Chris Clarkson - Dark Energy and Backreaction

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Chris Clarkson - Dark Energy and Backreaction

  1. 1. backreaction in the concordance model Chris Clarkson Astrophysics, Cosmology & Gravitation Centre University of Cape TownThursday, 26 January 12
  2. 2. key ingredients for cosmology • universe homogeneous and isotropic on large scales (>100 Mpc) • background dynamics determined by amount of matter + curvature present (+ a theory of gravity) • inflation lays down the seeds for structure formation from quantum fluctuations • works ... if we include a cosmological constant or ‘dark energy’ • matter + curvature not enoughThursday, 26 January 12
  3. 3. How does structure affect the background?Thursday, 26 January 12
  4. 4. How does structure affect the background?Thursday, 26 January 12
  5. 5. Thursday, 26 January 12
  6. 6. 3 issues •Averaging •Coarse-graining of structure, such that small-scale effects are hidden to reveal large scale geometry and dynamics. •Backreaction •Gravity gravitates, so local gravitational inhomogeneities may affect the cosmological dynamics. •Fitting •How do we fit an idealized model to observations made from one location in a lumpy universe, given that the ‘background’ does not in fact exist? (Averaging observables is not same as spatial average)Thursday, 26 January 12
  7. 7. scale of homogeneity of the dist 1 1 it seems 3natural to define the e HD ⇤ ↵N ⇧ D = N ⇧Jd x , As pointed out in matter flow. 3 3VD D Averaging is difficult one introduces an ambiguity inVD ⇤ Jd x is the Riemannian volume of D. The avereaging as m 3 expansions: the expansion oper measured by observers comoving •Define Riemannian averaging on ⌥. As mentionned before, the coer the domain D: domain D 1 the matter distribution, so that ↵D = ↵↵ D ⇤ ↵(t, xi )Jd3 x ,the space-time metri write down VD D In the following we will then ret scalar function ↵. One can then define the e⌥ective scale factor for: Riemannian volume ✏ element ⇣t aD where J ⇤ det(hij ) and VD ⇤ HD = . the Riemannian average over the spatial average implies wrt aD some foliation of spacetime that is well defined for any scala how can average of a tensor preserve Lorentz invariance? the function aD (t) obeying: Thursday, 26 January 12
  8. 8. scale of homogeneity of the dist 1 1 it seems 3natural to define the e HD ⇤ ↵N ⇧ D = N ⇧Jd x , As pointed out in matter flow. 3 3VD D Averaging is difficult one introduces an ambiguity inVD ⇤ Jd x is the Riemannian volume of D. The avereaging as m 3 expansions: the expansion oper measured by observers comoving •Define Riemannian averaging on ⌥. As mentionned before, the coer the domain D: domain D 1 the matter distribution, so that ↵D = ↵↵ D ⇤ ↵(t, xi )Jd3 x ,the space-time metri write down VD D In the following we will then ret scalar function ↵. One can then define the e⌥ective scale factor for: eg, to specify average Riemannian volume ✏ element energy density need ⇣t aD where J ⇤ det(hij ) and VD ⇤ full solution of the HD = . the Riemannian average over the spatial average implies wrt aD field equations some foliation of spacetime that is well defined for any scala how can average of a tensor preserve Lorentz invariance? the function aD (t) obeying: Thursday, 26 January 12
  9. 9. Backreaction is generic on small scales but on large scalesThursday, 26 January 12
  10. 10. Backreaction is generic on small scales but on large scales backreaction terms arise from averaging & non-linear EFEThursday, 26 January 12
  11. 11. Buchert ZalaletdinovThursday, 26 January 12
  12. 12. Buchert ZalaletdinovThursday, 26 January 12
  13. 13. Buchert backreaction from non-local variance of local expansion rate ZalaletdinovThursday, 26 January 12
