Power spectrum multipole analysis of BOSS

The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: Testing gravity with redshift-space distortions using the power spectrum multipoles

Florian Beutler, Shun Saito, Hee-Jong Seo, Jon Brinkmann, Kyle S. Dawson, Daniel J. Eisenstein, Andreu Font-Ribera, Shirley Ho, Cameron K. McBride, Francesco Montesano, Will J. Percival, Ashley J. Ross, Nicholas P. Ross, Lado Samushia, David J. Schlegel, Ariel G. Sánchez, Jeremy L. Tinker, Benjamin A. Weaver
Lawrence Berkeley National Lab, 1 Cyclotron Rd, Berkeley CA 94720, USA, Kavli Institute for the Physics and Mathematics of the Universe (WPI),Todai Institues for Advanced Study, The University of Tokyo, Chiba 277-8582, Japan, Department of Astronomy, University of California Berkeley, CA 94720, USA, Department of Physics, University of California Berkeley, CA 94720, USA, Berkeley Center for Cosmological Physics, LBL and Department of Physics, University of California, Berkeley, CA 94720, USA, Center for Cosmology and Astroparticle Physics, Department of Physics, The Ohio State University, OH 43210, USA, Apache Point Observatory, P.O. Box 59, Sunspot, NM 88349-0059,USA, Department of Physics and Astronomy, University of Utah, Salt Lake City, UT 84112, USA, Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138, USA, Department of Physics, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213, USA, McWilliams Center for Cosmology, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213, USA, Max-Planck-Institut für Extraterrestrische Physik, Giessenbachstrasse, 85748 Garching, Germany, Institute of Cosmology & Gravitation, Dennis Sciama Building, University of Portsmouth, Portsmouth, PO1 3FX, UK, Department of Physics, Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104, USA, Center for Cosmology and Particle Physics, New York University, New York, NY 10003 USA.
E-mail: fbeutler@lbl.gov

We analyse the anisotropic clustering of the Baryon Oscillation Spectroscopic Survey (BOSS) CMASS Data Release 11 (DR11) sample, which consists of galaxies in the redshift range and has a sky coverage of . We perform our analysis in Fourier space using a power spectrum estimator suggested by Yamamoto et al. (2006). We measure the multipole power spectra in a self-consistent manner for the first time in the sense that we provide a proper way to treat the survey window function and the integral constraint, without the commonly used assumption of an isotropic power spectrum and without the need to split the survey into sub-regions. The main cosmological signals exploited in our analysis are the Baryon Acoustic Oscillations and the signal of redshift space distortions, both of which are distorted by the Alcock-Paczynski effect. Together, these signals allow us to constrain the distance ratio , the Alcock-Paczynski parameter and the growth rate of structure at the effective redshift . We emphasise that our constraints are robust against possible systematic uncertainties. In order to ensure this, we perform a detailed systematics study against CMASS mock galaxy catalogues and N-body simulations. We find that such systematics will lead to uncertainty for if we limit our fitting range to - Mpc, where the statistical uncertainty is expected to be three times larger. We did not find significant systematic uncertainties for or . Combining our dataset with Planck to test General Relativity (GR) through the simple -parameterisation, where the growth rate is given by , reveals a tension between the data and the prediction by GR. The tension between our result and GR can be traced back to a tension in the clustering amplitude between CMASS and Planck.

surveys, cosmology: observations, dark energy, gravitation, cosmological parameters, large scale structure of Universe
pagerange: The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: Testing gravity with redshift-space distortions using the power spectrum multipolesCpubyear: 2013

1 introduction

Figure 1: Summary of different tests of General Relativity (GR) as a function of distance scale (bottom axis) and densities (top axis). The standard model of cosmology seems to run into problems (dark matter, dark energy) at large scales. Because these problems could indicate a breakdown of GR we need to test GR on large scales. Two probes which can do this are redshift space distortions (RSD) and lensing. While RSD measures the Newtonian potential , lensing measures the sum of the metric potentials . However, any modification of gravity needs to pass the very precise tests on smaller scales (Pound & Rebka experiment Pound & Rebka 1960, Gravity Probe A, Vessot et al. 1980, Hulse-Taylor binary pulsar Hulse & Taylor 1975, see Will 2006 for a complete list). Note that the error bars for Gravity Probe A and the Hulse-Taylor binary pulsar are smaller than the data points in this plot. In this analysis we perform a CDM consistency test (blue data point), where we use the CMASS-DR11 power spectrum multipoles together with Planck (Ade et al., 2013a) to tests GR on scales of Mpc (see section 9.1).

