiPAIDUhHUx9xIrvx9J8JPNJKH3qB0OUrOFV/T8SISznmru5MbpfzXiL+53Ui5Z13Y+qHkSI+ni3yIgZVAJNwYJ8KghUba4KwoPpWiIdIIKx0hDkdgj3/8iJplop2uVi6O8vXLtI4suAAHIICsEEF1MA1qIMGwOARPINX8GY8GS/Gu/Exa80Y6cw++APj8wf2DJoJ</latexit>
fNL = 0.9 ± 5.1 (1 )<latexit sha1_base64="MGqSx4th5hod5GGRxDRvEuYQa5w=">AAACEHicbVDLSsNAFJ34rPUVdelmsIgVNCRVUUGh4MaFSAX7gKaUyXTSDp1JwsxEKKGf4MZfceNCEbcu3fk3TtostPXAhcM593LvPV7EqFS2/W3MzM7NLyzmlvLLK6tr6+bGZk2GscCkikMWioaHJGE0IFVFFSONSBDEPUbqXv8q9esPREgaBvdqEJEWR92A+hQjpaW2uee3XY5UT/Dk9mYIL+GhbZ27EYcnluMewKLjStrlaL9tFmzLHgFOEycjBZCh0ja/3E6IY04ChRmSsunYkWolSCiKGRnm3ViSCOE+6pKmpgHiRLaS0UNDuKuVDvRDoStQcKT+nkgQl3LAPd2ZHi8nvVT8z2vGyj9rJTSIYkUCPF7kxwyqEKbpwA4VBCs20ARhQfWtEPeQQFjpDPM6BGfy5WlSK1nOkVW6Oy6UL7I4cmAb7IAicMApKINrUAFVgMEjeAav4M14Ml6Md+Nj3DpjZDNb4A+Mzx9Pa5ow</latexit>
Tightest limits today
Q
uantum
fluctuations
WMAP Collaboration, Bennett et al. (2013)
Planck Collaboration, Akrami et al. (1905.05697)
C
osm
ic
M
icrow
ave
Background
radiation

15. C
osm
ic
M
icrow
ave
Background
radiation
G
alaxies
Tightest limits tomorrow
Q
uantum
fluctuations
SNR2 ~ Number of modes ~ Volume
Galaxies & large-scale structure will give tightest limits in the future
Spoiler alert: Will reach (fNL) ' 1<latexit sha1_base64="dnH/MmjoXsWxe9QiW9iI+ZNwTgo=">AAACFXicbVDLSsNAFJ3UV62vqEsRBotQNyWpgi5cFNy4EKlgH9CEMplO2qEzSZyZCCVk5Uf4DW517U7cunbpnzhps7CtBy4czrmXe+/xIkalsqxvo7C0vLK6VlwvbWxube+Yu3stGcYCkyYOWSg6HpKE0YA0FVWMdCJBEPcYaXujq8xvPxIhaRjcq3FEXI4GAfUpRkpLPfPQkXTAUcXvORypoeDJ7U16okVOHqDdM8tW1ZoALhI7J2WQo9Ezf5x+iGNOAoUZkrJrW5FyEyQUxYykJSeWJEJ4hAakq2mAOJFuMnkjhcda6UM/FLoCBSfq34kEcSnH3NOd2a1y3svE/7xurPwLN6FBFCsS4OkiP2ZQhTDLBPapIFixsSYIC6pvhXiIBMJKJzezxeOpzsSeT2CRtGpV+7Rauzsr1y/zdIrgAByBCrDBOaiDa9AATYDBE3gBr+DNeDbejQ/jc9paMPKZfTAD4+sX6ZifJw==</latexit>

16. Galaxies form at peaks of the dark
matter distribution
Signature of multi-field inflation for galaxies
Kaiser (1984), Dalal et al. (2007), Top figure: J. Peacock
Position
Darkmatter
distribution
Galaxies Galaxies

17. Galaxies form at peaks of the dark
matter distribution
Signature of multi-field inflation for galaxies
Kaiser (1984), Dalal et al. (2007), Top figure: J. Peacock
Position
Darkmatter
distribution
Galaxies Galaxies
/
k2<latexit sha1_base64="f/JUgCJYeBnCTjeR1jVE2/WKT70=">AAACB3icbVDNSgMxGMzWv1r/Vj0KEiyCp7JbBXssePFYwbZCdy3ZbLYNzSYhyQpl6c2Lr+LFgyJefQVvvo1puwdtHQgMM9/Hl5lIMqqN5307pZXVtfWN8mZla3tnd8/dP+hokSlM2lgwoe4ipAmjnLQNNYzcSUVQGjHSjUZXU7/7QJSmgt+asSRhigacJhQjY6W+exzIIQ2kEtIIGCQK4TyICTNoko/u65O+W/Vq3gxwmfgFqYICrb77FcQCZynhBjOkdc/3pAlzpAzFjEwqQaaJRHiEBqRnKUcp0WE+yzGBp1aJYSKUfdzAmfp7I0ep1uM0spMpMkO96E3F/7xeZpJGmFMuM0M4nh9KMgZt5mkpMKaKYMPGliCsqP0rxENkyzC2uootwV+MvEw69Zp/XqvfXFSbjaKOMjgCJ+AM+OASNME1aIE2wOARPINX8OY8OS/Ou/MxHy05xc4h+APn8wfB9ZnX</latexit>
Multi-field inflation couples those
peaks to the background potential
Galaxies are modulated by the
background potential

