Analytic Gaussian covariance matrices for galaxy $N$-point correlation functions

We derive analytic covariance matrices for the $N$-point correlation functions (NPCFs) of galaxies in the Gaussian limit. Our results are given for arbitrary $N$ and projected onto the isotropic basis functions given by spherical harmonics and Wigner $3j$ symbols. A numerical implementation of the 4PCF covariance is compared to the sample covariance obtained from a set of lognormal simulations, Quijote dark matter halo catalogues, and MultiDark-Patchy galaxy mocks, with the latter including realistic survey geometry. The analytic formalism gives reasonable predictions for the covariances estimated from mock simulations with a periodic-box geometry. Furthermore, fitting for an effective volume and number density by maximizing a likelihood based on Kullback-Leibler divergence is shown to partially compensate for the effects of a nonuniform window function. Our result is recently shown to facilitate NPCF analysis on a realistic survey data.

Figure 1

Diagrammatic representation of the basic elements used as building blocks for the fully coupled (i.e., connected) covariance. Coupling between the overdensity fields across the unprimed and primed families [corresponding to density fields from the first and second NPCFs in Eq. (19)] is represented by a tripolar structure [diagrams 1–4, cf. Eqs. (20)–(23)]. Each tripolar structure depends on three vectors: ${\mathbf{r}}_i$, ${\mathbf{r}}_j^{\prime }$, and $\mathbf{s}$. We use dotted lines to represent the separation vector $\mathbf{s}$. The open circle attached to the end of the dotted lines can be connected to the one leg of the coupling kernel in diagram 5. Dashed lines depict primary vertices for ${\mathbf{r}}_0$ or ${\mathbf{r}}_0^{\prime }$ and solid lines are for endpoints with nonzero $i$ or $j$. Diagram 5 is the coupling kernel arising from the rotational average over the unit vectors $\mathbf{r}$, ${\mathbf{r}}^{\prime }$, and $\mathbf{s}$ [cf. second line in Eq. (26)]. In the $N=4$ case the coupling kernel has four legs. The lower-left diagram (with the cartoon telescope) shows our coordinate convention. $\mathbf{x}$ denotes the absolute coordinate; ${\mathbf{r}}_i$ and ${\mathbf{r}}_j^{\prime }$ are the relative coordinates for the unprimed and primed families, respectively. $\mathbf{s}$ is the separation vector between the two families.

Figure 2

Schematic for the fully coupled 4PCF covariance (i.e., the covariance of the connected 4PCF). We split the covariance into two cases. In Case I, the primary vertices (red dots, labeled by $r_0$ and $r_0^{\prime }$) from the primed and unprimed families are mutually coupled and all of the endpoints (labeled by $r_i$ and $r_i^{\prime }$) are coupled. In Case II, the primary vertices are each coupled to an endpoint from the opposite family.

Figure 3

Diagrammatic representation of a fully coupled covariance matrix with Case I shown in the upper panel and Case II in the lower panel (as in Fig. 2). Each case can be broken down into two elementary structures: (a) a tripolar structure arising from the contraction between overdensity fields from the primed and unprimed families, and (b) a coupling kernel given by the rotational average over $\mathbf{r}$, ${\mathbf{r}}^{\prime }$, and $\mathbf{s}$. Moreover, since the covariance involves two primary vertices (one from the primed and the other from the unprimed family), there are two dashed lines connected to either each other or a solid line. All three pieces are multiplied, summed over the angular momenta, and integrated over $s$. In this figure we use $G_i$ and $H_i$ to denote permutations. For $N=4$, the phase $(-1{)}^{\mathrm{\Sigma }(\mathrm{\Lambda })(1-{\mathcal{E}}_G)/2}$ or $(-1{)}^{\mathrm{\Sigma }({\mathrm{\Lambda }}^{\prime })(1-{\mathcal{E}}_H)/2}$ can be directly read off from the plot as one goes around clockwise: an even permutation in the ordering of angular momenta corresponds to a positive Levi-Civita symbol and always gives a positive phase, while an odd permutation can flip the sign of the phase for parity-odd correlators. Diagrams 2 and 3 in Fig. 1 can be distinguished from each other by reading the diagram clockwise (i.e., one cannot change one into the other by a 2D rotation in the page). The following steps are used to build the “snowflake” diagrams on the right-hand sides of the equation: (1) take the tripolar structures and multiply them with the coupling kernel, and then (2) perform an integral over the radial part $s$ of the separation vectors.

