Detectability of oscillatory-PDF primordial non-Gaussianity in the late-time large-scale structure

A specific class of primordial non-Gaussianity, generated by an axion-like spectator field coupled to the curvature perturbation, is fully characterized by a non-Gaussian one-point probability density function (PDF) of the primordial curvature perturbation: a Gaussian modulated by an oscillatory component with amplitude set by $A^2$ and frequency $f$. The only dedicated simulation study (Peña & Candlish 2022) found that this non-Gaussianity imprints at most percent-level, scale-dependent deviations in the late-time matter power spectrum, bispectrum, and trispectrum — marginal against sample variance — using small volumes, few realizations, and only highly symmetric configurations. The open problem is to determine whether this CMB-allowed class of primordial non-Gaussianity leaves any statistically significant, observationally usable imprint in the large-scale structure: through general (non-equilateral, non-square) polyspectrum configurations, through observables beyond polyspectra (the halo mass function, Minkowski functionals, and other statistical or topological measures of the density field), over a broader and finer survey of the $(f, A^2)$ parameter space with simulation ensembles of adequate volume, resolution, and realization count, and through a direct confrontation with actual cosmic-microwave-background and large-scale-structure data using the full primordial PDF rather than polyspectra alone.

Background

Inflation, the leading theory of the very early Universe, seeds today's cosmic structures through quantum fluctuations of a field called the inflaton. The resulting primordial curvature perturbation $\zeta$ is usually assumed to be a Gaussian random field, so that all its statistical information resides in the two-point correlation function; in Fourier space, its power spectrum. Many well-motivated extensions of the simplest single-field inflation models — multi-field dynamics, non-canonical kinetic terms, violations of slow roll — predict deviations from Gaussianity, collectively called primordial non-Gaussianity (PNG), and searching for PNG is a major route to discriminating inflationary models. The standard approach parametrizes PNG through the bispectrum and trispectrum, the Fourier-space analogues of the three- and four-point correlation functions, summarized by amplitudes such as $f_{\rm NL}$ and $g_{\rm NL}$.

A complementary characterization was introduced by X. Chen, G. A. Palma and collaborators in 2018 ('landscape tomography'): an axion-like isocurvature field with a sinusoidal potential, coupled linearly through its derivative to the curvature field. In the limit where all scales are observable, this scenario produces initial conditions fully specified by a non-Gaussian one-point probability density function (PDF) of $\zeta$: a Gaussian multiplied by an oscillatory modulation, with the oscillation amplitude controlled by $A^2$ and its frequency by $f$. Because the oscillatory factor is an even function of $\zeta$, all odd moments of the PDF vanish exactly as for a Gaussian, and the non-Gaussian signal first appears in the even $n$-point statistics. A companion paper (Palma, Scheihing Hitschfeld and Sypsas 2020) developed CMB and large-scale-structure (LSS) statistics beyond polyspectra for this class, such as the full PDF and its response to the initial conditions.

Cosmic microwave background (CMB) data constrain the model: applying the Planck bound on the local-type primordial trispectrum ($g_{\rm NL}$-like) to this PDF yields a limit $\beta \equiv A^2\,\sigma_\zeta^4 f_\zeta^{-4}\,\exp(-\sigma_\zeta^2/2f_\zeta^2) < 2.1\times 10^{-3}$ (with $\sigma_\zeta^2=1$ the variance of $\zeta$ and $f_\zeta$ the oscillation frequency of the primordial PDF), so that only the most extreme amplitude–frequency combinations are excluded; however, this constraint uses the trispectrum alone, and a constraint using the full PDF of CMB temperature fluctuations would be far more restrictive — a confrontation with actual data that has not been carried out.

Peña and Candlish (2022) performed the first and, so far, only dedicated simulation campaign on the late-time LSS consequences of this model. Drawing initial $\zeta$ samples from the oscillatory PDF, convolving with a standard $\Lambda$CDM transfer function, and evolving with the RAMSES full N-body code and the L-PICOLA approximate-companion mock generator (L-PICOLA implements the COmoving Lagrangian Acceleration method, an approximate fast substitute for full N-body gravity), they generated 65 cosmological volumes (box size 500 Mpc, $256^3$ particles, five realizations per model) spanning four non-Gaussianity levels and three oscillation frequencies, and measured the late-time matter power spectrum, bispectrum, and trispectrum at $z=0$ and $z=2.2$. Their findings: deviations from Gaussian initial conditions are at most at the percent level — the best case, model NG4f2 (which lies well inside the CMB-allowed region), shows a roughly 2% scale-dependent suppression in the bispectrum — and are marginal against the sample variance across realizations. Crucially, their full bispectrum analysis used only equilateral triangular configurations of the three wavenumbers, and their trispectrum only equal-side ('square', generally folded) quadrilateral configurations; a preliminary check of non-equilateral bispectrum triangles showed no significant difference, but the authors state that a complete treatment demands non-symmetric $n$-point functions, so the sensitivity of the vast space of non-symmetric configurations remains unquantified. They also note that the oscillatory PDF should modulate the halo mass function (the comoving number density of dark matter halos per unit mass) across a range of masses, and that Minkowski functionals — morphological descriptors of isodensity contours (enclosed volume, surface area, integrated mean curvature, and genus/Euler characteristic), standard statistics for testing Gaussianity of random fields — or other topological measures may be more sensitive probes than polyspectra, since this non-Gaussianity is not fully characterized by any finite set of $n$-point functions.

Problem Statement

Determine whether the class of primordial non-Gaussianity defined by an oscillatory modulation of the Gaussian probability density function of the primordial curvature perturbation (the Chen–Palma axion-spectator model, parametrized by oscillation frequency $f$ and amplitude $A^2$, and restricted to the region allowed by CMB trispectrum constraints, $\beta \equiv A^2\sigma_\zeta^4 f_\zeta^{-4}\exp(-\sigma_\zeta^2/2f_\zeta^2) < 2.1\times10^{-3}$) leaves a statistically significant, observationally usable imprint in the late-time large-scale structure. The single overarching objective — quantifying the detectability of this specific non-Gaussian PDF in LSS observables and in direct confrontation with CMB and LSS data — comprises the following jointly supporting parts left open by Peña & Candlish (2022): (1) measure the late-time matter bispectrum and trispectrum in general, non-symmetric $k$-space configurations (beyond the equilateral bispectrum and equal-side folded-square trispectrum so far analyzed; the only existing non-equilateral result is a preliminary bispectrum check reporting no significant difference), establishing whether any configuration family shows a signal stronger than the percent-level deviations already found; (2) quantify the imprint in observables beyond polyspectra — in particular the predicted oscillatory modulation of the halo mass function across halo masses, Minkowski functionals and other topological or full-PDF statistics of the evolved density field — and determine which observable is most sensitive to the oscillatory primordial PDF; (3) establish, over a broader and finer grid of $(f, A^2)$ within the CMB-allowed region and with simulation ensembles of sufficient volume, mass/spatial resolution, and realization count, whether the signals in (1) and (2) rise above sample variance to a level detectable by current or planned LSS surveys; and (4) confront the model with actual data using the full primordial PDF rather than polyspectra alone — CMB full-PDF constraints and LSS observables — yielding either a detection or an explicit exclusion region in the $(f, A^2)$ plane. Because the cited studies fix no operational detectability threshold, the criteria that define a statistically significant, observationally usable imprint — the survey, redshift, tracer, and significance level, together with the ensemble volume, mass/spatial resolution, and realization count required to reach it — are left to the solver's explicit, justified choice.

References

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