Mon. Not. R. Astron. Soc. 000, 1–3 (2009) Printed 26 June 2009 (MN LATEX style file v2.2)

Measuring the Power spectrum P(k) from Sloan DigitalSky Survey galaxy clusters

P. Arriagada11Departamento Astronomıa y Astrofısica, Pontificia Universidad Catolica de Chile, Av. Vicuna Mackenna 4860, Stgo., Chile

June, 2009.

ABSTRACT

Using a sample from de Sloan Digital Sky Survey (SDSS), we obtained an estimationof the variation of counts in cells as described by Gaztanaga et al. (1995) which is thenused to measure the power spectrum of the sample (Padilla & Baugh 2002). We haveroughly recovered the shape of the variance as well as the shape of the power spectrumfor scales between 7 and 30 Mpc h−1, although there are still some caveats regardingthe amplitude. We yet see no turnover points, but this statement is not final, sinceour measurements are still far from conclusive.

Key words: Power spectrum– galaxy clusters.

1 INTRODUCTION

The large scale structure of the Universe can most efficientlybe traced by the most massive virialized, most strongly cor-related and brightest systems : rich clusters of galaxies, giv-ing important advantages regarding observations and simu-lations. Using these rare objects, it is easier to map a largeportion of the distant Universe, as the number of redshiftsrequired to do so is quite low compared to galaxies by them-selves. Since they are virialized systems associated with highpeaks, their correlations evolve slowly in time, allowing sim-ulations to be almost insensitive to the epoch they are anal-ysed. Moreover, being strongly correlated, there is no needfor a huge sample to find these correlations. Finally, the mostimportant characteristic is the relation between the distribu-tion of mass and that of light of such systems is expected tobe governed by rather simple biasing schemes (e.g., Kaiser1984; Bardeen et al. 1986; Mo & White 1996)

To explore clustering properties of galaxy clusters, themost powerful statistic is the power spectrum of density fluc-tuations, P (k). Its direct relation to theoretical quantitiesmakes it an excellent tool to put constraints between differ-ent models of structure formation and cosmological modelsin general.

Power spectra of galaxy clusters has been previouslymeasured using optically selected cluster surveys (Peacock& West 1992; Einasto et al. 1993; Jing & Valdarnini 1993;Einasto et al. 1997; Retzlaff et al. 1998; Tadros et al. 1998).These are found to have a turnover at some value of kbetween 0.03 and 0.05 h/Mpc. Other measurements us-ing Abell clusters with richness > 1 found no signs ofthis turnover (Miller & Batuski 2000). The power spec-trum obtained from the REFLEX X-ray survey (Schuecker

et al. 2001) shows a broad maximum within the comovingwavenumber range 0.022 6 k 6 0.030.

Why is this turnover so important? Physically, the scaleof this controversial turnover is closely linked to the horizonscale where matter-radiation equality occurs, introducing aspecific scale into an otherwise almost scale-invariant pri-mordial power spectrum. Therefore it helps to discriminatebetween the different scenarios of cosmic structure forma-tion that have in discussion.

In this paper, we will try to measure the galaxy clusterpower spectrum by determining the variance of counts incells. In Section 2 we present the SDSS cluster data. In Sec-tions 3 and 4 we present the method of variance of countsin cells and the derivation of the power spectrum from thisvariance. Finally Section 5 shows results, a small discussionand future work.

2 THE SDSS GALAXY CLUSTER SAMPLE

We have chosen a volume-limited sample of galaxy clustersfrom the Sloan Digital Sky Survey (SDSS DR6, Adelman-McCarthy, et al. 2008) cutting the sample to redshifts largerthan 20000 km/s and lower than 30000 km/s with massesM > 5 × 1012M⊙ as shown in figure 1. From this sample,the largest rectangular and homogeneously crowded portionof the sky was selected, leaving us with 345 clusters in a 584deg2 region centered at 185 deg of right ascension and 57deg of declination.

c© 2009 RAS

2 Pamela Arriagada1

Figure 1. Vertical lines show the upper and lower limits in red-shift (cz). Beyond these limits, completeness is lost due to a under-sampling of galaxies because they don’t fall within our visual an-gle, or because of Malmquidst bias, where fainter galaxies are lost.

The horizontal line marks the lower limit in mass(luminosity),where below it less massive(fainter) clusters are lost.

3 VARIANCE OF CLUSTER COUNTS IN

CELLS

We make use of the method developed by Efstathiou et al.(1990) and then modified by Gaztanaga et al. (1995) whichdivides space into Nsh concentric shells centered on the ob-sever, and estimates each shell’s mean density its fluctua-tions, thus, accounting for selection effects automatically.Each of these shells has a width of 2R, and are further di-vided into a maximum of Ms spherical cells of radius R. Foreach of these cells, we count the number of clusters Nc andfor each shell, we calculate the mean number of clusters ina cell Ns(R) and the moments mJ(R)

mJ =

Ms∑c=1

(Nc − Ns) (1)

where c is the number of cth cell. Then, the correspondingconnected moments kJ = NJ

s ξJ corrected for Poisson shonoise are given by

k2 = m2 − Ns, (2)

k3 = m3 − 3m2 + 2Ns, (3)

k4 = m4 − 3m2

2 − 6m3 + 11m2 − 6Ns. (4)

Then, we have that ξ2 = k2/N2s is the variance, ξ3 = k3/N3

s

is the skewness and ξ4 = k4/N4s is the kurtosis for each shell.