  14. 14. Buchert backreaction from non-local variance of local expansion rate ZalaletdinovThursday, 26 January 12
  15. 15. Buchert backreaction from non-local variance of local expansion rate Zalaletdinov macroscopic Ricci correlation tensorThursday, 26 January 12
  16. 16. Another view of the averaging problem on l ity d ay e nd ati ua to Hubble radius i nfl eq averaging gives corrections hereThursday, 26 January 12
  17. 17. Another view of the averaging problem on l ity d ay e nd ati ua to Hubble radius i nfl eq model = flat FLRW + averaging gives corrections here perturbationsThursday, 26 January 12
  18. 18. Another view of the averaging problem on l ity d ay nd ati howfl do e weua to remove backreaction bits Hubble radius in eq to get to ‘real’ background? smoothed background today is not same background as at end of inflation ? model = flat FLRW + averaging gives corrections here perturbationsThursday, 26 January 12
  19. 19. Fitting isn’t obvious either a single line of sight gives D(z) { averaged on the Hubble rate along the sky to give line of sight smooth ‘model’Thursday, 26 January 12
  20. 20. Aren’t the corrections just ~10-5?Thursday, 26 January 12
  21. 21. Aren’t the corrections just ~10-5? •No. Of course not. Λ is bs [Buchert, Kolb ...]Thursday, 26 January 12
  22. 22. Aren’t the corrections just ~10-5? •No. Of course not. Λ is bs [Buchert, Kolb ...] •Yes. Absolutely. Those guys are idiots. [Wald, Peebles ...]Thursday, 26 January 12
  23. 23. Aren’t the corrections just ~10-5? •No. Of course not. Λ is bs [Buchert, Kolb ...] •Yes. Absolutely. Those guys are idiots. [Wald, Peebles ...] •Well, maybe. I’ve no idea what’s going on. [everyone else ... ?]Thursday, 26 January 12
  24. 24. Aren’t the corrections just ~10-5? •No. Of course not. Λ is bs [Buchert, Kolb ...] •Yes. Absolutely. Those guys are idiots. [Wald, Peebles ...] •Well, maybe. I’ve no idea what’s going on. [everyone else ... ?] •Corrections from averaging enter Friedmann and Raychaudhuri equations •is this degenerate with ‘dark energy’? •can we separate the effects [if there are any]? •or ... is it dark energy? neat solution to the coincidence problemThursday, 26 January 12
  25. 25. Could it be dark energy? ? ‘average’ expansion can accelerate while local one decelerates everywhere regions with faster expansion dominate the volume over time so, the average expansion will increaseThursday, 26 January 12
  26. 26. Could it be dark energy? ? ‘average’ expansion can accelerate while local one decelerates everywhere regions with faster expansion dominate the volume over time so, the average expansion will increaseThursday, 26 January 12
  27. 27. Non-Newtonian in natureThursday, 26 January 12
  28. 28. Non-Newtonian in natureThursday, 26 January 12
  29. 29. Non-Newtonian in nature curvature!Thursday, 26 January 12
  30. 30. What to compute? •general formalisms lack quantitative computability •perturbative methods don’t give coherent picture •but they do give predictionsThursday, 26 January 12
  31. 31. Different aspectsThursday, 26 January 12
  32. 32. Different aspects •Non-linear perturbations [Newtonian vs non-Newtonian]Thursday, 26 January 12
  33. 33. Different aspects •Non-linear perturbations [Newtonian vs non-Newtonian] •Relativistic corrections •linear and non-linearThursday, 26 January 12
  34. 34. Different aspects •Non-linear perturbations [Newtonian vs non-Newtonian] •Relativistic corrections •linear and non-linear •backreaction from averagingThursday, 26 January 12
  35. 35. Different aspects •Non-linear perturbations [Newtonian vs non-Newtonian] •Relativistic corrections •linear and non-linear •backreaction from averaging •corrections to observablesThursday, 26 January 12
