1. Cross-correlation of CMB & LSS : recentmeasurements, errors and prospects astro-ph/0701393 WMAP vs SDSS Enrique Gaztañaga Consejo Superior de Investigaciones Cientificas, CSIC Instituto de Ciencias del Espacio (ICE), www.ice.csic.es (Institute for Space Studies) Institut d'Estudis Espacials de Catalunya, (IEEC-CSIC) Santiago, 21-23rd March, 2007

2. Higher orders and ISW I- Perturbation theory and Higher order correlations II- CMB & LSS: ISW effect III- Error analysis in CMB-LSS cross-correlation

3. Tiempo Energia atoms HOW DID WE GET HERE? Two driving questions in Cosmology: - Background: Evolution of scale factor a(t). + Friedman Eq. (Gravity?) + matter-energy content H 2 (z) = H 2 0 [  M (1+z) 3 +  R (1+z) 4 +  K (1+z) 2 +  DE (1+z) 3(1+w) ] r(z) =  dz/H(z) Dark Matter and Dark Energy! - Structure Formation: origin of structure (IC) + gravitational instability + matter-energy content  ’’ + H  ’ - 3/2  m H 2  = 0 + galaxy formation (SFR)

4. Where does Structure in the Universe come From? How did galaxies/star/molecular clouds form? time Small Initial overdensed seed background Overdensed region Collapsed region Perturbation theory:  =  b  ( 1 +  ) =>  = (  -  b ) =  b    b  V  /M = 

7. Jeans Instability (linear regime)   L  x  D  0  x   L  x  a  0  x  EdS a = 0.1 EdS  Open a = 1/(1+z) a = 1 (now) a = 0.01 a = 10 z = 0 (now) z = 9 Another handle on Dark Energy (DE): -Friedman Eq. (Expansion history) can not separate gravity from DE -Growth of structure could: models with equal expansion history yield difference D(z) (EG & Lobo 2001), astro-ph/0303526 & 0307034) -how do you measure D(z) from observations?

8. Problem I Argue that the linear growth equation : Has the following solutions: Show that: (2)

9. Non-linear evolution

10. Spherical collapse model: In this case we can solve fully the non-linear evolution: results In a strongly non-linear collapse Critical density  c = 1.68 Another handle on DE: -Models with equal expansion history yield difference D(z) and difference  c (EG. & Lobo, astro-ph/0303526 & 0307034)

11. Weakly non-linear Perturbations : Solved problem!? RPT (Crocce & Sccocimarro 2006) vertices  L    L   angular average Leading order contribution in  corresponds to the spherical collapse. EdS

12. Observations require an statistical approach: instead of  Evolution of (rms) variance  2 = instead of  Or power spectrum P(k)= =>  2 = ∫ dk P(k) k 2 W(k) dk Initial Gaussian distribution of density fluctuations:   p (V) = = 0 for all p ≠2  p Perturbations due to gravity generate non-Gaussian statistics  p   3 = S 3  2 2 with S 3 (m)= 34/7 (time & Cosmo invariant) IC problem: Linear Theory  a    2 = = D 2 Normalization  8 2  To find D(z) -> Compare rms at two times or find evolution invariants

16. Predictions of Inflation - Flat universe - scale invariance IC: n~1 + CDM transfer funcion: P(k) = k n T(k) => Gaussian IC

17. Local spectral index P(k) ~ k n (initial spectrum + transfer function)   2 [r]= ∫ dk P(k) k 2 W(k) dk ~ r -(n+3) n ~ -2 =>  2 [r] ~ r -1 (1D fractal ) equal power on all scales (  m ~ 0.2) n ~ -1 =>  2 [r] ~ r -2 (2D fractal ) less power on large scales (  m ~ 1.0) n ~ 1 n ~ -1 CMB Superclusters Clusters Galaxies 88 mm SCDM  CDM n ~ -2 n ~ -1 n ~ -2 Horizon @ Equality  m ~0.2  m ~1.0

18. Interest of Higher order PT or correlations: - Gaussian IC? - non-linearities: mode coupling - non-linearities= non-gaussianities - cosmic time invariants: do not depend much on cosmic history (cosmological parameters) - bias: how light traces mass => measure mass

19. Weakly non-linear Perturbation Theory: Solved problem! vertices  L    L   angular average Leading order contribution in  corresponds to the spherical collapse.