18. Galaxies form at peaks of the dark
matter distribution
Signature of multi-field inflation for galaxies
Kaiser (1984), Dalal et al. (2007), Top figure: J. Peacock
Position
Darkmatter
distribution
Galaxies Galaxies
Enhancement of the power
spectrum of galaxies ~ fNL/k2
‘Scale-dependent galaxy bias’
(local) Primordial non-Gaussianity
Simone Ferraro (Berkeley)
7
galaxy power spectrum
fNL = 100
fNL = 0
N. Dalal ++ (2007)
INFLATION
single field Gaussian fluctuations
multi-field Non-Gaussian fluctuations
bias(k)
Generates scale dependent bias:
Wavenumber k (~ 1/scale)
Powerspectrumofgalaxies
fNL=100
fNL=0
/
k2<latexit sha1_base64="f/JUgCJYeBnCTjeR1jVE2/WKT70=">AAACB3icbVDNSgMxGMzWv1r/Vj0KEiyCp7JbBXssePFYwbZCdy3ZbLYNzSYhyQpl6c2Lr+LFgyJefQVvvo1puwdtHQgMM9/Hl5lIMqqN5307pZXVtfWN8mZla3tnd8/dP+hokSlM2lgwoe4ipAmjnLQNNYzcSUVQGjHSjUZXU7/7QJSmgt+asSRhigacJhQjY6W+exzIIQ2kEtIIGCQK4TyICTNoko/u65O+W/Vq3gxwmfgFqYICrb77FcQCZynhBjOkdc/3pAlzpAzFjEwqQaaJRHiEBqRnKUcp0WE+yzGBp1aJYSKUfdzAmfp7I0ep1uM0spMpMkO96E3F/7xeZpJGmFMuM0M4nh9KMgZt5mkpMKaKYMPGliCsqP0rxENkyzC2uootwV+MvEw69Zp/XqvfXFSbjaKOMjgCJ+AM+OASNME1aIE2wOARPINX8OY8OS/Ou/MxHy05xc4h+APn8wfB9ZnX</latexit>
Multi-field inflation couples those
peaks to the background potential
Galaxies are modulated by the
background potential

19. DES
SDSS
DESI
Euclid
MegaMapper?
Puma?
LSST
SPHEREx
Subaru HSC
Galaxy surveys
2020 2022 2030
Funding by DOE, ESA, Heising-Simons, Moore Foundation, NASA, NSF, Simons Foundation, …

20. LSST — now NSF Vera C. Rubin Observatory
LSST Project/NSF/AURA 01/31/2020
- Cerro Pachón, Chile (2,663 m / 8,737 ft)
- 8.4m / 27-ft mirror
- Cover entire southern sky every few nights
- 10 year survey over 18,000 deg2
- 37 billion stars and galaxies
- First light 2021, full operations 2022-2032

21. LSST Project/NSF/AURA 02/03/2020

22. (1) Observational systematics
- Dust extinction in our galaxy (affects galaxy spectra)
- Galaxy/star confusion
- Noise & observation conds. can vary across different patches
(2) Sample variance
- Few long-wavelength modes fit into observed volume
Two types of challenges on large scales
Milky Way dust

23. (1) Observational systematics
- Dust extinction in our galaxy (affects galaxy spectra)
- Galaxy/star confusion
- Noise & observation conds. can vary across different patches
(2) Sample variance
- Few long-wavelength modes fit into observed volume
Two types of challenges on large scales
Milky Way dust
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.
Galaxy catalog
Distribution of DM
Cross-correlating galaxy catalog with the
distribution of dark matter can help with both
MS & Seljak (2018)

24. Imagine you come up with a new image compression algorithm
Is it better than JPEG?
How to avoid sample variance?

25. Method 1
a. Ask people to rate JPEG-compressed
images
b. Ask other people to rate other images
compressed with new algorithm
c. Compare ratings to find winner
…
…
Subject to sample variance (error of the mean)
JPEG JPEG
New algorithm New algorithm

26. Original
JPEG New algorithm
a. Ask people to rate same image compressed with JPEG & new algorithm
b. Compare ratings 1-by-1 for each image
Method 2
No sample variance (can tell winner with 1 image)!

