Probing Dark Energy with Baryon Acoustic Oscillations from Future Large Galaxy Redshift Surveys

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1 Probing Dark Energy with Baryon Acoustic Oscillations from Future Large Galaxy Redshift Surveys Hee-Jong Seo (Steward Observatory) Daniel J. Eisenstein (Steward Observatory) Martin White, Edwin Sirko, David Spergel CITA. Jan p.1/42

2 Dark energy Observational implications SNIa redshift magnitude relation (Perlmutter et al. 1998, Riess et al. 1998) - Open? - Flat & an energy responsible for the acceleration? CMBR implies nearly Flat Universe. Matter estimated 1 4 of critical density. 3/4 of total energy, Negative pressure, Smooth and inert, Very little is known. Q: What is energy density versus redshift over time? measuring growth rate in structure formation - weak lensing surveys measuring expansion rate - SNIa surveys, the BAO from galaxy redshift surveys CITA. Jan p.2/42

3 Outline Physics of acoustic oscillations Physics of the standard ruler test Error forecasts on D A and H for future surveys Degradations of the baryonic signature due to nonlinear effects Quantification of the degradation Fitting formula Reconstruction Baryonic signature in future galaxy redshift surveys can provide measurements of D A and H to excellent precision. CITA. Jan p.3/42

4 Baryon acoustic oscillations (BAO) Primordial overdensity peak of dark matter, gas, photons at origin. CITA. Jan p.4/42

5 Baryon acoustic oscillations (BAO) Overpresured peak a spherical sound wave at c s c/ 3 CITA. Jan p.4/42

6 Baryon acoustic oscillations (BAO) Overpresured peak a spherical sound wave at c s c/ 3 CITA. Jan p.4/42

7 Baryon acoustic oscillations (BAO) Overpresured peak a spherical sound wave at c s c/ 3 CITA. Jan p.4/42

8 Baryon acoustic oscillations (BAO) At recombination (at z 1000), Optically thick optically thin Baryons decouple from photons in the CMB. Sound speed of gas decreases. The traveling wave stalls. CITA. Jan p.4/42

9 Baryon acoustic oscillations (BAO) A spherical peak at the distance that the wave has travelled before the recombination This is called the sound horizon scale at recombination (150 Mpc). CITA. Jan p.4/42

10 Baryon acoustic oscillations (BAO) WMAP3 from Hinshaw et al The sound horizon scale can be measured from the CMB itself. CITA. Jan p.4/42

11 Baryon acoustic oscillations (BAO) We expect to see the same features in the matter distribution and recently observed in large galaxy redshift surveys (e.g., Eisenstein et al. 2005; Cole et al. 2005). CITA. Jan p.4/42

12 A Standard Ruler Test to derive D A & H D A & H as geometric traces of the expansion history r = c z H CITA. Jan p.5/42

13 A Standard Ruler Test to derive D A & H D A & H as geometric traces of the expansion history r = (1 + z)d A θ D A = z cdz H(z) CITA. Jan p.5/42

14 A Standard Ruler Test to derive D A & H D A & H as geometric traces of the expansion history Knowing r D A and H separately measured STANDARD RULER TEST CITA. Jan p.5/42

15 Dark Energy to cosmological distances We characterize dark energy by the equation of state w X (z) = p X z ρx Parametrization : e.g., w X (z) = w 0 + w 1 z or w X (z) = w 0 + w a z/(1 + z) (For ΛCDM, w 0 = 1, w 1 = 0) Integrate Energy density of dark energy as a function of time Hubble parameter Integrate Angular diameter distance Going from D A to w 1 : 3 derivatives Going from H to w 1 : 2 derivatives CITA. Jan p.6/42

16 D A & H from galaxy redshift surveys Observables : ρ(z, θ) or δ(z, θ) two-point correlation function ( δδ ) or power spectrum map (z, θ) to (r H, r /D A ) or (k /H, k D A ) the acoustic peak Silk dampling The physical scale of the BAO (i.e., r or k) is well constrained by future CMB data detecting the BAO in the galaxy redshift surveys measures H and D A. CITA. Jan p.7/42

