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
atoms
HOW DID WE GET HERE?
Two driving questions in Cosmology:
- Background: Evolution
of scale factor a(t).
+ Friedman Eq. (Gravity?)
+ matter-energy content
H2(z) = H20 [ 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
d’’ + H d’ - 3/2 m H2 d = 0
+ galaxy formation (SFR)
Tiempo
Energia
Where does Structure in the Universe come From?
How did galaxies/star/molecular clouds form?
time
Overdensed region
Small Initial
overdensed
seed
background
Collapsed region
Perturbation theory:
r = rb ( 1 + d)

=> Dr = (r - rb ) = rb d
rb = M / V => DM /M = d
Jeans Instability (linear regime)
dL (x,t) = D(t) d0(x)
EdS
dL (x,t) = a(t) d0(x)
Another handle on
Dark Energy (DE):
EdS
Open
-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?
z=9
z = 0 (now)
L
a = 1/(1+z)
a = 0.01
a = 0.1
a = 1 (now)
a = 10
Problem I
Argue that the linear growth equation:
Has the following solutions:
Show that:
(2)
Non-linear evolution
Spherical collapse model:
In this case we can solve
fully the non-linear evolution: results
In a strongly non-linear collapse
Critical density dc = 1.68
Another handle on DE:
-Models with equal expansion history yield
difference D(z) and difference dc (EG. &
Lobo, astro-ph/0303526 & 0307034)
Weakly non-linear Perturbations: Solved problem!? RPT (Crocce & Sccocimarro 2006)
EdS
vertices
angular
average
d = dL + n2 dL2 + ...
Leading order contribution in d corresponds to the spherical collapse.
Observations require an statistical approach:
Evolution of (rms) variance
Or power spectrum
x2 = < d2>
instead of d
P(k)= < d2(k)> => x2 = ∫ dk P(k) k2 W(k) dk
IC problem: Linear Theory d = a d0
x2 = < d2> = D2 < d02>
Normalization s8 2  < d2(R=8)>
To find D(z) -> Compare rms at two times
or find evolution invariants
Initial Gaussian distribution of density fluctuations:
xp (V) = < dP> = 0 for all
p ≠2
Perturbations due to gravity generate non-Gaussian statistics
-> x3 = S3 x22
with
xp
S3(m)= 34/7 (time & Cosmo invariant)
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)
x2[r]= ∫ dk P(k) k2 W(k) dk ~ r-(n+3)
n ~ -2 => x2 [r] ~ r -1 (1D fractal ) equal power on all scales (m~0.2)
n ~ -2
n ~ -1 => x2 [r] ~ r -2 (2D fractal ) less power on large scales (m~1.0)
CMB
n~1
n ~ -1
Superclusters
Clusters
Galaxies
m
Horizon @ Equality
s8
SCDM
n ~ -1
LCDM
n ~ -2
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
angular
average
d = dL + n2 dL2 + ...
Leading order contribution in d corresponds to the spherical collapse.
Spherical collapse model:
In this case we can solve
fully the non-linear evolution: results
In a strongly non-linear collapse
Critical density dc = 1.68
Another handle on DE:
-Models with equal expansion history yield
difference D(z) and difference dc (EG. &
Lobo, astro-ph/0303526 & 0307034)
Weakly non-linear Perturbation Theory (Spherical average)
d = dL + n2 dL2 + ...
d3 = dL3 + 3 n2 dL4 + ...
Gaussian Initial conditions
< d3 > = < dL3 > + 3 n2 < dL4 > + ...
< dL3 > = a3 <d03>= 0
< dL4 > = < dL 2 >2
< d3 > = 3 n2 < dL2 >2 + ...
S 3  < d3 > / < d2 >2 = 3 n2 = 34 / 7
gravity?
High order statistics -> vertices of non-linear growth!
Test in N-body simulations
3-pt funct N3 = (106)3 !!
Weakly non-linear
Perturbation Theory
Gaussian Initial conditions: connected
correlations are zero, except 2-pt=> All
correlations are built from 2-pt!
Tree level= dominant
=
Tree level:
Loops(higher order corrections):
=
F2
F3
F2
F3
Weakly non-linear
Perturbation Theory
Tree level
P(k) ~ kn
3
1
r23
a=q
2
r12
Depends on local spectral index P(k) ~ kn (not on m)
x2[r]= ∫ dk P(k) k2 W(k) dk ~ r-(n+3)
n ~ -2 => x2 [r] ~ r -1 (1D fractal ) equal power on all scales (m~0.2)
n ~ -1 => x2 [r] ~ r -2 (2D fractal ) less power on large scales (m~1.0)
n ~ -2
n ~ -1
n ~ -2
n ~ -2n ~ -1
n ~ -1
Where does Structure in the Universe come From?
