Robert N. Cahn’s research while affiliated with Lawrence Berkeley National Laboratory and other places

Isotropic functions of positions r1, r2, . . . , rN, i.e. functions invariant under simultaneous rotations of all the coordinates, are conveniently formed using spherical harmonics and Clebsch-Gordan coefficients. An orthonormal basis of such functions provides a formalism suitable for analyzing isotropic distributions such as those that arise in cosmology, for instance in the clustering of galaxies as revealed by large-scale structure surveys. The algebraic properties of the basis functions are conveniently expressed in terms of 6-j and 9-j symbols. The calculation of relations among the basis functions is facilitated by “Yutsis” diagrams for the addition and recoupling of angular momenta.

A tetrahedron is the simplest shape that cannot be rotated into its mirror image in three-dimension (3D). The 4-point correlation function (4PCF), which quantifies excess clustering of quartets of galaxies over random, is the lowest order statistic sensitive to parity violation. Each galaxy defines one vertex of the tetrahedron. Parity-odd modes of the 4PCF probe an imbalance between tetrahedra and their mirror images. We measure these modes from the largest currently available spectroscopic samples, the 280 067 luminous red galaxies (LRGs) of the Baryon Oscillation Spectroscopic Survey (BOSS) twelfth data release (DR12) LOWZ ($\bar{z} = 0.32$ ) and the 803 112 LRGs of BOSS DR12 CMASS ($\bar{z} = 0.57$ ). In LOWZ, we find 3.1σ evidence for a non-zero parity-odd 4PCF, and in CMASS we detect a parity-odd 4PCF at 7.1σ. Gravitational evolution alone does not produce this effect; parity-breaking in LSS, if cosmological in origin, must stem from the epoch of inflation. We have explored many sources of systematic error and found none that can produce a spurious parity-odd signal sufficient to explain our result. Underestimation of the noise could also lead to a spurious detection. Our reported significances presume that the mock catalogues used to calculate the covariance sufficiently capture the covariance of the true data. We have performed numerous tests to explore this issue. The odd-parity 4PCF opens a new avenue for probing new forces during the epoch of inflation with 3D large-scale structure; such exploration is timely given large upcoming spectroscopic samples such as Dark Energy Spectroscopic Instrument and Euclid.

We show how the galaxy four-point correlation function can test for cosmological parity violation. The detection of cosmological parity violation would reflect previously unknown forces present at the earliest moments of the Universe. Recent developments both in rapidly evaluating galaxy N-point correlation functions and in determining the corresponding covariance matrices make the search for parity violation in the four-point correlation function possible in current and upcoming surveys such as those undertaken by Dark Energy Spectroscopic Instrument, the Euclid satellite, and the Vera C. Rubin Observatory. We estimate the limits on cosmic parity violation that could be set with these data.

We present an analytic treatment of the self-similar collapse of a uniform density ellipsoid to linear order in the deviations from sphericity. First we obtain a self-consistent closed-form solution for the evolution of an isolated ellipsoid, and then impose the effects of an external forcing. This model describes the evolution under gravity of a pre-stellar core of molecular gas embedded in a much larger and lower-density filament. We compare with numerical solutions for the collapse without the limitation of small deviations. These show how the external perturbing force producing the initial deviation from sphericity is eventually surpassed by the anisotropic forces generated by the collapsing ellipsoid itself. This model should be useful in interpreting the way in which environment shapes the evolution of pre-stellar cores.

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.

A tetrahedron is the simplest shape that cannot be rotated into its mirror image in 3D. The 4-Point Correlation Function (4PCF), which quantifies excess clustering of quartets of galaxies over random, is the lowest-order statistic sensitive to parity violation. Each galaxy defines one vertex of the tetrahedron. Parity-odd modes of the 4PCF probe an imbalance between tetrahedra and their mirror images. We measure these modes from the largest currently available spectroscopic samples, the 280,067 Luminous Red Galaxies (LRGs) of BOSS DR12 LOWZ ($\bar{z} = 0.32$) and the 803,112 LRGS of BOSS DR12 CMASS ($\bar{z} = 0.57$). In LOWZ we find $3.1\sigma$ evidence for a non-zero parity-odd 4PCF, and in CMASS we detect a parity-odd 4PCF at $7.1\sigma$. Gravitational evolution alone does not produce this effect; parity-breaking in LSS, if cosmological in origin, must stem from the epoch of inflation. We have explored many sources of systematic error and found none that can produce a spurious parity-odd \textit{signal} sufficient to explain our result. Underestimation of the \textit{noise} could also lead to a spurious detection. Our reported significances presume that the mock catalogs used to calculate the covariance sufficiently capture the covariance of the true data. We have performed numerous tests to explore this issue. The odd-parity 4PCF opens a new avenue for probing new forces during the epoch of inflation with 3D LSS; such exploration is timely given large upcoming spectroscopic samples such as DESI and Euclid.

We present a new algorithm for efficiently computing the N-point correlation functions (NPCFs) of a 3D density field for arbitrary N. This can be applied both to a discrete spectroscopic galaxy survey and a continuous field. By expanding the statistics in a separable basis of isotropic functions built from spherical harmonics, the NPCFs can be estimated by counting pairs of particles in space, leading to an algorithm with complexity $\mathcal {O}(N_\mathrm{g}^2)$ for Ng particles, or $\mathcal {O}\left(N_\mathrm{FFT}\log N_\mathrm{FFT}\right)$ when using a Fast Fourier Transform with NFFT grid-points. In practice, the rate-limiting step for N > 3 will often be the summation of the histogrammed spherical harmonic coefficients, particularly if the number of radial and angular bins is large. In this case, the algorithm scales linearly with Ng. The approach is implemented in the encore code, which can compute the 3PCF, 4PCF, 5PCF, and 6PCF of a BOSS-like galaxy survey in ∼ 100 CPU-hours, including the corrections necessary for non-uniform survey geometries. We discuss the implementation in depth, along with its GPU acceleration, and provide practical demonstration on realistic galaxy catalogs. Our approach can be straightforwardly applied to current and future datasets to unlock the potential of constraining cosmology from the higher-point functions.

We show that the galaxy 4-Point Correlation Function (4PCF) can test for cosmological parity violation. The detection of cosmological parity violation would reflect previously unknown forces present at the earliest moments of the Universe. Recent developments both in rapidly evaluating galaxy N-Point Correlation Functions (NPCFs) and in determining the corresponding covariance matrices make the search for parity violation in the 4PCF possible in current and upcoming surveys such as those undertaken by Dark Energy Spectroscopic Instrument (DESI), the Euclid satellite, and the Vera C. Rubin Observatory (VRO).

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 of Cahn & Slepian (2020), recently shown to facilitate efficient NPCF estimation. 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 non-uniform window function.