IJAAInternational Journal of Astronomy and Astrophysics2161-4717Scientific Research Publishing10.4236/ijaa.2013.33029IJAA-36511ArticlesPhysics&Mathematics Window Effect in the Power Spectrum Analysis of a Galaxy Redshift Survey akahiroSato1*GertHütsi2*GenNakamura1*KazuhiroYamamoto1*Tartu Observatory, T?revere, Estonia Graduate School of Physical Sciences, Hiroshima University, Higashi-Hiroshima, Japan* E-mail:kazuhiro@hiroshima-u.ac.jp(AS);kazuhiro@hiroshima-u.ac.jp(GH);kazuhiro@hiroshima-u.ac.jp(GN);kazuhiro@hiroshima-u.ac.jp(KY);230820130303243256July 7, 2013August 9, 2013 August 16, 2013© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

We investigate the effect of the window function on the multipole power spectrum in two different ways. First, we consider the convolved power spectrum including the window effect, which is obtained by following the familiar (FKP) method developed by Feldman, Kaiser and Peacock. We show how the convolved multipole power spectrum is related to the original power spectrum, using the multipole moments of the window function. Second, we investigate the deconvolved power spectrum, which is obtained by using the Fourier deconvolution theorem. In the second approach, we measure the multipole power spectrum deconvolved from the window effect. We demonstrate how to deal with the window effect in these two approaches, applying them to the Sloan Digital Sky Survey (SDSS) luminous red galaxy (LRG) sample.

Cosmology; Large-Scale Structure of Universe1. Introduction

One of the most fundamental problems in cosmology is the origin of an accelerated expansion of the Universe [1,2]. A hypothetical energy component, dark energy, may explain the accelerated expansion [3]. Modification of the gravity theory is an alternative way to explain it. In either case, this problem seems to be deeply rooted in the nature of fundamental physics, which has attracted many researchers. Dark energy surveys which aim at measuring redshifts of huge number of galaxies are in progress or planned [4,5]. These surveys provide us with a chance to test the hypothetical dark energy, as well as the gravity theory on the scales of cosmology. A key for distinguishing between the dark energy and modified gravity theory is a measurement of the evolution of cosmological perturbations.

Galaxy redshift surveys provide promising ways of measuring the dark energy properties. Here, a measurement of the baryon acoustic oscillations in the galaxy distribution plays a key role. Also, the spatial distribution of galaxies is distorted due to the peculiar motions, which is called the redshift-space distortion. The Kaiser effect is the redshift-space distortion in the linear regime of the density perturbations. It is caused by the bulk motion of galaxies [6]. The measurement of the Kaiser effect is thought to be useful for testing the general relativity and other modified gravity theories [7-9]. In these analyses, measuring the multiple power spectrum in the distribution of galaxies plays a key role (cf. [10,11]).

The multipole power spectrum is useful for measuring the redshift-space distortion [12-18]. The usefulness of the quadrupole power spectrum to constrain modified gravity models is demonstrated in Refs. [19,20], as well as the dark energy model [21]. An estimator of the quadrupole power spectrum is developed in Ref. [16]. However, the disadvantage of the method is not being compatible with the use of the fast Fourier transform (FFT). In the present paper, we consider different estimators of the quadrupole power spectrum which allows the use of the FFT. In this method, a full sample of a wide survey area is divided into smaller subsamples with equal areas. This approach was taken in Refs. [14,15]. In this case, the effect of the window function is crucial as we will show in the present paper. Thus, it must be properly taken into account when comparing the observational data with theoretical predictions.

The convolved power spectrum includes the effect of the window function [22-25]. In the first half of the present paper, we consider the convolved power spectrum. We develop a theoretical formula to incorporate the window effect into the multipole power spectra for the first time. We apply this formula to the Sloan Digital Sky Survey (SDSS) luminous red galaxy (LRG) sample from the data release (DR) 7, and investigate the behavior of the window function and its effect on the monopole and quadrupole spectra. We demonstrate how the window effect modifies the monopole spectrum and the quadrupole spectrum. In the second half, we consider the deconvolved power spectrum, which is developed in Ref. [26], and compare it with the results of the first approach.