The key to understand the dynamical properties of the Universe, its past and its future, is the understanding of gravity. Today’s dominant theory of the origin of the Universe, the Big Bang model, is based on Albert Einstein’s General Relativity (GR). The crucial idea behind GR, the connection between space and time into space-time first allowed us to talk about curved space and expanding space, terms which do not exist in Newton’s gravity.

GR is a very powerful theory, which makes many testable predictions, like the deflection of light or gravitational waves. Despite the successes of our current understanding of gravity, there are several problems, which motivated scientists to search for alternative formulations or expansions of GR. One problem, which we will not pursue any further in this paper, is that GR cannot be combined with the other fundamental forces, since GR is not formulated as a quantum field theory. Another problem is that the motions of galaxies and galaxy clusters cannot be explained by GR and baryonic matter alone, but require the introduction of a new form of matter, so-called dark matter (Zwicky, 1937; Kahn & Woltjer, 1959; Freeman et al., 1970; Rubin & Ford, 1970), which nobody has yet observed directly. While the issue of dark matter has existed since the 1930s, in the late 1990s Type Ia supernova surveys made the intriguing discovery that the expansion of the Universe is accelerating (Riess et al., 1998; Perlmutter et al., 1998). This required the introduction of yet another ÓdarkÓ component, so-called dark energy, which would counteract the gravitational force leading to an accelerated expansion. The question now is whether these problems indicate a breakdown of GR or whether there are additional unknown components of the Universe. While the problems of GR on cosmological scales (dark matter and dark energy) gave birth to many new models of gravity (see e.g. Clifton et al. 2011; Capozziello & Laurentis 2013; Jain & Khoury 2010), so far none of these models has been able to convince scientists that it is time to abandon GR.

Given that it is on cosmological scales where GR runs into trouble, it is on cosmological scales where we have to test GR. Figure 1 shows different tests of GR at different scales (see e.g. Will 2006). One interesting observable, which allows us to test GR on cosmic scales, is redshift space distortions (RSD) (Sargent & Turner, 1977; Kaiser, 1987; Hamilton, 1998). RSD are peculiar velocities of galaxies due to gravitational interaction. The line-of-sight component of this additional velocity cannot be easily separated from the Hubble flow and contaminates our measurement of the cosmic expansion. This makes the observed galaxy clustering anisotropically distorted, since the line-of-sight direction becomes “special”. This is what we call RSD. The anisotropic pattern of RSDs in galaxy clustering allows us to extract information on the peculiar velocities which are directly related to the Newton potential through the Euler equation. Given the amount of matter in the Universe, GR makes a clear and testable prediction for the amplitude of this anisotropic signal. In the last decade, galaxy redshift surveys became large enough to test this prediction (Peacock et al., 2001; Hawkins et al., 2003; Tegmark et al., 2006; Guzzo et al., 2008; Yamamoto, Sato & Huetsi, 2008; Blake et al., 2011a; Beutler et al., 2012; Reid et al., 2012; Samushia et al., 2013; Chuang et al., 2013a; Nishimichi & Oka, 2013).

In addition to the RSD signal, the galaxy power spectrum and correlation function carry geometric information. The measurement of the Baryon Acoustic Oscillation scale in the distribution of galaxies has become one of the most powerful probes of cosmology, together with the Cosmic Microwave Background (Ade et al., 2013a). The BAO scale has now been detected at several different redshifts (Eisenstein et al., 2005; Beutler et al., 2011; Blake et al., 2011b; Padmanabhan et al., 2012; Anderson et al., 2012; Slosar et al., 2013; Anderson et al., 2013b). Most notably the ongoing BOSS survey (Schlegel et al., 2009) reduced the measurement uncertainty on the BAO scale to  (Anderson et al., 2013b), which is still considerably larger than the expected systematic bias (Eisenstein & White, 2004; Padmanabhan & White, 2009; Mehta et al., 2011). Measuring the galaxy clustering along the line-of-sight and perpendicular to the line-of-sight allows us to perform an Alcock-Paczynski (AP) test (Alcock & Paczynski, 1979; Matsubara & Suto, 1996; Ballinger, Peacock & Heavens, 1996) with both the RSD and BAO signals. The Alcock-Paczynski test describes a distortion in an otherwise isotropic feature in the galaxy clustering when the assumed fiducial cosmological model used to transfer the measured redshifts into distances deviates from the true cosmology. This anisotropic signal may appear degenerate with the RSD signal in a featureless power spectrum. Using the BAO signal we can break this degeneracy and exploit all three signals, RSD, BAO and the Alcock-Paczynski effect for cosmological parameter constraints.