27. Method 1: Measure galaxy power spectrum
Seljak (2009), McDonald & Seljak (2009), MS & Seljak (2018)
Wavenumber k (~ 1/scale)
Multi-field inflation
Measured galaxy power
(=Gaussian realization with
mean given by dashed)
Sample variance
(error on the mean)
Single field inflation

28. Method 1: Measure galaxy power spectrum
Seljak (2009), McDonald & Seljak (2009), MS & Seljak (2018)
Wavenumber k (~ 1/scale)
Multi-field inflation
Measured galaxy power
(=Gaussian realization with
mean given by dashed)
Sample variance
(error on the mean)
Single field inflation
Cannot tell if single field or multi-field inflation because of sample variance

29. Method 2: Compare 1-by-1 to dark matter
Seljak (2009), McDonald & Seljak (2009), MS & Seljak (2018)
Wavenumber k (~ 1/scale)
Multi-field inflation
Measured dark matter power
Measured galaxy power
Sample variance
(error on the mean)
Single field inflation

30. Method 2: Compare 1-by-1 to dark matter
Seljak (2009), McDonald & Seljak (2009), MS & Seljak (2018)
Sample variance cancels so we detect multi-field inflation
Wavenumber k (~ 1/scale)
Multi-field inflation
Measured dark matter power
Measured galaxy power
Sample variance
(error on the mean)
Single field inflation

31. Method 2: Compare 1-by-1 to dark matter
Seljak (2009), McDonald & Seljak (2009), MS & Seljak (2018)
Sample variance cancels so we detect multi-field inflation
Wavenumber k (~ 1/scale)
Multi-field inflation
Measured dark matter power
Measured galaxy power
Sample variance
(error on the mean)
Single field inflation
Wavenumber k (~ 1/scale)

32. Use gravitational lensing
How to measure the distribution of dark matter?

33. Dark matter also distorts the Cosmic Microwave Bg.
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-

34. 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

35. Cosmic Microwave Background (CMB)
Hot and cold blobs: Picture of the Universe 13.6996 bn years ago
T = 2.7 K, T/T ⇠ 10 5
<latexit sha1_base64="g5rqHxeTKPCg/l8FrvHHuKmp5zA=">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</latexit>

36. Statistics of the CMB before lensing
Lensed
Normally distributed as far as we can tell

37. Statistics of the CMB before lensing
Lensed
Variance
Power spectrum

38. 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

39. 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

40. 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

41. 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

42. Measured CMB 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

43. CMB experiments
Today 2022 2030
LiteBIRD
POLARBEAR
Funding by DOE, Heising-Simons, JAXA, NASA, NSF, Simons Foundation, …

44. Future CMB lensing
Signal
Planck
Simons Observatory
CMB-S4
PICO
Derived from foreground deprojection forecasts by Colin Hill

45. Future CMB lensing
Signal
Planck
Simons Observatory
CMB-S4
PICO
4x
20x
50x
Derived from foreground deprojection forecasts by Colin Hill

46. The Simons Observatory
- Cerro Toco, Atacama desert, Chile
- 5200m / 17,100 ft
- Currently home to Atacama Cosmology
Telescope, POLARBEAR, Simons Array,
CLASS
- 60,000 detectors
- 6 spectral bands at 27-280 GHz
- Science observations 2022-2027
- 260+ researchers at 40+ institutions
in 10+ countries

47. Primordial non-Gaussianity with Simons + LSST
MS & Seljak (2018), Simons Observatory Science White Paper (2019)
LSST gold
LSST optimistic
Excluded
by
Planck
(1σ)
M
ulti-field
inflation
Single
field
inflation
Largest angular scale
~ 3-5x better than Planck
Includes factor ~2 from sample variance cancellation

48. 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 galaxies
Gold sample Optimistic sample
At low z, use clustering redshifts (Gorecki+ 2014)
At high z, add Lyman-break dropout galaxies (extrapolated from HSC observations)
Details: MS & Seljak (2018)
~1 billion galaxies each at z = 0-0.5, 0.5-1,1-2, 2-4
CMB lensing kernel

49. Lyman-break dropout galaxies
Young star-forming galaxies that have lots of neutral hydrogen
Photons with enough energy ionize that and don’t get out of galaxy
Figure:Ellis(1998)
Galaxyspectrum(flux)
Observed wavelength in Å
low-energy photons
high-en.
photons

50. Lyman-break dropout galaxies
Young star-forming galaxies that have lots of neutral hydrogen
Photons with enough energy ionize that and don’t get out of galaxy
Figure:Ellis(1998)
Galaxyspectrum(flux)
Observed wavelength in Å
low-energy photons
high-en.
photons
U B V I
U-dropout

51. Lyman-break dropout galaxies
Young star-forming galaxies that have lots of neutral hydrogen
Photons with enough energy ionize that and don’t get out of galaxy
Figure:Ellis(1998)
HSC found 0.5M at z=4-7 in 100 deg2, so expect ~100M with LSST
Also MegaMapper
Ono, Ouchi+ (2018)
Wilson & White (2019), Ferraro et al. (1903.09208), Schlegel et al. (1907.11171)
Galaxyspectrum(flux)
Observed wavelength in Å
low-energy photons
high-en.
photons
U B V I
U-dropout

52. Other CMB experiments
Today 2022 2030
LiteBIRD
POLARBEAR
Funding by DOE, Heising-Simons, JAXA, NASA, NSF, Simons Foundation, …