17 Why baryon oscillations? If matter power spectrum is a simple power law, CITA. Jan p.8/42

18 Why baryon oscillations? If matter power spectrum is a simple power law, CITA. Jan p.8/42

19 Why baryon oscillations? If matter power spectrum is a simple power law, CITA. Jan p.8/42

20 Why baryonic oscillations? With a distinct signature, CITA. Jan p.9/42

21 Why baryonic oscillations? With a distinct signature, CITA. Jan p.9/42

22 Why baryonic oscillations? With a distinct signature, CITA. Jan p.9/42

23 Why baryonic oscillations? With a distinct signature, CITA. Jan p.9/42

24 Section summary Future CMB data will measure the characteristic scale of the BAO to excellent precision. Then, using the baryon oscillations in galaxy redshift surveys as a standard ruler, we can measure cosmological distances that geometrically trace the expansion history of the Universe, which in turn depends on dark energy properties. CITA. Jan p.10/42

25 How good are the constraints? Assume Gaussian random density field. Propagate observational errors on power spectrum to errors on D A and H. Measure errors on D A and H σ P (z = 3) CITA. Jan p.11/42

26 How good are the constraints? Assume Gaussian random density field. Propagate observational errors on power spectrum to errors on D A and H. Measure errors on D A and H σ P (z = 3) 5% in D A or H CITA. Jan p.11/42

27 How good are the constraints? When the density field follows Gaussian ramdom distribution, the statistical error on the observed power spectrum at k is σ P P = P + 1 n P When the power spectrum is averaged over a wavenumber bin, the error decreases by the square root of the number of the independent modes within the bin, N modes. When averaged over k and µ, ( σ P P = np Nmodes np where N modes = 2πV survey k 2 k µ/(2(2π) 3 ) σ P decreases as V survey or n increases ) CITA. Jan p.12/42

28 Astrophysical complications In reality, the later-time density field is non-gaussian. And this degrades the baryonic signature on small scales. Nonlinear structure growth Also there are additional effect that may degrade the baryonic signature. Redshift distortions Galaxy bias * Results from N-body Simulations CITA. Jan p.13/42

29 Nonlinear structure growth Primordial perturbation Structure formation Galaxies Small halos Groups,Clusters, Super clusters Large halos With structure growth z = 0.3 z = 1 z = 3 z = 49 CITA. Jan p.14/42

30 Nonlinear structure growth Primordial perturbation Structure formation Galaxies Small halos Groups,Clusters, Super clusters Large halos Erases features on nonlinear scales (σ R 1) Nonlinear scale advances to smaller k (Larger scale) as structures grow hierarchically. We ignored k smaller than k max with conservative choice of k max (σ R 0.5) (Seo & Eisenstein 2003). In reality, remnant nonlinearity may not be simple. CITA. Jan p.14/42

31 Cosmological N-body simulations We ran a total volume of 6.85h 3 Gpc 3 using the Hydra code (Couchman, Thomas, & Pearce 1995) in collisionless P 3 M mode. Ω m = 0.27, Ω X = 0.73, Ω b = 0.046, h = 0.72 and n = σ 8 = 0.9 at z = 0 Volume of h 3 Mpc 3 per box, grids, dark matter particles ( M sun /particle) Initial random density fields at z = 49 Output at z = 3 (4h 3 Gpc 3 ), z = 1 and z = 0.3 (6.85h 3 Gpc 3 ) CITA. Jan p.15/42

32 Nonlinear growth from N-body results z = 49 & z = 3 : 4h 3 Gpc 3 z = 1 & z = 0.3 : 6.85h 3 Gpc 3 Seo & Eisenstein 2005 Baryonic peaks survive on large scales! Mild nonlinearity within k max (σ R 0.5). CITA. Jan p.16/42

33 Nonlinear growth from N-body results Subtract nonlinear growth effect on the broadband power Seo & Eisenstein 2005 CITA. Jan p.17/42

34 Section summary Baryonic peaks on large scales survive well despite mild nonlinearity of gravity. Nonlinear growth gradually erases baryonic peaks, gradually proceeding from small scales to larger scales with time. More peaks survive at high redshift. However, we still lack a quantitative description of the nonlinear effects on the baryonic signature. CITA. Jan p.18/42

35 Modeling nonlinear effects on the BAO Two-point correlation function The characteristic separation of 150 Mpc of pairs is blurred by structure formation. As a result, the baryonic peak is harder to centroid. CITA. Jan p.19/42