  36. 36. Canonical Cosmology •compute everything as power series in small parameter ε gµ⇥ = gµ⇥ + ⇥ ¯ (1) gµ⇥ + ⇥ 2 (2) gµ⇥ + · · · ‘real’ spacetime second-order first-order ‘background’ spacetime perturbation correction FLRW fit to ‘background’ observables - SNIa etcThursday, 26 January 12
  37. 37. Canonical Cosmology •compute everything as power series in small parameter ε gµ⇥ = gµ⇥ + ⇥ ¯ (1) gµ⇥ + ⇥ 2 (2) gµ⇥ + · · · ‘real’ spacetime second-order first-order ‘background’ spacetime perturbation correction FLRW how do these ? fit in? fit to ‘background’ observables - SNIa etcThursday, 26 January 12
  38. 38. so, ....Thursday, 26 January 12
  39. 39. so, .... is it big or is it small ... ?Thursday, 26 January 12
  40. 40. so, .... is it big or is it small ... ? (I don’t know either)Thursday, 26 January 12
  41. 41. Simulations can’t see it periodic BC’s and no horizon give Olbers paradox - cancels backreactionThursday, 26 January 12
  42. 42. Simulations can’t see it periodic BC’s and no horizon give Olbers paradox - cancels backreactionThursday, 26 January 12
  43. 43. Perturbation theory metric to second-order Bardeen equation no real backreaction from first-order perturbations - average of perturbations vanish by assumptionThursday, 26 January 12
  44. 44. Perturbation theory metric to second-order second-order potentials induced by first-order scalars vectors and tensors give measure of relativistic correctionsThursday, 26 January 12
  45. 45. Perturbation theory metric to second-order second-order potentials induced by first-order scalars vectors and tensors give measure of relativistic correctionsThursday, 26 January 12
  46. 46. Vectors ~1%Thursday, 26 January 12
  47. 47. Thursday, 26 January 12
  48. 48. induced tensors bigger than primordial [today]Thursday, 26 January 12
  49. 49. backreaction as correction to the background •second-order modes give non-trivial ‘backreaction’ •Hubble rate depends on •What is it?Thursday, 26 January 12
  50. 50. backreaction as correction to the background •second-order modes give non-trivial ‘backreaction’ •Hubble rate depends on •What is it?Thursday, 26 January 12
  51. 51. backreaction as correction to the background •second-order modes give non-trivial ‘backreaction’ •Hubble rate depends on •What is it?Thursday, 26 January 12
  52. 52. backreaction as correction to the background •second-order modes give non-trivial ‘backreaction’ •Hubble rate depends on •What is it? determines amplitude of backreactionThursday, 26 January 12
  53. 53. backreaction is concerned with the homogeneous, average contributions what does this depend on?Thursday, 26 January 12
  54. 54. scaling behaviour d lity y en ua da to eq Hubble radius at first-order scale increasingThursday, 26 January 12
  55. 55. amplitude of second-order contributionsThursday, 26 January 12
  56. 56. amplitude of second-order contributionsThursday, 26 January 12
  57. 57. amplitude of second-order contributionsThursday, 26 January 12
  58. 58. amplitude of second-order contributions large equality scale suppresses backreaction - but overcomes factor(s) of DeltaThursday, 26 January 12
  59. 59. backreaction - expansion rate •second-order modes give non-trivial backreaction •Hubble rate depends on •UV divergent terms don’t contribute on average [Newtonain] •well defined and well behaved backreaction •but, this is only well behaved because of the long radiation era •what would we do if the equality scale were smaller?Thursday, 26 January 12
