higher orders and isw

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

Higher orders and ISW

I- Perturbation theory and Higher order correlations

II- CMB & LSS: ISW effect

III- Error analysis in CMB-LSS cross-correlation

Tiempo Energia

atomsHOW DID WE GET HERE?

Two driving questions in Cosmology:

- Background: Evolution of scale factor a(t).

+ Friedman Eq. (Gravity?) + matter-energy content H2(z) = H2

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 H2 = 0 + galaxy formation (SFR)

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 =

Jeans Instability (linear regime) LxD0x

Lxa0x

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?

Problem IArgue that the linear growth equation:

Has the following solutions:

Show that:

(2)

Non-linear evolution

Spherical collapse model:In this case we can solvefully the non-linear evolution: resultsIn a strongly non-linear collapse

Critical densityc = 1.68

Another handle on DE:

-Models with equal expansion history yield

difference D(z) and difference c (EG. &

Lobo, astro-ph/0303526 & 0307034)

Weakly non-linear Perturbations: Solved problem!? RPT (Crocce & Sccocimarro 2006)

vertices

LL

angular average

Leading order contribution in corresponds to the spherical collapse.

EdS

Observations require an statistical approach:

Evolution of (rms) variance 2 = < 2> instead ofinstead of

Or power spectrum P(k)= < 2(k)> => 2 = ∫ dk P(k) k2 W(k) dk

Initial Gaussian distribution of density fluctuations:

p p (V) = < (V) = < PP>> == 0 0 for allfor all p ≠2p ≠2

Perturbations due to gravity generate non-Gaussian statistics pp

3

= S3

22 with S3(m)= 34/7 (time & Cosmo invariant)

IC problem: Linear Theory a

2 = < 2> = D2 < 2>

Normalization8 2 < 2(R=8)>

To find D(z) -> Compare rms at two times or find evolution invariants

Predictions of Inflation

- Flat universe

- scale invariance IC: n~1

+ CDM transfer funcion: P(k) = kn T(k)

=> Gaussian IC

Local spectral index P(k) ~ kn (initial spectrum + transfer function)

2[r]= ∫ dk P(k) k2 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

m

SCDM

CDM

n ~ -2

n ~ -1

n ~ -2

Horizon @ Equality

m~0.2 m~1.0

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

Weakly non-linear Perturbation Theory: Solved problem!

vertices

LL

angular average

Leading order contribution in corresponds to the spherical collapse.

Spherical collapse model:In this case we can solvefully the non-linear evolution: resultsIn a strongly non-linear collapse

Critical densityc = 1.68

Another handle on DE:

-Models with equal expansion history yield

difference D(z) and difference c (EG. &

Lobo, astro-ph/0303526 & 0307034)

LL

Weakly non-linear Perturbation Theory (Spherical average)

LL

L

L

La

<0

LL

Gaussian Initial conditions

L

S

High order statistics -> vertices of non-linear growth!

gravity?

Test in N-body simulations

3-pt funct N3 = (106)3 !!

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!

Weakly non-linear Perturbation Theory

Tree level

P(k) ~ kn

r12

r23

1

2

3

n ~ -2

n ~ -1

n ~ -2

n ~ -1

Depends on local spectral index P(k) ~ kn (not on m)

2[r]= ∫ dk P(k) k2 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

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

Hogg & Blanton

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Bias: lets take a very simple model.

rare peaks in a Gaussian field (Kaiser 1984, BBKS)

Linear bias “b”: (peak) = b (peak) = b (mass)(mass) with b= with b=

SC:c

2 2 (peak) = b(peak) = b22 2 2 (m) (m)

Threshold

Biasing: does light trace mass?

On large scales 2-ptStatistics is linear

gb m

gbm

bD0

m L D0

Gravity vs Galaxy formation

Gravity

Bias

Biasing: does light trace mass?

Local approximation gF[ m]

gb

mbm

gbm

bL

g

bmbb

2

m

gbbbg

mLL

Gravity vs Galaxy formation

c

2bb

c

3bb

Lis Gaussian

m is not

Bias: rare peaks in a Gaussian field (Kaiser 1984. BBKS)

Linear bias “b”: (peak) = b (peak) = b (mass) with b= (mass) with b= for SCc

2 2 (peak) = b(peak) = b22 2 2 (m) (m)

Non-linear bias: b2= b2 ( bk= bk )

Bias S33 S416 (Skkk-2 ) -> Close to DM!!

Gravity S334/7-(n+3) ~ 3 S420Threshold

How to separate one from the other?

How to separate Bias from Gravity? QG= (Qm+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

- 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)

b1= 0.95 0.12 b2 = -0.3 0.1 ( -0.4 <c2< -0.2) -0.4 <c2< -0.2)

Work in progress (by galaxy type and color)

-measure of normalization: 0.8 < 0.8 < 88 < 1.0 < 1.0

=> Future applications?

Comparison with 2dfGRS

Gravity vs Galaxy formation

Bias & Higher: conclusionLocal approximation works on larrge scales gF[ m]

For P(k) or 2-pt statistics:Linear theory works on scales > 10 MpcBut 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 notDepend on D(z). Can separate b1 from b2!