53. 10x better than Planck, even with LSST gold (conservative)
Largest angular scale
M
ulti-field
inflation
Single
field
inflation
Excluded
by
Planck
(1σ)
Simons Observatory
CMB-S4
PICO
Simons Observatory: Science White Paper (1808.07445), Decadal White Paper (1907.08284)
CMB-S4: Science Case Paper (1907.04473), Decadal White Paper (1908.01062)
PICO: NASA Mission Study (1902.10541), Decadal White Paper (1908.07495)
Other CMB experiments

54. All forecasts assume SO + Planck
Include systematic error budget
All quoted errors are 1σ
Other science goals of Simons Observatory

55. All forecasts assume SO + Planck
Include systematic error budget
All quoted errors are 1σ
Other science goals of Simons Observatory
SO can detect any particle
with spin that decoupled
after the start of the QCD
phase transition (at 2σ)

56. All forecasts assume SO + Planck
Include systematic error budget
All quoted errors are 1σ
Other science goals of Simons Observatory

57. All forecasts assume SO + Planck
Include systematic error budget
All quoted errors are 1σ
Other science goals of Simons Observatory

58. Deviations from a cosmological constant
Lensing = line of sight integral so cannot resolve time dependence
Galaxy clustering amplitude depends on galaxy type (unknown)
Either alone cannot measure dark energy as function of time
But joint analysis of lensing + clustering can!
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
also use simulations to test approximations for the likelihood of the lensing re-

59. Test acceleration of the Universe and dark energy with 1-2% precision
(currently >7%)
Deviations from a cosmological constant
relations
maps of the reconstructed
w-redshift tracers of the
handle on the growth of
ussianity, curvature, and
t the science obtainable
SO reconstructed lensing
particularly suited for this
l observe a large number
covering a large fraction
ernel. The LSST galaxy
bed in Sec. 2.
cast the cross-correlation
ng field,  (using the min-
d the galaxy overdensity

Cgg
L )1/2
, to be ⇢ 80%
a maximum correlation of
e LSST galaxies are opti-
atch the CMB lensing ker-
15; Schmittfull and Seljak
timistic LSST sample will
oe cient to ⇢ 80% for
rrelation of ⇢ ' 90% for
n coe cients imply that
n be useful (Seljak 2009),
to-noise ratio calculation
rstate the potential con-
0 1 2 3 4 5 6 7
z
-0.04
0
0.04
(8(z))/8(z)
fsky = 0.1
SO Baseline + LSST gold; fsky = 0.4
SO Goal + LSST gold
SO Goal + LSST optimistic
Figure 29. The relative uncertainty on 8, defined in six tomo-
graphic redshift bins (z=0–0.5, 0.5–1, 1–2, 2–3, 3–4, 4–7), as a func-
tion of redshift. The forecast assumes a joint analysis of C
L , Cg
L
and Cgg
L , where  is SO CMB lensing and g denotes LSST galax-
ies binned in tomographic redshift bins. We use multipoles from
Lmin = 30 to Lmax corresponding to kmax = 0.3hMpc 1 in each
tomographic bin. We marginalize over one linear galaxy bias pa-
rameter in each tomographic bin and over ⇤CDM parameters. We
also include Planck CMB information – with a ⌧ prior, (⌧) = 0.01
– and BAO measurements (DESI forecast). The filled bands andMS & Seljak (2018), Simons Observatory Science White Paper (2019), Figure: B. Yu
Varianceoffluctuations
Redshift z

60. - Getting redshift errors down to LSST requirement is crucial
- Reducing redshift errors further would help (clustering redshifts, …)
More careful forecast
Cawthon (2018)
0 1 2 3 4 5 6 7
z
10 2
10 1
(8)/8
lmax = 1000
No prior
p(z0) = 0.03(1 + z)
p( z) = 0.03(1 + z)
p(both) = 0.03(1 + z)
p(z0) = 0.003(1 + z)
p( z) = 0.003(1 + z)
p(both) = 0.003(1 + z)
FIG. 17. Constraints on 8 for our fiducial LSST/CMB-S4 analysis when adding priors on redshift
lmax = 1000. Right: constraints when having lmax = 2000. Each curve adds either a prior on z0, on
the curves with priors to the fiducial case of no priors, and the opposite extreme of no redshift unc
analysis. The priors of 0.003(1 + z) and 0.03(1 + z) come from the LSST DESC SRD requireme
prior of 0.0004(1 + z) is a plausible future achievement by clustering redshifts at low z found in [5