  60. 60. on we wish to probe, and some subtleties arise because we have tohen we examineto the Hubble we areat second-order Change averagedHubble as a function of redshift as a fractional in twothe backgroundthe rate. the rate rate interested things:ed Friedmann equation. In the Friedmann Friedmann equation, and theinterested FIG. 1: The ˘ Hubble rate the Hubble rate H calculated from the ensemble-averaged D equationchange are left shows the Hubble rat we to Hubble eld equations asBothresult of and EdS models are considered, and the di erentcomponents indicated directly, H . a concordance averaging, which are three new averaging schemes, D if we put R = 0, H still doesn’t have a UV divergence. of dark energy, it is common to think of these as e⇥ective fluid or S D hese two agree up to rate Normalised Hubble perturbative order, when we take the ensemble as function of redshifte given by the generalised Friedmann equation. For a given domain from averaging Friedmannveraged Hubble rate we might expect to find. When we present HD equation ⇧ ˘ 2 HD ⌅ HD (98) Friedmann equation and then taken the square-root. This does not (55) directly (but note that if we square Eq (55), take the ensemble equality scale domainhe Hubble rate as calculated directly from the Friedmann equation).ce’ of the Hubble rate, which may be defined by Clarkson, Ananda & Larena, 0907.3377 2[HD ] = HD 2 HD . (99) ⌥⌃ ⌥⌃ ˘ FIG. 2: Plots for H D with RD = 1/kequality , RS = 0, with the variance included, as a function of redshif Thursday, 26 January 12
  61. 61. effective fluidThursday, 26 January 12
  62. 62. effective fluid •amplitudes apply to effective fluid approach [Baumann etal] and Green and Wald [PPN] •they claim backreaction is small from thisThursday, 26 January 12
  63. 63. great!Thursday, 26 January 12
  64. 64. great! backreaction is small ...Thursday, 26 January 12
  65. 65. backreaction - acceleration rate •other quantities are much stranger •time derivative of the Hubble rate represented in the deceleration parameter •UV divergent terms do not cancel outThursday, 26 January 12
  66. 66. backreaction - acceleration rate •other quantities are much stranger •time derivative of the Hubble rate represented in the deceleration parameter •UV divergent terms do not cancel outThursday, 26 January 12
  67. 67. backreaction - acceleration rate •other quantities are much stranger •time derivative of the Hubble rate represented in the deceleration parameter •UV divergent terms do not cancel outThursday, 26 January 12
  68. 68. backreaction - acceleration rate •other quantities are much stranger •time derivative of the Hubble rate represented in the deceleration parameter •UV divergent terms do not cancel out dominates backreactionThursday, 26 January 12
  69. 69. divergent termsThursday, 26 January 12
  70. 70. divergent termsThursday, 26 January 12
  71. 71. divergent termsThursday, 26 January 12
  72. 72. divergent termsThursday, 26 January 12
  73. 73. divergent termsThursday, 26 January 12
  74. 74. oh no!Thursday, 26 January 12
  75. 75. oh no! backreaction is huge ... but wait!Thursday, 26 January 12
  76. 76. use observables - spatial averaging is meaningless! •we fit our cosmology to an all-sky average of observed quantities •distance-redshift, number-count redshift, etc. •if averaging/backreaction is significant then these should fit to the wrong model (should use LTB metric!) •can be computed using Kristian-Sachs approachThursday, 26 January 12
  77. 77. expectation value observed deceleration governed by cut-offThursday, 26 January 12
  78. 78. expectation value observed deceleration governed by cut-offThursday, 26 January 12
  79. 79. expectation value observed deceleration governed by cut-offThursday, 26 January 12
  80. 80. UV cutoffThursday, 26 January 12
  81. 81. UV cutoff •modes below equality scale dominate effect - not physicalThursday, 26 January 12
  82. 82. UV cutoff •modes below equality scale dominate effect - not physical •where should they be cut off?Thursday, 26 January 12
