what galaxy surveys really measure. - uni-bielefeld.de · 2013. 4. 26. · introduction the cmb cmb...

. . . . . .

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......What Galaxy Surveys Really Measure.

Ruth DurrerUniversite de Geneve

Department de Physique Theorique et Center for Astroparticle Physics

Bielefeld, Kosmologietag 2013

Ruth Durrer (Universite de Geneve, DPT & CAP) What Galaxy Surveys Really Measure. Bielefeld, April 2013 1 / 29

. . . . . .

Outline

...1 Introduction

...2 What are very large scale galaxy catalogs really measuring?Matter fluctuations per redshift binVolume perturbations

...3 The angular power spectrum and the correlation function of galaxy densityfluctuations

The transversal power spectrumThe radial power spectrum

...4 Alcock-Paczynski test

...5 Conclusions


. . . . . .

Introduction

The CMB

CMB sky as seen by Planck

The Planck Collaboration:Planck results 2013 XV

2 10 500

1000

2000

3000

4000

5000

6000

D `[µ

K2 ]

90 18

500 1000 1500 2000 2500

Multipole moment, `

1 0.2 0.1 0.07Angular scale


. . . . . .

Introduction

M. Blanton and the Sloan Digital Sky Survey Team.Ruth Durrer (Universite de Geneve, DPT & CAP) What Galaxy Surveys Really Measure. Bielefeld, April 2013 4 / 29

. . . . . .

Galaxy power spectrum from the Sloan Digital Sky Survey (BOSS)14 L. Anderson et al.

Figure 8. The CMASS DR9 power spectra before (left) and after (right) reconstruction with the best-fit models overplotted. The vertical dotted lines showthe range of scales fitted (0.02 < k < 0.3hMpc1), and the inset shows the BAO within this k-range, determined by dividing both model and data by thebest-fit model calculated (including window function convolution) with no BAO. Error bars indicate

p

Cii

for the power spectrum and the rms error calculatedfrom fitting BAO to the 600 mocks in the inset (see Section 4.2 for details).

an estimate of the “redshift-space” power, binned into bins in k ofwidth 0.04hMpc

1.

6.2 Fitting the power spectrum

We fit the observed redshift-space power spectrum, calculated asdescribed in Section 6, with a two component model comprising asmooth cubic spline multiplied by a model for the BAO, followingthe procedure developed by Percival et al. (2007a,c, 2010). Themodel power spectrum is given by

P (k)m = P (k)smooth Bm

(k/↵), (32)

where P (k)smooth is a smooth model that fits the overall shapeof the power spectrum, and the BAO model Bm(k), calculated forour fiducial cosmology, is scaled by the dilation parameter ↵ asdefined in Eq. 21. The calculation of the BAO model is describedin detail below. This scaling of the acoustic signal is identical tothat used in the correlation function fits, although the differing non-linear prescriptions in (Eqns 23 & 32) means that the non-linearBAO damping is treated in a subtly different way.

Each power spectrum model to be fitted is convolved with thesurvey window function, giving our final model power spectrum tobe compared with the data. The window function for this convolu-tion is the normalised power in a Fourier transform of the weightedsurvey coverage, as defined by the random catalogue, and is calcu-lated using the same Fourier procedure described in Section 6 (e.g.Percival et al. 2007c). This is then fitted to express the windowfunction as a matrix relating the model power spectrum evaluatedat 1000 wavenumbers, k

n

, equally spaced in 0 < k < 2hMpc

1,to the central wavenumbers of the observed bandpowers k

i

:

P (ki

)fit =

X

n

W (ki

, kn

)P (kn

)m W (ki

, 0). (33)

The final term W (ki

, 0) arises because we estimate the averagegalaxy density from the sample, and is related to the integral con-straint in the correlation function. In fact this term is smooth (as

the power of the window function is smooth), and so can be ab-sorbed into the smooth component of the fit, and we therefore donot explicitly include this term in our fits.

To model the overall shape of the galaxy clustering powerspectrum we use a cubic spline (Press et al. 1992), with nine nodesfixed empirically at k = 0.001, and 0.02 < k < 0.4 withk = 0.05, matching that adopted in Percival et al. (2007c, 2010).This model was tested in these papers, but we show in Section B3that it also provides an excellent fit to the overall shape of the DR9CMASS mock catalogues, and that there is no evidence for devia-tions for the fits to the data.

