CONSTRAINTS AND TENSIONS IN MG CFHTLENS AND OTHER DATA SETS PARAMETERS FROM PLANCK, INCLUDING INTRINSIC ALIGNMENTS SYSTEMATICS.

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1 CONSTRAINTS AND TENSIONS IN MG PARAMETERS FROM PLANCK, CFHTLENS AND OTHER DATA SETS INCLUDING INTRINSIC ALIGNMENTS SYSTEMATICS 1 Mustapha Ishak The University of Texas at Dallas Jason Dossett INAF Osservatorio Astronomico di Brera, Italy

2 2015 IS THE 100 TH ANNIVERSARY OF EINSTEIN S GENERAL RELATIVITY 2

3 MODIFIED GROWTH EQUATIONS Flat Perturbed FLRW Metric. ds 2 = a(τ) 2 [ (1 + 2Ψ)dτ 2 +(1 2Φ)dx i dx i ] Modified Growth Equations k 2 Φ = 4πGa 2 i ρ i i Q(k, a) k 2 (Ψ R(k, a) Φ) = 12πGa 2 i ρ i (1 + w i )σ i Q(k, a). k 2 (Ψ + Φ) = 8πGa 2 i ρ i i Σ(k, a) 12πGa 2 i ρ i (1 + w i )σ i Q(k, a), = or D Q(1 + R) 2

4 EVOLVING THE MODIFIED GRAVITY PARAMETERS: BINNING METHODS Both Traditional binning (P1) and Hybrid Method D (P2) evolve in redshift as: X z1 k X z2 k X z 1 X(k, z) = 1+X z 1 (k) 2 + X z 2 (k) X z1 (k) 2 z div z TGR z tanh z z div z tw Scale Dependence + 1 X z 2 (k) 2 tanh z z TGR z tw, Redshift bins Scale bins 0.0 <z 1 1 <z <k 0.01 Q 1, Σ 1 Q 3, Σ <k< Q 2, Σ 2 Q 4, Σ 4 Traditional Binning Method (P1) Hybrid Method (P2) X z1 k X z1 (k) = X z2 (k) = { X1 if k<k c X 2 if k k c, { X3 if k<k c X 4 if k k c. X z1 (k) = X 1 e k/k c + X 2 (1 e k/k c ) X z2 (k) = X 3 e k/k c + X 4 (1 e k/k c ), X z1 k X 1 X 1 X 2 k c k X 2 k c k

5 EVOLVING THE MODIFIED GRAVITY PARAMETERS: FUNCTIONAL EVOLUTION (P3) In this evolution method we assume scale independent evolution. The parameters evolve in terms of the scale factor as: X(a) =(X 0 1) a s +1 As a function of redshift with s = 3 X z X z

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7 GALAXY INTRINSIC ALIGNMENTS (IA) AS A CONTAMINANT TO WEAK LENSING (WL) SIGNAL Contaminates WL signal by up to 15-20%. Ref 2 pt. IA biases cosmological parameters at 10%-50% level The measured correlation function = sum of GG, GI and II signals. Used a model for IA that is parameterized by an amplitude A CFHTLenS 7

8 DATA SETS USED CMB temperature anisotropy power-spectrum from Planck Surveyor Low-l WMAP Polarization data Weak lensing tomography shear-shear cross correlations from the CFHTLenS Galaxy power spectrum from the WiggleZ survey ISW-galaxy cross correlations of Ho et al. (2008). BAO data from 6dF, SDSS DR7, and BOSS DR9. 8

9 RESULTS IA: CORRELATIONS WITH MG PARAMETERS We find only weak to moderate correlations between MG parameters and the IA parameter. Both scale dependent parameterizations show most correlation in low-z, high-k bins (bin probed most by lensing data). Correlation table Binning parameterization (P1) Q 1 Q 2 Q 3 Q 4 Σ 1 Σ 2 Σ 3 Σ 4 A CF HT LenS σ Ω m Hybrid parameterization (P2) A CF HT LenS σ Ω m Correlation table Functional parameterization (P3) Q 0 Σ 0 A CF HT LenS σ Ω m

10 FIG. 6: 68% and 95% 2-D confidence contours for the intrinsic alignment amplitude parameter A CF HT LenS ROW: the theory is fixed to GR and the constraints obtained are in good agreement with those of [73] thou to more precise recent data. To the left are the results for the IA-optimized red galaxy sample of [73]. T GR RESULTS IA: COMPARING DIFFERENT LENSING DATASETS. P1 GR P2 P1 P3 10

11 RESULTS P1 FIG. 1: 68% and 95% 2-D confidence contours for the parameters Q i and Σ i from parameterization P1 for redshift and scale dependence of the MG parameters. All of the constraints for this evolution method are fully consistent with GR at the 68% level. 95% confidence limits on MG parameters evolved using form P1 Q 1 [0.49,2.56] Σ 1 [0.97,1.14] Q 2 [0.05,3.08] Σ 2 [0.84,1.22] Q 3 [0.30,1.78] Σ 3 [0.97,1.06] Q 4 [0.28,2.88] Σ 4 [0.90,1.12] 11