  83. 83. UV cutoff •modes below equality scale dominate effect - not physical •where should they be cut off? •inflation scale [last mode to leave the Hubble scale]? backreaction >>>1Thursday, 26 January 12
  84. 84. UV cutoff •modes below equality scale dominate effect - not physical •where should they be cut off? •inflation scale [last mode to leave the Hubble scale]? backreaction >>>1 •dark matter free-streaming scale? [~ pc] backreaction >>1Thursday, 26 January 12
  85. 85. UV cutoff •modes below equality scale dominate effect - not physical •where should they be cut off? •inflation scale [last mode to leave the Hubble scale]? backreaction >>>1 •dark matter free-streaming scale? [~ pc] backreaction >>1 •‘spherical scale’ [Kolb]Thursday, 26 January 12
  86. 86. UV cutoff •modes below equality scale dominate effect - not physical •where should they be cut off? •inflation scale [last mode to leave the Hubble scale]? backreaction >>>1 •dark matter free-streaming scale? [~ pc] backreaction >>1 •‘spherical scale’ [Kolb] •virial ‘scale’ ~ Mpc’s [Baumann etal] - gives O(1) backreactionThursday, 26 January 12
  87. 87. wtf? amplitude of backreaction determined purely from cut-off virial scales only sensible cutoff O(1) backreaction ignore them - maybe gauge or unphysical ?Thursday, 26 January 12
  88. 88. fourth-order perturbation theory ... ?Thursday, 26 January 12
  89. 89. Hubble rate at fourth-orderThursday, 26 January 12
  90. 90. Hubble rate at fourth-orderThursday, 26 January 12
  91. 91. Hubble rate at fourth-orderThursday, 26 January 12
  92. 92. Hubble rate at fourth-orderThursday, 26 January 12
  93. 93. ok ... so,Thursday, 26 January 12
  94. 94. ok ... so, backreaction could be ... anything ...Thursday, 26 January 12
  95. 95. key questions •convergence of perturbation theory function of equality scale •why are we so lucky? •with less radiation, scales up to ‘virial scale’ must contribute to backreaction - how would we compute this? •model with no radiation era might be best model to explore backreactionThursday, 26 January 12
  96. 96. What is the background? Do observations measure the background? What is ‘precision cosmology’?Thursday, 26 January 12
  97. 97. Interfering with dark energy •until we understand backreaction precision cosmology not secure •what are we measuring? •Zalaletdinov’s gravity predicts decoupled geometric and dynamical curvature •removes constraints on dynamical DE •evidence for acceleration only provided by SNIaThursday, 26 January 12
  98. 98. a single line of sight gives D(z) { •We average this over the sky, and (re)construct model - alternative aspect to ‘backreaction’ •average depends on z, so ‘looks’ spherically symmetric and inhomogeneous •would remove copernican constraints on void models! •no need for LTB solution, kSZ constraints removed...Thursday, 26 January 12
  99. 99. Conclusions Confusions •Why are second-order perturbations so large? • •tells us that perturbation theory must be relativistic, not Newtonian •role of UV divergence must be understood to decide whether backreaction is small - higher order or resummation methods needed? must include tensors! •do we need relativistic N-body replacement? •what is the background? what is ‘precision cosmology’? •‘void models’ may be mis-interpretation of backreaction ...Thursday, 26 January 12
  100. 100. Thursday, 26 January 12
  101. 101. Thursday, 26 January 12
  102. 102. Thursday, 26 January 12
  103. 103. Curvature test for the Copernican Principle • in FLRW we can combine Hubble rate and distance data to find curvature 2 [H(z)D (z)] 1 k = [H0 D(z)]2 ⇥ dL = (1 + z)D = (1 + z) dA 2 • independent of all other cosmological parameters, including dark energy model, and theory of gravity • tests the Copernican principle and the basis of FLRW (‘on-lightcone’ test) ⇥ C (z) = 1 + H 2 DD D 2 + HH DD = 0 Clarkson, Basset & Lu, PRL 100 191303Thursday, 26 January 12