To calculate our fiducial BAO model, we start with a linearmatter power spectrum P (k)lin, calculated using CAMB (Lewis etal. 2000), which numerically solves the Boltzman equation describ-ing the physical processes in the Universe before the baryon-dragepoch. We then evolve using the HALOFIT prescription (Smithet al. 2003), giving an approximation to the evolved power spec-trum at the effective redshift of the survey. To extract the BAO, thispower spectrum is fitted with a model as given by Eq. 32, where weadopt a fixed BAO model (BEH) calculated using the Eisenstein &Hu (1998) fitting formulae at the same fiducial cosmology. Divid-ing P (k)lin by the best-fit smooth power spectrum component fromthis fit produces our BAO model, which we denote BCAMB.

We damp the acoustic oscillations to allow for non-linear ef-fects

Bm

= (BCAMB 1)ek

22nl/2

+ 1, (34)

where the damping scale

nl

is a fitted parameter. We assumea Gaussian prior on

nl

with width ±2h1Mpc, centred on

8.24h1Mpc for pre-reconstruction fits and 4.47h1

Mpc forpost-reconstruction fits, matching the average recovered valuesfrom fits to the 600 mock catalogs with no prior. The exact width ofthe prior is not important, but if we do not include such a prior, thenthe fit can become unstable with respect to local minima at extremevalues.

c 2011 RAS, MNRAS 000, 2–33

from Anderson et al. ’12

SDSS-III (BOSS)power spectrum.

Galaxy surveys ≃matter density fluctuations,biasing and redshift spacedistortions.


. . . . . .

Introduction

The observed Universe is well approximated by a ΛCDM model,ΩΛ ≃ 0.72, Ωm = Ωcdm +Ωb ≃ 0.28, Ωb ≃ 0.04.

20 L. Anderson et al.

Figure 18. BAO in the power spectrum measured from the reconstructedCMASS data (solid circles with 1 errors, lower panel) compared with un-reconstructed BAO recovered from the SDSS-II LRG data (solid circleswith 1 errors, upper panel). Best-fit models are shown by the solid lines.The SDSS-II data are based on the sample and power spectrum calculated inReid et al. (2010) and analysed by Percival et al. (2010); it has been shiftedto match the fiducial cosmology assumed in this paper. Clearly the CMASSerrors are significantly smaller than those of the SDSS-II data, and we alsobenefit from reconstruction, reducing the the BAO damping scale.

Figure 19. A plot of the distance-redshift relation from various BAO mea-surements from spectroscopic data sets. We plot D

V

(z)/rs

times the fidu-cial r

s

to restore a distance. Included here are this CMASS measurement,the 6dF Galaxy Survey measurement at z = 0.1 (Beutler et al. 2011), theSDSS-II LRG measurement at z = 0.35 (Padmanabhan et al. 2012a; Xuet al. 2012; Mehta et al. 2012), and the WiggleZ measurement at z = 0.6(Blake et al. 2011a). The latter is a combination of 3 partially covariant datasets. The grey region is the 1 prediction from WMAP under the assump-tion of a flat Universe with a cosmological constant (Komatsu et al. 2011).The agreement between the various BAO measurements and this predictionis excellent.

Figure 20. The BAO distance-redshift relation divided by the best-fit flat,CDM prediction from WMAP (

m

= 0.266, h = 0.708; note thatthis is slightly different from the adopted fiducial cosmology of this paper).The grey band indicates the 1 prediction range from WMAP (Komatsuet al. 2011). In addition to the SDSS-II LRG data point from Padmanabhanet al. (2012a), we also show the result from Percival et al. (2010) using acombination of SDSS-II DR7 LRG and Main sample galaxies as well as2dF Galaxy Redshift Survey data; because of the overlap in samples, weuse a different symbol. The BAO results agree with the best-fit WMAPmodel at the few percent level. If

m

h2 were 1 higher than the best-fit WMAP value, then the prediction would be the upper edge of the greyregion, which matches the BAO data very closely. For example, the dashedline is the best-fit CMB+LRG+CMASS flat CDM model from § 9, whichclearly is a good fit to all data sets. Also shown are the predicted regionsfrom varying the spatial curvature to

K

= 0.01 (blue band) or varyingthe equation of state to w = 0.7 (red band).

place the acoustic peak at other nearby locations and particularlyat smaller scales is rejected at 8 .