20. Spherical collapse model: In this case we can solve fully the non-linear evolution: results In a strongly non-linear collapse Critical density  c = 1.68 Another handle on DE: -Models with equal expansion history yield difference D(z) and difference  c (EG. & Lobo, astro-ph/0303526 & 0307034)

21.  L    L   Weakly non-linear Perturbation Theory (Spherical average)    L     L      L     L     L   a  <  0     L   L    Gaussian Initial conditions      L      S               High order statistics -> vertices of non-linear growth! gravity?

22. Test in N-body simulations 3-pt funct N 3 = (10 6 ) 3 !!

23. Weakly non-linear Perturbation Theory Loops(higher order corrections): F2 F3 Tree level= dominant Tree level: F2 F3 = = Gaussian Initial conditions: connected correlations are zero, except 2-pt=> All correlations are built from 2-pt!

24. Weakly non-linear Perturbation Theory Tree level P(k) ~ k n r 12 r 23  1 2 3

25. n ~ -2 n ~ -1 n ~ -2 n ~ -1 Depends on local spectral index P(k) ~ k n (not on  m )   2 [r]= ∫ dk P(k) k 2 W(k) dk ~ r -(n+3) n ~ -2 =>  2 [r] ~ r -1 (1D fractal ) equal power on all scales (  m ~ 0.2) n ~ -1 =>  2 [r] ~ r -2 (2D fractal ) less power on large scales (  m ~ 1.0) n ~ -1 n ~ -2

26. Where does Structure in the Universe come From? How did galaxies/star/molecular clouds form? IC + Gravity + Chemistry = Star/Galaxy (tracer of mass?) time Initial overdensed seed background Overdensed region Collapsed region H2 dust STARS D.Hughes

28. Hogg & Blanton

30. Bias: lets take a very simple model. rare peaks in a Gaussian field (Kaiser 1984, BBKS)  (peak) = b  (mass) with b=  Linear bias “b”:  (peak) = b  (mass) with b=  SC:  c    2 (peak) = b 2  2 (m) Threshold

31. Biasing: does light trace mass? On large scales 2-pt Statistics is linear  g  b  m     g   b   m     b  D   0      m  L  D  0     Gravity vs Galaxy formation Gravity Bias

32. Biasing: does light trace mass? Local approximation  g  F[  m ]  g  b  m  b   m     g   b   m     b   L      g   b   m     b  b 2  m      g   b     b   b  g      m  L    L      Gravity vs Galaxy formation  c 2  b   b    c 3  b   b    L  is Gaussian  m is not

33. Bias: rare peaks in a Gaussian field (Kaiser 1984. BBKS)  (peak) = b  (mass) with b=  Linear bias “b”:  (peak) = b  (mass) with b=  for SC  c   2 (peak) = b 2  2 (m)  Non- linear bias:  b 2 = b 2 ( b k = b k )  Bias S 3  3 S 4  16 ( S k  k k-2 ) -> Close to DM!! Gravity S 3  34/7-(n+3) ~ 3 S 4  20 Threshold How to separate one from the other?

34. How to separate Bias from Gravity? Q G = (Q m +C)/B Using scale or shape (configurational) dependence of 3-pt function: Fry & EG 1993; EG & Frieman 1994; Frieman & EG 1994; Fry 1994; Scoccimarro 1998; Verde etal 2001 B>1 B<1 C CGF model: Bower etal 1993

35. - Gravity @ work ( astro-ph/0501637 & astro-ph/0506249 ) -3pt correlation can be used to understand biasing: this is independent of normalization or cosmological parameters -1st mesurement of galaxy bias (c2 and b) with 3pt function (away from b=1 and c2=0, Verde etal 2001) -0.4 <c2< -0.2) b 1 = 0.95  0.12 b 2 = -0.3  0.1 ( -0.4 <c2< -0.2) Work in progress (by galaxy type and color) 0.8 <  8 < 1.0 -measure of normalization: 0.8 <  8 < 1.0 => Future applications? Comparison with 2dfGRS Gravity vs Galaxy formation

36. Bias & Higher: conclusion Local approximation works on larrge scales  g  F[  m ] For P(k) or 2-pt statistics: Linear theory works on scales > 10 Mpc But amplitude (b1) is unknown: degeneracy between D(z) or sigma8 and b1! For 3-pt statistics: Need higher bias coeffcients (b1, b2, b3…) But can define invariables (S3, Q3) that do not Depend on D(z). Can separate b1 from b2! => Need to find b1, b2, b3….