In this analysis we are going to use the CMASS sample of BOSS galaxies that will be included in the Sloan Digital Sky Survey (SDSS) Data Release 11 (DR11), which will become publicly available together with the final data (DR12) at the end of 2014. We use this dataset to constrain the growth of structure and the geometry of the Universe simultaneously. We measure the growth rate via the parameter combination and the geometry of the Universe via and at an effective redshift of . The BAO signal and the AP effect constrain the geometry, i.e., and , thereby isolating the anisotropy in the clustering amplitude due to the RSD. The growth rate, is constrained by this RSD signal. We will make our analysis in Fourier space using the power spectrum monopole and quadrupole. The power spectrum multipoles are measured using a new power spectrum estimator suggested by Yamamoto et al. (2006). The popular power spectrum estimator suggested by Feldman, Kaiser & Peacock (1993) (from here on FKP estimator) cannot be used to make angle-dependent measurements in BOSS because of the plane parallel approximation that this estimator implicitly makes (see section 3 for details).

Since the power spectrum quadrupole is more sensitive to window function effects than the more commonly used monopole, we suggest a new way of including the window function into the power spectrum analysis. In order to robustly constrain the RSD and AP-test parameters, we model the anisotropic galaxy power spectrum using perturbation theory (PT) which fairly reflects a series of recent theoretical progresses. Our PT model accurately describes non-linear issues such as gravitational evolution, mapping from real to redshift space, and local and non-local galaxy bias. We also perform a detailed study of possible systematic uncertainties and quantify a systematic error for our parameter constraints. Our analysis has been done blind, meaning that all model tests and the set-up of the fitting conditions are investigated using mock data and only at the final stage do we fit the actual CMASS-DR11 measurements. The CMASS-DR11 constraints on , and are the most precise constraints to date using this technique.

This paper is organised as follows. In section 2 we describe the BOSS CMASS-DR11 dataset. In section 3 we describe the power spectrum estimator used in our analysis and in section 4 we describe the mock catalogues together with the derivation of the covariance matrix. We then discuss the measurement of window function effects including the integral constraint in section 5. In section 6 we discuss our model for the power spectrum multipoles, together with the modelling of the Alcock-Paczynski effect. We perform a detailed study of possible systematic uncertainties in section 7, followed by the data analyses in section 8. We use our data constraints for cosmological tests in section 9 and conclude in section 10. The appendix gives detailed derivations of equations used in our analysis.

The fiducial cosmology used to turn redshifts into distances assumes a flat CDM universe with . The Hubble constant is set to km/s/Mpc, with our fiducial model using .

2 The BOSS CMASS-DR11 dataset

Figure 2: The CMASS-DR11 North Galactic Cap (top) and South Galactic Cap (bottom) sky coverage. The grey region indicates the final footprint of the survey (DR12). The colours indicate the completeness in the regions included in our analysis.

BOSS, as part of SDSS-III (Eisenstein et al., 2011; Davis et al., 2013) is measuring spectroscopic redshifts of million galaxies (and quasars) making use of the SDSS multi-fibre spectrographs (Bolton et al., 2012; Smee et al., 2013). The galaxies are selected from multi-colour SDSS imaging (Fukugita et al., 1996; Gunn et al., 1998; Smith et al., 2002; Gunn et al., 2006; Doi et al., 2010) and cover a redshift range of - , where the survey is split into two samples called LOWZ ( - ) and CMASS ( - ). In this analysis we are only using the CMASS sample. The survey is optimised for the measurement of the BAO scale and hence covers a large cosmic volume (Mpc) with a density of Mpc, high enough to ensure that shot noise is not the dominant error contribution at the BAO scale (White et al., 2011). Most CMASS galaxies are red with a prominent Å  break in their spectral energy distribution. Halo Occupation studies have shown that galaxies selected like the CMASS galaxies are mainly central galaxies residing in dark matter halos of , with a - satellite fraction (White et al., 2011). CMASS galaxies are highly biased (), which boosts the clustering signal including BAO in respect to the shot noise level.

The CMASS-DR11 sample covers in the North Galactic Cap (NGC) and in the South Galactic Cap (SGC); the total area of represents a significant increase from CMASS-DR9, which covered in total. The sample used in our analysis includes galaxies in the NGC and galaxies in the SGC. Figure 2 shows the footprint of the survey in the two regions, where the grey area indicates the expected footprint of DR12.

We include three different incompleteness weights to account for shortcomings of the CMASS dataset (see Ross et al. 2012a and Anderson et al. 2013b for details): A redshift failure weight, , a fibre collision weight, and a systematics weight, , which is a combination of a stellar density weight and a seeing condition weight. Each galaxy is thus counted as


We will discuss these weights in more detail in section 3.3.

