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Calculating the local-type fNL for slow-roll inflation with a non-vacuum initial state

Jonathan Ganc

TL;DR

The paper investigates how a non-vacuum initial state during slow-roll inflation, parameterized by Bogoliubov coefficients $\alpha_k$, $\beta_k$ and the occupation number $N_k$ with phase $\theta_k$, can enhance the local-type non-Gaussianity $f_{ ext{NL}}$ in the CMB. It derives the primordial bispectrum, propagates it through the CMB transfer function with a full 2D projection, and computes the observable $f_{ ext{NL}}$ via an optimal estimator under various $N_k$ and $\theta_k$ scenarios, including a constant $N_k$ and a cutoff form $N_k\approx N_{k,0}e^{-k^2/k_{\text{cut}}^2}$. In the conservative limit $\theta_k \approx k\eta_0$, $f_{ ext{NL}} \lesssim 1.6\, (\epsilon/0.01)$, effectively undetectable, while allowing a constant $\theta_k$ with $N_k=O(1)$ yields larger values (e.g., $f_{ ext{NL}} \approx 28\,(\epsilon/0.01)$ or negative up to $-6.4\,(\epsilon/0.01)$), potentially detectable by Planck or future satellites. The results show that sizable local $f_{ ext{NL}}$ can arise without violating the single-field consistency relation, emphasizing the importance of initial-state effects in interpreting CMB non-Gaussianity.

Abstract

Single-field slow-roll inflation with a non-vacuum initial state has an enhanced bispectrum in the local limit. We numerically calculate the local-type fNL signal in the CMB that would be measured for such models (including the full transfer function and 2D projection). The nature of the result depends on several parameters, including the occupation number N_k, the phase angle θ_k between the Bogoliubov parameters, and the slow-roll parameter ε. In the most conservative case, where one takes θ_k \approx η_0 k (justified by physical reasons discussed within) and ε\lesssim 0.01, we find that 0 < fNL < 1.52 (ε/0.01), which is likely too small to be detected in the CMB. However, if one is willing to allow a constant value for the phase angle θ_k and N_k=O(1), fNL can be much larger and/or negative (depending on the choice of θ_k), e.g. fNL \approx 28 (ε/0.01) or -6.4 (ε/0.01); depending on ε, these scenarios could be detected by Planck or a future satellite. While we show that these results are not actually a violation of the single-field consistency relation, they do produce a value for fNL that is considerably larger than that usually predicted from single-field inflation.

Calculating the local-type fNL for slow-roll inflation with a non-vacuum initial state

TL;DR

The paper investigates how a non-vacuum initial state during slow-roll inflation, parameterized by Bogoliubov coefficients , and the occupation number with phase , can enhance the local-type non-Gaussianity in the CMB. It derives the primordial bispectrum, propagates it through the CMB transfer function with a full 2D projection, and computes the observable via an optimal estimator under various and scenarios, including a constant and a cutoff form . In the conservative limit , , effectively undetectable, while allowing a constant with yields larger values (e.g., or negative up to ), potentially detectable by Planck or future satellites. The results show that sizable local can arise without violating the single-field consistency relation, emphasizing the importance of initial-state effects in interpreting CMB non-Gaussianity.

Abstract

Single-field slow-roll inflation with a non-vacuum initial state has an enhanced bispectrum in the local limit. We numerically calculate the local-type fNL signal in the CMB that would be measured for such models (including the full transfer function and 2D projection). The nature of the result depends on several parameters, including the occupation number N_k, the phase angle θ_k between the Bogoliubov parameters, and the slow-roll parameter ε. In the most conservative case, where one takes θ_k \approx η_0 k (justified by physical reasons discussed within) and ε\lesssim 0.01, we find that 0 < fNL < 1.52 (ε/0.01), which is likely too small to be detected in the CMB. However, if one is willing to allow a constant value for the phase angle θ_k and N_k=O(1), fNL can be much larger and/or negative (depending on the choice of θ_k), e.g. fNL \approx 28 (ε/0.01) or -6.4 (ε/0.01); depending on ε, these scenarios could be detected by Planck or a future satellite. While we show that these results are not actually a violation of the single-field consistency relation, they do produce a value for fNL that is considerably larger than that usually predicted from single-field inflation.

Paper Structure

This paper contains 6 sections, 32 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Slow-roll inflation with a non-Bunch-Davies initial state
  3. The observed value of $f_{\text{NL}}$
  4. Results
  5. Discussion and conclusion
  6. Explicit Bispectrum

Figures (1)

  • Figure 1: For very large $k_{\text{cut}}$, where $N_k \approx N_{k,0} = \text{const}$, and fine-tuned values of $\theta_k=\text{const}$, $f_{\text{NL}}$ can be large. The maximum values (upper solid line) are achieved for $\cos \theta = (1 - 12 N - 12 N^2) / 6 \sqrt{N (N+1)} (1+2N)$, and the minimum values (lower dashed line) are achieved for $\cos \theta=-1$.