37. Higher orders and ISW I- Perturbation theory and Higher order correlations II- CMB & LSS: ISW effect III- Error analysis in CMB-LSS cross-correlation

38. Observations require an statistical approach: instead of  Evolution of (rms) variance  2 = instead of  IC problem: Linear Theory  a     2 = = D 2 Normalization  8 2  To find D(z) -> Compare rms at two times or find evolution invariants

39. Where does Structure in the Universe come From? Perturbation theory:  =  b  ( 1 +  ) =>  = (  -  b ) =  b    b  V  /M =   With   ’’ + H  ’ - 3/2  m H 2  = 0  in EdS linear theory:  a   Gravitation potential:  = - G M /R =>  = G  M / R = GM/R  in EdS linear theory:  a   =>   = GM (  R  GM (    R     is constant even when fluctuations grow linearly! We can mesure  today an at CMB: should be the same!

40.  T/T=(SW)=  /c 2 PRIMARY CMB ANISOTROPIES Sachs-Wolfe (ApJ, 1967)  T/T(n) = [  (n) ] i f Temp. F. = diff in N.Potential (SW) ii ff  = GM (  R  /c 2 CMB & LSS

41. Problem II Calculate the rms temperature fluctuation in the CMB due to the Sachs-Wolfe effect as a function sigma_8 (the linear rms density fluctuations on a sphere of radius 8 Mpc/h) and the value of Omega_m (fraction of matter over the critical density). Does the result depend on the cosmological constant (ie Omega_Lambda)? ii ff

42. PRIMARY & SECONDARY CMB ANISOTROPIES Sachs-Wolfe (ApJ, 1967)  T/T(n) = [ 1/4  (n) + v.n +  (n) ] i f Temp. F. = Photon-baryon fluid AP + Doppler + N.Potential (SW) ii ff In EdS (linear regime) D(z) = a, and therfore d  d   Not in  dominated universe ! SZ- Inverse Compton Scattering -> Polarization + Integrated Sachs-Wolfe (ISW) + lensing + Rees-Sciama + SZ 2 ∫ i f d  d  d  (n)

43. CMB Noise ISW map, z< 4Early map, z~1000 Primary CMB signal becomes a contaminant when looking for secondary (ISW, SZ, lensing) signal. The solution is to go for bigger area. But we are limited by having a single sky. Noise ! Signal Crittenden

44. Cross-correlation idea Crittenden & Turok (PRL, 1995) Both  T and  (g) are proportional to local mass fluctuations  (m)

45. Problem III (1) Assuming that galaxies trace the mass, demostrate that in the linear regime and for small angles (~<10 deg), the angular galaxy-galaxy correlation and the galaxy-temperature correlation (induced by ISW effect) are: sight (2) How does the above expressions change with linear bias?

46. ISW in equations... Limber approximation

49. APM WMAP  APM APM WMAP WMAP  APM WMAP 0.7 deg FWHM 5.0 deg FWHM

50. Possible ISW contaminants: -Primary CMB (noise) -Extincion/Absorption (of dust) in our galaxy (CMB and LSS contaminants) -Dust emission in galaxies/clusters -SZ effect -RS effect -CMB lensing by LSS structures -Magnification bias - … ?

51. APM Significance: P= 1.2% null detection -> w TG = 0.35 ± 0.13  K (68% CL) @ 4-10 deg ->   = 0.53-0.86 ( 2-sigma) Pablo Fosalba & EG, (astro-ph/0305468)

52. Significance (null detection): SDSS high-z: P= 0.3% for < 10 deg. (P=1.4% for 4-10 deg) SDSS all: P= 4.8% Combined: P=0.1 - 0.03% (3.3 - 3.6 sigma) P. Fosalba, EG, F.Castander (astro-ph/0307249, ApJ 2003 )   = 0.69-0.87 ( 2-sigma)