3 The power spectrum estimator

In this section we describe the power spectrum estimator we use to measure the multipole power spectrum from the CMASS-DR11 sample. We carefully address how to incorporate the incompleteness weights. Before explaining the estimator itself, we summarise different approximations commonly used in galaxy clustering analysis.

3.1 Commonly used approximations

Here we discuss different approximations used in galaxy clustering statistics, and if used in our analysis we discuss their impact on our measurement:

  1. Distant observer approximation: Here one assumes that a displacement (e.g. caused by redshift space distortions) is much smaller than the distance, , to the galaxy itself. This approximation is commonly used for the volume element in the Jacobian mapping from real to redshift space. We assume the distant observer approximation when modelling the galaxy power spectrum in section 6.1.

  2. Local plane parallel approximation: Here one assumes that the position vectors of a galaxy pair can be treated as parallel, meaning


    where and . This approximation is only valid for a galaxy pair with a small angular separation and hence will break down on large scales (Papai & Szapudi, 2008). It has been shown, however, that the local plane parallel approximation is a very good approximation for most galaxy samples even when they cover a large fraction of the sky (Samushia, Percival & Raccanelli, 2011; Beutler et al., 2011; Yoo & Seljak, 2013). Most of the anisotropic galaxy clustering measurements adopt this assumption including our analysis, where it is introduced in eq. 9.

  3. (Global) plane parallel approximation (or flat-sky approximation): Here one assumes that the line-of-sight vector is the same for all galaxies in the survey, meaning


    where is the global line-of-sight vector. This approximation is included in the FKP estimator suggested by Feldman, Kaiser & Peacock (1993). Since the line-of-sight vector only appears in the calculation of the cosine angle to the line-of-sight, , the monopole power spectrum is not affected by this approximation. The higher order multipoles are strongly affected, except for very narrow angle surveys (Blake et al., 2011a). The invalidity of the plane parallel approximation for the geometry of the CMASS sample (Yoo & Seljak, 2013) motivated the use of the power spectrum estimator suggested by Yamamoto et al. (2006) in our analysis.

3.2 The Yamamoto et al. (2006) power spectrum estimator

The multipole power spectrum of a galaxy distribution can be calculated as (Feldman, Kaiser & Peacock, 1993; Yamamoto et al., 2006)


where is the Legendre polynomial, and


where is the galaxy density, is the density of the random catalogue and is the ratio of real galaxies to random galaxies. The shot noise term is given by


In our notation, quantities marked with a () include all completeness weights, like where . In CMASS-DR11, the completeness weights increase the average galaxy density by about 111In our analysis we have for the NGC and for the SGC, while the actually observed values are and , respectively.. Whenever we have to write the weighting explicitly, we use the completeness weight . The random galaxies follow the redshift distribution of the weighted galaxy catalogue, , which means that the randoms do not need a completeness weight. In addition to the completeness weight we employ a minimum variance weight, , which applies to the data and random galaxies (see eq. 21).

Most power spectrum studies in the past employed a Fast Fourier Transform (FFT) to solve the double integral in eq. 4. Such an approach however, requires the (global) plane parallel approximation (see section 3 for the definition), which for wide-angle surveys like BOSS, introduces significant bias into the higher order multipoles of the power spectrum (see e.g. Yoo & Seljak 2013). The monopole of the power spectrum is unaffected by this assumption, because it does not require an explicit knowledge of the angle to the line-of-sight. Yamamoto et al. (2006) suggested a power spectrum estimator which does not use the plane parallel approximation, for the price of significantly higher computation time. This is the estimator we employ in this analysis.

Using the relation , the integrals in eq. 4 can be written as


where the local plane parallel approximate has been used. If we define


the power spectrum estimate is given by (Yamamoto et al., 2006; Blake et al., 2011a)


where the represents the complex conjugate. The normalisation is given by


and the shot noise for each multipole is defined as


Note that because for and any constant , the shot noise term will vanish for the quadrupole () and hexadecapole () if the window function is isotropic. In order to minimise the additional shot noise contribution from the random catalogue to the power spectrum and its error, we generate a very large (i.e., dense) random catalogue with .

Figure 3: The measured CMASS-DR11 monopole (top) and quadrupole (bottom) power spectra. The black data points are the measurement of the North Galactic Cap (NGC) and the red data points are the measurement of the South Galactic Cap (SGC) of CMASS-DR11. The black data points have been shifted by Mpc to the right for clarity. The error bars are the diagonal of the covariance matrix. Because of the smaller volume in the SGC the error bars are larger by a factor of . The solid black and red lines represent the best fitting power spectra for the NGC (black) and SGC (red) respectively (fitting range - Mpc, see section 8.1). The red and black lines are based on the same cosmology and only differ in the effect of the window function (see section 5). The lower two panels show the difference between the measured monopole and the best fitting monopole (middle panel) and the measured quadrupole and the best fitting quadrupole (bottom panel), both relative to the diagonal elements of the covariance matrix. We fit the monopole and quadrupole simultaneously. The best fitting is (NGC + SGC) for bins and free parameters (see Table 2). The contribution to from the monopole and quadrupole alone is given in the middle and lower panel, for comparison.

