Galaxy power spectrum multipoles covariance in perturbation theory

We compute the covariance of the galaxy power spectrum multipoles in perturbation theory, including the effects of nonlinear evolution, nonlinear and nonlocal bias, radial redshift-space distortions, arbitrary survey window, and shot noise. We rewrite the power spectrum FKP estimator in terms of the usual windowed galaxy fluctuations and the fluctuations in the number of galaxies inside the survey volume. We show that this leads to a stronger supersample covariance than assumed in the literature and causes a substantial leakage of Gaussian information. We decompose the covariance matrix into several contributions that provide an insight into its behavior for different biased tracers. We show that for realistic surveys, the covariance of power spectrum multipoles is already dominated by shot noise and super survey mode coupling in the weakly nonlinear regime. Both these effects can be accurately modeled analytically, making a perturbative treatment of the covariance very compelling. Our method allows for the covariance to be varied as a function of cosmology and bias parameters very efficiently, with survey geometry entering as fixed kernels that can be computed separately using fast fourier transforms (FFTs). We find excellent agreement between our analytic covariance and that estimated from BOSS DR12 Patchy mock catalogs in the whole range we tested, up to $k=0.6\mathrm{h}/\mathrm{Mpc}$. This bodes well for application to future surveys such as DESI and Euclid.

Figure 1

The trispectrum contribution to the covariance can be split into a regular part ${\mathrm{T}}_0$ [only contribution in an infinite universe or for periodic boundary conditions, Eq. (25)] and a BC part ${\mathrm{T}}_{\mathrm{BC}}$ [Eq. (26)] that describes the coupling of small-scale modes ${\mathbf{k}}_i$ to the large-scale mode $\mathit{\epsilon }$.

Figure 2

Fractional (total) contribution of power spectrum multipoles to the diagonal elements of the Gaussian part of the auto (cross) covariance matrix. We used the BOSS DR12 window for $0.5<z<0.75$ to calculate these contributions. Dashed lines represent negative contributions. Notice that the quadrupole autocovariance is almost entirely made up by leakage of monopole power at high-$k$.

Figure 3

Fractional contribution of the ordinary trispectrum (${\mathrm{T}}_0$) and the total supersurvey mode contribution ($\mathrm{BC}+\mathrm{LA}$) to the non-Gaussian part ($\mathrm{BC}+\mathrm{LA}+{\mathrm{T}}_0$) of the autocovariance matrix of the monopole (top) and quadrupole (bottom). The contribution from the supersurvey modes (referred to as SSC in the literature) can be broken down into two parts: the beat-coupling terms in the trispectrum (BC) and the LA effect which is caused due to fluctuations in the total number of galaxies ${\mathrm{N}}_{\mathrm{g}}$ inside the survey volume. Because the supersurvey modes (which are in the linear regime) dominate over ${\mathrm{T}}_0$ at high-$k$, PT can be used to model the non-Gaussian covariance fairly accurately.

Figure 4

SN contributions to the power spectrum covariance involving the trispectrum; see Eq. (89).

Figure 5

Enhancement in the diagonal elements (top panel) of the power spectrum monopole autocovariance due to using the FKP shot noise estimator as compared to the true shot noise estimator. The bottom panel shows the enhancement of the nondiagonal elements for $k^{\prime }=0.3h{\mathrm{Mpc}}^{-1}$.

Figure 6

Impact of the SN estimator (top: FKP shot noise, bottom: true shot noise) on varying the $P_0$ parameter in the FKP weights [Eq. (6)], relative to the case with $P_0={10}^3(\mathrm{Mpc}/h{)}^3$ in blue. It is more optimal to use a higher $P_0$ value at low-$k$ and vice versa in the FKP weights as the variance is lower.

Figure 7

Individual contributions to the multipole covariance matrix elements. Top (bottom) subpanel shows diagonals (a particular row). In the case when the fields are continuous [i.e., no shot noise (SN)], we show the Gaussian part (G) of covariance in blue and the non-Gaussian part (NG) in orange. The dashed lines represent the additional contribution of SN to the corresponding terms coming from discreteness of galaxies. The black line is the sum total of all the components. Before reaching the nonlinear regime, the diagonal covariance is already dominated by shot noise and therefore can be accurately modeled analytically.

Figure 8

Diagonal elements of the autocovariance (left panels) and cross-covariance (right panels) matrix of power spectrum multipoles obtained from the BOSS DR12 Patchy mocks (orange band) and the corresponding predictions from our perturbative calculation (dashed black). We find an excellent agreement even though no parameter fitting to the Patchy mocks was performed in our analysis.

Figure 9

Rows of the reduced auto and cross-covariance matrix of power spectrum multipoles obtained from the Patchy mocks (orange and blue) and the corresponding row from our perturbative calculation (dashed red and dashed black). Even though 2048 Patchy mocks were generated for the BOSS DR12 analysis, there is still a large numerical noise in the Patchy cross-covariance.

Figure 10

Constraints on cosmological parameters (for fixed linear spectrum shape) when changing the galaxy power spectrum multipoles covariance matrix: covariance from the BOSS DR12 Patchy mocks (blue) and from our perturbative calculation (orange). The data vector in this analysis is taken as the theoretical model computed at the fiducial cosmological parameters. The results for the full BOSS parameter analysis will be presented in an upcoming work [97].

Figure 11

Cumulative S/N ratio given by Eq. (105) for the monopole and the quadrupole power spectra. The dashed line for each case corresponds to using the covariance matrix containing only the Gaussian contribution, and the solid line corresponds to using the full matrix which includes the non-Gaussian contribution (see Fig. 7 for the decomposition of the covariance). Even though nondiagonal elements sourced by non-Gaussian effects are small, they do have an overall impact on the S/N for $k>0.1h{\mathrm{Mpc}}^{-1}$ and should be appropriately modeled.

Figure 12

A schematic diagram corresponding to Eq. (3) showing a wide-angle survey where the LOS changes significantly along the survey. This configuration space diagram is shown to emphasize that keeping the LOS fixed is a good approximation for the small-scale modes (${\mathbf{k}}_1$, ${\mathbf{k}}_2$) but not a good approximation when calculating integrals involving the beat mode ($\mathit{\epsilon }$).

Figure 13

Fractional change in the cumulative S/N ratio of the monopole power on using the FKP SN estimator relative to using the true SN estimator [see Eq. (93)].

Figure 14

Same as Fig. 8 but for the $0.2<z<0.5$ bin instead of the $0.5<z<0.75$ bin.