Window Effect in the Power Spectrum Analysis of a Galaxy Redshift Survey ()
1. 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
(1)
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
(2)
(3)
(4)
where 
denotes the mean number density of the galaxies, and 
is the two-point correlation function. These relations lead to
(5)
We introduce the Fourier coefficient of 
by
(6)
where 
is the weight function (Throughout this paper, we assume
). The expectation value of
is
(7)
with
(8)
and
(9)
where we used
(10)
Here, 
is the window function and 
is the shotnoise. The estimator of the convolved power spectrum is taken
(11)
whose expectation value is
(12)
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
(13)
where 
is the volume of the shell in the 
-space. Similarly, a higher multipole power spectrum can be obtained [16]. Using the quantity
(14)
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])
(15)
with
(16)
The expectation value of Equation (15) is
(17)
where we defined
(18)
By adopting the distant observer approximation, we have
(19)
and
(20)
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
(21)
Let us consider the limit
, then we have
(22)
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
(23)
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
(24)
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
(25)
(26)
(27)
Assuming the following formula within the distant observer approximation,
Figure 1. A sketch of the configuration of the vectors and coordinate variables.
(28)
and
, Equation (20) yields
(29)
Using (25), (26), and (23), we can write Equation (29) as
(30)
(31)
where
. Using the relation
(32)
we obtain
(33)
(34)
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,
(35)
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),
(36)
We consider the window function expanded in the form of Equation (28). Mimicking the method to obtain the multipole power spectrum, we introduce
(37)
and use the following estimator for the multipole moment of the window function,
(38)
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,
(39)
(40)
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
Figure 2. Angular distribution of the SDSS LRG sample. In the present paper, we consider the three cases of the division of the full sample into subsamples. This figure shows the three cases of the division, where the full sample is divided into 18 subsamples (upper panel), 32 subsamples (middle panel) and 72 subsamples (lower panel), with mean area of 397 square degrees, 223 square degrees and 99 square degrees per patch, respectively.
Table 1. Values of the best fitting parameters for 
and 
in Equations (39) and (40), respectively.