Analytical approximations for curved primordial tensor spectra

TL;DR

This work addresses how spatial curvature during inflation shapes the primordial tensor spectrum. By extending a potential-independent analytic framework to tensor modes and modeling the background with a two-phase KD→USR history, the authors derive closed-form solutions for tensor perturbations in curved spacetimes and construct a template for the tensor power spectrum with curvature-shifted wavevectors $k_\pm$. The resulting spectrum exhibits oscillations and large-scale suppression or enhancement (depending on the sign of $K$) that translate into distinctive $B$-mode signatures in the CMB, offering a potential route to constrain primordial curvature with upcoming observations. The approach provides a unified, curvature-focused interpretation that complements numerical analyses and scalar analyses, with broad applicability to $K\Lambda$CDM and potential extensions to higher-order corrections and mixed scalar–tensor correlations.

Abstract

We build upon previous analytical treatments of scalar perturbations in curved inflationary universes to obtain analytical templates for the primordial tensor power spectrum in models with non-zero primordial spatial curvature. These templates are derived without assuming a particular inflaton potential, thereby isolating the universal imprints of curvature on tensor modes. Our results predict characteristic large-scale features -- including low-$\ell$ cut-offs and oscillatory patterns -- that are consistent with numerical solutions and provide a clear physical interpretation of how curvature modifies the underlying dynamics. In particular, we show that curvature effects manifest mathematically as systematic shifts in the dynamically relevant wavevectors, mirroring the behaviour previously identified in the scalar power spectrum. These features translate into distinctive signatures in the large-angle $B$-mode polarisation spectrum, offering a potential discriminant for spatial curvature in forthcoming CMB observations.

Figures (6)

• Figure 1: Left: Primordial tensor power spectrum $\mathcal{P}_{\mathcal{T}}$ corresponding to the range of allowed values of the transition time $\eta_{t}$ for closed universes $K = +1$. Oscillations and a generic suppression of power are visible at low-$k$. Only integer values of comoving $k$ with $k \geq 3$ are allowed hence the jagged spectra, here the dots indicate the first $100$ comoving $k$ and for clarity we include the continuous spectrum. Right: the corresponding low-$\ell$ effects on the CMB unlensed $B$-mode power spectrum. The power law $K\Lambda$CDM spectrum is highlighted in dashed grey. There is no appreciable deviation from the traditional power spectrum at higher $k$ and $\ell$ values.
• Figure 2: Same as Figure \ref{['fig:ClosedTensorSpectra']}, but for open universes, $K = -1$. The primordial tensor power spectra exhibit a mild enhancement of power at low $k$, in contrast to the suppression observed in the closed case, notably dependent on the choice of transition time $\eta_{t}$. As $\eta_t \to \eta_{\text{max}}$, the leading order term in $\mathcal{P}_{\mathcal{T}}$ is an exponential, hence the exponential growth observed. At $k=\sqrt{10/3}$, $k_{-}$ becomes imaginary and introduces a cutoff to the spectrum. Right: The corresponding CMB $B$-mode spectra display a small excess of power at the low $\ell$, while converging to the standard $K\Lambda$CDM prediction at higher $\ell$.
• Figure 3: Simulated closed universes $(K=+1)$ CMB $B$-mode polarisation maps of four chosen values of transition time $\eta_t \in \{(0.03,0.1,0.5,1)\times \eta_{\text{max}} \}$ where $\eta_{\text{max}} = \pi/4$, and one for fiducial closed $K\Lambda$CDM. Each image is a $15^{\circ}\times 5^{\circ}$ flat-sky projection capturing the low multipole ($< 500$) signatures of the $B$-mode polarisation. The polarisation maps have the colourbar truncated to $\pm7\times 10^{-5} \mu K^2$.
• Figure 4: Simulated open universes $(K=-1)$ CMB $B$-mode polarisation maps of four chosen values of transition time $\eta_t \in \{(0.03,0.1,0.5,1)\times \eta_{\text{max}} \}$ where $\eta_{\text{max}} = \pi/4$, and one for fiducial open $K\Lambda$CDM. Each image is a $15^{\circ}\times 5^{\circ}$ flat-sky projection capturing the low multipole ($< 500$) signatures of the $B$-mode polarisation. The polarisation maps have the colourbar truncated to $\pm 7 \times 10^{-5} \mu K^2$.
• Figure 5: Simulated closed universe $(K=+1)$ CMB $B$-mode polarisation map differences between closed $K\Lambda$CDM spectra and the analytic spectra for the given values of transition time $\eta_{t}$. We generate the polarisation maps taken from the analytical spectra $\mathcal{D}^{BB}_{\ell,K\Lambda CDM} -\mathcal{D}^{BB}_{\ell,\eta_{t}}$, where $\eta_t \in \{(0.03,0.1,0.5,1)\times \eta_{\text{max}} \}$ and $\eta_{\text{max}} = \pi/4$. Each image is a $17^{\circ}\times 17^{\circ}$ flat-sky projection to better capture the low multipole ($< 500$) signatures of the $B$-mode polarisation. The labels also identify the colourbar scale for each polarisation map. Each map has the colourbar truncated to order $\mathcal{O}(10^{-7}) \mu K^{2}$.