Fig. 18 repeats this comparison with the power spectrum fromthe SDSS-II LRG analysis presented in Reid et al. (2010) and Per-cival et al. (2010). This analysis did not use reconstruction, but onecan see good agreement in the BAO and significant improvementin the error bars with the CMASS sample.

In Fig. 19, we plot DV

(z) constraints from measurements ofthe BAO from various spectroscopic samples. In addition to theSDSS-II LRG value at z = 0.35 (Padmanabhan et al. 2012a) andthe CMASS consensus result at z = 0.57, we also plot the z =

0.1 constraint from the 6dF Galaxy Survey (6dFGS) (Beutler et al.2011) and a z = 0.6 constraint from the WiggleZ survey (Blakeet al. 2011a). WiggleZ quotes BAO constraints in 3 redshift bins,but these separate constraints are weaker and there are significantcorrelations between the redshift bins. We choose here to plot theiruncorrelated data points for 0.2 < z < 1.0. Each data point here isactually a constraint on D

V

(z)/rs

, and we have multiplied by ourfiducial r

s

to get a distance.As described further in Mehta et al. (2012), the WMAP curve

on this graph is a prediction, not a fit, assuming a flat CDM cos-mology. For each value of

m

h2 and

b

h2, one can predict a soundhorizon, and the angular acoustic scale measured by WMAP plusthe assumptions about spatial curvature and dark energy equationof state then provide a very precise breaking of the degeneracy be-tween

m

and H0 and hence a unique DV

(z)/rs

. Taking the 1range of

m

h2 and

b

h2 produces the grey band in Fig. 19. Thereis excellent agreement between all four BAO measurements and theWMAP CDM prediction.

c 2011 RAS, MNRAS 000, 2–33

from Anderson et al. ’12 from Planck Collaboration, XVI


. . . . . .

Introduction

We have to take fully into account that all observations are made on our pastlightcone, for example, we see density fluctuations which are further away from us,further in the past;

the measured redshift is not only perturbed by velocities but also by thegravitational potential;

not only the number of galaxies but also the volume is distorted;

the angles we are looking into are not the ones into which the photons from agiven galaxy arriving at our position have been emitted.

For small galaxy catalogs, these effects are not very important, but when we goout to z ∼ 1 or more, they become relevant. Already for SDSS which goes out toz ≃ 0.2 (main catalog) or even z ≃ 0.6 (BOSS).

But of course much more for future surveys like DESspec, bigBOSS and Euclid.

Whenever we convert a measured redshift or angle into a length scale, we makean assumption about the underlying cosmology.

D(z) =∫ z

0

dz′

H(z′)


. . . . . .

What are very large scale galaxy catalogs really measuring?

Following C. Bonvin & RD [arXiv:1105.5080]; F. Montanari & RD [arXiv:1206.3545]see also Challinor & Lewis, [arXiv:1105:5092].Relativistic corrections to galaxy surveys are also discussed in:J. Yoo el al. 2009; J. Yoo 2010

For each galaxy in a catalog we measure

(θ, ϕ, z) = (n, z) + info about mass, spectral type...

We can count the galaxies inside a redshift bin and small solid angle, N(n, z) andmeasure the fluctuation of this count:

∆(n, z) =N(n, z)− N(z)

N(z).

ξ(θ, z, z′) = ⟨∆(n, z)∆(n′, z′)⟩ , n · n′ = cos θ .

This quantity is directly measurable ⇒ gauge invariant.


. . . . . .


Density fluctuation per redshift bin dz and per solid angle dΩ as δz(n, z).

δz(n, z) =ρ(n, z)− ρ(z)

ρ(z)=

N(n,z)V (n,z) −

N(z)V (z)

N(z)V (z)

This together with the volume fluctuations, results in the directly observed numberfluctuations

∆(n, z) = δz(n, z) +δV (n, z)

V (z)

Both these terms are in principle measurable and therefore gauge invariant. We wantto express them in terms of standard gauge invariant perturbation variables.


