Contributions to the Dark Matter 3-Pt Function from the Radiation Era

TL;DR

The paper investigates how nonlinear growth after horizon re-entry, including radiation-era dynamics and relativistic corrections, affects the dark matter bispectrum and how these non-primordial contributions compare to primordial non-Gaussianity signals parameterized by $f_{\rm NL}$. It develops a second-order perturbative framework and a numerical method that solves the second-order equations with first-order sources, bridging Newtonian gravity with radiation and general relativistic effects. Key findings show that radiation-era corrections on short-wavelength modes can mimic $f_{\rm NL}$ of order ~4, while relativistic (GR) corrections become important for longer wavelengths at the few-$f_{\rm NL}$ level; the total correction is well approximated by the sum of radiation and GR pieces, even in $\Lambda$CDM. The work highlights implications for interpreting large-scale structure data, emphasizing that radiation- and relativistic-induced bispectrum corrections can bias or obscure primordial non-Gaussian signals unless properly modeled, and outlines gauge-related caveats and observational considerations necessary to translate these results into robust constraints.

Abstract

We consider the contribution to the three-point function of matter density fluctuations from nonlinear growth after modes re-enter the horizon, and discuss effects that must be included in order to predict the three-point function with an accuracy comparable to primordial nongaussianities with f_NL ~ few. In particular, we note that the shortest wavelength modes measured in galaxy surveys entered the horizon during the radiation era, and, as a result, the radiation era modifies their three-point function by a magnitude equivalent to f_NL ~ O(4). On longer wavelengths, where the radiation era is negligible, we find that the corrections to the nonlinear growth from relativistic effects become important at the level f_NL ~ few. We implement a simple method for numerically calculating the three-point function, by solving the second-order equations of motion for the perturbations with the first order perturbations providing a source.

Figures (9)

• Figure 1: The kernel $x_1^{-1} x_2^{-1} Q(x_1, x_2)$ for $k$ fixed at $k=11 k_{\rm eq}$. The left plot is from our numeric computation, and is indistinguishable by eye from the PT limit given in equation (\ref{['eq:ptkernel']}). The right plot is the difference between the numeric result and the PT result, rescaled by $a_{\rm eq}$, because by equation (\ref{['eq:BGappx']}), $a_{\rm eq}$ controls the size of the radiation correction.
• Figure 2: The log-enhanced difference between the PT kernel and the kernel from using Newtonian gravity with a background radiation component. The analytic result is given in equation (\ref{['eq:f2rad']}).
• Figure 3: The correction $|\delta Q|$ for $k_1=k_2=yk_3, k_3 \equiv k$ with $y$ fixed at ${{\frac{1}{2}}}, 1, 10$, from left to right, respectively. The correction is shown for a) a matter-only universe ($a_{\rm eq} =0$), red, dotted, b) the exact numeric correction, black, solid, and c) the difference in Newtonian gravity between a universe with matter and one with matter + radiation (blue, dot-dashed, for which the log-enhanced piece of equation (\ref{['eq:f2rad']}) vanishes when $y={{\frac{1}{2}}}$; the size is independent of $k$ because it is the difference between two results within Newtonian gravity). The black dashed line is the exact radiation effect, defined as the difference between the exact numeric correction and the matter-only correction. The exact numeric calculation has $a_{\rm eq} =3 \times 10^{-4}$, in which case $k_{\rm eq}/H \approx 80$.
• Figure 4: Comparison between exact numeric corrections and an analytic approximation $\delta Q_2 \equiv \delta Q_{2,rad} + \delta Q_{2,GR}$. The corrections $|\delta Q|$ for $k_1=k_2=yk_3\equiv yk$ with $y$ fixed at ${{\frac{1}{2}}}, 1, 10$, for left, center, right, respectively. In the top row, the exact result (analytic approximation) is shown in solid, black (blue, dashed). The difference between the exact and approximate results for $\delta Q$ are shown in the second row, divided by their sum in quadrature.
• Figure 5: The correction $|\delta Q|$ relative to the primordial contribution with $f_{\rm NL}=1$ on equilateral triangles as a function of redshift $z$. The wavenumber in the left plot is, from top to bottom, $k =k_{\rm eq} \times \{30, 10, 4,3, 2.3 \}$. As $k$ decreases from $30 k_{\rm eq}$, the GR corrections grow and approximately cancel the radiation corrections near $k\sim 2 k_{\rm eq}$. At smaller $k$, the correction from GR is well-approximated by the matter-only analytic result (\ref{['eq:GRMD']}). The right plot shows the exact result relative to $\delta Q_{f_{\rm NL} =1}$ for $k=0.4 k_{\rm eq}$, where essentially all the correction is from GR. The reason the correction in the left plot decreases at large $z$ is that $\delta/\phi$ gets smaller in the past, so the contribution from $f_{\rm NL}$ is bigger in terms of $\delta$.