Figure 4

Covariance calculation for the 4PCF in the limit of zero separation (i.e., where $\xi$ becomes a Dirac delta function). Left column: in Case I, this limit implies that the two tetrahedra overlap at their origin with $\mathbf{s}\to 0$, ${\mathbf{r}}_1^{\prime }\to {\mathbf{r}}_{G1}$, ${\mathbf{r}}_2^{\prime }\to {\mathbf{r}}_{G2}$, and ${\mathbf{r}}_3^{\prime }\to {\mathbf{r}}_{G3}$. Right column: the same limit in Case II implies that the two tetrahedra also overlap but with one of the primary vertices sitting on an endpoint from the other family. Consequently, we find $\mathbf{s}\to {\mathbf{r}}_{G1}$, $\mathbf{s}\to -{\mathbf{r}}_{H1}^{\prime }$, $\mathbf{s}\to {\mathbf{r}}_{H2}^{\prime }-{\mathbf{r}}_{G2}$, and $\mathbf{s}\to {\mathbf{r}}_{H3}^{\prime }-{\mathbf{r}}_{G3}$.

Figure 5

Comparison of the analytic and sample covariance matrices for a set of lognormal simulations. The first and second panels show the comparison of the correlation matrix (defined by ${\mathsf{M}}_{ij}={\mathsf{C}}_{ij}/\sqrt{{\mathsf{C}}_{ii}{\mathsf{C}}_{jj}}$) for angular momenta $\{\mathrm{\Lambda },{\mathrm{\Lambda }}^{\prime }\}=\{000,000\}$ in real space. Panel (a) gives the model prediction for the fully coupled 4PCF correlation matrix, and the panels above and to the left show the (disconnected) Gaussian 4PCF model in real space. The horizontal and vertical axes indicate 120 radial bins, ordered so that $r_1<r_2<r_3$. This gives rise to the block structure in the matrix and the sawtooth shape of the correlation functions. Panel (b) shows the correlation matrix estimated from 1000 log-normal mocks, with the extended panels showing the measured full 4PCF from the log-normal mocks in real space. Panel (c) shows a comparison of the diagonal elements of the two covariance matrices; we note that the values (vertical axis) are logarithmically scaled.

Figure 6

Left: half-inverse test comparing the log-normal simulations and the analytic covariance, both of which are shown in Fig. 5. If the covariance matrices agree, both the mean and the off-diagonal elements should be noisy fluctuations around zero. For clarity, we plot only the lower triangle, and give the standard deviation for the off-diagonal elements (${\sigma }_{\text{nondiag}}$), for the diagonal elements (${\sigma }_{\mathrm{diag}}$), and for all elements combined (${\sigma }_{\mathrm{all}}$). Right: comparison between the eigenvalues of the analytic covariance (solid curve) and log-normal mock covariance (dotted curve).

Figure 7

Same as Fig. 5, but comparing the analytic and log-normal covariances in redshift space. Here we show the results of the half-inverse test in the same format as Fig. 6. The model also works well in redshift space in that it shows comparable coupling structure for the correlation matrices and the diagonal elements of the covariances. However, we do observe a residual in the diagonal elements of the matrix for the half-inverse test.

Figure 8

Same as Fig. 5, but for the Quijote halo catalog in real space, using 100 simulations. For the simulations with non-negligible non-Gaussianity, the model can adequately predict various features of the correlation matrix, with a good match for the diagonal elements of the covariance as well.