. . . . . .

Matter fluctuations per redshift bin

We consider a photon emitted from a galaxy (S), moving in direction n into ourtelescope. The observer (O) receives the photon redshifted by a factor

1 + z =(n · u)S

(n · u)O.

To first order in perturbation theory one finds (in longitudinal gauge)

δz(1 + z)

= −[(

n · V +Ψ)(n, z) +

∫ χ(z)

0(Φ + Ψ)dχ

]With this, the density fluctuation in redshift space becomes

δz(n, z) = Dg(n, z) + 3(V · n)(n, z) + 3(Ψ + Φ)(n, z) + 3∫ χS

0(Ψ + Φ)(n, z(χ))dχ

This quantity is gauge invariant and therefore, in principle, measurable. But when wecount numbers of galaxies per solid angle and per redshift bin, we also have toconsider volume perturbations.


. . . . . .



1 + z =(n · u)S

(n · u)O.


δz(1 + z)

= −[(

n · V +Ψ)(n, z) +

∫ χ(z)

0(Φ + Ψ)dχ

]

With this, the density fluctuation in redshift space becomes


0(Ψ + Φ)(n, z(χ))dχ



. . . . . .



1 + z =(n · u)S

(n · u)O.


δz(1 + z)

= −[(

n · V +Ψ)(n, z) +

∫ χ(z)

0(Φ + Ψ)dχ

]With this, the density fluctuation in redshift space becomes


0(Ψ + Φ)(n, z(χ))dχ



. . . . . .

The total galaxy density fluctuation per redshift bin, per sold angle

Putting the density and volume fluctuations together one obtains the galaxy numberdensity fluctuations

.

.

∆(n, z) = Dg +Φ+Ψ+1H

[Φ + ∂χ(V · n)

]+

(HH2 +

2χ(z)H

)(Ψ+ V · n +

∫ χ(z)

0dχ(Φ + Ψ)

)

+1

χ(z)

∫ χ(z)

0dχ[2 − χ(z)− χ

χ∆Ω

](Φ + Ψ).

( C. Bonvin & RD ’11)


. . . . . .

The total galaxy density fluctuation per redshift bin, per sold angle

Putting the density and volume fluctuations together one obtains the galaxy numberdensity fluctuations

.

.

∆(n, z) = Dg +Φ+Ψ+

1H

[Φ +

∂χ(V · n)]

+

(HH2 +

2χ(z)H

)(Ψ+

V · n +

∫ χ(z)

0dχ(Φ + Ψ)

)

+1χS

∫ χ(z)

0dχ[2(Φ + Ψ)−

χ(z)−χχ

∆Ω(Φ + Ψ)

].

( C. Bonvin & RD ’11)


. . . . . .


(From Gaztanaga et al. 2008)The correlation function is not isotropic ⇒ redshift space distortions.


. . . . . .

Anisotropic clustering as seen in the BOSS survey

(from Reid et al. ’12)Anisotropic clustering in CMASS galaxies 5

rσ (Mpc/h)

r π (Mpc/h)

−100 −50 0 50 100

−100

−50

0

50

100

rσ (Mpc/h)

r π (Mpc/h)

−50 −40 −30 −20 −10 0 10 20 30 40 50−50

−40

−30

−20

−10

0

10

20

30

40

50

Figure 3. Left panel: Two-dimensional correlation function of CMASS galaxies (color) compared with the best fit model described in Section 6.1 (black lines).Contours of equal ξ are shown at [0.6, 0.2, 0.1, 0.05, 0.02, 0]. Right panel: Smaller-scale two-dimensional clustering. We show model contours at [0.14, 0.05,0.01, 0]. The value of ξ0 at the minimum separation bin in our analysis is shown as the innermost contour. The µ ≈ 1 “finger-of-god” effects are small on thescales we use in this analysis.

in Figure 4. The effective redshift of weighted pairs of galaxies inour sample is z = 0.57, with negligible scale dependence for therange of interest in this paper. For the purposes of constraining cos-mological models, we will interpret our measurements as being atz = 0.57.

