Joint likelihood function of cluster counts and $n$-point correlation functions: Improving their power through including halo sample variance

Naive estimates of the statistics of large-scale structure and weak lensing power spectrum measurements that include only Gaussian errors exaggerate their scientific impact. Nonlinear evolution and finite-volume effects are both significant sources of non-Gaussian covariance that reduce the ability of power spectrum measurements to constrain cosmological parameters. Using a halo model formalism, we derive an intuitive understanding of the various contributions to the covariance and show that our analytical treatment agrees with simulations. This approach enables an approximate derivation of a joint likelihood for the cluster number counts, the weak lensing power spectrum and the bispectrum. We show that this likelihood is a good description of the ray-tracing simulation. Since all of these observables are sensitive to the same finite-volume effects and contain information about the nonlinear evolution, a combined analysis recovers much of the “lost” information. For upcoming weak lensing surveys, we estimate that a joint analysis of power spectrum, number counts and bispectrum will produce an improvement of about 30–40% in determinations of the matter density and the scalar amplitude. This improvement is equivalent to doubling the survey area.

Figure 1

Comparison of our analytical prediction for the lensing power spectrum covariance [Eqs. (34) and (20)] with the ray-tracing simulation results obtained from 1000 realizations. Here we considered a single source redshift $z_s=1$. For the analytical predictions, the Gaussian term (PP), the trispectrum term (T) and the HSV are included. The error bars for the simulation results are obtained from the variance of the 1000 realizations, which correspond to $1\sigma$ scatters of power spectrum estimation for 25 square degrees, the area of each ray-tracing simulation. Left panel: We included only the 1-halo term for the HSV calculation. Right panel: We further included the 2-halo term contribution for the HSV effect. The analytical predictions are in fairly good agreement with the simulation results, over a wide range of multipoles, if the HSV effect is included in the analytical prediction. Comparing the left and right panels shows that including the 2-halo term of the HSV effect improves the agreement at the transition regime between the 1- and 2-halo terms, in the range of $l\simeq$ a few hundreds to ${10}^3$.

Figure 2

Correlation coefficient matrix for the power spectrum, defined as $r[P({\ell }_1),P({\ell }_2)]\equiv \text{Cov}[P({\ell }_1),P({\ell }_2)]/\sqrt{\text{Var}[P({\ell }_1)]\text{Var}[P({\ell }_2)]}$. Note $r(\ell ,\ell )=1$ for the diagonal elements by definition. Left panel: comparison between the simulation (the upper-right matrix elements) and the halo model (the lower-left elements). The visual agreement is very good. Right panel: the relative error (its absolute value) of the halo model prediction to the simulation result for the correlation coefficient matrix. The upper-right and lower-left elements show the results with and without the halo sample covariance contribution in the halo model calculation. Including the halo sample covariance term greatly improves the agreement with simulations.

Figure 3

Comparison of the halo model prediction with the simulation results for the lensing bispectrum covariance for equilateral triangle configurations, as a function of the side length. For the analytical model, we included the standard Gaussian and non-Gaussian contributions arising from a combination of the correlation functions up to the 6-point correlation function [PPP, BB, PT, P6, from Eq. (a15)] and also included the HSV contribution [Eq. (34)]. For the HSV contribution, we included all 1-, 2- and 3-halo terms, but only the $1=\text{halo}$ term gives an important contribution.

Figure 4

Similarly to Fig. 15, but for the bispectrum covariance: $r[B({\ell }_1),B({\ell }_2)]\equiv \text{Cov}[B({\ell }_1),B({\ell }_2)]/\sqrt{\text{Var}[B({\ell }_1)]\text{Var}[B({\ell }_2)]}$. Note that we considered the equilateral triangles, and the matrix elements are shown as a function of the side length. The halo model is in a good agreement with the simulation if the halo sample covariance is included.

Figure 5

Cross-covariance between the lensing power spectrum, $P_{\kappa }(l)$, and the bispectrum of equilateral triangle configuration, $B_{\text{eq}}(l,l,l)$, against multipoles. Similarly to the previous plot, for the halo model prediction, we included the standard contributions [PB, P5, from Eq. (a16)] and the HSV contribution [Eq. (34)].