  104. 104. Using age data to reconstruct H(z) Shafieloo & Clarkson, PRDThursday, 26 January 12
  105. 105. Consistency Test for ΛCDM For flat ΛCDM mod-d for all curvatures, where H(z) is given by the Friedmann equation, els the slope of the distance data curve must satisfyonsistency Test for ΛCDM For flat ΛCDM mod- lies outside the n-σ er H(z) 2 H 2 m 1/ + Ωk (1 + z)3 + (1 − Ωm ). + Rearranging for D=(z)of= (1 + z)3Ωm (1 +z)2 + ΩDE exp 3satisfy w(zdeviations from Λ, asthe A litmus test for flat ΛCDM slope 0 Ωthe distance data curve must z 1 ) dz ,z) = 1/ wemhavez)3 + (1 − Ωm ). Rearranging for + z below for the general Ωm Ω (1 + 0 1 Ωk The 2 If, on the other hand we .have usual procedure is to postulate a several parameter form for w(z) and calculateernative method is to reconstruct w(z) by1 − D reconstructingfit for all suitable para directly (z) the luminosity-distance Ωm = w(z). Writing D(z) = (H ./c)(1 + z)−1d (z), we hav 1 − D (z) 2 [9? ] dL (z) Ωm = inverted to yield + z) (5) 3 − 1]D (5) 20 that is good evidence may be [(1 + z) [(1 . 3 − 1]D (z)2 (z) L is that only one choic 2(1 + z)(1 + Ωk D )D − (1 + z) Ωk D + 2(1 + z)Ωk DD − 3(1 + Ωk D2 ) D 2 2 2hin the flat ΛCDMflat ΛCDMwe measure D if we2 measure D to provide e Within the paradigm, if paradigm, (z) at L (z) = 0 (z) at 2 [Ω + (1 + z)Ω ] D 2 − (1 + Ω D )} D . 3 {(1 + z)e z and calculate the rhs kof this equation, we should m k some z and calculate the rhs of this equation,parameterisation ery we should in the same answer independently of the for flat LCDM w(z) that ].could ne this is constant redshift equation of state [? ofe-redshift curve D(z), we can reconstruct the dark energyof of the redshift Typicall in obtain the same answer independently surement. Differentiating Eq. (5)ansatz used in [5], ed ansatz for D(z), such as the Pade we then find that through various ensur measurement. Differentiating Eq. (5) we then find that nated. L (z) = ζD (z) + 3(1(1 + z)D α (1 − z) − 1 2 ] α + z)2 − (z)[1 + D (z)+ DL (z) = = ζD (z) √ 3(1 + z)2 D (z)[1 − D (z)2 ]the test To L (z) flat ΛCDMγmodels. 2 − α − β −(6) . + +z+ Testing = 0 for all β(1 + z) + 1 γ strate that it will wo have used then shorthandw(z) from Eq ΛCDM models.in Eq. (4) toisthe lume data, and the calculatesfor = 2[(1 + z)3 − 1]. a reconstruction method (6) g = 0 ζ all flat (3). Such Note morew(z) directly because we are fitting directly to the observable, and 3 can spot small The this is completely independent of the value of Ωm . simulated data. vari soslate to dramatic a test for Λ as Zunckel & Clarkson, PRL, arXiv:0807.4304;Note errmayWe this as variations in w(z). Unfortunately,param- + z) to larger and substit use have used the shorthand ζ a this also leads − 1]. reported follows: take = 2[(1 calculated form this is directly. It is also Sahni L (z) which errors = 0 within .the e see data. If etal 0807.3548 value of Ω sedparameterising it completely not clear, however,= 0 theL (z)on our understand by that for D(z) and fit to the independent of me taken most seriously. What is very nice about this method is that, if done in small ref w(z) in a given useis independent of bins atΛ as z; this is not the case param- We may bin this as a test for lower follows: take a for parameterigrateeterised form for D(z)dand Bothto the data. If strong degeneracies over redshift when calculating (z). fit methods suffer from L (z) = 0 Thursday, 26 January 12