This paper is organized as follows: In Section 2, we briefly review the power spectrum analysis and the window effect, where the convolved power spectrum is introduced. In Section 3, using the multipole moments of the window function, we derive the main formula to describe how the convolved multipole power spectrum is related to the original power spectrum. Then, a method to measure the multipole moments of the window function is presented. We also apply the method to the SDSS LRG DR 7. In Section 4, the method for measuring the deconvolved power spectrum is reviewed. Then, a comparison of the two approaches is given. Section 5 is devoted to summary and conclusions. In the appendix, we give a brief review of a theoretical model, which we adopted. Throughout this paper, we use units in which the velocity of light equals 1, and adopt the Hubble parameter with.

2. Basic Formulas of the FKP Method

Let us first summarize the power spectrum analysis developed by Feldman, Kaiser and Peacock ([27], hereafter FKP). With this formulation we obtain the convolved power spectrum, including the window effect. We denote the number density field of galaxies by, where is the three-dimensional coordinate in the (fiducial) redshift space, is the unit directional vector, and is the comoving distance of a fiducial cosmological model. According to Ref. [27], we introduce the fluctuation field

where, with being the location of the th object; similarly, is the density of a synthetic catalog that has a mean number density times that of the galaxy catalog. In the present paper, we adopt. The synthetic catalog is a set of random points without any correlation, which can be constructed through a random process by mimicking the selection function of the galaxy catalog. For and, we assume

where denotes the mean number density of the galaxies, and is the two-point correlation function. These relations lead to

We introduce the Fourier coefficient of by

where is the weight function (Throughout this paper, we assume). The expectation value of

is

with

and

where we used

Here, is the window function and is the shotnoise. The estimator of the convolved power spectrum is taken

whose expectation value is

Hereafter, we omit, for simplicity.

3. Convolved Power Spectrum

In this section, using the multipole moments of the window function, we drive the main formulas for the convolved multipole power spectrum, Equations (33) and (34), which describe the relations between the convolved multipole power spectrum and the original multipole spectrum. We exemplify the behavior of the multipole moments of the window function and the convolved spectra, using the SDSS LRG sample from the DR 7.

3.1. Formulation

The estimator of the monopole power spectrum should be taken as

where is the volume of the shell in the -space. Similarly, a higher multipole power spectrum can be obtained [16]. Using the quantity

where is the Legendre polynomial, and is the unit wavenumber vector, the estimator for the higher multipole power spectrum should be taken as (cf. [16])

with

The expectation value of Equation (15) is

where we defined

By adopting the distant observer approximation, we have

and

where is the unit vector along the line of sight. We consider the shell in the Fourier space whose outer (inner) radius is. The volume of the shell is where and, then

Let us consider the limit, then we have

Note that our definition of the multipole spectrum is different from the conventional one by the factor [12,13].

Now we introduce the coordinate variables to describe and. For and, we adopt

respectively. As we consider the power spectrum and the window function averaged over the longitudinal variable around the axis of the direction, we may choose so that without loss of generality. Then, we choose the coordinate variable to describe as

where and are the angle coordinates around so as to be the polar axis. The matrix of the right hand side of Equation (24) denotes the rotation around the y-axis. See Figure 1 for the configuration. Note that

Assuming the following formula within the distant observer approximation,

and, Equation (20) yields

Using (25), (26), and (23), we can write Equation (29) as

where. Using the relation

we obtain

These formulas describe how the convolved spectra, and, are modified due to the window effect, compared with the original spectrum. Using Equations (33) and (34), we define the quantity,

which is the correction factor connecting the original spectrum and the convolved power spectrum.

3.2. Measurement of the Multipole Moments of the Window Function

In this subsection, we explain a method to measure the multipole moment of the window function. The window function can be evaluated using the random catalog in a similar way of evaluating the power spectrum. Similar to the case of the power spectrum, we need to subtract the shotnoise contribution. Then, we adopt the following estimator for the window function, corresponding to the right hand side of Equation (8),

We consider the window function expanded in the form of Equation (28). Mimicking the method to obtain the multipole power spectrum, we introduce

and use the following estimator for the multipole moment of the window function,

In the present work, we use the SDSS public data from the DR7 [28]. Our LRG sample is restricted to the redshift range -. In order to reduce the sidelobes of the survey window we remove some noncontiguous parts of the sample, which leads us to 7150 deg2 sky coverage with the total number LRGs. The data reduction is the same as that described in Refs. [19,20,29,30]. In this subsection, we show general features of the window function of the LRG sample. In our approach, division of the full sample into subsamples is necessary because the line of sight direction is approximated by one direction, and the distant observer approximation is required. Each subsample is distributed in a narrow area. We consider the three cases of the division, which are demonstrated in Figure 2. The full sample is divided into 18, 32, and 72 subsamples, respectively. In those divisions of the full sample, each subsample has almost the same survey area, 398, 223, and 99 square degrees, respectively. Figure 2 shows the cases divided into 18 subsamples, 32 subsamples and 72 subsamples. Figure 3 shows and as a function of, which are obtained by averaging the results over all subsamples. As demonstrated in Figure 3, and can be fitted in the form,

where the best fitting parameters, , , and, which depend on the division of the full sample, are given in Table 1.