3.2 Covariance Matrices

The matrix describing the expected covariance of our measure-ments of ξ"(s) in bins of redshift space separation depends in lineartheory only on the underlying linear matter power spectrum, thebias of the galaxies, the shot-noise (often assumed Poisson) and thegeometry of the survey. We use 600 mock galaxy catalogs, basedon Lagrangian perturbation theory (LPT) and described in detail inManera et al. (2012), to estimate the covariance matrix of our mea-surements. We compute ξ"(si) for each mock in exactly the sameway as from the data (Sec. 3.1) and estimate the covariance matrixas

C"1"2i j =1

599

600∑

k=1

(ξk"1 (si) − ξ"1 (si)

) (ξk"2 (s j) − ξ"2 (s j)

), (7)

where ξk" (si) is the monopole (" = 0) or quadrupole (" = 2) correla-tion function for pairs in the ith separation bin in the kth mock. ξ"(s)is the mean value over all 600 mocks. The shape and amplitude ofthe average two-dimensional correlation function computed fromthe mocks is a good match to the measured correlation functionof the CMASS galaxies; see Manera et al. (2012) and Ross et al.(2012) for more detailed comparisons. The square roots of the di-agonal elements of our covariance matrix are shown as the error-bars accompanying our measurements in Fig. 4. We will examinethe off-diagonal terms in the covariance matrix via the correlation

matrix, or “reduced covariance matrix”, defined as

C"1"2,redi j = C"1"2i j /

√C"1"1ii C"2"2j j , (8)

where the division sign denotes a term by term division.In Figure 5 we compare selected slices of our mock covari-

ance matrix (points) to a simplified prediction from linear theory(solid lines) that assumes a constant number density n = 3 × 10−4

(h−1 Mpc)−3 and neglects the effects of survey geometry (see, e.g.,Tegmark 1997). Xu et al. (2012) performed a detailed compari-son of linear theory predictions with measurements from the LasDamas SDSS-II LRG mock catalogs (McBride et al. prep), andshowed that a modified version of the linear theory covariance witha few extra parameters provides a good description of the N-bodybased covariances for ξ0(s). The same seems to be true here aswell. The mock catalogs show a deviation from the naive lineartheory prediction for ξ2(s) on small scales; a direct consequence isthat our errors on quantities dependent on the quadrupole are largerthan a simple Fisher analysis would indicate. We verify that thesame qualitative behavior is seen for the diagonal elements of thequadrupole covariance matrix in our smaller set of N-body simu-lations used to calibrate the model correlation function. This com-parison suggests that the LPT-based mocks are not underestimatingthe errors on ξ2, though more N-body simulations (and an account-ing of survey geometry) would be required for a detailed check ofthe LPT-based mocks.

The lower panels of Figure 5 compare the reduced covari-ance matrix to linear theory, where we have scaled the Cred

i j pre-diction from linear theory down by a constant, ci. This compar-ison demonstrates that the scale dependences of the off-diagonalterms in the covariance matrix are described well by linear the-ory, but that the nonlinear evolution captured by the LPT mockscan be parametrized simply as an additional diagonal term. Finally,

c© 0000 RAS, MNRAS 000, 1–1


. . . . . .

The angular power spectrum of galaxy density fluctuations

For fixed z, we can expand ∆(n, z) in spherical harmonics,

.

.

∆(n, z) =∑ℓm

aℓm(z)Yℓm(n), Cℓ(z, z′) = ⟨aℓm(z)a∗ℓm(z

′)⟩.

ξ(θ, z, z′) = ⟨∆(n, z)∆(n′, z′)⟩ = 14π

∑ℓ

(2ℓ+ 1)Cℓ(z, z′)Pℓ(cos θ)


. . . . . .


If we convert the measuredξ(θ, z, z′) to a power spectrum,we have to introduce a cosmology,to convert angles and redshiftsinto length scales. r(z, z′, θ) =√

χ2(z) + χ2(z′)− 2χ(z)χ(z′) cos θ.