The final power spectrum is then calculated as the average over spherical k-space shells


where is the volume of the k-space shell and is the number of modes in that shell. In our analysis we use Mpc.

The method described above has a bias at larger scales arising from the discreteness of the gridding in k-space (Blake et al., 2011a). The effect can be estimated by comparing a model power spectrum with a gridded model power spectrum, where the gridded model power spectrum is defined as


This should be averaged following eq. 17 and compared to a model power spectrum of the form


The final estimate of the power spectrum is then given by


where on the right hand side is the measured power spectrum and is the measured power spectrum after being corrected for the discrete gridding in k space. In our case this correction is () at Mpc and () at Mpc for the monopole and quadrupole, respectively. We show the measurement of the power spectrum monopole and quadrupole for CMASS-DR11 NGC (black) and SGC (red) in Figure 3.

3.3 The Poisson shot noise

Here we are going to discuss the impact of the CMASS incompleteness weighting on the shot noise term. In principle, any arbitrary constant weight applied to observed galaxies should not change the shot noise term, since no information is added. For example if one decides to up-weight each galaxy by a constant factor, e.g. the average incompleteness of the survey, the shot noise term should not change. In CMASS we have several different kinds of weights, and here we argue that some of these weights use extra information, in a sense that they should reduce the shot noise:

  1. Fibre collision, and redshift failure, weight: Galaxies which did not get a redshift due to fibre collision or redshift failure are still included in the galaxy catalogue by double counting the nearest galaxy (see Ross et al. 2012a for details). For each missing galaxy we know its angular position exactly. Even though the procedure to use the redshift of the closest galaxy is incorrect for some fraction of the missing galaxies (Guo, Zehavi & Zheng, 2012) it means we effectively put extra galaxies into the survey in a non-random fashion, which should reduce the shot noise term. We hence include the fibre collision as well as the redshift failure weights in the shot noise term.

  2. Systematic weights, : The CMASS sample shows correlations between the galaxy density and the proximity to a star as well as between the galaxy density and the seeing conditions for a particular observation. These correlations are removed using galaxy specific weights (systematic weights). Here we know only statistically that there were missed galaxies, but never know exactly where. To correct for these correlations we up-weight observed galaxies depending on their proximity to stars and the seeing condition for that particular observation. The correction is not random, but it is linked to a Poisson process (e.g. the existence of another galaxy around that star). Therefore we argue that the systematic weights should not reduce the shot noise. We also note that the systematic weights are much smaller than the fibre collision and redshift failure weight and hence the impact to the shot noise term is small.

The shot noise term defines how the galaxy density field enters in the minimum variance weight, , and hence the arguments discussed above result in a minimum variance weight of the form:


A detailed derivation can be found in appendix A. Since the systematic weights employed in our analysis are very small, our definition of is almost identical to the commonly used


If we were to assume that the systematic weights, , reduce the shot noise, eq. 21 and eq. 22 would be identical. The value of defines the power spectrum amplitude at which the error is minimised. In this analysis we use Mpc, which corresponds to Mpc and evaluate the density in redshift bins.

Several studies in recent years reported deviations from the pure Poisson shot noise assumption (Casas-Miranda et al., 2002; Seljak, Hamaus & Desjacques, 2009; Manera & Gaztanaga, 2010; Hamaus et al., 2010; Baldauf et al., 2013). Even though we discussed our definition of the shot noise term at length in this section, the parameter constraints we derive in this paper are fairly independent of the precise definition, since for all parameter constraints we are marginalising over a constant offset, (see section 6.1).

4 CMASS-DR11 mock catalogues

Figure 4: The power spectrum monopole (top) and quadrupole (bottom) of the QPM mock catalogues (grey lines) for the North Galactic Cap (left) and the South Galactic Cap (right), relative to an Eisenstein & Hu (1998) no-BAO monopole power spectrum. We plot the power spectrum without the shot noise subtraction, since this way, the scatter closely represents the diagonal of the covariance matrix. The red lines show the mean of all mock catalogues with the error representing the variance around the mean. The blue lines show the measured CMASS-DR11 power spectra.