61. B. Yu, Knight et al. (incl. MS, 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,
Massless neutrinosVarianceoffluctuations
Fix other cosmological params
Marginalize over other cosmological params
⌃m⌫ = 60 meV<latexit sha1_base64="sKmtVB7Atjife+RVgA4jtsUMngg=">AAACFnicbVDLSsNAFJ34rPUVdSnIYBFcSEmqqAuFghuXFe0DmhAm02k7dGYSZiZCCd35EX6DW127E7duXfonTtosbOuBC4dz7uXee8KYUaUd59taWFxaXlktrBXXNza3tu2d3YaKEolJHUcskq0QKcKoIHVNNSOtWBLEQ0aa4eAm85uPRCoaiQc9jInPUU/QLsVIGymwD7x72uMI8sATyfW5A70TjyPdlzzlpDEK7JJTdsaA88TNSQnkqAX2j9eJcMKJ0JghpdquE2s/RVJTzMio6CWKxAgPUI+0DRWIE+Wn4z9G8MgoHdiNpCmh4Vj9O5EirtSQh6Yzu1HNepn4n9dOdPfST6mIE00EnizqJgzqCGahwA6VBGs2NARhSc2tEPeRRFib6Ka2hDzLxJ1NYJ40KmX3tFy5OytVr/J0CmAfHIJj4IILUAW3oAbqAIMn8AJewZv1bL1bH9bnpHXBymf2wBSsr18MYJ8z</latexit>
Redshift z
Sum of neutrino masses

62. B. Yu, Knight et al. (incl. MS, 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,
Massless neutrinosVarianceoffluctuations
Fix other cosmological params
Marginalize over other cosmological params
⌃m⌫ = 60 meV<latexit sha1_base64="sKmtVB7Atjife+RVgA4jtsUMngg=">AAACFnicbVDLSsNAFJ34rPUVdSnIYBFcSEmqqAuFghuXFe0DmhAm02k7dGYSZiZCCd35EX6DW127E7duXfonTtosbOuBC4dz7uXee8KYUaUd59taWFxaXlktrBXXNza3tu2d3YaKEolJHUcskq0QKcKoIHVNNSOtWBLEQ0aa4eAm85uPRCoaiQc9jInPUU/QLsVIGymwD7x72uMI8sATyfW5A70TjyPdlzzlpDEK7JJTdsaA88TNSQnkqAX2j9eJcMKJ0JghpdquE2s/RVJTzMio6CWKxAgPUI+0DRWIE+Wn4z9G8MgoHdiNpCmh4Vj9O5EirtSQh6Yzu1HNepn4n9dOdPfST6mIE00EnizqJgzqCGahwA6VBGs2NARhSc2tEPeRRFib6Ka2hDzLxJ1NYJ40KmX3tFy5OytVr/J0CmAfHIJj4IILUAW3oAbqAIMn8AJewZv1bL1bH9bnpHXBymf2wBSsr18MYJ8z</latexit>
Redshift z
Can measure neutrino mass using
z<7 lever arm, without need for
optical depth to CMB
Independent from usual probes
With CMB-S4 and BAO, can reach
~25 meV uncertainty (1σ)
Guaranteed >2σ signal (>60meV
from oscillation expts.)
Sum of neutrino masses

63. Within the SO collaboration we currently
prepare the analysis pipeline for cross-
correlation with LSST
(L3.2) CMB lensing cross-correlations
V. Boehm, A. Challinor, G. Fabbian, S. Ferraro, M.
Madhavacheril, E. Schaan, N. Sehgal, B. Sherwin,
A. van Engelen, …
(LT) Likelihood and Theory
E. Calabrese, M. Gerbino, V. Gluscevic, R. Hlozek,
A. Lewis, …
Simons Observatory Analysis Working Groups
I’m also a member of the CMB-S4, PICO and LSST collaborations,
where related efforts are currently underway

64. Other CMB lensing research interests (ask me later)
(1) Covariance for joint analysis of CMB and CMB magnification
(2) Estimate unlensed CMB using galaxies
(3) New bias 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)
Sherwin & MS (2015)
Böhm, MS & Sherwin (2016)
Beck, Fabbian & Errard (2018)
Böhm, Sherwin, Liu, Hill, MS & Namikawa (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

65. II. Galaxy clustering: Theory & Analysis
(1) Modeling galaxy clustering
(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

66. (1) Modeling galaxy clustering
(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
II. Galaxy clustering: Theory & Analysis

67. If we had the data today, would not be able to make cosmology
inference because some components are not ready yet
- Theoretical modeling of galaxy clustering
- Simplified in forecasts
- Good enough for current data but not next-generation data
- Relation between galaxies and dark matter (‘galaxy bias’)
- Redshift space: Redshift errors, redshift space distortions, fingers of God
- Galaxy formation/baryonic physics: stellar feedback, blackhole feedback,
radiative transfer, magnetic fields, …
Data will be amazing. Let’s make the theory & analysis adequate.
Motivation