(Figure by F. Montanari)

Observed

0 2 4 6 8 100.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

Θ @ degD

ΞHΘ

LΘ

2

z1 =z2 =0.7

True Wm = 0.24

Wrong Wm = 0.3

Wrong Wm = 0.5

0 50 100 150 200 250 3000.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

r @ Mpc hD

Ξ@r

HΘ,z

1,z

2LD

Θ2

z1 =z2 =0.7


. . . . . .


True Wm = 0.24

Wrong Wm = 0.3

Wrong Wm = 0.5

0.010 0.1000.0500.020 0.2000.030 0.3000.015 0.1500.070

4.

4.5

5.

5.5

6.

6.5

7.

7.5

k @ h MpcD

DHk

Lk

@Mp

ch

D

z =0

(Figure by F. Montanari)

∆(k)/k = k2P(k)


. . . . . .

The transversal power spectrum

The transverse power spectrum, z′ = z (from Bonvin & RD ’11)

20 40 60 80 100

10-5

10-4

0.001

0.01

0.1

l

lHl+

1L2Π

Cl

From top to bottom z = 0.1, z = 0.5, z = 1 and z = 3.


. . . . . .


Contributions to the transverse power spectrum at redshift z = 0.1, ∆z = 0.01(from Bonvin & RD ’11)

20 40 60 80 100

10-5

10-4

0.001

0.01

0.1

l

lHl+

1L2Π

Cl

CDDℓ (red), Czz

ℓ (green), 2CDzℓ (blue), CDoppler

ℓ (cyan, C lensingℓ (magenta) Cgrav

ℓ (black).


. . . . . .


Contributions to the transverse power spectrum at redshift z = 3,∆z = 0.3(from Bonvin & RD ’11)

20 40 60 80 100

5´10-71´10-6

5´10-61´10-5

5´10-51´10-4

l

lHl+

1L2Π

Cl

CDDℓ (red), Czz

ℓ (green), 2CDzℓ (blue), C lensing

ℓ (magenta).


. . . . . .


Contributions to the transversal power spectrum as function of the redshift, ℓ = 20,∆z = 0 (from Bonvin & RD ’11)

0.5 1.0 1.5 2.0 2.5 3.010-7

10-6

10-5

10-4

0.001

0.01

0.1

z

lHl+

1L2Π

Cl

CDDℓ (red), Czz

ℓ (green), 2CDzℓ (blue), C lensing

ℓ (magenta), CDopplerℓ (cyan),

Cgravℓ (black).


. . . . . .


20 40 60 80 1001´10-6

5´10-61´10-5

5´10-51´10-4

5´10-4

l

lHl+

1L2Π

Cl

0 < z < 2

CDDℓ (red), C lensing

ℓ (magenta).


. . . . . .

The transversal correlation function

0 5 10 15 20

- 0.002

0.000

0.002

0.004

100 200 300 400 500 600 700

Θ @ degD

Ξg

HΘL

Θ2

r H z , z , Θ L @ Mpc h D

(fromMontanari & RD ’12)

θ2ξ(θ, z, z)

blue CDDℓ (real space),

green flat space approximation for redshift space distortions,red CDD

ℓ , Czzℓ and 2CDz

ℓ (fully positive!).


. . . . . .

The radial power spectrum

0.105 0.110 0.115 0.120 0.125 0.130

10-4

0.001

0.01

0.1

z’

lHl+

1L2Π

Cl

0.50 0.55 0.60 0.65 0.70 0.7510-7

10-6

10-5

10-4

0.001

0.01

z’

lHl+

1L2Π

Cl

1.0 1.1 1.2 1.3 1.4 1.5

10-6

10-5

10-4

0.001

z’

lHl+

1L2Π

Cl

3.0 3.2 3.4 3.6 3.8

5´10-61´10-5

5´10-51´10-4

5´10-4

0.001

z’

lHl+

1L2Π

Cl

The radial power spectrum Cℓ(z, z′)for ℓ = 20Left, top to bottom: z = 0.1, 0.5, 1,top right: z = 3

Standard terms (blue), C lensingℓ (magenta),

CDopplerℓ (cyan), Cgrav

ℓ (black),


. . . . . .