36 Modeling nonlinear effects on the BAO Let s consider a pair of objects initially separated by 150 Mpc. CITA. Jan p.19/42

37 Modeling nonlinear effects on the BAO Structure formation moves particles around. By 10 Mpc at z = 0.3. CITA. Jan p.19/42

38 Modeling nonlinear effects on the BAO u 12 u 12 is also the difference between the displacements of individual particles. CITA. Jan p.19/42

39 Distribution of u 12 At z = 0.3 The distributions are close to Gaussian with σ = 8.15h 1 Mpc at z = 0.3. Eisenstein, Seo, & White 2006 CITA. Jan p.20/42

40 The model Gaussian displacement distribution ξ lin (r) No free parameter! z = 0.3 z = 1 Dashed: linear ξ Solid black: from N-body results Red: the model ξ Eisenstein, Seo, & White 2006 CITA. Jan p.21/42

41 The model In Fourier space, the corresponding exponential factor P lin (k) z = 0.3 z = 1 Red: the model P/P lin. Data points: from N-body results. Eisenstein, Seo, & White 2006 CITA. Jan p.21/42

42 Redshift distortions Again, what we observe is galaxy distribution at (z,θ). Without peculiar velocities, (z c, θ) (r H, r /D A ) With peculiar velocities,(z, θ)=(z c +v/c, θ) (r H, r /D A ) On large scales, CITA. Jan p.22/42

43 Redshift distortions CITA. Jan p.22/42

44 Redshift distortions On large scales (linear scales), angle-dependent multiplicative change in power by (1 + βµ 2 ) 2 On small scales, the thermal velocity suppress the power (a.k.a. the finger-of- God effect). Solid lines: real space Dashed line: redshift space CITA. Jan p.23/42

45 Redshift distortions Redshift distortions decrease contrasts and introduce noises within k max CITA. Jan p.23/42

46 Modeling redshift distortions on the BAO CITA. Jan p.24/42

47 Modeling redshift distortions on the BAO The characteristic separation of 150 Mpc (100h 1 Mpc) of pairs is further blurred by large-scale peculiar velocity that distorts the true distance to the galaxies. CITA. Jan p.24/42

48 Modeling redshift distortions on the BAO CITA. Jan p.24/42

49 Modeling redshift distortions on the BAO u 12 u 12,real CITA. Jan p.24/42

50 Distribution of u 12 in redshift space At z = 0.3 The distributions along the lineof-sight are close to Gaussian with σ = 13.6h 1 Mpc at z = 0.3. (across the line-of-sight σ = 8.15h 1 Mpc at z = 0.3. Eisenstein, Seo, & White 2006 CITA. Jan p.25/42

51 The model z = 0.3 z = 1 Eisenstein, Seo, & White 2006 Red: the model P/P lin Data points: from N-body results CITA. Jan p.26/42

52 Redshift evolution of rms displacement Solid line: Σ = (12.4h 1 Mpc)G(z) Dashed line:σ = (12.4h 1 Mpc)G(z)(1 + f), where f Ω 0.6 m Data points: real space - radial direction, redshift space - radial direction line-of-sight direction CITA. Jan p.27/42

53 Section summary We can explain the gradual degradation of the baryonic signature due to nonlinear growth and redshift distortions by: we estimate the amount of differential motions of pairs initially separated by the characteristic scales of the sound horizon at the recombination. We can now quantify the amount of degradation due to various nonlinear effects. Estimating the Lagrangian displacement fields with a reasonable precision does not require a simulation as large as estimating the degradation in the BAO does. CITA. Jan p.28/42

54 Galaxy (halo) bias Lagrangian displacements are dominated by bulk flows and super cluster formation. The motions within halos adds only small portion to Σ, i.e, 10%. Generated power spectrum with simple halo bias model from the N-body results. CITA. Jan p.29/42

55 Bias at z = 0.3 (central + satellite galaxies) Real space Redshift space Red: the model P bias /P lin Data points: from N-body results Eisenstein, Seo, & White 2006 Halo bias adds only small amount of additional nonlinear degradation. CITA. Jan p.30/42