53. Data Compilation EG, Manera, Multamaki (astro-ph/ 0407022, MNRAS 2006) RADIO (NVSS) &X-ray (HEAO) Boughm & Crittenden (astro- ph/0305001). WMAP team Nolta et al., astro-ph/0305097 z =0.8-1.1 (tentative < 2.5  ) APM Fosalba & EG astro-ph/0305468 z=0.15-0.3 (tentative < 2.5  ) SDSS Fosalba, EG, Castander, astro- ph/0307249 SDSS team Scranton et al 0307335 Pamanabhan (2005) Cabre etal 2006 z=0.3-0.5 (detection > 4  ) 2Mass Afshordi et al 0308260 Rassat etal 06 z=0.1 (tentative < 2.  ) QSO Giannantonio etal 06 (tentative < 2.5  ) Coverage: z= 0.1 - 1.0 Area 4000 sqrdeg to All sky Bands: X-ray,Optical, IR, Radio Sytematics: Extinction & dust in galaxies.  m = 0.20  8 =0.9 High!? LSS!?

54. b=1 S/N^2 = fsky*(2l+1) /[1+ C l (TT)*C l (GG)/C l (TG)^2] ~  8 =0.9

55. S/N^2 = fsky*(2l+1) /[1+ C l (TT)*C l (GG)/C l (TG)^2]  8 =0.9

56. Compilation EG, Manera, Multamaki (MNRAS 2006) Marginalized over: -h=0.6-0.8 -relative normalization of P(k) Normalize to sigma8=1 for CM Bias from Gal-Gal correlation With SNIa:   = 0.71 +/- 0.13  m = 0.29 +/- 0.04 Prob of NO detection: 3/100,000 With SNIa+ flat prior:   = 0.70 +/- 0.05 w= 1.02 +/- 0.17   = 0.4-1.2  m = 0.18- 0.34 Corasantini, Giannantonio, Melchiorri 05

57. has info about structure growth at redshift of sample  galaxy bias tells about growth rates at lens redshifts  (2.5s-1) s = d log(N(m))/dm Relative magnitude of the two terms is redshift, scale and galaxy population dependent Cosmic Magnification and the ISW effect EG

58. More Information The total signal to noise remains large at high redshifts but The high redshift signal is strongly correlated with the low redshift signal

59. Higher orders and ISW I- Perturbation theory and Higher order correlations II- CMB & LSS: ISW effect III- Error analysis in CMB-LSS cross-correlation

60. Error Analysis Consider 4 methods: 1.Gaussian errors in Harmonic space (TH) + transform into configurational space 2. New errors in Configurational space (TC) 3. Jack-Knife errors (JK) 4. Simulations (MC1 and MC2)

61. Error Analysis Consider 4 methods: 1.Gaussian errors in Harmonic space (TH) transform into configurational space

62. Problem IV (1) Assuming that both the galaxy (G) and temperature (T) CMB fluctuations in the sky are Gaussian random fields show that for an all sky survey (f_sky=1) the expected variance in the galaxy-temperature angular cross-correlation spectrum (C^TG) at multipole “l” is: Where C^TT and C^GG are the corresponding temperature-temperature and galaxy-galaxy angular spectrum. (2) Argue under what approximations the above expression is valid when we only have measurements over a fraction f_sky of the whole sky. (3) Argue why the above expression is dominated by the second term. How does the S/N change with bias in this case? And with sigma_8?

63. Error Analysis Consider 4 methods: 2. New errors in Configurational space (TC)

64. Poors-man Boostrap? EACH SIMULACION PRODUCES A JK ERROR AND JK Cij 3.

65. 4. All sky Montecarlo simulations Simulate both CMB and LSS as gaussian fields with the corresponding c_l spectrum for TT, GG and also TG: Boughn, Crittenden & Turok 1998

66. Input vs 1000 sim 10% sky z=0.33

67. All sky z=0.33 Input vs sim

68. 10% sky z=0.33 Input vs sim

70. JK= 0.207 ± 0.041 (true=0.224) JK= 0.193 ± 0.045 (true=0.202) JK= 0.170 ± 0.049 (true=0.167) JK= 0.113 ± 0.039 (true=0.107) Comparison of JK errors with MC errors

71. Error in the error

73. ERROS in C_L This wildly used Eq. only works for Binned data!

74. ERROS in C_L -Can propagate diagonal errors in C_l to w(  ) -Thid is surprising for f<1: transfer to off-diagonal elements -Bin C_l data to get diagonal errors.

