Cosmic variance in inflation with two light scalars

TL;DR

The paper analyzes how a light spectator scalar coupled to the inflaton in quasi-single-field inflation induces long-short mode coupling, leading to cosmic variance in sub-volume statistics when $m_{\sigma} \lesssim 0.1\,H$. By computing higher-order contact diagrams from non-derivative self-interactions and employing an in-in formalism with a long-short mode split, the authors show that sub-volume statistics (power spectrum, $f_{\rm NL}$) vary, but the squeezed-limit scaling of the bispectrum remains tied to the mass via $\nu$, allowing robust mass inference despite cosmic variance. The work derives general expressions for $n$-point functions and demonstrates that the locally observed squeezed-limit bispectrum preserves its $k_L/k_S$ scaling across sub-volumes, while global amplitudes and tilt can fluctuate; this reveals an initial-conditions problem where local data may not uniquely specify the global inflationary Lagrangian. The results emphasize that, although cosmic variance obscures some details of the hidden-field potential, the mass parameter can still be diagnosed from squeezed-limit observations, and they outline directions to extend the analysis to exchange diagrams and derivative interactions.

Abstract

We examine the squeezed limit of the bispectrum when a light scalar with arbitrary non-derivative self-interactions is coupled to the inflaton. We find that when the hidden sector scalar is sufficiently light ($m\lesssim0.1\,H$), the coupling between long and short wavelength modes from the series of higher order correlation functions (from arbitrary order contact diagrams) causes the statistics of the fluctuations to vary in sub-volumes. This means that observations of primordial non-Gaussianity cannot be used to uniquely reconstruct the potential of the hidden field. However, the local bispectrum induced by mode-coupling from these diagrams always has the same squeezed limit, so the field's locally determined mass is not affected by this cosmic variance.

Figures (6)

• Figure 1: Interaction terms in quasi-single field inflation: a transfer vertex between $\varphi$ (solid line) and $\sigma$ (dashed line) with dimensionless coupling strength $\rho/H$, and the cubic self-interaction of $\sigma$ with dimensionless coupling $\mu/H$.
• Figure 2: The four-point interaction vertex. As before solid lines represent the $\varphi$ field while dashed lines are $\sigma$.
• Figure 3: The figure shows the shift between the observed and global power spectrum parameters $\frac{ \left|\Delta r \right|}{r} = \frac{\left|r^{\text{obs}}-r\right|}{r}$ and $\frac{\left|\Delta n_s\right|}{n_s} = \frac{\left|n_s^{\text{obs}}-n_s\right|}{n_s}$ due to long and short modes coupling in a scenario where the large volume is weakly non-Gaussian (and the non-linear terms are only quadratic in the Gaussian field). The shift in parameters for sub-volumes one, two or three standard deviations from the mean value are shown (darkest to lightest regions respectively). The statistics of the large volume have $f_{\text{NL}}=1$. The radial lines of the mesh indicate lines of constant $\nu$ (from vertical line for $\nu=1.5$, i.e. $\frac{m}{H}=0$ to almost horizontal line for $\nu=1.46$, i.e. $\frac{m}{H}\simeq 0.34$), with a separation of $\Delta\nu=0.01$. The curved lines of the mesh are lines of constant $\langle\zeta_L^2\rangle$, or close to lines of constant $N_{\text{extra}}$ (since $\zeta_L$ depends on $n_s$, $\Delta_{\zeta}^2$ and $N_{\text{extra}}$, there is a degeneracy). The value $\nu=1.495$ corresponds to $\frac{m}{H}\simeq 0.12$. We set $k_0$ to be the largest CMB observable mode $k_0=0.008$ Mpc$^{-1}$, and values for the amplitude of the scalar spectrum and the spectral index at $k_*=0.05$ Mpc$^{-1}$ are taken from Ade:2015lrj. Non-gaussian corrections to the power spectrum of the large volume have been taken into account to plot $\frac{\left| \Delta r\right|}{r}$, although they are not large in the scenario plotted here.
• Figure 4: The vertex showing the correction to the power spectrum due to a long mode. The long mode, in grey, is outside the horizon and classical.
• Figure 5: The four-point interaction from the cubic interaction. As before solid lines represent the $\varphi$ field while dashed lines are $\sigma$.