56 How good are the constraints? When the density field follows Gaussian ramdom distribution, the statistical error on the observed power spectrum at k is σ P P = P + 1 n P When the power spectrum is averaged over a wavenumber bin, the error decreases by the square root of the number of the independent modes within the bin, N modes. When averaged over k and µ, ( σ P P = np Nmodes np where N modes = 2πV survey k 2 k µ/(2(2π) 3 ) σ P decreases as V survey or n increases ) CITA. Jan p.31/42

57 Fisher information matrix Error propagation of Gaussian observation error to cosmological parameters p i. χ 2 of observables = k (P model(k) P obs (k)) 2 /2σ 2 P (k). The covariance matrix of parameter p i is derived from Cov 1 ij = F ij = 2 χ 2 p i p j CITA. Jan p.32/42

58 Fisher information matrix F ij kmax k0 V survey P ( k) P ( k) (P ( k) + 1/n) 2 p i p j d k 2(2π) 3 Tegmark (1997), Seo & Eisenstein (2003) CITA. Jan p.33/42

59 Fisher information matrix F ij kmax 0 V survey P ( k) P ( k) (P ( V eff ( k) + 1/n) 2 p i p d k) k j 2(2π) 3 CITA. Jan p.33/42

60 Fisher information matrix F ij kmax 0 V survey P ( k) P ( k) (P ( k) + 1/n ) 2 p i p j d k 2(2π) 3 CITA. Jan p.33/42

61 Fisher information matrix F ij kmax 0 V survey P ( k) P ( k) (P ( k) + 1/n) 2 p i p j d k 2(2π) 3 CITA. Jan p.33/42

62 Fisher information matrix F ij kmax 0 V survey P ( k) P ( k) (P ( k) + 1/n) 2 p i p j d k 2(2π) 3 P ( k) p i when p i = D A or H P (k) k k p i Nonlinear effects damp the BAO contribution to this quantity CITA. Jan p.33/42

63 Fisher information matrix F ij 0 kmax V survey P ( k) P ( k) (P ( k) + 1/n) 2 p i p j d k 2(2π) 3 with linear P ( k) in P ( k) p i. In Seo & Eisenstein 2003, we excluded nonlinear regime by a conservative choice of k max ( σ R 0.5). CITA. Jan p.33/42

64 Fisher information matrix F ij kmax 0 V survey P ( k) P ( k) (P ( k) + 1/n) 2 p i p j d k 2(2π) 3 We can now include nonlinear effect in P ( k) in P ( k) p i. CITA. Jan p.33/42

65 Fisher information matrix F ij exp 0 V survey P (k, µ) P (k, µ) (P (k, µ) + 1/n) 2 p i p j [ ] k 2 Σ 2 k 2 µ 2 (Σ 2 Σ2 ) d k 2(2π) 3 where P (k, µ) in the derivatives is a linear power spectrum. In Seo & Eisenstein 2007, we implemented the Fisher matrix with the Lagrangian Displacement field. Nonlinear growth, redshift distortions, and bias effect on the BAO is simply parameterized by Σ and Σ. CITA. Jan p.33/42

66 Fisher information matrix CMB data (Planck Temperature and Polarization) + redshift surveys (V survey, number density, Σ) Cosmological parameters p i : 6 parameters (Ω m h 2, Ω b h 2, τ, n s, T/S, As) & D A tocmb & [D A, H, Unknown growth functions, β, P shot ] at each redshift Ignore distance information from growth rate and suppress distance information from the shape of the power spectrum. Subtract any non-baryonic distance information: F ij F ij where F is a Fisher matrix with no baryon. Now the remaining Fisher matrix contains mainly distance information from the BAO. F 1 ij, p i, p j ; Ω m h 2, D A s, H s CITA. Jan p.34/42

67 A physically motivated 1 and 2-D fitting formula Once we isolate the distance information from the BAO only, the process is basically the problem of centroiding a peak in the presence of noise. Assuming the peak in correlation function is a delta function damped by the Silk damping effect and Nonlinear damping, P b (k) sin ks o ks o exp ( (kσ s ) 1.4 ) exp ( k 2 Σ 2 nl/2). F ln so = V survey 0 1 (P (k) + n 1 ) 2 [ Pb (k) ln s o ] 2 4πk 2 dk 2(2π) 3. Leading term of P b(k) ln s o constant cos ks o exp ( (kσ s ) 1.4 ) exp ( k 2 Σ 2 nl/2) We approximate cos 2 ks o inside the integral as 1 2. CITA. Jan p.35/42

