bwdem – 06/04/2005doing cosmology with galaxy clusters cosmology with galaxy clusters: testing the...
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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
bwdem – 06/04/2005doing cosmology with galaxy clusters
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?
bwdem – 06/04/2005doing cosmology with galaxy clusters
1. How clusters help pinpoint Dark Energy: that “little extra mile”
WMAP 2003
Allen et al. 2004
bwdem – 06/04/2005doing cosmology with galaxy clusters
Constraints from future SZe surveys
Carlstrom, Holder & Reese 2002
bwdem – 06/04/2005doing cosmology with galaxy clusters
Constraints from cosmic shear
Jain & Taylor 2004
Effective depth to measure w of a shear survey, compared to other observations
Ferguson & Bridle 2005
bwdem – 06/04/2005doing cosmology with galaxy clusters
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
bwdem – 06/04/2005doing cosmology with galaxy clusters
• CMB
mm
m
mm
aC
YaT
2
,
12
1
),(),(
Bennett et al. ApJS 148:1 (2003)
bwdem – 06/04/2005doing cosmology with galaxy clusters
• 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
bwdem – 06/04/2005doing cosmology with galaxy clusters
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
bwdem – 06/04/2005doing cosmology with galaxy clusters
Therefore, the total temperature fluctuations are:
where:
),()()()2(12
40
*02/3
3
kFY
kdaaa
kmk
Im
SWmm
),())()(3
1),( 000 kgkjgkF r
bwdem – 06/04/2005doing cosmology with galaxy clusters
• 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
bwdem – 06/04/2005doing cosmology with galaxy clusters
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.
bwdem – 06/04/2005doing cosmology with galaxy clusters
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).
bwdem – 06/04/2005doing cosmology with galaxy clusters
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.
bwdem – 06/04/2005doing cosmology with galaxy clusters
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
bwdem – 06/04/2005doing cosmology with galaxy clusters
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 :
bwdem – 06/04/2005doing cosmology with galaxy clusters
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
bwdem – 06/04/2005doing cosmology with galaxy clusters
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
bwdem – 06/04/2005doing cosmology with galaxy clusters
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