bwdem – 06/04/2005doing cosmology with galaxy clusters cosmology with galaxy clusters: testing the...

bwdem – 06/04/2005doing cosmology with galaxy clusters

Cosmology with galaxy clusters:

testing the evolution of dark energy

Raul Abramo – Instituto de Física – Universidade de São Paulo

BWDE/M – Joinville

Miller et al. 2005

Gemini


Summary

• Galaxy clusters are the largest virialized structures in the Universe. They are rich is gas (both hot and ionized as well as cold gas) as well as cold dark matter.

• The hot gas can be detected through the Sunyaev-Zeldovich effect, i.e., inverse compton scattering of the CMB photons off the hot intracluster electrons. This non-thermal effect leaves a characteristic imprint on the (originally) planckian spectrum.

• Hot gas can also be detected, at high concentrations, through its X-ray emission (brehmsstrahlung).

• The total (baryons + CDM) amount of matter in a cluster can be detected through weak (e.g. shear) and strong (e.g. Einstein ring) lensing.

• Cold gas can also be detected through the hydrogen absorption spectra if a background quasar is available.

• Clusters are great tools to study Cosmology! What about Dark Energy?


1. How clusters help pinpoint Dark Energy: that “little extra mile”

WMAP 2003

Allen et al. 2004


Constraints from future SZe surveys

Carlstrom, Holder & Reese 2002


Constraints from cosmic shear

Jain & Taylor 2004

Effective depth to measure w of a shear survey, compared to other observations

Ferguson & Bridle 2005


2. Cluster Polarization and DE

• The map of the CMB temperature fluctuations are essencially the portrait of a two-dimensional surface (the LSS) of radius R=r . [: conformal time; d= dt/a(t) ]

• In the presence of DE, the large-scale gravitational potential decays when DE starts to dominate. CMB photons propagating to us through space fall in and out of these time-changing potentials, gaining (or losing) energy. [Integrated Sachs-Wolfe effect – ISWe]

(z>>1)

(z<1)

• It is hard to actually use the ISWe to extract information about the equation of state, since we cannot know in principle what in the CMB is due to the SW (intrinsic, at the LSS) effect and what is due to the ISWe.

• We can use the Sunyaev-Zeldovich effect, which induces a polarization on the CMB photons that scatter off free electrons in galaxy clusters, to determine the ISWe. We can also use this polarization signal to make a tomography of the 3D spectrum of fluctuations!

n

ISW lndl

T

nT

ˆ0

),ˆ(2

)ˆ(

Kamionkowski & Loeb 1997, Cooray & Baumann 2002


• CMB

mm

m

mm

aC

YaT

2

,

12

1

),(),(

Bennett et al. ApJS 148:1 (2003)


• What goes into the alm that we measure today:

)(),(3

1)(

),( **

0

mrmrSW

m YdYT

Tda

1. Sachs-Wolfe effect:

r

kmrk

SWm kjY

kda

00

0*

2/3

3

)()()(3

1

)2(12

4


2. ISWe: ),()(),(),ˆ(

2)ˆ(

0

ˆ0

xgxln

dlT

nT

n

ISW

0

1200

0

0

22

0|)(|)(

)()()(

)12(

2),(

ll

lgdkljldlkg

),()()()2(12

40

*02/3

3

kgY

kda

kmk

Im

Crittenden & Turok 1997


Therefore, the total temperature fluctuations are:

where:

),()()()2(12

40

*02/3

3

kFY

kdaaa

kmk

Im

SWmm

),())()(3

1),( 000 kgkjgkF r


• Consider now a cluster at the direction c and at redshift zc. What kind of CMB would an observer in that cluster observe?

r 0

rc

c 0

)()(ˆ ccccc znr

• Spherical harmonic at cluster:

),()()()2(12

4),ˆ( *

02/3

3

ckmrki

kccm kFYekd

na c


Hot ionized gas in clusters affects CMB photons through inverse Compton effect:

• How can we in fact measure the alm‘s or the Cl‘s in clusters? Sunyaev-Zeldovich effect Sunyaev & Zeldovich 80,81

Sazonov & Sunyaev 99

The scattering also polarizes the CMB photons according to the quadrupole of the CMB temperature seen by the cluster. If we ignore the ISWe, the Q and U polarization modes for a cluster at position () are given by:

2sincos)()(1.0),,(

cossincossincoscos)(1.0),,( 222222

yx

yx

xfxU

xfxQ

• x=h/kT ,• f(x) is a spectral correction (of order 1),• is the optical depth of the cluster,• x and y are linear combinations (eigenvalues) of the components of the

quadrupole (a20, a21, a22) at the location of the cluster.