In our analysis we use mock catalogues which follow the same selection function as the CMASS-DR11 sample. The catalogues are produced using quick particle-mesh (QPM) N-body simulations (White, Tinker & McBride, 2013) with particles in a box. These simulations have been found to better describe the clustering of CMASS galaxies compared to the previous version of CMASS mock catalogues (Manera et al., 2012), especially at small scales (McBride et al. in prep., ). Each simulation started from 2LPT initial conditions at and evolved to the present using time steps of in , where is the scale factor. The fiducial cosmology assumes flat CDM with , , and . We use the simulation output at , where the simulation generated a sub-sample of the N-body particles and a halo catalogue using the friends-of-friends algorithm with a linking length of times the mean inter-particle spacing. The halo catalogue is then extended to lower masses by appointing a set of the sub-sampled particles as halos and assigning them a mass using the peak-background split mass function. The halos are then populated by galaxies using the Halo Occupation Distribution (HOD) formalism with the occupation functions (see e.g. Tinker et al. 2013)


where we use , , , and (Jeremy Tinker, private communication). In section 7 we will modify the HOD parameters to test possible systematic effects in our modelling of the power spectrum multipoles. For more details about the QPM mock catalogues see McBride et al. in prep. () and White, Tinker & McBride (2013).

4.1 The covariance matrix

Figure 5: The correlation matrix for the NGC (left) and SGC (right) of CMASS-DR11. The colour indicates the level of correlation, where red represents high correlation, blue represents high anti-correlation and green represents no-correlation. The correlation between the bins in the monopole is shown in the lower left hand corner, while the correlation between the -bins in the quadrupole is shown in the upper right hand corner. The upper left hand corner and the lower right hand corner show the cross-correlations.

We measure the power spectrum monopole and quadrupole for each of the QPM mocks, using the estimator introduced in section 3. The power spectrum monopoles and quadrupoles are shown in Figure 4 together with the mean (red) and the CMASS-DR11 measurements (blue). We can see that the mock catalogues closely reproduce the data power spectrum multipoles for the entire range of wavenumbers relevant for this analysis.

The covariance matrix is then given by


where represents the number of mock realisations. We estimate the covariance matrices for the NGC and SGC separately, i.e. treat them as statistically independent samples. This covariance matrix contains the monopole as well as the quadrupole, and the elements of the matrices are given by , where is the number of bins in each multipole power spectrum. Our -binning yields for the fitting range - - Mpc, and hence the dimensions of the covariance matrices become () for the NGC and SGC. The mean of the power spectrum is defined as


The mock catalogues automatically incorporate the window function and integral constraint effect present in the data. Figure 5 shows the correlation matrix for CMASS-DR11 NGC (left) and SGC (right), where the correlation coefficient is defined as


The lower left hand corner shows the correlation between bins in the monopole, the upper right hand corner shows correlations between the bins in the quadrupole and the upper left hand corner and lower right hand corner show the correlation between the monopole and quadrupole. Most of the correlation matrix is coloured green, indicating no or a small level of correlation. This is expected for the linear power spectrum since each Fourier mode evolves independently. For larger wave-numbers non-linear effects will introduce correlations between bins, while for very small wave-numbers window function effects can introduce correlations.

As the estimated covariance matrix is inferred from mock catalogues, its inverse, , provides a biased estimate of the true inverse covariance matrix, due to the skewed nature of the inverse Wishart distribution (Hartlap et al., 2007). To correct for this bias we rescale the inverse covariance matrix as


where is the number of power spectrum bins. With these covariance matrices we can then perform a standard minimisation to find the best fitting parameters.

Figure 6: Relative error using the diagonal elements of the covariance matrix of the power spectrum multipoles in CMASS-DR11. The upper three dashed lines show the quadrupole error and the lower three solid lines show the error in the monopole. Because of the larger volume, the error in the NGC of CMASS-DR11 (black lines) is about a factor of smaller than the error in the SGC (red lines). The power spectrum error for the entire CMASS-DR11 sample (blue lines) shows an error of in the monopole and in the quadrupole at Mpc.

In Figure 6 we show the diagonal elements of the covariance matrix for the monopole and quadrupole power spectrum. We find an error of in the monopole and in the quadrupole at Mpc. This represents the most precise measurement of the galaxy power spectrum multipoles ever obtained.

5 The survey window function

The power spectrum estimator we discussed in section 3 is not actually estimating the true galaxy power spectrum, but rather the galaxy power spectrum convolved with the survey window function:


The window function, has the following two effects: (1) It mixes the modes with different wave-numbers and introduces correlations and (2) it changes the amplitude of the power spectrum at small . First we discuss the first term of eq. 29, the convolution of the true power spectrum with the window function. The second term of eq. 29, the so-called integral constraint, will be discussed in the next subsection. We present the full derivation of the equations of this section in Appendix B and restrict the discussion here to the main results.