The radial correlation function

0.05 0.10 0.15

-14

-12

-10

- 8

- 6

- 4

- 2

0

Dz

Ξg

HDzL

Dz2

´10

6

(fromMontanari & RD ’12)

z = 2,z = 1,z = 0.7,z = 0.3.

(∆z)2ξ(θ, z −∆z/2, z +∆z/2)Purely negative for ∆z >∼ 0.01.


. . . . . .

Cosmological parameters from the angle redshift correlation function3

Figure 1. Top panel shows the redshift distribution in the spec-troscopic and narrow band photo-z survey (violet). For the nar-row band case we show how the true redshift distributions givenby Eq. (1) look like if we divide the volume in eight consecutiveredshift bins. Bottom panel shows the same but for a broadbandphotometric survey divided in five bins.

2.1.2 Narrow Band Photometric survey

This case intends to be representative of a configuration suchas the one proposed for the PAU survey where a set of nar-row band filters is expected to deliver “low-resolution” spec-tra in a redshift range actually broader than the one con-sidered here (Benıtez et al. 2009; Gaztanaga et al. 2012).Hence our narrow-band photo-z survey has accurate photo-metric redshifts of σz = 0.004, in the same redshift range ofthe spectroscopic case (0.45 < z < 0.65). The bias (b = 2)and the shot-noise cases considered match those of Sec. 2.1.1(and are given in Table 1).

In turn the bin configurations assumed for the 2D to-mography are also the same as for the spectroscopic sur-vey given in Table 2, but with bin limits that now refer to

Number of bins ∆z ∆r (h−1 Mpc)

1 0.20 4684 0.05 113 - 1228 0.025 56 - 6116 0.0125 28 - 3120 0.010 22 - 25

Table 2. Bin configurations used for the 2D tomography in thecase of the spectroscopic and the narrow band photometric surveyin a redshift range of 0.45 < z < 0.65. We show the number ofradial bins and their range of widths in redshift and comovingdistance.

photometric redshifts. Thus the true redshift distribution ofgalaxies in each bin is no longer a top hat, but rather hasa small overlap with the nearest neighbouring bins due tothe photo-z error, as described in Eq. (18) below. In the toppanel of Fig. 1 we show this effect for the particular case of8 bins.

2.1.3 Broad Band Photometric survey

On the other hand we consider a photometric survey thatuses broad-band filters such as DES 5, Pan-Starrs 6 or thefuture imaging component of Euclid 7. These surveys areexpected to achieve photometric redshift estimates with ac-curacies σz ∼ 5%/(1 + z) (Banerji et al. 2008; Ross et al.2011). In what follows we do not consider a possible red-shift evolution of the photometric error but instead assumea conservative value of σz = 0.1.

Typically optical photo-z surveys are fainter and samplea much larger number of galaxies than spectroscopic ones,hence we assume a broader redshift range, 0.4 < z < 1.4,and only a low shot-noise case as given in Table 1. For theredshift range assumed this implies ∼ 150 × 106 galaxies.Table 3 show the bin configurations we have considered forthis case. While in the previous cases we have assumed thebias is constant with redshift (because of the narrow redshiftrange), for the broadband photometric survey we introducean evolution following (Fry 1996),

b(z) = 1 + (b! − 1)D(z!)D(z)

(2)

where b! = 2 is the bias at z! = 1. In turn for the evolutionof bias we have always assume the fiducial cosmology.

2.2 Spatial (3D) power spectrum

Since we are only interested in quasi-linear scales we as-sume the following simple model for the 3D galaxy powerspectrum in redshift space,

Pg(k, µ, z) = (b+ fµ2)2 D2(z)P0(k)e−k2σ2

t (z)µ2

, (3)

