Separate Universe Simulations

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.Comment: 5 pages, 2 figures, submitted to MNRAS. References added, typos corrected. Added a paragraph on DE perturbation

The angle-averaged squeezed limit of nonlinear matter N-point functions

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, $P^{-1}d^nP(k|\delta_L)/d\delta_L^n|_{\delta_L=0}$, with $n=N-2$. By performing N-body simulations with a homogeneous overdensity superimposed on a flat Friedmann-Robertson-Lema\^itre-Walker (FRLW) universe using the \emph{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 $k\simeq 0.2 - 0.5 \,h\,{\rm Mpc}^{-1}$ 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.Comment: 32 pages, 5 figures. Submitted to JCA

Position-dependent power spectrum of the large-scale structure: a novel method to measure the squeezed-limit bispectrum

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., $k_3\ll k_1\approx k_2$. 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 squeezed-limit bispectrum from large-scale structure observations such as galaxy redshift surveys.Comment: 23 pages, 6 figures; dependence on cosmological parameters added; JCAP accepte

Separating the Universe into the Real and Fake

The separate universe technique provides a means of establishing consistency relations between short wavelength observables and the long wavelength matter density fluctuations within which they evolve by absorbing the latter into the cosmological background. We extend it to cases where non-gravitational forces introduce a Jeans scale in other species like dynamical dark energy or massive neutrinos. The technique matches the synchronous gauge matter density fluctuations to the local expansion using the acceleration equation and accounts for the temporal nonlocality and scale dependence of the long wavelength response of small scale matter observables, e.g. the nonlinear power spectrum, halo abundance and the implied halo bias, and $N$-point correlation functions. Above the Jeans scale, the local Friedmann equation relates the expansion to real energy densities and a curvature that is constant in comoving coordinates. Below the Jeans scale, the curvature evolves and acts like a fake density component. In all cases, the matter evolution on small scales is correctly modeled as we illustrate using scalar field dark energy with adiabatic or isocurvature initial conditions across the Jeans scale set by its finite sound speed.Comment: 14 pages, 4 figure

Response approach to the squeezed-limit bispectrum: application to the correlation of quasar and Lyman-α\alpha forest power spectrum

The squeezed-limit bispectrum, which is generated by nonlinear gravitational evolution as well as inflationary physics, measures the correlation of three wavenumbers, in the configuration where one wavenumber is much smaller than the other two. Since the squeezed-limit bispectrum encodes the impact of a large-scale fluctuation on the small-scale power spectrum, it can be understood as how the small-scale power spectrum "responds" to the large-scale fluctuation. Viewed in this way, the squeezed-limit bispectrum can be calculated using the response approach even in the cases which do not submit to perturbative treatment. To illustrate this point, we apply this approach to the cross-correlation between the large-scale quasar density field and small-scale Lyman-$\alpha$ forest flux power spectrum. In particular, using separate universe simulations which implement changes in the large-scale density, velocity gradient, and primordial power spectrum amplitude, we measure how the Lyman-$\alpha$ forest flux power spectrum responds to the local, long-wavelength quasar overdensity, and equivalently their squeezed-limit bispectrum. We perform a Fisher forecast for the ability of future experiments to constrain local non-Gaussianity using the bispectrum of quasars and the Lyman-$\alpha$ forest. Combining with quasar and Lyman-$\alpha$ forest power spectra to constrain the biases, we find that for DESI the expected $1-\sigma$ constraint is ${\rm err}[f_{\rm NL}]\sim60$. Ability for DESI to measure $f_{\rm NL}$ through this channel is limited primarily by the aliasing and instrumental noise of the Lyman-$\alpha$ forest flux power spectrum. The combination of response approach and separate universe simulations provides a novel technique to explore the constraints from the squeezed-limit bispectrum between different observables.Comment: 20 pages, 4 figures; matches JCAP accepted versio

Orderly Spanning Trees with Applications

We introduce and study the {\em orderly spanning trees} of plane graphs. This algorithmic tool generalizes {\em canonical orderings}, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an {\em orderly pair} for any connected planar graph $G$, consisting of a plane graph $H$ of $G$, and an orderly spanning tree of $H$. We also present several applications of orderly spanning trees: (1) a new constructive proof for Schnyder's Realizer Theorem, (2) the first area-optimal 2-visibility drawing of $G$, and (3) the best known encodings of $G$ with O(1)-time query support. All algorithms in this paper run in linear time.Comment: 25 pages, 7 figures, A preliminary version appeared in Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2001), Washington D.C., USA, January 7-9, 2001, pp. 506-51

Position-dependent correlation function from the SDSS-III Baryon Oscillation Spectroscopic Survey Data Release 10 CMASS Sample

We report on the first measurement of the three-point function with the position-dependent correlation function from the SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS) Data Release 10 CMASS sample. This new observable measures the correlation between two-point functions of galaxy pairs within different subvolumes, $\hat{\xi}({\rm r},{\rm r}_L)$, where ${\rm r}_L$ is the location of a subvolume, and the corresponding mean overdensities, $\bar{\delta}({\rm r}_L)$. This correlation, which we call the "integrated three-point function", $i\zeta(r)=\langle\hat{\xi}({\rm r},{\rm r}_L)\bar{\delta}({\rm r}_L)\rangle$, measures a three-point function of two short- and one long-wavelength modes, and is generated by nonlinear gravitational evolution and possibly also by the physics of inflation. The $i\zeta(r)$ measured from the BOSS data lies within the scatter of those from the mock galaxy catalogs in redshift space, yielding a ten-percent-level determination of the amplitude of $i\zeta(r)$. The tree-level perturbation theory in redshift space predicts how this amplitude depends on the linear and quadratic nonlinear galaxy bias parameters ($b_1$ and $b_2$), as well as on the amplitude and linear growth rate of matter fluctuations ($\sigma_8$ and $f$). Combining $i\zeta(r)$ with the constraints on $b_1\sigma_8$ and $f\sigma_8$ from the global two-point correlation function and that on $\sigma_8$ from the weak lensing signal of BOSS galaxies, we measure $b_2=0.41\pm0.41$ (68% C.L.) assuming standard perturbation theory at the tree level and the local bias model.Comment: 30 pages, 11 figures. revised version submitted to JCA