5.1 The convolution of the power spectrum with the window function

Figure 7: The window function multipoles of the NGC of CMASS-DR11 required in eq. 32 and calculated using eq. 33. The window function multipoles are plotted as a function of for fixed values of (black dashed lines). Note that the window function multipoles are not symmetric under and (see eq. 33).

Window function effects in the measured power spectrum do not necessarily represent a problem, since the survey window function is known in principle. One possible way to handle the window function is to deconvolve the measured power spectrum to get the true galaxy power spectrum (Baugh & Efstathiou, 1993; Lin et al., 1996; Sato, Huetsi & Yamamoto, 2011; Sato et al., 2013). Here we follow the more common procedure to convolve each model power spectrum (i.e., ) with the survey window function and derive a model , which is then compared to the measured power spectrum. However, the straightforward implementation of eq. 29 mode-by-mode would lead to a complexity of , where is the total number of modes. For most practical cases this is impossible to evaluate. Therefore, most studies in the past evaluated eq. 29 as a convolution with the spherically averaged window function, (see e.g. Laix & Starkman 1997; Percival et al. 2001; Cole et al. 2005; Percival et al. 2007; Ross et al. 2012b)


which assumes an isotropic power spectrum. The spherically averaged window function is defined as


with . In our analysis we want to measure anisotropic signals in the power spectrum (AP effect and RSD), and hence the assumption of an isotropic power spectrum seems contradictory.

In a recent analysis, Sato et al. (2013) suggested splitting the survey into sub-regions (see also Hemantha et al. 2013), which are small enough that the plane parallel approximation can be applied. In this case the window function can be calculated using FFTs. However, the window function effect on the power spectrum in any sub-region will be larger than in the original survey, and there is a trade-off between keeping the window function(s) compact and making the plane parallel approximation work. These problems become especially prominent for the higher order multipoles. In addition to the enhanced window function effects, splitting the survey will discard large scale modes.

In this section we will present a treatment of the convolution of the power spectrum with the window function without any assumptions regarding isotropy and without the need to split the survey into sub-regions. Our approach has a complexity of only . We believe that our approach is more rigorous and allows a more efficient use of the available data, compared to the methods discussed above.

Figure 8: A model monopole (black dashed lines) and quadrupole (red dashed lines) power spectra using Planck cosmological parameters and the linear Kaiser effect. The solid lines show the same models convolved with the CMASS-DR11 window functions for the NGC (left) and the SGC (right). The black dotted lines show the convolved monopole power spectra where the quadrupole contribution to the monopole has been ignored and the red dotted lines shows the convolved quadrupole power spectra where the monopole contribution to the quadrupole has been ignored (see eq. 32). The bottom panels show the same power spectra relative to the original linear power spectra including the Kaiser effect (dashed lines in the top panels).

We can express eq. 29 in terms of the wavevector amplitude , the cosine of the angle to the line-of-sight and the azimuthal angle :


where the window function is now expanded into the Legendre multipole space, and analytical integration over the angles yields


In this equation represents the spherical Bessel function of order and (for a detailed derivation of this equation see appendix B). We plot the different window function multipoles for CMASS-DR11 in Figure 7. Eq. 33 shows that there are cross terms between different multipoles, meaning that there is a contribution from e.g. the monopole to the convolved quadrupole. In other words, the survey window may induce an anisotropic signal in the convolved power spectrum even without the RSD or AP effect. These cross terms are neglected in the simplified treatment of eq. 30.

The normalisation for the window function is given by


In Figure 8 we show linear model monopole and quadrupole power spectra before (dashed lines) and after (solid lines) the convolution with the CMASS-DR11 window functions. The dotted lines show the convolved monopole power spectra ignoring the quadrupole contribution in eq. 32 (black dotted line) and the convolved quadrupole power spectra ignoring the monopole contribution (red dotted line). While the quadrupole contribution to the monopole seems negligible, there is a small monopole contribution to the quadrupole. All window function effects seem quite small in CMASS-DR11, because of the very compact window function. Whether the full treatment of eq. 32 and eq. 33 is needed, or whether one of the approximations discussed in the beginning of this section can be employed, needs to be tested for each galaxy survey.

5.2 The integral constraint

Figure 9: The window function monopole (solid lines) and quadrupole (dashed lines) for the NGC (black lines) and SGC (red lines) calculated using eq. 37. The NGC multipoles of CMASS-DR11 peak at smaller wave-numbers and show weaker oscillations, which is a result of the larger sky coverage (see Figure 2). The window function multipoles shown in this figure are needed for the integral constraint calculation in eq. 36.