5 www.darkenergysurvey.org6 pan-starrs.ifa.hawaii.edu7 www.euclid-imaging.net

c© 2012 RAS, MNRAS 000, 1–10

7

0 5 10 15 200

2000

4000

6000

Low Shot Noise

0 5 10 15 20

High Shot Noise

0 2 4 60

1

2

3

4

0 2 4 6

Figure 2. Spectroscopic survey & bias fixed. Top panels show FoMΩm (2D) and FoMΩm (3D) as a function of the number of bins inwhich we divide the survey for the analysis (left panel for a low shot-noise survey and right to a high shot noise). Dashed line correspondsto the 3D analysis, dotted to the 2D tomography using only auto-correlations and solid to auto plus cross correlations. Different colorscorrespond to different minimum scales, as detailed in the bottom panel inset labels. Bottom panels show the ratio of FoMΩm(2D) (autoplus cross) and FoMΩm(3D) as a function of the bin width ∆r normalized by the minimum scale assumed in the 3D analysis. Remarkablythe recovered constraints from full tomography match the 3D ones for ∆r ∼ λ3D

min for all λ3Dmin. We note that different lines in the bottom

panels are truncated differently merely because we have done the three kmax cases down to the same minimum ∆r.

same result for all the cases studied in this paper, as long asPg does not change abruptly with redshift. Thus from nowon we will only refer to the 3D results in the whole survey.

This picture changes for the 2D tomography. Here thetransverse information is fixed once !max is set (2!+1 modesper ! value up to !max). As we increase the number of nar-rower bins Nz (with fixed total redshift range) we have sev-eral effects:

(i) Decreasing the number of galaxies per bin increasesthe shot noise per bin

(ii) Increasing the number of bins so that they arethinner proportionally increases the signal auto powerspectrum in each bin (there is less signal power suppressiondue to averaging along the radial direction).

(iii) When we split a wide redshift bin in two, we double

the number of angular auto power spectra (transversemodes). This results in a larger FoM because the signalto noise in each bin remains nearly constant (the shotnoise and signal in each bin both increase proportionately).This gain is illustrated by the dotted line in Fig. 2,which corresponds to the FoM produced by just usingauto-correlations. For even narrower redshift bins the binswill become correlated and the gain will saturate, but thisis not yet the case in our results as the redshift bins are stilllarge compared to the clustering correlation length. In thelimit in which all modes of interest are very small comparedto shell thickness and they are statistically equivalent, fora single power spectrum amplitude parameter one expects

c© 2012 RAS, MNRAS 000, 1–10

(From Asorey et al. 2012)


. . . . . .

Example: Alcock-Paczynski test

(Alcock & Paczynski ’79)Consider a comoving scale L in the sky.Horizontally it is projected to the angle θL = L

(1+z)DA(z).

Radially its ends are at a slightly different redshifts, ∆zL = LH(z).

∆zL

θL= (1 + z)DA(z)H(z) = F (z) ≡

∫ z

0

H(z)H(z′)

dz′

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ΘBAO @radians]

DzBAO

1 2 3 4z tr

0.05

0.10

0.15

0.20


. . . . . .

Example: Alcock-Paczynski test

F (z)AP ≡ ∆zL/θL measured from the theoretical power spectrum (with Euclid-likeredshift accuracies) F (z) ≡

∫ z0

H(z)H(z′)dz′.

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0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

- 0.04

- 0.02

0.00

0.02

0.04

z tr

FA

PHz

LF

HzL-

1

solid errors:angular resolution 0.02o

dashed errors:angular resolution 0.05o

violet: linear P(k)

(from Montanari & RD ’12)


. . . . . .

Conclusions

So far cosmological LSS data mainly determined ξ(r), or equivalently P(k). These1d functions are easier to measure (less noisy) but they require an inputcosmology converting redshift and angles to length scales,

r =√

χ(z)2 + χ(z′)2 − 2χ(z)χ(z′) cos θ .

But future large & precise 3d galaxy catalogs like Euclid will be able to determinedirectly the measured 3d correlation function ξ(θ, z, z′) and Cℓ(z, z′) from the data.

These 3d quantities will of course be more noisy, but they also contain moreinformation.

These spectra are not only sensitive to the matter distribution (density) but also tothe velocity via (redshift space distortions) and to the perturbations of spacetimegeometry (lensing) .

The spectra depend sensitively and in several different ways on dark energy(growth factor, distance redshift relation), on the matter and baryon densities, bias,etc. Their measurements provide a new route to estimate cosmologicalparameters.

An example is the Alcock-Paczynski test.————————


what galaxy surveys really measure. - uni-bielefeld.de · 2013. 4. 26. · introduction the cmb cmb...

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