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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.

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 [

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 [

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 [

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. [

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.

Let us first summarize the power spectrum analysis developed by Feldman, Kaiser and Peacock ([

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.

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.

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 [

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

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

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.

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 [^{2} 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

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

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