3.3. Measurement of the Convolved Power Spectrum

Let us demonstrate the convolved multiple power spectrum using the SDSS LRG sample from DR7. Figure 4

Table 1. Values of the best fitting parameters for and in Equations (39) and (40), respectively.

ReferencesS. Perlmutter, et al., “Measurements of Ω and Λ from 42 High-Redshift Supernovae,” The Astrophysical Journal, Vol. 517, No. 2, 1999, p. 565. doi:10.1086/307221A. G. Riess, et al., “Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant,” The Astrophysical Journal, Vol. 116, No. 3, 1998, p. 1009. doi:10.1086/ 300499P. J. E. Peebles and B. Ratra, “The Cosmological Constant and Dark Energy,” Reviews of Modern Physics, Vol. 75, No. 2, 2003, pp. 559-606. doi:10.1103/RevModPhys.75.559A. Albrecht, et al., “Report of the Dark Energy Task Force,” 2006. arXiv:astro-ph/0609591J. A. Peacock, et al., “ESA-ESO Working Group on Fundamental Cosmology,” 2006. arXiv astro-ph/0610906N. Kaiser, “Clustering in Real Space and in Redshift Space,” Monthly Notices of the Royal Astronomical Society, Vol. 227, 1987, pp. 1-21.E. V. Linder, “Redshift Distortions as a Probe of Gravity,” Astroparticle Physics, Vol. 29, No. 5, 2008, pp. 336-339. doi:10.1016/j.astropartphys.2008.03.002L. Guzzo, et al., “A Test of the Nature of Cosmic Acceleration Using Galaxy Redshift Distortions,” Nature, Vol. 451, 2008, pp. 541-544. doi:10.1038/nature06555R. Reyes, et al., “Confirmation of General Relativity on Large Scales from Weak Lensing and Galaxy Velocities,” Nature, Vol. 464, 2010. pp. 256-258. doi:10.1038/nature08857T. Okumura, et al., “Large-Scale Anisotropic Correlation Function of SDSS Luminous Red Galaxies,” The Astrophysical Journal, Vol. 676, No. 2, 2008, p. 889. doi:10.1086/528951A. Cabre and E. Gaztanaga, “Clustering of Luminous Red Galaxies—I. Large-Scale Redshift-Space Distortions,” Monthly Notices of the Royal Astronomical Society, Vol. 393, No. 4, 2009, pp. 1183-1208. doi:10.1111/j.1365-2966.2008.14281.xS. Cole, K. Fisher and D. H. Weinberg, “Fourier Analysis of Redshift Space Distortions and the Determination of Omega,” Monthly Notices of the Royal Astronomical Society, Vol. 267, 1994, pp. 785-799.A. J. S. Hamilton, “The Evolving Universe,” Kluwer Academic Publishers, Dordrecht, 1998. doi:10.1007/978-94-011-4960-0_17P. J. Outram, et al., “The 2dF QSO Redshift Survey—VI. Measuring Λ and β from Redshift-Space Distortions in the Power Spectrum,” Monthly Notices of the Royal Astronomical Society, Vol. 328, No. 1, 2001, pp. 174-184. doi:10.1046/j.1365-8711.2001.04852.xP. J. Outram, et al., “The 2dF QSO Redshift Survey—XIII. A Measurement of Λ from the Quasi-Stellar Object Power Spectrum, PS(k//, k⊥),” Monthly Notices of the Royal Astronomical Society, Vol. 348, No. 3, 2004, pp. 745-752. doi:10.1111/j.1365-2966.2004.07348.xK. Yamamoto, M. Nakamichi, A. Kamino, B. A. Bassett and H. Nishioka, “A Measurement of the Quadrupole Power Spectrum in the Clustering of the 2dF QSO Survey,” Publications of the Astronomical Society of Japan, Vol. 58, 2006, pp. 93-102. 