12 RESULTS CONT D P2 FIG. 2: 68% and 95% 2-D confidence contours for the parameters Q i and Σ i from parameterization P2 for redshift and scale dependence of the MG parameters. As you can see in the first bin, there a tension with the GR value of 1. However, contrary to the marginalized 1-D constraints given in Table III the GR point is still within the 95% confidence region. 95% confidence limits on MG parameters evolved using form P2 Q 1 [0.38,3.43] Σ 1 [1.03,1.37] Q 2 [0.00,2.86] Σ 2 [0.75,1.07] Q 3 [0.28,2.46] Σ 3 [0.93,1.14] Q 4 [0.05,1.99] Σ 4 [0.86,1.14] 12

13 RESULTS CONT D P3 FIG. 3: 68% and 95% 2-D confidence contours for the parameters Q 0, Σ 0,andR 0 from the scale independent parameterization, P3, for the MG parameters. These constraints are consistent with GR a the 95% level, but a tension is evident. The tension is evident when viewing these plots is not easily seen using the 1-D constraints given in Table V. Thisisduetothenon-Gaussianity of the probability distribution for these parameters as further Fig % confidence limits on MG parameters evolved using form P3 Q 0 [0.77,1.99] Σ 0 [0.79,1.16] R 0 [-0.23,1.18] 13

14 TENSIONS BETWEEN THE DATA SETS We have seen indications of tensions in the MG parameter space for P2 and P3. Known tension between CMB and weak lensing, notably in constraints on σ 8. For P3 we get a bimodal σ 8, hinting the tension in MG parameter space is likely related to known tension between the data sets. 14

15 SUMMARY We find a 40-53% improvement on figure of merit for the MG parameters over previous results. The intrinsic alignment amplitude shows weak to moderate correlation with the MG parameters (Q 2 & Σ 2 most correlated). GR & P3 show a clear IA signal for the optimized early-type galaxy sample GR is consistent with the data at the 95% CL when considering 2D contours. A clear tension is present in the parameter Σ apparently related to the known tension between CMB and weak lensing. 15

16 EVOLVING THE MODIFIED GRAVITY PARAMETERS: BINNING METHODS Both Traditional binning and Hybrid Method evolve in redshift as Traditional Binning Method X z1 k X(k, z) = 1+X z 1 (k) + X z 2 (k) X z1 (k) P (k) 10 4 { X1 if k<k X z1 (k) = c X 2 if k k c, { X3 if k<k X z2 (k) = c X if k k c. X z1 k X z2 k X z 1 z div z TGR z Scale Dependence Hybrid Method X z1 (k) = X 1 e k/k c + X 2 (1 e k/k c ) X z2 (k) = X 3 e k/k c + X 4 (1 e k/k c ), X z1 k D tanh z z div z tw + 1 X z 2 (k) 2 tanh z z TGR z tw, X 1 X 2 k c X 1 X 2 k c GR Hybrid Method Traditional Binning k k k

17 As usual, the shear cross correlation functions ξ+, (θ) kl GG between bins k, l are given by +, (θ) GG = 1 dl lj 0,4 (lθ)p ( κ kl (l), ) (9) 2π ξ kl 0 where J n is the n th -order Bessel function of the first kind, l is the modulus of the two-dimensional wave vector, and Pκ kl is the convergence cross-power spectra between bins k and l is given by [75] P kl κ (l) = χh 0 g k (χ) 1 a(χ) ˆξ kl +, (θ) =ξ kl +, (θ) II C kl GI (l) = χh 0 C kl II (l) = χh ( l ) dχ g k (χ)g l (χ) P φ,φ f K (χ), χ, (10) χh χ dχ p k (χ ) f K(χ χ) f K (χ, ) + ξ kl +, (θ) GI dχ g k(χ)p l (χ)+g l (χ)p k (χ) f K (χ) 0 + ξ kl +, (θ) GG. ( l ) F I P φ,δ0 f K (χ), χ, (13) dχ p k(χ)p l (χ) ( [f K (χ)] 2 FI 2 l ) P δ0,δ 0 f K (χ), χ, (14) where δ 0 is the matter overdensity today and F I is a cosmology dependent factor given by: F I = A CFHTLenS C 1 ρ crit Ω m. (15) 17 Above, ρ crit is the critical density of the universe today, C 1 is a constant with a value h 2 M 1 Mpc 3, and A CFHTLenS is a nuisance parameter that we will marginalize over in our likelihood analysis.

18 THE CONSISTENCY RELATION BETWEEN THE EXPANSION HISTORY AND THE GROWTH RATE OF LARGE SCALE STRUCTURE (MI, UPADHYE, AND SPERGEL, PRD 2006, ASTRO- PH 2005) 18

19 Results: Equations of state found using two different combinations of simulated data sets. Solid contours are for fits to the [Supernova + CMB] data combination, while dashed contours are for fits to [Weak Lensing + CMB] data combination. (MI, Upadhye, and Spergel, Phys.Rev. D74 (2006) , astro-ph-2005) The significant difference (inconsistency) between the equations of state found using these two combinations is a due to the DGP model in the simulated data. In this simulated case, The inconsistency tells us that we are in presence of modified gravity rather than GR+Dark Energy. 19