How did galaxies/star/molecular clouds form?
time
Overdensed region
Initial
overdensed
seed
background
Collapsed region
IC + Gravity+ Chemistry = Star/Galaxy (tracer of mass?)
dust
H2
STARS
D.Hughes
Hogg & Blanton
QuickTi me™ and a
TIFF (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”:
d (peak) = b d(mass) with
-> x2 (peak) = b2 x2 (m)
b= n/s (SC: n=dc/s
Threshold n
Biasing: does light trace mass?
On large scales 2-pt
Statistics is linear
dg= b dm
dm = dL = D d0
Bias
Gravity vs Galaxy
formation
< dg2 > = b2 < dm2 >2 = b2 D2 < d02 >2
Gravity
Biasing: does light trace mass?
Local approximation
dg= F[ dm]
dg= b dm + b2 dm2
dm = dL + n2 dL 2
dL
dm
is Gaussian
is not
< dg2 > = b2 < dm2 >2 = b2 < dL2 >2
< dg3 > = b3 < dm3 > + 3 b2b2 < dm4 > + ...
< dg3 > = b3 ( 3 n2 + 3 b2/b) < dg2 >2 + ...
Gravity vs Galaxy
formation
c 2  b2 / b
c 3  b3 / b
Bias:
rare peaks in a Gaussian field (Kaiser 1984. BBKS)
Linear bias “b”: d (peak) = b d(mass)
with b= n/s (for SC n=dc/s)
-> x2 (peak) = b2 x2 (m)
Non-linear bias: -> b2= b2
Bias S3 = 3
( bk= bk ) ->
S4 = 16
Gravity S3 = 34/7-(n+3) ~ 3
(Sk = kk-2 ) -> Close
to DM!!
S4 ~ 20
Threshold n
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
C
B>1
CGF model: Bower etal 1993
Comparison with 2dfGRS
- Gravity @ work (astro-ph/0501637 & astro-ph/0506249)
-3pt correlation can be used to understand biasing:
this is independent of normalization or cosmological
parameters
Gravity vs Galaxy
formation
-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)
Work in progress (by galaxy type and color)
-measure of normalization:
=> Future applications?
0.8 < s8 < 1.0
Bias & Higher: conclusion
Local approximation works on larrge scales dg= F[ dm]
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….
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
x 2 = < d 2>
instead of d
IC problem:
Linear Theory d = a d0
=>
x2 = < d2> = D2 < d02>
Normalization s8 2  < d2(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:
r = rb ( 1 + d) => Dr = (r - rb ) = rb d

rb = M / V => DM /M = d

With :
d’’ + H d’ - 3/2 m H2 d = 0
in EdS linear theory: d
= a d0
Gravitation potential:
F = - G M /R => DF = G DM / R = GM/R d
in EdS linear theory: d = a d0 => DF = GM (d/ R) = GM (d0/ R0)
Df is constant even when fluctuations grow linearly!
We can mesure Df today an at CMB: should be the same!
!!
PRIMARY CMB
ANISOTROPIES
Sachs-Wolfe (ApJ, 1967)
DT/T(n) = [F (n) ]if
Temp. F. = diff in N.Potential (SW)
QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture.
Fi
DT/T=(SW)= DF /c2
Ff
DF = GM (d/ 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)?
QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture.
Fi
Ff
PRIMARY & SECONDARY CMB
ANISOTROPIES
Sachs-Wolfe (ApJ, 1967)
DT/T(n) = [ 1/4 dg (n) + v.n
+ F (n) ]if
Temp. F. = Photon-baryon fluid AP + Doppler + N.Potential (SW)
QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture.
SZ- Inverse Compton Scattering
-> Polarization
Fi
+ Integrated Sachs-Wolfe (ISW) + lensing + Rees-Sciama + SZ
∫
2 if dt dF/dt (n)
In EdS (linear regime) D(z) = a , and therfore dF/dt = 0
Not in L dominated universe !
Ff
CMB Noise
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.
Signal
Crittenden
ISW map, z< 4
Noise
!
Early map, z~1000
Cross-correlation idea
Crittenden & Turok (PRL,
1995)
Both DT and d (g) are proportional to local mass fluctuations d (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
WMAP
APM
APM
5.0 deg
FWHM
WMAP
WMAP
APM
0.7 deg
FWHM
WMAP
5.0 deg
0.7 deg
FWHM
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
-…?