Figure 9

Same as Fig. 5, but for the Quijote halo catalog in real space. Here, we plot the cross covariance with angular momenta $\{\mathrm{\Lambda },{\mathrm{\Lambda }}^{\prime }\}=\{000,101\}$.

Figure 10

Half-inverse test for the analytic covariance and sample covariance of the Quijote halo catalog in real space, in the format of Fig. 6. Left: angular momenta $\mathrm{\Lambda }=\{000\}$ and ${\mathrm{\Lambda }}^{\prime }=\{000\}$. Right: cross covariance with angular momenta $\mathrm{\Lambda }=\{000\}$ and ${\mathrm{\Lambda }}^{\prime }=\{101\}$. For comparison we show the full matrix with $\{\mathrm{\Lambda },{\mathrm{\Lambda }}^{\prime }\}=\{000,000\}+\{000,101\}+\{101,000\}+\{101,101\}$. The standard deviations for the off-diagonal elements (${\sigma }_{\text{nondiag}}$), the diagonal elements (${\sigma }_{\mathrm{diag}}$), and all of the elements (${\sigma }_{\mathrm{all}}$) are given in the insets.

Figure 11

Same as Fig. 7, but for the Quijote halo catalog with angular momenta $\{\mathrm{\Lambda },{\mathrm{\Lambda }}^{\prime }\}=\{000,101\}$ including RSD. The analytic covariance well describes the structure of the sample covariance in this scenario.

Figure 12

Log-likelihood for the parameters $\overline{n}$ and $V_{\mathrm{eff}}$ obtained from fitting the analytic covariance to the sample covariance of 1000 Patchy mocks (including redshift-space effects and nonuniform survey geometry). The likelihood is constructed using the KL divergence, as in Eq. (47).

Figure 13

Same as Fig. 5 but for 999 Patchy mocks. These include both RSD and survey geometry. The third panel shows a comparison of the diagonal elements for the Patchy covariance (grey dashed curve), and analytic covariance with and without optimization (red solid curve and black dotted curve, respectively).

Figure 14

Left: half-inverse test for the model applied to the Patchy NGC mocks for angular momenta $\{\mathrm{\Lambda },{\mathrm{\Lambda }}^{\prime }\}=\{000,000\}$, as in Fig. 6. Right: comparison of the eigenvalues for the theoretical covariance before optimization (dotted black curve), after optimization (solid red curve), and from the Patchy mocks (dashed grey curve).

Figure 15

Comparison of correlation matrices estimated from Patchy NGC mocks (left) and model (right). Here we display 11 different choices of $\mathrm{\Lambda }$, with each submatrix being the correlation between angular momentum sets $\{\mathrm{\Lambda },{\mathrm{\Lambda }}^{\prime }\}=\{{\ell }_1{\ell }_2{\ell }_3,{\ell }_1^{\prime }{\ell }_2^{\prime }{\ell }_3^{\prime }\}$. The shot noise and volume entering the analytic covariance are optimized using 13 choices of $\mathrm{\Lambda }$ (those involving the first angular momentum being less than or equal to one). Overall, we find reasonably good agreement between the Gaussian model and the sample covariance. We see some differences in the off-diagonal terms, and these differences increase with increasing angular momenta. The diagonal terms are relatively consistent with each other, mostly as a result of the parameter fitting.

Figure 16

Diagrams for the partially coupled covariance. This figure is analogous to Fig. 2, but gives the terms necessary to model the disconnected 4PCF covariance.

Figure 17

Comparison of the numerical and analytic implementations of the bin-averaged $f$ integral, setting ${\ell }_1={\ell }_2={\ell }_3=0$, and using a damped power-law power spectrum. We evaluate the integral at radial bin centers $a$ and $b$ as given in the legend, and their units are $h^{-1}\mathrm{Mpc}$.