77. CMB dataLSS data WMAP 3 rd year SDSS DR4 -5200 sq deg (13% sky) -Selection of subsamples with different redshift distribution -3 magnitude subsamples with r=18-19, r=19-20 and r=20-21 with 10 6 – 10 7 galaxies -high redshift Luminous Red Galaxy (Eisentein et al. 2001) -Mask avoids holes, trails, bleeding, bright stars and seeing>1.8 V-band (61 Hz) HEALPix tessellation Kp0 mask

78. Jack-knife errors Covariance matrix   distribution Singular Value Decomposition (SVD) Redshift selection function r=20-21 z c =0 z 0 =0.2 z m =0.3 LRG z c =0.37 z 0 =0.45 z m =0.5 20-21 LRG

79. r=20-21 S/N=3.6 LRG S/N=3. S/N total=4.7

80. For a flat universe, with bias, sigma8 and w=-1 fix.... dark energy must be... 68% 0.80-0.85 95% 0.77-0.86

81. Can we obtain information about w? Contour: 1, 2 sigma 1 dof

82. The Science Case for the Dark Energy Survey

83. The Dark Energy Survey We propose to make precision measurements of Dark Energy –Cluster counting, weak lensing, galaxy clustering and supernovae –Independent measurements by mapping the cosmological density field to z=1 –Measuring 300 million galaxies –Spread over 5000 sq-degrees using new instrumentation of our own design. –500 Megapixel camera –2.1 degree field of view corrector –Install on the existing CTIO 4m

84. DARK ENERGY SURVEY (DES) Science Goal: measure w=p/ , the dark energy equation of state, to a precision of  w ≤ 5%, with Cluster Survey Weak Lensing Galaxy Angular Power Spectrum Supernovae

85. Science Goals to Science Objective To achieve our science goals: –Cluster counting to z > 1 –Spatial angular power spectra of galaxies to z = 1 –Weak lensing, shear-galaxy and shear-shear –2000 z<0.8 supernova light curves We have chosen our science objective: –5000 sq-degree imaging survey Complete cluster catalog to z = 1, photometric redshifts to z=1.3 Overlapping the South Pole Telescope SZ survey 30% telescope time over 5 years –40 sq-degree time domain survey 5 year, 6 months/year, 1 hour/night, 3 day cadence

86. DES Dark Energy Constraints Method/PriorUniformWMAPPlanck Galaxy Clusters: abundance w/ WL mass calibration 0.13 0.09 0.10 0.08 0.04 0.02 Weak Lensing: shear-shear (SS) galaxy-shear (GS) + galaxy-galaxy (GG) SS+GS+GG SS+bispectrum 0.15 0.08 0.03 0.07 0.05 0.03 0.04 0.03 0.02 0.03 Galaxy angular clustering0.360.200.11 Supernovae Ia0.340.150.04 Forecast statistical constraints on constant equation of state parameter w models (DES DETF white paper, astro-ph/0510346) ● 4 Dark Energy Techniques – Galaxy clusters – Weak lensing – Angular power spectrum – Type Ia supernovae ● Statistical errors on constant w models typically σ(w) = 0.05-0.1 ● Complementary methods – Constrain different combinations of cosmological parameters – Subject to different systematic errors

87. DES Instrument Project OUTLINE Science and Technical Requirements Instrument Description Cost and Schedule Prime Focus Cage of the Blanco Telescope We plan to replace this and everything inside it

88. Zmax=2 Dz=0.08

89.  8 =1.0  8 =0.9 ISW predictions

90. Detailed CONCLUSIONS - #>800 simulations for 5% error accuracy - Diagonal errors in w(  ) are accurate to  <20 deg -Survey geometry important for  deg (f<0.1): useTC method! -MC1 is 10% low -JK is OK within 10% -Uncertainty in error is about 20% because of sampling -S/N and fit in harmonic space equivalent to configuration space. -Can propagate diagonal errors in C_l to w(  ) -The above is surprising for f<1: transfer to off-diagonal elements -Bin C_l data to get diagonal errors. -Bias to large Omega_DE for large errors -S/N is quite model depended.

91. GENERIC CONCLUSION -Cross-correlation povides a new observational tool to challenge understanding of DE -4-5 sigma detection of the effect (prospers are not so much better than this: up to 11 sigma). This is higher than previously forcasted (JK errors). -need to improve on current analysis tools and simlations to get more realistic. -Signal is very hard to explain with EDS. - LCDM is OK: on low side even with large  8 or large  .