APM
Significance:
P= 1.2% null detection
-> wTG = 0.35 ± 0.13 mK
(68% CL) @ 4-10 deg
-> L = 0.53-0.86 ( 2-sigma)
Pablo Fosalba & EG, (astro-ph/0305468)
P. Fosalba, EG, F.Castander
(astro-ph/0307249, ApJ 2003)
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)
L = 0.69-0.87 ( 2-sigma)
Data Compilation
EG, Manera, Multamaki (astro-ph/ 0407022, MNRAS 2006)
Coverage:
z= 0.1 - 1.0
Area 4000 sqrdeg to All sky
Bands: X-ray,Optical, IR, Radio
m= 0.20 s8=0.9
Sytematics: Extinction
& dust in galaxies.
High!?
RADIO (NVSS) &X-ray (HEAO)
Boughm & Crittenden (astroph/0305001). WMAP team Nolta et al.,
astro-ph/0305097
z =0.8-1.1 (tentative < 2.5 s)
APM Fosalba & EG astro-ph/0305468
z=0.15-0.3 (tentative < 2.5 s)
SDSS Fosalba, EG, Castander, astroph/0307249
SDSS team Scranton et al 0307335
Pamanabhan (2005)
Cabre etal 2006 z=0.3-0.5 (detection
> 4 s!)
2Mass Afshordi et al 0308260
Rassat etal 06
z=0.1 (tentative < 2. s)
QSO Giannantonio etal 06 (tentative < 2.5s)
LSS!?
s8=0.9
S/N^2 = fsky*(2l+1) /[1+ Cl(TT)*Cl(GG)/Cl(TG)^2] ~
b=1
S/N^2 = fsky*(2l+1) /[1+ Cl(TT)*Cl(GG)/Cl(TG)^2]
s8=0.9
Compilation
EG, Manera, Multamaki
(MNRAS 2006)
Prob of NO
detection: 3/100,000
Corasantini, Giannantonio, Melchiorri 05
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:
L = 0.71 +/- 0.13
m= 0.29 +/- 0.04
With SNIa+ flat prior:
L = 0.70 +/- 0.05
w= 1.02 +/- 0.17
L = 0.4-1.2
m= 0.18- 0.34
Cosmic Magnification and the ISW effect
EG
• tells about growth
• has info about structure
growth at redshift of sample rates at lens redshifts
•  (2.5s-1)
•  galaxy bias
s = d log(N(m))/dm
Relative magnitude of the two terms is redshift,
scale and galaxy population dependent
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
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 Analysis
Consider 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 Analysis
Consider 4 methods:
2. New errors in Configurational space (TC)
3.
Poors-man Boostrap?
EACH SIMULACION
PRODUCES A
JK ERROR AND
JK Cij
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
10% sky
z=0.33
Input
vs
1000 sim
All sky
z=0.33
Input
vs
sim
10% sky
z=0.33
Input
vs
sim
Comparison of JK errors with MC errors
JK= 0.207 ± 0.041
(true=0.224)
JK= 0.170 ± 0.049
(true=0.167)
JK= 0.193 ± 0.045
(true=0.202)
JK= 0.113 ± 0.039
(true=0.107)
Error in the error
ERROS in C_L
This wildly used
Eq. only works for
Binned data!
ERROS in C_L
-Can propagate diagonal
errors in C_l to w(q)
-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
V-band (61 Hz)
-3 magnitude subsamples with r=18-19, r=19-20 and r=20-21
HEALPix tessellation with 106 – 107 galaxies
Kp0 mask
-high redshift Luminous Red Galaxy (Eisentein et al. 2001)
-Mask avoids holes, trails, bleeding, bright stars and seeing>1.8
Jack-knife errors
Redshift selection function
20-21
LRG
Covariance matrix
c2 distribution
Singular Value Decomposition
(SVD)
r=20-21 zc=0 z0=0.2 zm=0.3
LRG zc=0.37 z0=0.45 zm=0.5
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 sigma
1 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)
Science Goal: measure w=p/r, the
dark energy equation of state, to a
precision of Dw ≤ 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
●
●
●
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
Forecast statistical constraints on
constant equation of state parameter w models
(DES DETF white paper, astro-ph/0510346)
Method/Prior
Uniform WMAP
Planck
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
0.05
0.03
0.03
0.04
0.03
0.02
0.03
Galaxy angular clustering
0.36
0.20
0.11
Supernovae Ia
0.34
0.15
0.04
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=2
Dz=0.08
s8=0.9
s8=1.0
ISW predictions
Detailed CONCLUSIONS
- #>800 simulations for 5% error accuracy
- Diagonal errors in w(q) are accurate to q <20 deg
-Survey geometry important for q>10 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(q)
-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 s8 or large L.