68 A physically motivated 1 and 2-D fitting formula F ij = V survey A dµ f i (µ)f j (µ) dk k2 exp [ 2(kΣ s ) 1.4] ( ) 2 exp P (k) 1 P np 0.2 R(µ) where f 1 (µ) = µ 2 1 and f 2 (µ) = µ 2 [ ] k 2 (1 µ 2 )Σ 2 k 2 µ 2 Σ 2 From Seo & Eisenstein 2007 C-prgram available in eisenste/acousticpeak/bao_forecast.html CITA. Jan p.35/42

69 Photometric redshift survey How does the redshift error affect the result? Deep wide-field multicolor imaging surveys will offer photometric redshifts over a wide field σ z (or Σ z ) increases σ DA and especially σ H increase Power spectra over a range of D A overlap Features are smeared 4% in (1+z) seems safe Modes with k σ r 1 strongly suppressed (P (k, µ) exp ( k 2 µ 2 Σ 2 z)) σ DA increases and we lose information on H. CITA. Jan p.36/42

70 Photometric redshift survey P (k, µ) exp ( k 2 µ 2 Σ 2 z) not only decreases a signal but also decreases a noise. F ij = V survey A dµ f i (µ)f j (µ) dk k2 exp [ 2(kΣ s ) 1.4] ( ) 2 exp P (k) 1 P np 0.2 R(µ) [ ] k 2 (1 µ 2 )Σ 2 k 2 µ 2 Σ 2 With 3% error in (1 + z), one gets 4% in D A for 0.6 < z < 1.0 for every 1000 deg 2. CITA. Jan p.36/42

71 Distance errors from N-body results Applied Lagrangian displacement distributions to a construct a model P m (k) that is used to fit the N-body data Jacknknife subsampling of the data χ 2 analysis measures a mean and a distance error Distance errors Fisher matrix calculations and χ 2 analysis using the N-body results agree to excellent precision (< 13%). From N-body results at z = 0.3 σ distance = 0.6% for 6.85h 3 Gpc 3 σ distance = 1.57% for 1h 3 Gpc 3 From the Fisher matrix, σ distance = 1.50% for 1h 3 Gpc 3 CITA. Jan p.37/42

72 Fractional errors of D A and H 3π steradian of skys with z = 0.1. Black: the cosmic variance limit Blue: full nonlinear degradation with a reasonable shot noise. Red: after reconstruction with a reasonable shot noise. CITA. Jan p.38/42

73 Reconstruction Recover the erased portion of baryonic acoustic oscillations by undoing the Lagrangian displacement (Eisenstein, Seo, Sirko, & Spergel 2006). Overdensity Overdensity CITA. Jan p.39/42

74 Reconstruction The Lagrangian displacement ( q) is estimated from the nonlinear density fields by applying the linear continuity equation to the nonlinear density fields (δ = divergence of q) Overdensity Overdensity CITA. Jan p.39/42

75 Reconstruction The Lagrangian displacement ( q) is estimated from the nonlinear density fields by applying the linear continuity equation to the nonlinear density fields (δ = divergence of q) CITA. Jan p.39/42

76 Future surveys CITA. Jan p.40/42

77 Future surveys w X = w 0 + w a z/(1 + z) CITA. Jan p.41/42

78 Summary We can measure the cosmological distance scale using the baryonic signature in galaxy redshift surveys as a standard ruler. Baryonic oscillations on large scales survive well while nonlinear growth and redshift distortions obscure the features. Galaxy (halo) bias adds only small additional degradation. Using Lagranginan displacement distributions, we can quantify the nonlinear effects on the baryonic signature in galaxy redshift surveys. By reversing the Lagrangian displacement distributions, we can reconstruct a portion of the baryonic signature degraded by the various nonlinearities We improved error forecasts by implementing the Fisher matrix calculation with the Lagrangian displacement distributions. We provide a physical motivated fitting formula for the covariance matrix of D A and H. CITA. Jan p.42/42

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