Abstract. The influence of large-scale density fluctuations on structure formation on small scales is described by the three-point correlation function (bispectrum) in the so-called "squeezed configurations," in which one wavenumber, say k 3 , is much smaller than the other two, i.e., k 3 ≪ k 1 ≈ k 2 . This bispectrum is generated by non-linear gravitational evolution and possibly also by inflationary physics. In this paper, we use this fact to show that the bispectrum in the squeezed configurations can be measured without employing three-point function estimators. Specifically, we use the "position-dependent power spectrum," i.e., the power spectrum measured in smaller subvolumes of the survey (or simulation box), and correlate it with the mean overdensity of the corresponding subvolume. This correlation directly measures an integral of the bispectrum dominated by the squeezed configurations. Measuring this correlation is only slightly more complex than measuring the power spectrum itself, and sidesteps the considerable complexity of the full bispectrum estimation. We use cosmological N -body simulations of collisionless particles with Gaussian initial conditions to show that the measured correlation between the position-dependent power spectrum and the long-wavelength overdensity agrees with the theoretical expectation. The position-dependent power spectrum thus provides a new, efficient, and promising way to measure the squeezedlimit bispectrum from large-scale structure observations such as galaxy redshift surveys.

The large-scale statistics of observables such as the galaxy density are chiefly determined by their dependence on the local coarse-grained matter density. This dependence can be measured directly and efficiently in N-body simulations by using the fact that a uniform density perturbation with respect to some fiducial background cosmology is equivalent to modifying the background and including curvature, i.e., by simulating a "separate universe". We derive this mapping to fully non-linear order, and provide a step-by-step description of how to perform and analyse the separate universe simulations. This technique can be applied to a wide range of observables. As an example, we calculate the response of the non-linear matter power spectrum to long-wavelength density perturbations, which corresponds to the angle-averaged squeezed limit of the matter bispectrum and higher n-point functions. Using only a modest simulation volume, we obtain results with percent-level precision over a wide range of scales.

We show that in a certain, angle-averaged squeezed limit, the N -point function of matter is related to the response of the matter power spectrum to a long-wavelength density perturbation, P −1 d n P (k|δ L )/dδ n L | δ L =0 , with n = N − 2. By performing N-body simulations with a homogeneous overdensity superimposed on a flat Friedmann-Robertson-Lemaître-Walker (FRLW) universe using the separate universe approach, we obtain measurements of the nonlinear matter power spectrum response up to n = 3, which is equivalent to measuring the fully nonlinear matter 3− to 5−point function in this squeezed limit. The sub-percent to few percent accuracy of those measurements is unprecedented. We then test the hypothesis that nonlinear N -point functions at a given time are a function of the linear power spectrum at that time, which is predicted by standard perturbation theory (SPT) and its variants that are based on the ideal pressureless fluid equations. Specifically, we compare the responses computed from the separate universe simulations and simulations with a rescaled initial (linear) power spectrum amplitude. We find discrepancies of 10% at k 0.2 − 0.5 h Mpc −1 for 5− to 3−point functions at z = 0. The discrepancy occurs at higher wavenumbers at z = 2. Thus, SPT and its variants, carried out to arbitrarily high order, are guaranteed to fail to describe matter N -point functions (N > 2) around that scale.

We present a public code to generate a mock galaxy catalog in redshift space assuming a log-normal probability density function (PDF) of galaxy and matter density fields. We draw galaxies by Poisson-sampling the log-normal field, and calculate the velocity field from the linearised continuity equation of matter fields, assuming zero vorticity. This procedure yields a PDF of the pairwise velocity fields that is qualitatively similar to that of N-body simulations. We check fidelity of the catalog, showing that the measured two-point correlation function and power spectrum in real space agree with the input precisely. We find that a linear bias relation in the power spectrum does not guarantee a linear bias relation in the density contrasts, leading to a cross-correlation coefficient of matter and galaxies deviating from unity on small scales. We also find that linearising the Jacobian of the real-to-redshift space mapping provides a poor model for the two-point statistics in redshift space. That is, non-linear redshift-space distortion is dominated by non-linearity in the Jacobian. The power spectrum in redshift space shows a damping on small scales that is qualitatively similar to that of the well-known Fingers-of-God (FoG) effect due to random velocities, except that the log-normal mock does not include random velocities. This damping is a consequence of non-linearity in the Jacobian, and thus attributing the damping of the power spectrum solely to FoG, as commonly done in the literature, is misleading.

Cosmic background neutrinos have a large velocity dispersion, which causes the evolution of longwavelength density perturbations to depend on scale. This scale-dependent growth leads to the well-known suppression in the linear theory matter power spectrum that is used to probe neutrino mass. In this paper, we study the impact of long-wavelength density perturbations on small-scale structure formation. By performing separate universe simulations where the long-wavelength mode is absorbed into the local expansion, we measure the responses of the cold dark matter (CDM) power spectrum and halo mass function, which correspond to the squeezed-limit bispectrum and halo bias. We find that the scale-dependent evolution of the long-wavelength modes causes these quantities to depend on scale and provide simple expressions to model them in terms of scale and the amount of massive neutrinos. Importantly, this scale-dependent bias reduces the suppression in the linear halo power spectrum due to massive neutrinos by 13 and 26% for objects of biasb = 2 andb 1, respectively. We demonstrate with high statistical significance that the scale-dependent halo bias cannot be modeled by the CDM and neutrino density transfer functions at the time when the halos are identified. This reinforces the importance of the temporal nonlocality of structure formation, especially when the growth is scale dependent.