68. Must connect dark matter distribution to galaxy distribution
Modeling galaxy clustering
Simplest model: Linear ‘bias’ relation
Can prove this is correct on very
large scales (Peebles, Kaiser, …)
But breaks down on smaller scales
Need nonlinear corrections
g(x) = b1 m(x)<latexit sha1_base64="la12iyi+0jxUS9VUv3RuVNhfCT8=">AAACGHicbVDLSsNAFJ3UV62vqEs3g0Wom5pUQRcKBTcuK9gHNCFMJpN26GQSZiZiCf0MN/6KGxeKuO3Ov3HSZqGtBy4czrmXe+/xE0alsqxvo7Syura+Ud6sbG3v7O6Z+wcdGacCkzaOWSx6PpKEUU7aiipGeokgKPIZ6fqj29zvPhIhacwf1DghboQGnIYUI6UlzzxzAsIU8gY1J0Jq6IfZ0+QU3kDfs2FhRb8tz6xadWsGuEzsglRBgZZnTp0gxmlEuMIMSdm3rUS5GRKKYkYmFSeVJEF4hAakrylHEZFuNntsAk+0EsAwFrq4gjP190SGIinHka878xPlopeL/3n9VIVXbkZ5kirC8XxRmDKoYpinBAMqCFZsrAnCgupbIR4igbDSWVZ0CPbiy8uk06jb5/XG/UW1eV3EUQZH4BjUgA0uQRPcgRZoAwyewSt4Bx/Gi/FmfBpf89aSUcwcgj8wpj+yT5+K</latexit>

69. This approach has been extensively studied in the past
Most of the analyses use n-point functions. Disadvantages:
- Sample variance, compromise on resolution/size of simulation
- On small scales hard to disentangle different sources of nonlinearity
- Overfitting (smooth curves, many parameters)
- Only few lowest n-point functions explored in practice
- Difficult to isolate and study the noise
Modeling galaxy clustering
Review by Desjacques, Jeong, Schmidt (2018): Large-Scale Galaxy Bias

70. New setup: Field level comparison
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>
Pert.m
odel
MS, Simonović, Assassi & Zaldarriaga (2019)

71. Benefits of using 3D fields rather than summary statistics
+ No sample variance, can use small volumes with high resolution
+ No overfitting (6 parameters describe >1 million 3D Fourier modes)
+ ‘All’ n-point functions measured simultaneously
+ Easy to isolate the noise
+ Applicable to field-level likelihood and initial condition reconstruction
Benefits
MS, Simonović, Assassi & Zaldarriaga (2019)

72. Benefits of using 3D fields rather than summary statistics
+ No sample variance, can use small volumes with high resolution
+ No overfitting (6 parameters describe >1 million 3D Fourier modes)
+ ‘All’ n-point functions measured simultaneously
+ Easy to isolate the noise
+ Applicable to field-level likelihood and initial condition reconstruction
Benefits
MS, Simonović, Assassi & Zaldarriaga (2019)
Questions we studied
1. How well does perturbative bias model work?
2. How correlated is the galaxy density field with the initial conditions?
3. What are the properties of the noise?

73. Simulation
Ran N-body code on local cluster with ~2000 CPUs
15363 = 3.6B particles in a 3D cubic box
30723 = 29B grid points for long-range force computation
4000 time steps
5 realizations
~ 1M CPU hours
MS, Simonović, Assassi & Zaldarriaga (2019)

74. Comparison with linear model
Reasonable prediction on large scales
Missing structure on small scales
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>
MS, Simonović, Assassi & Zaldarriaga (2019)

75. Nonlinear model
Include all nonlinear terms allowed by symmetries
(= Effective Field Theory)
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>
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>
MS, Simonović, Assassi & Zaldarriaga (2019)
Desjacques, Jeong & Schmidt (2018): Review of Large-Scale Galaxy Bias

76. 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>
Comparison with nonlinear 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>
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 linear model
Simulation
(= truth)
MS, Simonović, Assassi, Zaldarriaga (2019)

77. Power spectrum of the noise (model error)
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
White noise error is crucial to avoid biasing cosmology parameters
MS, Simonović, Assassi, Zaldarriaga (2019)

78. Tried many other nonlinear bias operators
3-6x larger model error. Reason: Bulk flows, need ‘shifted’ operators
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 perturba
Errorpowerspectrum
MS, Simonović, Assassi, Zaldarriaga (2019)
Increase in wavenumber corresponds to 8-30x larger volume

79. Model must account for bulk flows to get small model error
Bulk flows
Model accounts for bulk flows
✏(x) ⌘ truth model<latexit sha1_base64="j8JWHHDkGfzWSFy/jB9sc9v3ags=">AAACOXicbVDLSgNBEJz1Gd9Rj14Gg6AHw64KevAQ8eJRwaiQDWF20jGD81hnesWw5Hv8CL/Bq+LRm3j1B5yNOWi0YKC6qpvuqSSVwmEYvgZj4xOTU9Olmdm5+YXFpfLyyoUzmeVQ50Yae5UwB1JoqKNACVepBaYSCZfJzXHhX96BdcLoc+yl0FTsWouO4Ay91CofxZA6IY3evN+iMdxm4o7SuA0SWStWDLtW5Wgz7Pbp9qiujK/7rXIlrIYD0L8kGpIKGeK0VX6L24ZnCjRyyZxrRGGKzZxZFFxCfzbOHKSM37BraHiqmQLXzAdf7dMNr7Rpx1j/NNKB+nMiZ8q5nkp8Z3GlG/UK8T+vkWHnoJkLnWYImn8v6mSSoqFFbrQtLHCUPU8Yt8LfSnmXWcbRp/trS6KKTKLRBP6Si51qtFvdOdur1A6H6ZTIGlknmyQi+6RGTsgpqRNOHsgTeSYvwWPwFrwHH9+tY8FwZpX8QvD5BQAFrsc=</latexit>
Model doesn’t account for bulk flows
MS, Simonović, Assassi, Zaldarriaga (2019)