  106. 106. Consistency Test for ΛCDM For flat ΛCDM mod-d for all curvatures, where H(z) is given by the Friedmann equation, els the slope of the distance data curve must satisfyonsistency Test for ΛCDM For flat ΛCDM mod- lies outside the n-σ er H(z) 2 H 2 m 1/ + Ωk (1 + z)3 + (1 − Ωm ). + Rearranging for D=(z)of= (1 + z)3Ωm (1 +z)2 + ΩDE exp 3satisfy w(zdeviations from Λ, asthe A litmus test for flat ΛCDM slope 0 Ωthe distance data curve must z 1 ) dz ,z) = 1/ wemhavez)3 + (1 − Ωm ). Rearranging for + z below for the general Ωm Ω (1 + 0 1 Ωk The 2 If, on the other hand we .have usual procedure is to postulate a several parameter form for w(z) and calculateernative method is to reconstruct w(z) by1 − D reconstructingfit for all suitable para directly (z) the luminosity-distance Ωm = w(z). Writing D(z) = (H ./c)(1 + z)−1d (z), we hav 1 − D (z) 2 [9? ] dL (z) Ωm = inverted to yield + z) (5) 3 − 1]D (5) 20 that is good evidence may be [(1 + z) [(1 . 3 − 1]D (z)2 (z) L is that only one choic 2(1 + z)(1 + Ωk D )D − (1 + z) Ωk D + 2(1 + z)Ωk DD − 3(1 + Ωk D2 ) D 2 2 2hin the flat ΛCDMflat ΛCDMwe measure D if we2 measure D to provide e Within the paradigm, if paradigm, (z) at L (z) = 0 (z) at 2 [Ω + (1 + z)Ω ] D 2 − (1 + Ω D )} D . 3 {(1 + z)e z and calculate the rhs kof this equation, we should m k some z and calculate the rhs of this equation,parameterisation ery we should in the same answer independently of the for flat LCDM w(z) that ].could ne this is constant redshift equation of state [? ofe-redshift curve D(z), we can reconstruct the dark energyof of the redshift Typicall in obtain the same answer independently surement. Differentiating Eq. (5)ansatz used in [5], ed ansatz for D(z), such as the Pade we then find that through various ensur measurement. Differentiating Eq. (5) we then find that nated. L (z) = ζD (z) + 3(1(1 + z)D α (1 − z) − 1 2 ] α + z)2 − (z)[1 + D (z)+ DL (z) = = ζD (z) √ 3(1 + z)2 D (z)[1 − D (z)2 ]the test To L (z) flat ΛCDMγmodels. 2 − α − β −(6) . + +z+ Testing = 0 for all β(1 + z) + 1 γ strate that it will wo have used then shorthandw(z) from Eq ΛCDM models.in Eq. (4) toisthe lume data, and the calculatesfor = 2[(1 + z)3 − 1]. a reconstruction method (6) g = 0 ζ all flat (3). Such Note morew(z) directly because we are fitting directly to the observable, and 3 can spot small The this is completely independent of the value of Ωm . simulated data. vari soslate to dramatic a test for Λ as Zunckel & Clarkson, PRL, arXiv:0807.4304;Note errmayWe this as variations in w(z). Unfortunately,param- + z) to larger and substit use have used the shorthand ζ a this also leads − 1]. reported follows: take = 2[(1 calculated form this is directly. It is also Sahni L (z) which errors = 0 within .the e see data. If etal 0807.3548 value of Ω sedparameterising it completely not clear, however,= 0 theL (z)on our understand by that for D(z) and fit to the independent of me taken most seriously. What is very nice about this method is that, if done in small ref w(z) in a given useis independent of bins atΛ as z; this is not the case param- We may bin this as a test for lower follows: take a for parameterigrateeterised form for D(z)dand Bothto the data. If strong degeneracies over redshift when calculating (z). fit methods suffer from L (z) = 0 Thursday, 26 January 12
  107. 107. A litmus test for flat ΛCDM these are better fits to constitution data than LCDM with Arman Shafieloo, PRDThursday, 26 January 12
  108. 108. A litmus test for flat ΛCDM these are better fits to constitution data than LCDM should be zero! with Arman Shafieloo, PRDThursday, 26 January 12
  109. 109. A litmus testno dependence on Omega_m for flat ΛCDM these are better fits to constitution data than LCDM should be zero! with Arman Shafieloo, PRDThursday, 26 January 12
  110. 110. other tests( )Thursday, 26 January 12
  111. 111. other tests( )Thursday, 26 January 12
  112. 112. Conclusions • none.Thursday, 26 January 12

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