For the whole sample, we need to estimate the mean valueof each, in particular, of the variance 〈ξ2〉

〈ξ2〉 =1

K

Nsh∑s=1

N4s

Var(k2)ξ2, (5)

K =

Nsh∑s=1

N4s

Var(k2)(6)

where

Var(k2) =k4 − 2k3 + k2 − k2

2

Ms

(7)

By this method, and using K as the weights for eachaverage one can also calculate an estimate of the variance of〈ξ2〉

Var〈ξ2〉 =〈ξ2

2〉 − 〈ξ2〉2

Nsh

(8)

Finally, we obtain different values of 〈ξ2〉 and Var〈ξ2〉for different cell radii R which will then be used to estimatethe power spectrum P (k), explained in detail in the nextsection.

4 THE POWER SPECTRUM

We measure the power spectrum using the transformationdescribed by Padilla & Baugh (2002) which estimates P (k)from the variance of counts in cells, measured previously asdetailed in Section 2. This method is based on the followingapproximation: the variance of counts in a spherical cell ofradius R is given by:

〈ξ2(R)〉 =1

2π2

∫∞

0

dkk2P (k)W 2(k) (9)

where

W (k) =3

(kR)3[sin(kR) − kR cos(kR)] (10)

is the Fourier transform of a spherical top hat smooth-ing window. The integral in equation 13 is sharply peakedaround a characteristic wavenumber keff for a particularsmoothing scale R, thus the variance 〈ξ2(R)〉 can be relatedto the power spectrum at keff

〈ξ2(R)〉 = ∆2(keff) (11)

where ∆2(k) = k3P (k)/2π2. Now, considering a power lawspectrum P (k) ∝ kn, with logarithmic slope n then, keff

keff =1

R[9I(n)]

1n+3 (12)

with

I(n) =

∫∞

0

dxxn−4[sin(x) − x cos(x)]2 (13)

The accuracy of this method is demonstrated in Padilla& Baugh (2002) applying this transformation to the vari-ance computed from integrating an input power spectrum(see their figure 7). They conclude that the power spectrumshape is best obtained when n is inferred from the local slopeof the variance.

5 DISCUSSION AND FUTURE WORK

Using the method described in Section 2 we have calculatedthe variance of counts in cells of our SDSS sample, whichis shown in figure 2 (filled triangles). As a comparison, thevariance calculated by Gaztanaga et al. (1995) (crosses). Bysimple visual inspection of our variance it seems that it fol-lows the same shape as the one from the APM sample, butwith higher amplitude at all scales, which makes us suspectthat we might be counting less clusters per cell that there

c© 2009 RAS, MNRAS 000, 1–3

Measuring the Power Spectrum P(k) from SDSS galaxy clusters 3

Figure 2. The variance ξ2 estimated from counts in cells in theSDSS clusters (triangles) compared with the variance in APMclusters (crosses).

Figure 3. Estimate of the power spectrum of SDSS clusters ob-

tained by applying the method described in Padilla & Baugh(2002). It assumes a slope of n = −1 .

really are leaving us with a larger contribution from shotnoise.

In spite of this problem, and to check that we are onthe right track, we have measured the power spectrum P (k)from out variance estimation following the Padilla et al.(2002) prescription using n = −1, which is shown in fig-ure 3. If we compare this power spectrum with that ob-tained by Padilla et al. (2002) using the APM variance ofcounts in cells, we see that they also follow the same shape,and again, the amplitudes are being overestimated. Part ofthis discrepancy could be due to the fact that the recoveredspectrum from variance estimations can be a factor of twolarger than the true spectrum, as mentioned by Padilla etal. (2002). Moreover, they also show discrepancies in ampli-tudes between two measurements of the P (k), so it is safeto say that is very sensitive to the method of estimation,the shape of the survey volume and the uncertainties of themean cluster density.

From our measurements and assuming a fixed slope of-1, we see no clear turnover point in our spectrum, so in thefuture we would like to be able to reproduce the amplitudeof variance of counts in cells calculated with APM surveysample to obtain the slope. The next step after tweakingour calculations would be to enlarge out chosen portion ofthe sky. Our ultimate goal is to have an accurate and reliablemeasurement of the power spectrum using SDSS clusters.

REFERENCES

Padilla, N.D., Baugh, C.M., 2002, MNRAS, 329, 431Gaztanaga, E., Croft, R.A.C.,Dalton,G.B., 1995, MNRAS,276, 336

Schuecker, P., Bohringer, H., Guzzo, L., Collins, C. A.,Neumann, D. M., Schindler, S., Voges, W., De Grandi,S., Chincarini, G., Cruddace, R., Muller, V., Reiprich, T.H., Retzlaff, J., Shaver, P., 2001, A&A, 368, 86S

Tadros, H., Efstathiou, Ge., Dalton, G., 1998, MNRAS,296, 995T

c© 2009 RAS, MNRAS 000, 1–3