These clusters appear to us as “mini-quadrupoles” through the polarized CMB photons.

However, in the presence of DE we know that the quadrupole evolved quite a lot, recently (z<~2) because of the ISWe.

Therefore, with independent data (thermal SZe, X-ray) about the optical depth of a given cluster, the polarization of the CMB in the direction of clusters is a witness to the redshift-dependent quadrupole of the CMB.

This means that, if the CMB quadrupole is constant, the degree and orientation of the SZe-induced polarization of clusters along a given line of sight does not vary with redshift (except for Cosmic Variance).


Computing the Cl ’s for redshift bins and averaging over their positions will give:

),(|)(|2

|),ˆ(|)( 220

32ckccmc kFk

k

dknaC

Effect is stronger for the quadrupole (l=2).

If we measure the change of the quadrupole with time, we set limits over Fl (k,c) and the growth function, g(z). This is a strong test of DE.


The time variation for the low multipoles (l<10) is very sensitive to the decay of the gravitational potentials:

z

zg

z

C

)(

~2

wmm

z

zzHzH

zzHdz

H

zzHdz

zHzg

3330

0 20

2

)1)(1()1()(

)'1)('('

1

)'1)('('

)1(1)(

We can estimate how this test determines the parameters m and w through the function g’(z):

4.0,3.0,2.0,1.0m


Baumann & Cooray 2004

• Efficacy of the observations

Assuming a survey of clusters down to 1014 MO, with sensitivity for polarization of 0.1 K, over 104 deg2 :


2. Polarization tomography

Let’s return to the expression for the harmonic components, and let me assume for simplicity that the ISWe is small, so only the SWe remains:

)()()()2(

)( *02/3

3

34

ckmrki

k

icm kjYe

kdra c

where

Notice that the harmonic components are essentially functions of the position of the cluster, and are quite similar to a Fourier transform. Let’s invert this:

.||,ˆ ccccc rnr

),()'()()1(2

)()()'()2(

22

ˆ'

2/3

3

cmcccc

cmnmcrkic

kGkjkjdkkd

raYkjerd

c

c

Where)()()(

3

1),( **

kmkmckcm YYkjdkGk

'kkk


Using the completeness of Bessel’s functions and doing some algebra we get:

)'

,'()1(2

),()'(2

)1(2

),()'()()1(2

)()()'()2(

2

22

22

ˆ'

2/3

3

kkG

kkGkk

kkkd

kGkjkjdkkd

raYkjerd

m

m

cmcccc

cmnmcrkic

c

c

Summing over m=-l,...,l we obtain:

)(cos)cos2()1)(12(6

)()()'()2( ˆ

'2/3

3

Pjd

raYkjerd

kk

mcmnmc

rkicc

c

Where )'ˆˆ(',ˆˆcos kkkkkk


Therefore, if we can measure with some accuracy the quadrupole in galaxy clusters up to relatively high redshifts (z3), then we will be able to reconstruct the three-dimensional density field inside our LSS volume – and not only the two-dimensional spectrum on the LSS surface!

For the quadrupole, the weight function which multiplies the spectrum is approximately peaked at cos = ± 1 .

-0.75 -0.5 -0.25 0 0.25 0.5 0.75 1

0

0.1

0.2

0.3

0.4

Therefore, up to an approximation which can be easily improved, we can reconstruct the 3-dimensional spectrum of fluctuations by measuring the harmonic components in space (through cluster polarization). The final result:

2

22ˆ222/3

3

)()()()2(5

6128.0

mcmnmc

rkicWk

raYkjerd

c

c

bwdem – 06/04/2005doing cosmology with galaxy clusters cosmology with galaxy clusters: testing the...

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