One-Loop Tensor Power Spectrum from a Non-Minimally Coupled Spectator Field during Inflation

Abstract

We compute the full one-loop corrections to the primordial tensor power spectrum in an inflationary scenario with a non-minimally coupled spectator field, using the in-in formalism. We derive semi-analytic results for the scalar-sourced one-loop tensor spectrum and the effective tensor-to-scalar ratio, $r_{\mathrm{eff}}$ . We consider two representative coupling functions: a localized Gaussian dip (Model G), which leads to moderate loop corrections, and a rapidly oscillatory coupling (Model O), which can yield much larger loop contributions. For Model G, we find a $\mathcal{O}(1)$ correction to $r_{\mathrm{eff}}$ while Model O can significantly enhance $r_{\mathrm{eff}}$ by several orders of magnitude (relative to the tree-level value). We further calculate the energy density of primordial gravitational waves. Assuming that primordial black holes with mass $10^{-12}M_{\odot}$ generated in this scenario, constitute all of the dark matter, we find that the results are several orders of magnitude lower than the sensitivities of Taiji/TianQin/LISA.

Figures (5)

• Figure 1: One-loop Feynman diagrams correspond to $P_{h1}$ and $P_{h2}$, respectively.
• Figure 2: The non-minimal coupling functions $f(x)$ in Eq. \ref{['equ:modelG']} and Eq. \ref{['equ:modelO']}. The horizontal axis represents the dimensionless time $x=p_*\tau$. Here we set $\Delta=0.1$, $\Lambda=0.01$, $\xi=0.001$ and the values of $A_G$ and $A_O$ are chosen for PBHs making up all of the dark matter. The vertical red dashed line marks the lower limit of the time integral, $x_0=-1$ (earlier contributions are neglected).
• Figure 3: The one-loop corrections to the tensor power spectrum for (a) Model G and (b) Model O. The vertical axis displays the loop contributions normalized by the tree-level spectrum $P_{h0}$ and the scaling factor $(r/0.01)^{-1}$, while the horizontal axis represents the dimensionless wavenumber $\tilde{q} \equiv q/p_*$. The solid blue curves correspond to the bubble diagram contribution ($P_{h2}$), and the dashed orange curves represent absolute value of the seagull diagram contribution ($|P_{h1}|$). The horizontal black dashed line marks unity ($P_{\text{loop}} \approx P_{\text{tree}}$). Note the varying vertical scales: Model O exhibits a strong enhancement ($\sim 10^{12}$), whereas Model G shows a relative milder feature.
• Figure 4: The effective tensor-to-scalar ratio for Model G and Model O as a function of the tree-level value. The gray shaded region denotes the $0.036$ bound given by BICEP:2021xfz at $95\%$ C.L. and the black dotted line refers to $r_{\mathrm{eff}} = r_0$.
• Figure 5: The present-day spectral energy density of primordial GWs as a function of frequency $f$ for Model G (blue curves) and Model O (orange curves). The dashed lines are the primordial GWs and the solid lines denote scalar-induced GWs generated by Model G and Model O Meng:2022low. The parameters for inflation are fixed at $\Delta = 0.1$, $\Lambda=0.01$, $\xi =0.001$. The characteristic scales are $p_*=7 \times 10^{11}\text{Mpc}^{-1}$, $p_*=9 \times 10^{9}\text{Mpc}^{-1}$ for Model G and Model O respectively so that the inflation can generate PBHs of $10^{-12}M_{\odot}$. The amplitude of the coupling function is chosen so that these PBHs make up all the dark matter Meng:2022low. The $4$-year power-law integrated sensitivity curves are also shown for TianQin TianQin:2015yph, TaijiHu:2017mde and LISA LISA:2017pwj.