Cross-correlation of CMB

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
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 (1z)3 ?R (1z)4 ?K(1z)2
  ?DE (1z)3(1w)
• 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 Wm H2 d 0
• galaxy formation (SFR)

Tiempo Energia
4

• 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) gt Dr (r - rb ) rb d
• rb M / V gt DM /M d

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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)
• 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?

EdS
Open
z 9
z 0 (now)
L
a 1/(1z)
a 0.1
a 1 (now)
a 0.01
a 10
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Problem I
Argue that the linear growth equation Has the
following solutions
Show that

• (2)

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Non-linear evolution
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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)

11
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.
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Observations require an statistical approach
Evolution of (rms) variance x2 lt d2gt
instead of d Or power spectrum P(k) lt d2(k)gt
gt x2 ? dk P(k) k2 W(k) dk
IC problem Linear Theory d a d0 x2 lt d2gt
D2 lt d02gt Normalization s8 2 º lt d2(R8)gt
To find D(z) -gt Compare rms at two times or
find evolution invariants
Initial Gaussian distribution of density
fluctuations xp (V) lt dPgt 0 for all
p ?2
Perturbations due to gravity generate
non-Gaussian statistics xp -gt x3 S3 x22
with S3(m) 34/7 (time Cosmo invariant)
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Predictions of Inflation

• Flat universe
• scale invariance IC n1
• CDM transfer funcion P(k) kn T(k)
• gt Gaussian IC

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Local spectral index P(k) kn (initial
spectrum transfer function) x2r ? dk P(k)
k2 W(k) dk r-(n3) n -2 gt x2 r r -1
(1D fractal ) equal power on all scales
(Wm0.2) n -1 gt x2 r r -2 (2D fractal )
less power on large scales (Wm1.0)
n -2
CMB Superclusters Clusters
Galaxies
n 1
Wm
n -1
Horizon _at_ Equality
s8
SCDM
n -1
LCDM
n -2
Wm1.0
Wm0.2
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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 gt measure mass

19
Weakly non-linear Perturbation Theory Solved
problem!
vertices
angular average
d dL n2 dL2 ...
Leading order contribution in d corresponds to
the spherical collapse.
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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)

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Weakly non-linear Perturbation Theory
(Spherical average)
d dL n2 dL2 ...
d3 dL3 3 n2 dL4 ...
Gaussian Initial conditions
lt dL3 gt a3 ltd03gt 0
lt d3 gt lt dL3 gt 3 n2 lt dL4 gt ...
lt dL4 gt lt dL 2 gt2
lt d3 gt 3 n2 lt dL2 gt2 ...
S 3 º lt d3 gt / lt d2 gt2 3 n2 34 / 7
gravity?
High order statistics -gt vertices of non-linear
growth!
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Test in N-body simulations
3-pt funct N3 (106)3 !!
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Weakly non-linear Perturbation Theory
Gaussian Initial conditions connected
correlations are zero, except 2-ptgt All
correlations are built from 2-pt!
Tree level dominant

Tree level F2 F3
Loops(higher order corrections) F2 F3

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Weakly non-linear Perturbation Theory
Tree level
P(k) kn
3
1
r23
aq
r12
2
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Depends on local spectral index P(k) kn (not
on Wm) x2r ? dk P(k) k2 W(k) dk r-(n3) n
-2 gt x2 r r -1 (1D fractal ) equal power
on all scales (Wm0.2) n -1 gt x2 r r -2
(2D fractal ) less power on large scales (Wm1.0)
n -2
n -2
n -1
n -1
n -2
n -1
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• 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
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Hogg Blanton
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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 b
n/s (SC ndc/s -gt x2 (peak) b2 x2 (m)
Threshold n
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Biasing does light trace mass? On large scales
2-pt Statistics is linear
Gravity
??g???b ?m ???
Bias
??m ? ?L ???D??0 ? ???
Gravity vs Galaxy
formation
????g???????b??????m??????? b? D??????0???????
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Biasing does light trace mass? Local
approximation ??g???F ?m
??g???b ?m ????b????m? ???
??L??is Gaussian ??m is not