Figure 6

Correlation coefficient matrix between the power spectrum and the bispectrum of equilateral triangle configurations, defined as $r[P({\ell }_1),B({\ell }_2)]\equiv \text{Cov}[P({\ell }_1),B({\ell }_2)]/\sqrt{\text{Var}[P({\ell }_1)]\text{Var}[B({\ell }_2)]}$. The different elements run over the multipole bins or the triangle side lengths. The upper-left and -right panels show the simulation result and the halo model predictions. The lower-left and -right panels show the relative errors when the halo model with and without the halo sample variance is used.

Figure 7

Cross-covariance between the angular number counts of halos and the lensing power spectrum $P_{\kappa }(l)$ as a function of multipoles. For the number counts, we included all the halos that are in the light cone up to $z_s=1$ over area 25 sq degrees (area of the ray-tracing simulation) and have masses greater than $M={10}^{14}{M}_{\odot }/h$. The error bars for the simulation results are the variance estimated from the 1000 realizations. For the halo model prediction, we used Eq. (35) to compute the contributions arising from the product of the number counts and the power spectrum (NP) and the HSV effect. The difference between the left and right panels is whether the halo model prediction includes the 2-halo term of the HSV effect (right) or not (left). The halo model prediction, with the 2-halo HSV effect, well reproduces the simulation result over the range of multipoles.

Figure 8

Similar to the previous plot, the cross-covariance between the angular halo number counts and the bispectrum of equilateral triangle configurations, $B_{\kappa ,\text{eq}}(l,l,l)$. The halo model computation is based on Eq. (36). The difference between the left and right panels is whether we included the 2- and 3-halo terms for the HSV contribution in the halo model prediction (right panel) or not (left).

Figure 9

The likelihood distribution of the angular number counts of halos with $M\ge {10}^{14}{M}_{\odot }/h$, in the light cone up to $z_s=1$ and with area 25 sq degrees. The histogram shows the distribution measured from the 1000 ray-tracing simulations. The red-color, solid curve shows the halo model prediction computed assuming the Gaussian likelihood function [Eq. (38)], where we used the halo model to compute the mean and the variance. For comparison, the green-color, solid curve shows the Gaussian distribution that has the same mean value and variance as those of the simulations. The mean, variance, skewness and excess kurtosis values measured from the simulations are also given.

Figure 10

Similarly to the previous figure, but for the lensing power spectrum $P_{\kappa }(l)$. The different panels show the distributions for different multipole bins as indicated. Again the halo model prediction (red, solid curve) well reproduces the width of the simulation distribution, if the HSV contribution is included. The skewness and kurtosis measured from simulations are small compared to the width of the distribution.

Figure 11

Similar to the previous plot, but for the lensing bispectrum of equilateral triangle configurations. The different panels are for different-size triangle configurations. Compared to Figs. 9, 10, the simulation distribution shows a larger asymmetry and therefore larger skewness and kurtosis values. The halo model nevertheless well reproduces the width of the distribution.

Figure 12

A joint distribution of the angular number counts of halos with $M\ge {10}^{14}{M}_{\odot }$ and the lensing power spectrum $P_{\kappa }(l)$. The different panels show the results for different multipole bins as denoted. Each cross symbol denotes one realization out of the 1000 realizations. The blue-color contour in each panel shows the 68% percentile of the distribution, which is estimated by binning the 1000 simulations into a two dimensional histogram. The red-color contour shows the halo model prediction for the joint distribution, computed based on Eq. (38). For comparison, the green contour shows the multivariate Gaussian distribution that has the same mean and variances as those of the simulation distribution.

Figure 13

Similar to the previous figure, but the joint distribution for the halo number counts and the lensing bispectrum of equilateral triangle configurations, $B_{\kappa ,\text{eq}}(l,l,l)$. The halo model well reproduces the distribution seen in the simulations.

Figure 14

Similar to the previous figure, but the joint distribution for the lensing power spectrum and bispectrum.

Figure 15

Forecast error on the amplitude of scalar perturbations $A_S$ as a function of the maximum observed multipole $l_{\max }$, when marginalizing over ${\mathrm{\Omega }}_m$, and keeping all other parameters fixed. We here assumed a hypothetical survey that is characterized by area 15000 sq degrees and the redshift distribution following [46] with $⟨z_s⟩=3z_0=1$. We included the galaxy shape noise contamination, assuming ${\sigma }_{\epsilon }=0.22$ and $n_{\text{gal}}=20{\text{arcmin}}^{-2}$ for the number density and the rms intrinsic ellipticities. The solid-red curve shows the error when including the power spectrum information alone. The solid-green and dashed curves show the results when further including the bispectrum measurements of equilateral triangle configuration and the cluster number counts.