Scale-dependent bias of galaxies and mu-type distortion of the cosmic microwave background spectrum from single-field inflation with a modified initial state

TL;DR

This work investigates how a non‑Bunch‑Davies initial state for a single‑field inflaton, encoded by a Bogoliubov transformation with occupation $N$ and phase $\theta_k$, affects observable non‑Gaussian signatures. The authors show that the squeezed‑limit bispectrum is enhanced by a factor of $k_1/k$, producing a distinctive $k^{-3}$ scale‑dependent halo bias and enabling a cross‑correlation between large‑scale CMB temperature and small‑scale μ‑type distortions, amplified by $k_D(z_i) r_L$. They quantify the μT cross‑signal for both local and modified initial‑state bispectra, finding that the modified state can yield a dramatically larger $C_l^{\mu T}$ (and S/N) than the local form, potentially detectable by absolutely calibrated experiments like PIXIE and possibly by Planck/LiteBIRD with precise relative calibration. The results provide a concrete observational handle on the inflationary initial state and motivate further theoretical work on initial‑state foundations, BEFT approaches, and folded‑limit signatures.

Abstract

We investigate the phenomenological consequences of a modification of the initial state of a single inflationary field. While single-field inflation with the standard Bunch-Davies initial vacuum state does not generally produce a measurable three-point function (bispectrum) in the squeezed configuration, allowing for a non-standard initial state produces an exception. Here, we calculate the signature of an initial state modification in single-field slow-roll inflation in both the scale-dependent bias of the large-scale structure (LSS) and mu-type distortion in the black-body spectrum of the cosmic microwave background (CMB). We parametrize the initial state modifications and identify certain choices of parameters as natural, though we also note some fine-tuned choices that can yield a larger bispectrum. In both cases, we observe a distinctive k^-3 signature in LSS (as opposed to k^-2 for the local-form). As a non-zero bispectrum in the squeezed configuration correlates a long-wavelength mode with two short-wavelength modes, it induces a correlation between the CMB temperature anisotropy on large scales with the temperature-anisotropy-squared on very small scales; this correlation persists as the small-scale anisotropy-squared is processed into mu-type distortions. While the local-form mu-distortion turns out to be too small to detect in the near future, a modified initial vacuum state enhances the signal by a large factor owing to an extra factor of k_1/k. For example, a proposed absolutely-calibrated experiment, PIXIE, is expected to detect this correlation with a signal-to-noise ratio greater than 10, for an occupation number of about 0.5 in the observable modes. Relatively calibrated experiments such as Planck and LiteBIRD should also be able to measure this effect, provided that the relative calibration between different frequencies meets the required precision. (Abridged)

Figures (6)

• Figure 1: Scale-dependent halo-bias from single-field inflation with a non-standard initial state, using a smoothing scale of $R=1~h^{-1}$ Mpc. The occupation number is $N=0.5$, the slow-roll parameter $\epsilon=0.01$, and the initial conformal time $|\eta_0|=1.0\times10^6\ \text{Mpc}$ (the results are insensitive to the exact choice of $\eta_0$, so long as it is large). The bottom three (thicker) lines show the more natural case where $\theta_k\approx k \eta_0$, while the top two (thinner) lines show the case when $\theta_k\approx\text{const}$ is chosen to give the maximal halo bias. The dashed lines show the new approximations given by Eqs. \ref{['eq:n_not_const_approx']} and \ref{['eq:n_const_approx']}, while the dot-dashed line shows the approximation used in Ganc:2011dy (which is equal to Eqs. \ref{['eq:n_not_const_approx']} and \ref{['eq:n_const_approx']} without the last lines).
• Figure 2: Scale-dependent bias from the local-form bispectrum (dot-dashed line) versus the modified initial-state case described herein (solid line). The parameters here are the same as in Fig. \ref{['fig:thet-not-const_v_thet-const']}, with $f_{\text{NL}}=1$ for the local-form bispectrum. The difference in scaling between the models is quite evident.
• Figure 3: $C_l^{\mu T}$, the cross-power spectrum of the $\mu$-type distortion and the CMB temperature anisotropy, from the local-form bispectrum with $f_{\rm NL}=1$. The solid line shows $C_l^{\mu T}$ using the full radiation transfer function, while the dot-dashed line shows it using the Sachs-Wolfe approximation. The amplitude of $C_l^{\mu T}$ is linearly proportional to $f_{\rm NL}$.
• Figure 4: Signal-to-noise ratio of $C_l^{\mu T}$ from the local-form bispectrum with $f_{\rm NL}=1$. The bottom two lines show $C_l^{\mu T}$ using the full radiation transfer function, while the dot-dashed line shows it using the Sachs-Wolfe approximation. The solid, dashed line is for $\theta_b=0$ (ideal) and $1.6^\circ$ (PIXIE), respectively. The noise level is $C_l^{\mu\mu,N}=4\pi\times 10^{-16}e^{l^2\theta_b^2/(8\ln 2)}$ for all cases (i.e., the r.m.s. uncertainty of $\mu$ averaged over the full sky is $10^{-8}$). The signal-to-noise is proportional to $f_{\rm NL}$ and is inversely proportional to the r.m.s. uncertainty of $\mu$.
• Figure 5: A comparison of the shapes of $C_l^{\mu T}$ (the cross-power spectra of the $\mu$-type distortion and the CMB temperature anisotropy) from the local-form bispectrum (solid line) and the modified state bispectrum (dot-dashed line). The amplitudes of the graphs are scaled so they can both appear in the same figure; therefore, the overall amplitude is arbitrary.