??m?????L?????????L ? ? ???
????g???????b??????m??????? b??????L???????
????g?????? b3?????m3??????? 3 b?b2
???m4??????????? ? ????g??????
b??????????????b??b?????g?????????????
Gravity vs Galaxy
formation
? c 2 ?? ?b??/ b ???
? c 3 ?? ?b3?/ b ???
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Bias rare peaks in a Gaussian field (Kaiser
1984. BBKS) Linear bias b d (peak) b
d(mass) with b n/s (for SC ndc/s) -gt
x2 (peak) b2 x2 (m) Non-linear bias -gt b2
b2 ( bk bk ) -gt Bias S3 3 S4
16 (Sk kk-2 ) -gt Close to DM!!
Gravity S3 34/7-(n3) 3 S4 20
Threshold n
How to separate one from the other?
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How to separate Bias from Gravity? QG (QmC)/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

Blt1
C
Bgt1
CGF model Bower etal 1993
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Comparison with 2dfGRS

• - Gravity _at_ 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 b1 and c20, Verde etal
  2001)
• b1 0.95 ? 0.12 b2 -0.3 ? 0.1 ( -0.4 ltc2lt
  -0.2)
• Work in progress (by galaxy type and color)
• measure of normalization 0.8 lt s8 lt 1.0
• gt Future applications?

Gravity vs Galaxy
formation
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Bias Higher conclusion

• Local approximation works on larrge scales
  ??g???F ?m
• For P(k) or 2-pt statistics
• Linear theory works on scales gt 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!
• gt 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
Evolution of (rms) variance x2 lt d2gt
instead of d
IC problem Linear Theory d a d0 gt x2 lt
d2gt D2 lt d02gt Normalization s8 2 º lt
d2(R8)gt To find D(z) -gt Compare rms at two
times or find evolution invariants
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• Where does Structure in the Universe come From?

• Perturbation theory
• r rb ( 1 d) gt Dr (r - rb ) rb d
• rb M / V gt DM /M d
• With d H d - 3/2 Wm H2 d 0 in EdS
  linear theory d a d0
• Gravitation potential
• F - G M /R gt DF G DM / R GM/R d
• in EdS linear theory d a d0 gt 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!

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PRIMARY CMB ANISOTROPIES Sachs-Wolfe (ApJ, 1967)
DT/T(n) F (n) if Temp. F. diff in
N.Potential (SW)
Ff
Fi
DT/T(SW) DF /c2
DF GM (d/ R) /c2
CMB LSS
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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)?
Ff
Fi
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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)
Ff
SZ- Inverse Compton Scattering -gt 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 !
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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.
Noise!
Signal
Crittenden
ISW map, zlt 4
Early map, z1000
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Cross-correlation idea
Crittenden Turok (PRL, 1995)
Both DT and d (g) are proportional to local
mass fluctuations d (m)
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Problem III
(1) Assuming that galaxies trace the mass,
demostrate that in the linear regime and for
small angles (lt10 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?
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ISW in equations...
Limber approximation
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APM
APM
5.0 deg FWHM
0.7 deg FWHM
WMAP?APM
WMAP?APM
WMAP
WMAP
0.7 deg FWHM
5.0 deg FWHM
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• 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
• ?

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APM
Significance P 1.2 null detection -gt wTG
0.35 0.13 mK (68 CL) _at_ 4-10 deg -gt WL
0.53-0.86 ( 2-sigma)
Pablo Fosalba EG, (astro-ph/0305468)
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P. Fosalba, EG, F.Castander (astro-ph/0307249,
ApJ 2003)
Significance (null detection) SDSS high-z P
0.3 for lt 10 deg. (P1.4 for 4-10 deg) SDSS
all P 4.8 Combined P0.1 - 0.03 (3.3 -
3.6 sigma)
WL 0.69-0.87 ( 2-sigma)
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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 Sytematics Extinction dust in galaxies.
Wm 0.20 s80.9
High!?
RADIO (NVSS) X-ray (HEAO) Boughm Crittenden
(astro-ph/0305001). WMAP team Nolta et al.,
astro-ph/0305097 z 0.8-1.1
(tentative lt 2.5 s) APM Fosalba EG
astro-ph/0305468 z0.15-0.3 (tentative lt 2.5
s) SDSS Fosalba, EG, Castander, astro-ph/0307249
SDSS team Scranton et al 0307335
Pamanabhan (2005) Cabre etal 2006 z0.3-0.5
(detection gt 4 s!) 2Mass Afshordi et al
0308260 Rassat etal 06 z0.1 (tentative lt 2.
s) QSO Giannantonio etal 06 (tentative lt 2.5s)
LSS!?
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s80.9
S/N2 fsky(2l1) /1 Cl(TT)Cl(GG)/Cl(TG)2

b1
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S/N2 fsky(2l1) /1 Cl(TT)Cl(GG)/Cl(TG)2
s80.9
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Compilation EG, Manera, Multamaki (MNRAS 2006)
Prob of NO detection 3/100,000
Corasantini, Giannantonio, Melchiorri 05