Here we discuss the second term of eq. 29. If we go to our original power spectrum estimator (section 3), we can see that for the mode at we have by design of the random catalogue:


By setting the mode to zero, we assume that the average density of our survey is equal to the average density of the Universe. The existence of sample variance tells us that this assumption must introduce a bias in our power spectrum estimate, which is known as integral constraint. The effect is that we underestimate the power in modes with wavelength approaching the size of our survey. So even neglecting the window function, we do not measure the true underlying power spectrum, but rather a power spectrum with the property for (see e.g. Peacock & Nicholson 1991). This is the reason for the second term in eq. 29. It represents the subtraction of the component which spreads to larger , because of the convolution with the window function. Similar to what we did with the window function in the last section, we express the second term in eq. 29 in terms of amplitude , the cosine of the angle to the line-of-sight and the azimuthal angle :




This window function is normalised to


which is equivalent to eq. 34. In Figure 9 we plot the window function multipoles for the NGC and SGC of CMASS-DR11. The NGC window function multipoles are more compact (concentrated to small ), which results in smaller window function effects in Figure 8. Later, when we fit the measured power spectrum multipoles, we calculate the integral constraint correction for each model multipole power spectrum and subtract it, following eq. 29. This allows a consistent comparison of model power spectra with our measurement.

6 Modelling the multipole power spectra

In this section we discuss our approach to modelling the multipole power spectra to be compared with the CMASS-DR11 measurement. In order to robustly extract information on RSD and AP from the anisotropic galaxy power spectrum in redshift space, it is crucial to prepare a theoretical template which takes account of the non-linear effects of gravitational evolution, galaxy bias, and RSD at a sufficiently accurate level. Particularly in terms of non-linear RSD, several different approaches to model the power spectrum or correlation function of the anisotropic galaxy clustering have been suggested in recent years  (Scoccimarro, 2004; Matsubara, 2008a, b; Carlson, White & Padmanabhan, 2009; Taruya, Nishimichi & Saito, 2010; Reid & White, 2011; Matsubara, 2011; Seljak & McDonald, 2011; Vlah et al., 2012; Wang, Reid & White, 2013; Matsubara, 2013; Taruya, Nishimichi & Bernardeau, 2013; Vlah et al., 2012; Blazek et al., 2013).

We are going to use perturbation theory (PT) for such non-linear corrections, which is physically well motivated and widely applicable. We first introduce the model of the anisotropic power spectrum in two-dimensional space, and then explain how to incorporate the AP effect.

6.1 PT approach to model the galaxy power spectrum in redshift space

Figure 10: These plots show the effect of different ingredients in the extended TNS (eTNS) model (see eq. 3940 and 41). All multipole power spectra are shown relative to an Eisenstein & Hu (1998) no-BAO monopole power spectrum. The black dashed line is the linear CAMB power spectrum, the magenta line is the eTNS model including all correction terms of section 6.1 and calculating , and using -loop perturbation theory. The grey shaded area along the magenta line shows the power spectrum errors of CMASS-DR11 (NGC). (left) The dash-dot green line shows the eTNS model setting , and . The dashed red line and the dotted blue line show the effect of the different bias terms. (right) The dashed red line shows the eTNS model using -loop perturbation theory and the dotted blue line shows the same model with a different damping . Comparing the dotted blue line and the solid magenta line, one can see that the difference between -loop and -loop PT calculation can be absorbed by to some extent.

Our model for the anisotropic galaxy power spectrum is based on Taruya, Nishimichi & Saito (2010) (TNS):


where denotes the cosine of the angle between the wavenumber vector and the line-of-sight direction. The overall exponential factor represents the suppression due to the Finger of God effect, and we treat as a free parameter.

The first three terms in the square bracket in eq. 39 describe an extension of the Kaiser factor. The density (), velocity divergence () and their cross-power spectra () are identical in linear theory, while in the quasi non-linear regime, the density power spectrum increases and velocities are randomised on small scales which damps the velocity power spectrum (Scoccimarro, 2004). Besides this fact, we need to relate the density and velocity fields for (dark) matter to those of galaxies. Here we assume no velocity bias, i.e., , but include every possible galaxy bias term at next-to-leading order using symmetry arguments (McDonald & Roy, 2009):


where is the linear matter power spectrum. Here we introduce five galaxy bias parameters: the renormalised linear bias, , nd-order local bias, , nd-order non-local bias, , rd-order non-local bias, , and the constant stochasticity term, . From now we will call the model in eq. 3940 and 41 extended TNS (eTNS) model. We evaluate the non-linear matter power spectra, , , with the RegPT scheme at -loop order (Taruya et al., 2012). The other bias terms are given by


where the symmetrised nd-order PT kernels, , , and are given by


If we additionally define