80. (1) Modeling galaxy clustering
(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
II. Galaxy clustering: Theory & Analysis

81. Application to data
Model was applied to SDSS BOSS data (~1 million galaxy spectra)
MCMC sampling of posteriors was enabled by fast evaluation of the
model power spectrum (reducing 2D loop integrals to 1D FFTs)
MS, Vlah & McDonald (2016)
McEwen, Fang, Hirata & Blazek (2016)
Cataneo, Foreman & Senatore (2017)
Simonović, Baldauf, Zaldarriaga et al. (2018)
D’Amico, Gleyzes, Kokron et al. (1909.05271)
Ivanov, Simonović & Zaldarriaga (1909.05277)
Tröster, Sanchez, Asgari et al. (2020)

82. Similar precision as Planck for some parameters
distribution for cosmological parameters extracted from
ctrum likelihood. We show results for four independent
CMB (Planck 2018)
Galaxies (SDSS)
Expansion rate rms of fluctuationsAmount of DM
Ivanov et al. (arXiv:1909.05277)
Similar precision
as CMB

83. Future: Dark Energy Spectroscopic Instrument DESI
https://www.desi.lbl.gov/
- 35 million galaxy spectra over 14,000 deg2
- 5,000 robots to position fibers that take spectra
- 5 year survey, first light October 2019
- 600 scientists from 82 institutions
- Funding by DOE, NSF, Heising-Simons, Moore, STFC, …
SI survey will provide a rich tool for cosmology. DESI will make
and the impact of dark energy using the baryon acoustic
ars, and the Lyman-α forest. Dark energy and dark matter will
ities and galaxy clustering. The DESI survey will probe these
m the clustering of intergalactic hydrogen at redshift 3 to large
s of low-redshift galaxies. In addition the study of dark energy,
tergalactic medium, the origin of black holes, the astrophysics
e mass/energy/chemical cycles within galaxies.
by
es,
most
I will map
4,000
dditional
-redshift

84. Future: Dark Energy Spectroscopic Instrument DESI
https://www.desi.lbl.gov/
- 35 million galaxy spectra over 14,000 deg2
- 5,000 robots to position fibers that take spectra
- 5 year survey, first light October 2019
- 600 scientists from 82 institutions
- Funding by DOE, NSF, Heising-Simons, Moore, STFC, …
SI survey will provide a rich tool for cosmology. DESI will make
and the impact of dark energy using the baryon acoustic
ars, and the Lyman-α forest. Dark energy and dark matter will
ities and galaxy clustering. The DESI survey will probe these
m the clustering of intergalactic hydrogen at redshift 3 to large
s of low-redshift galaxies. In addition the study of dark energy,
tergalactic medium, the origin of black holes, the astrophysics
e mass/energy/chemical cycles within galaxies.
by
es,
most
I will map
4,000
dditional
-redshift
Survey (r < 19.5)
usters, and Cross-
sars
t
Figure: O. Lahav

85. II. Galaxy clustering: Theory & Analysis
(1) Modeling galaxy clustering
(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

86. Probability distribution of DM halos / galaxies
Not normally distributed, highly skewed
DM halo number density
variance
Field value
Histogram
Information in the tails can improve precision of parameter estimates

87. Measuring skewness
Challenging: Can form too many Fourier mode triplets
Solution 1
In space of all triplets, expand in simple basis functions
Solution 2
Given parameter of interest, compute max. likelihood estimator (matched filter).
Sum over all triplets can be computed using a few 3D FFTs.
Regan, MS, Shellard & Fergusson (2011)
MS, Regan & Shellard (2013)
MS, Baldauf & Seljak (2015)
Moradinezhad Dizgah, Lee, MS & Dvorkin (2020)

88. Modeling skewness
Modeling skewness on mildly nonlinear scales is challenging
Large literature
Still no good model for galaxies in redshift space including
primordial non-Gaussianity
We should improve modeling and estimators for DESI & SPHEREx
Angulo, Foreman, MS & Senatore (2015)
Lazanu, Giannantonio, MS & Shellard (2016)
Lazanu, Giannantonio, MS & Shellard (2017)
+ many more papers by others, incl. Baldauf, Gil-Marin, Porciani, Scoccimarro, Sefusatti

89. (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
II. Galaxy clustering: Theory & Analysis

90. Getting initial from final conditions
Gravitational force
Initial conditions
Galaxies

91. Getting initial from final conditions
Gravitational force
Reconstruction
Initial conditions
Galaxies

92. Getting initial from final conditions
Gravitational force
Goal: Reconstruct initial conditions & measure their power spectrum
Reconstruction
Initial conditions
Galaxies