• Marginalized over
• h0.6-0.8
• -relative normalization of P(k)
• Normalize to sigma81 for CM
• Bias from Gal-Gal correlation

WL 0.4-1.2 Wm 0.18- 0.34
With SNIa WL 0.71 /- 0.13 Wm 0.29 /- 0.04
With SNIa flat prior WL 0.70 /- 0.05 w 1.02
/- 0.17
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Cosmic Magnification and the ISW effect
EG

• tells about growth rates at lens redshifts
• ? (2.5s-1)
• s d log(N(m))/dm

• has info about structure growth at redshift
  of sample
• ? galaxy bias

Relative magnitude of the two terms is redshift,
scale and galaxy population dependent
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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
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Higher orders and ISW

• I- Perturbation theory and Higher order
  correlations
• II- CMB LSS ISW effect
• III- Error analysis in CMB-LSS cross-correlation

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Error Analysis

• Consider 4 methods
• 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)

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Error Analysis

• Consider 4 methods
• Gaussian errors in Harmonic space (TH)
• transform into configurational space

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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_sky1) the expected variance in the
galaxy-temperature angular cross-correlation
spectrum (CTG) at multipole l is
Where CTT and CGG 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?
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Error Analysis
Consider 4 methods 2. New errors in
Configurational space (TC)
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3.
Poors-man Boostrap? EACH SIMULACION PRODUCES
A JK ERROR AND JK Cij
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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

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10 sky z0.33
Input vs 1000 sim
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All sky z0.33
Input vs sim
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10 sky z0.33
Input vs sim
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Comparison of JK errors with MC errors
JK 0.193 0.045 (true0.202)
JK 0.207 0.041 (true0.224)
JK 0.170 0.049 (true0.167)
JK 0.113 0.039 (true0.107)
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Error in the error
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ERROS in C_L This wildly used Eq. only works
for Binned data!
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• ERROS in C_L
• Can propagate diagonal
• errors in C_l to w(q)
• Thid is surprising for flt1 transfer to
  off-diagonal elements
• Bin C_l data to get diagonal errors.

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CMB data
LSS data
SDSS DR4
WMAP 3rd year
-5200 sq deg (13 sky) -Selection of subsamples
with different redshift distribution -3 magnitude
subsamples with r18-19, r19-20 and r20-21 with
106 107 galaxies -high redshift Luminous Red
Galaxy (Eisentein et al. 2001) -Mask avoids
holes, trails, bleeding, bright stars and
seeinggt1.8
V-band (61 Hz) HEALPix tessellation Kp0 mask
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Redshift selection function
Jack-knife errors
LRG
20-21
Singular Value Decomposition (SVD)
c2 distribution
r20-21 zc0 z00.2 zm0.3 LRG zc0.37
z00.45 zm0.5
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r20-21 S/N3.6
LRG S/N3.
S/N total4.7
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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
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Can we obtain information about w?
Contour 1, 2 sigma 1 dof
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The Science Case for the Dark Energy Survey
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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 z1
• 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

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DARK ENERGY SURVEY (DES)
Science Goal measure wp/r, the dark energy
equation of state, to a precision of Dw 5,
with

• Cluster Survey
• Weak Lensing
• Galaxy Angular Power Spectrum
• Supernovae

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Science Goals to Science Objective

• To achieve our science goals
• Cluster counting to z gt 1
• Spatial angular power spectra of galaxies to z
  1
• Weak lensing, shear-galaxy and shear-shear
• 2000 zlt0.8 supernova light curves
• We have chosen our science objective
• 5000 sq-degree imaging survey
• Complete cluster catalog to z 1, photometric
  redshifts to z1.3
• 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

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DES Dark Energy Constraints
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
  s(w) 0.05-0.1
• Complementary methods
• Constrain different combinations of cosmological
  parameters
• Subject to different systematic errors

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) SSGSGG SSbispectrum 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
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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

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Zmax2 Dz0.08
89
s80.9
ISW predictions
s81.0
90

• Detailed CONCLUSIONS
• gt800 simulations for 5 error accuracy
• - Diagonal errors in w(q) are accurate to q lt20
  deg
• Survey geometry important for qgt10 deg (flt0.1)
  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 flt1 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 s8 or
  large WL.