415, No. 3, 2011, pp. 2876-2891. http://pasj.asj.or.jp/v58/n1/580114/58012766.pdfC. Blake, et al., “The WiggleZ Dark Energy Survey: The Growth Rate of Cosmic Structure Since Redshift z = 0.9,” Monthly Notices of the Royal Astronomical Society, Vol. 415, No. 3, 2011, pp. 2876-2891.A. Taruya, S. Saito and T. Nishimichi, “Forecasting the Cosmological Constraints with Anisotropic Baryon Acoustic Oscillations from Multipole Expansion,” Physical Review D, Vol. 83, No. 10, 2011, Article ID: 103527. doi:10.1103/PhysRevD.83.103527K. Yamamoto, T. Sato and G. Hütsi, “Testing General Relativity with the Multipole Spectra of the SDSS Luminous Red Galaxies,” Progress of Theoretical Physics, Vol. 120, No. 3, 2008, pp. 609-614. doi:10.1143/PTP.120.609K. Yamamoto, G. Nakamura, G. Huetsi, T. Narikawa and T. Sato, “Constraint on the Cosmological f(R) Model from the Multipole Power Spectrum of the SDSS Luminous Red Galaxy Sample and Prospects for a Future Redshift Survey,” Physical Review D, Vol. 81, No. 10, 2010, Article ID: 103517. doi:10.1103/PhysRevD.81.103517K. Yamamoto, B. A. Bassett and H. Nishioka, “Dark Energy Reflections in the Redshift-Space Quadrupole,” Physical Review Letters, Vol. 94, No. 5, 2005, Article ID: 051301. doi:10.1103/PhysRevLett.94.051301W. Percival, et al., “Baryon Acoustic Oscillations in the Sloan Digital Sky Survey Data Release 7 Galaxy Sample,” Monthly Notices of the Royal Astronomical Society, Vol. 401, No. 4, 2010, pp. 2148-2168. doi:10.1111/j.1365-2966.2009.15812.xB. A. Reid, et al., “Thick Gas Discs in Faint Dwarf Galaxies,” Monthly Notices of the Royal Astronomical Society, Vol. 404, No. 1, 2010, pp. L60-L63. doi:10.1111/j.1745-3933.2010.00835.xW. Percival, et al., “The Shape of the Sloan Digital Sky Survey Data Release 5 Galaxy Power Spectrum,” The Astrophysical Journal, Vol. 657, No. 2, 2007, p. 645. doi:10.1086/510615S. Cole, et al., “The 2dF Galaxy Redshift Survey: PowerSpectrum Analysis of the Final Data Set and Cosmological Implications,” Monthly Notices of the Royal Astronomical Society, Vol. 362, No. 2, 2005, pp. 505-534. doi:10.1111/j.1365-2966.2005.09318.xT. Sato, G. Hütsi and K. Yamamoto, “Deconvolution of Window Effect in Galaxy Power Spectrum Analysis,” Progress of Theoretical Physics, Vol. 125, No. 1, 2011, pp. 187-197. doi:10.1143/PTP.125.187H. A. Feldman, N. Kaiser and J. A. Peacock, “PowerSpectrum Analysis of Three-Dimensional Redshift Surveys,” Astrophysical Journal, Vol. 426, No. 1, 1994, pp. 23-37. doi:10.1086/174036K. N. Abazajian, et al., “The Seventh Data Release of the Sloan Digital Sky Survey,” The Astrophysical Journal Supplement Series, Vol. 182, No. 2, 2009, p. 543. doi:10.1088/0067-0049/182/2/543G. Hütsi, “Acoustic Oscillations in the SDSS DR4 Luminous Red Galaxy Sample Power Spectrum,” Astronomy & Astrophysics, Vol. 449, No. 3, 2006, pp. 891-902. doi:10.1051/0004-6361:20053939G. Hütsi, “Power Spectrum of the SDSS Luminous Red Galaxies: Constraints on Cosmological Parameters,” Astronomy & Astrophysics, Vol. 459, No. 2, 2006, pp. 375-389. doi:10.1051/0004-6361:20065377J. M. Bardeen, J. R. Bond, N. Kaiser and A. S. Szalay, “The Statistics of Peaks of Gaussian Random Fields,” The Astrophysical Journal, Vol. 304, 1986, pp. 15-61.J. A. Peacock and S. J. Dodds, “Reconstructing the Linear Power Spectrum of Cosmological Mass Fluctuations” Monthly Notices of the Royal Astronomical Society, Vol. 267, 1994, pp. 1020-1034.