93. Many proposed algorithms
First by J. Peebles in late 1980’s, then for BAO by D. Eisenstein 2007
Renewed interest in last ~5 years
MS, Feng+ (2015), MS, Baldauf+ (2017)
Zhu, Yu+ (2017), Wang, Yu+ (2017)
Seljak, Aslanyan+ (2017), Modi, Feng+ (2018)
Shi+ (2018), Hada+ (2018), Modi, White+ (2019), Sarpa+ (2019), Schmidt+ (2019), Elsner+
(2019), Yu & Zhu (2019), Zhu, White+ (2019)
Also sampling
Jasche, Kitaura, Lavaux, Wandelt, …
Machine learning
Li, Ho, Villaescusa-Navarro, …
Theory
work by Eisenstein, Padmanabhan, White etc, later e.g. MS+ (2015), Cohn+ (2016), Hikage+
(2017), Wang+ (2018), Sherwin+ (2018)

94. True initial conditions
Halo density before reconstruction
MS, Baldauf & Zaldarriaga (2017)
Reconstruction

95. True initial conditions
Halo density before reconstruction
Reconstruction
MS, Baldauf & Zaldarriaga (2017)

96. No
reconstruction
Correlation with true initial conditions
100%
80%
60%
40%
20%
0%
rcc
Wavenumber

97. No
reconstruction
Correlation with true initial conditions
100%
80%
60%
40%
20%
0%
rcc
Standard
reconstruction
Wavenumber

98. No
reconstruction
Correlation with true initial conditions
MS, Baldauf & Zaldarriaga (2017), similar to Zhu, Yu+ (2017); noise-free DM
100%
80%
60%
40%
20%
0%
rcc
Standard
reconstruction
New reconstruction
Wavenumber

99. For SDSS/BOSS, standard reconstruction gave ~2x tighter
measurement of BAO scale and Hubble parameter (= 4x volume)
For DESI, more optimal BAO reconstruction gives
(a) 30-40% tighter Hubble parameter than standard rec. (= 2x volume)
(b) 70-120% tighter constraints on primordial features from some inflation models
(c) Unbiased and tighter constraints on compensated isocurvature perturbations
Reconstruction of the linear BAO scale
(a), (b) Preliminary forecasts by M. Ivanov, B. Wallisch, (c) Heinrich & Schmittfull (2019)
Large additional gains possible if we can also get broadband linear power spectrum.
•Thisdistanceof150Mpcisveryaccuratelycomputed
fromtheanisotropiesoftheCMB.
–0.4%calibrationwithcurrentCMB.
ImageCredit:E.M.Huff,theSDSS-IIIteam,andthe
SouthPoleTelescopeteam.GraphicbyZosiaRostomian

100. Use gradient descent to maximize posterior distribution of initial
conditions given observed galaxy density
Optimization in 1M+ dimensions
Gradients:
- Automatic differentiation of simulation code
- or analytical derivative of bias model (simpler)
Reconstruct by inverting forward model
From simulation or bias model
P( IC| g) =
L( g| IC)P( IC)
P( g)<latexit sha1_base64="nhsW14myFMFirXMZIHaJK2sdCWM=">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</latexit>
Normal distribution
(Gaussian ICs)
Seljak, Aslanyan et al. (2017)
Schmidt, Elsner et al. (2019)
Modi, White et al. (arXiv:1907.02330)

101. Recovering modes from the 21cm wedge
Modi, White et al. (arXiv:1907.02330)
Foregrounds destroy long modes: ‘21cm wedge’
Reconstruction inverting bias model with shifted
operators recovers these modes
Recon-
struction
Figure 1: The fraction of the tota
like survey at z = 2 (left), 4 (midd
a pessimistic and optimistic foreca
most visible at z = 2, is due to the
leading to a minimum baseline len
in the wedge are illustrated by th
below and to the right of these line
lose modes with kk < kmin
k where km
k
at intermediate k the shot-noise st
the instrument dominates.
3 Data: Hidden Valley s
To test the e cacy of our metho
trillion-particle N-body simulation
workhorse simulation will be HV10
box from Gaussian initial condition
resolution, one is able to resolve h
of the cosmological HI signal, whi
baryon acoustic oscillations.
The halos in the simulation we
in more detail in ref. [37]. We ma
Wavenumber k
Correlationwithtruedensity

102. Anticipate large influx of high-quality data over the next decade
Many synergies between datasets
CMB lensing and galaxy clustering can
- Rule out single field inflation
- Measure deviation from cosmological constant / standard growth at 1% level
- Provide independent and competitive measurement of neutrino mass
To exhaust scientific potential of the data, we must
- Develop adequate theoretical models
- Taylor our analysis methods to the datasets and make them optimal
- Test both very carefully with simulations
Conclusion
Figure 1. 2-d slices of the overdensity h(x) of simulated 1010.8
1011.8
h 1
M hal

103. Thank you

104. Backup slides