=> Need to find b1, b2, b3….

Higher orders and ISW

I- Perturbation theory and Higher order correlations

II- CMB & LSS: ISW effect

III- Error analysis in CMB-LSS cross-correlation

Observations require an statistical approach:

Evolution of (rms) variance 2 = < 2> instead ofinstead of

IC problem:

Linear Theory a

2 = < 2> = D2 < 2>

Normalization8 2 < 2(R=8)>

To find D(z) -> Compare rms at two times or find evolution invariants

Where does Structure in the Universe come From?

Perturbation theory:

= b( 1 + ) => = ( - b ) = b b V /M =

With’’ + H ’ - 3/2 m H2 = 0in EdS linear theory:

a

Gravitation potential:

= - G M /R => = G M / R = GM/R

in EdS linear theory: a => = GM (RGM (R

is constant even when fluctuations grow linearly!

We can mesure today an at CMB: should be the same!

T/T=(SW)=/c2

PRIMARY CMB ANISOTROPIES

Sachs-Wolfe (ApJ, 1967)

T/T(n) = [(n) ]if

Temp. F. = diff in N.Potential (SW)

i

f

= GM (R/c2

CMB & LSS

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

f

PRIMARY & SECONDARY CMB ANISOTROPIES

Sachs-Wolfe (ApJ, 1967)

T/T(n) = [ 1/4 (n) + v.n + (n) ]if

Temp. F. = Photon-baryon fluid AP + Doppler + N.Potential (SW)

i

f

In EdS (linear regime) D(z) = a , and therfore dd

Not in dominated universe !

SZ- Inverse Compton Scattering -> Polarization

+ Integrated Sachs-Wolfe (ISW) + lensing + Rees-Sciama + SZ

2 ∫if d dd(n)

CMB Noise

ISW map, z< 4 Early 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

Cross-correlation ideaCrittenden & Turok

(PRL, 1995)

Both Both T T and and (g) (g) are proportional to local mass fluctuations are proportional to local mass fluctuations

(m) (m)

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?

ISW in equations...

Limber approximation

APM

WMAPAPM

APM

WMAP

WMAPAPM

WMAP

0.7 deg FWHM

0.7 deg FWHM

5.0 deg FWHM

5.0 deg FWHM

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- … ?

APMSignificance:

P= 1.2% null detection

-> wTG = 0.35 ± 0.13 K (68% CL) @ 4-10 deg

-> = 0.53-0.86 ( 2-sigma)

Pablo Fosalba & EG, (astro-ph/0305468)

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)

Data CompilationEG, 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!?

b=1

S/N^2 = fsky*(2l+1) /[1+ Cl(TT)*Cl(GG)/Cl(TG)^2] ~ 8=0.9

S/N^2 = fsky*(2l+1) /[1+ Cl(TT)*Cl(GG)/Cl(TG)^2]

8=0.9

CompilationEG, 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

• 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

More Information

The total signal to noise remains large at high redshifts

but

The high redshiftsignal is strongly correlated with the low redshift signal

Higher orders and ISW

I- Perturbation theory and Higher order correlations

II- CMB & LSS: ISW effect

III- Error analysis in CMB-LSS cross-correlation

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)

Error AnalysisConsider 4 methods:1. Gaussian errors in Harmonic space (TH)

transform into configurational space

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?

Error AnalysisConsider 4 methods:

2. New errors in Configurational space (TC)

Poors-man Boostrap?

EACH SIMULACIONPRODUCES AJK ERROR ANDJK Cij

3.

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

Input vs1000 sim

10% sky z=0.33

All sky z=0.33

Input vssim

10% sky z=0.33

Input vssim

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

Error in the error

ERROS in C_L

This wildly used Eq. only works forBinned data!

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.

CMB data LSS data

WMAP 3rd 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 106 – 107 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 tessellationKp0 mask

Jack-knife errors

Covariance matrix

distribution

Singular Value Decomposition (SVD)

Redshift selection function

r=20-21 zc=0 z

0=0.2 z

m=0.3

LRG zc=0.37 z

0=0.45 z

m=0.5

20-21 LRG

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

S/N total=4.7

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

Can we obtain information about w?

Contour: 1, 2 sigma1 dof

The Science Case for the Dark Energy Survey

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

DARK ENERGY SURVEY (DES)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

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

DES Dark Energy Constraints

Method/Prior Uniform WMAP Planck

Galaxy Clusters: abundance w/ WL mass calibration

0.130.09

0.100.08

0.040.02

Weak Lensing: shear-shear (SS) galaxy-shear (GS) +

galaxy-galaxy (GG) SS+GS+GG SS+bispectrum

0.150.080.030.07

0.050.050.030.03

0.040.030.020.03

Galaxy angular clustering 0.36 0.20 0.11

Supernovae Ia 0.34 0.15 0.04

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

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

Zmax=2Dz=0.08

8=1.0

8=0.9ISW predictions

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.

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 .