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Slow-rolling down the curvature: a reassessment of the Planck constraints on $φ^2$ inflation in a closed universe

Enrico Specogna, Tatevik Vardanyan, William Giarè, Eleonora Di Valentino

TL;DR

The paper reassesses Planck constraints on spatial curvature by embedding a closed-universe inflationary scenario with a gauge-invariant primordial spectrum into Planck analyses. It implements the spectrum in CAMB as a flat-spectrum multiplied by a curvature correction factor, sampling $\{\Omega_{\mathcal{K}},\epsilon,\delta\}$ across Planck likelihoods. The results show that the inferred curvature moves closer to flatness compared with gauge-dependent approaches: for PL18, $\Omega_{\mathcal{K}}\approx-0.026^{+0.012}_{-0.011}$ (about $2.5\sigma$ from flat), while CamSpec+lensing yields $\Omega_{\mathcal{K}}\approx-0.0078^{+0.0060}_{-0.0032}$, reducing the Planck preference for negative curvature. The model improves the low-$\ell$ quadrupole fit but may degrade high-$\ell$ fits in some likelihoods, illustrating that inflationary consistency reshapes curvature inferences. The study emphasizes deriving primordial spectra within physically motivated models and outlines future work on tensor perturbations and alternative inflationary scenarios to robustly assess cosmic curvature with upcoming CMB data.

Abstract

We revisit the Cosmic Microwave Background (CMB) constraints on the spatial curvature of the Universe, assessing how they change when the curvature parameter and the primordial inflationary scalar spectrum are treated consistently within theoretically motivated frameworks. Instead of relying on the phenomenological parametrisation commonly used to capture curvature effects at the largest scales, we present a case study based on closed quadratic inflation, where the primordial spectrum is derived in full generality and in a gauge-invariant manner. Within this framework, we analyze both the $\texttt{plik}$ PR3 and $\texttt{CamSpec}$ PR4 Planck CMB likelihoods and find that the constraints on $Ω_{\mathcal{K}}$ shift towards spatial flatness. In $\texttt{plik}$ the preference for $Ω_{\mathcal{K}}<0$ decreases from $\gtrsim 3.5σ$ to $\sim 2.5σ$, while in $\texttt{CamSpec}$ it reduces to $\sim 2σ$. At large angular scales ($\ell < 10$), our model explains the low-$\ell$ power suppression anomaly, notably improving the fit to the quadrupole. However, the reduced preference for highly negative values of $Ω_{\mathcal{K}}$ only partially accounts for the lensing anomaly at high multipoles, worsening the fit to the $\texttt{plik}$ spectrum at small scales. By contrast, in the $\texttt{CamSpec}$ PR4 spectrum, where the lensing anomaly is less pronounced, the model yields an overall improvement. Our analysis highlights a key conceptual point: closed-inflation models tie the curvature parameter to the inflationary dynamics and the primordial spectrum, enforcing consistency conditions that do not necessarily allow for the large deviations from flatness seen in phenomenological parametrisations. In the case of quadratic inflation, these restrictions reduce the apparent evidence for negative curvature reported by earlier analyses, while allowing for a mildly closed geometry.

Slow-rolling down the curvature: a reassessment of the Planck constraints on $φ^2$ inflation in a closed universe

TL;DR

The paper reassesses Planck constraints on spatial curvature by embedding a closed-universe inflationary scenario with a gauge-invariant primordial spectrum into Planck analyses. It implements the spectrum in CAMB as a flat-spectrum multiplied by a curvature correction factor, sampling across Planck likelihoods. The results show that the inferred curvature moves closer to flatness compared with gauge-dependent approaches: for PL18, (about from flat), while CamSpec+lensing yields , reducing the Planck preference for negative curvature. The model improves the low- quadrupole fit but may degrade high- fits in some likelihoods, illustrating that inflationary consistency reshapes curvature inferences. The study emphasizes deriving primordial spectra within physically motivated models and outlines future work on tensor perturbations and alternative inflationary scenarios to robustly assess cosmic curvature with upcoming CMB data.

Abstract

We revisit the Cosmic Microwave Background (CMB) constraints on the spatial curvature of the Universe, assessing how they change when the curvature parameter and the primordial inflationary scalar spectrum are treated consistently within theoretically motivated frameworks. Instead of relying on the phenomenological parametrisation commonly used to capture curvature effects at the largest scales, we present a case study based on closed quadratic inflation, where the primordial spectrum is derived in full generality and in a gauge-invariant manner. Within this framework, we analyze both the PR3 and PR4 Planck CMB likelihoods and find that the constraints on shift towards spatial flatness. In the preference for decreases from to , while in it reduces to . At large angular scales (), our model explains the low- power suppression anomaly, notably improving the fit to the quadrupole. However, the reduced preference for highly negative values of only partially accounts for the lensing anomaly at high multipoles, worsening the fit to the spectrum at small scales. By contrast, in the PR4 spectrum, where the lensing anomaly is less pronounced, the model yields an overall improvement. Our analysis highlights a key conceptual point: closed-inflation models tie the curvature parameter to the inflationary dynamics and the primordial spectrum, enforcing consistency conditions that do not necessarily allow for the large deviations from flatness seen in phenomenological parametrisations. In the case of quadratic inflation, these restrictions reduce the apparent evidence for negative curvature reported by earlier analyses, while allowing for a mildly closed geometry.

Paper Structure

This paper contains 8 sections, 17 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. The Gauge-invariant Primordial power Spectrum in a Closed Universe
  3. Methodology and datasets
  4. Re-expressing $\mathcal{P}^c_\zeta$
  5. Planck Likelihoods and Data
  6. Results
  7. Discussion and Conclusion
  8. Constraints including low-redshift data

Figures (6)

  • Figure 1: Left panel: Variations of $C_\ell^{TT}$ when the slow-roll parameter $\epsilon$ is varied in the range $\epsilon = [0.01, 0.05]$ with fixed $\delta = 0.02$ and $\Omega_K = -0.01$. Right panel: Variations of $C_\ell^{TT}$ when the slow-roll parameter $\delta$ is varied in the range $\delta = [-0.03, 0.03]$ with fixed $\epsilon = 0.02$ and $\Omega_K = -0.01$.
  • Figure 2: The CMB TT spectrum according to Eq.(\ref{['eq:spec_param']}) (upper figure) and the one calculated with the standard CAMB implementation (lower figure). The lower $\Omega_{\mathcal{K}}$, the stronger the suppression of the low-$\ell$ spectrum.
  • Figure 3: Two-dimensional marginalized constraints at 68% and 95% CL on $\Omega_{\mathcal{K}},\, \epsilon,\, \delta$ in the gauge-invariant closed-universe model, from different Planck likelihood combinations (PL18, PL18+lensing, CamSpec, and CamSpec+lensing).
  • Figure 4: One-dimensional marginalized constraints at 68% CL on $\Omega_{\mathcal{K}}$ from PL18 and CamSpec likelihoods, comparing the standard CAMB implementation of curvature with our gauge-invariant closed-universe model.
  • Figure 5: Comparison of the best-fit CMB temperature spectra at low multipoles: $\Lambda$CDM (orange, solid line), the standard curvature implementation (blue, solid line), and our gauge-invariant positive-curvature model (green, dash-dotted line). The data points are the Planck 2018 TT spectrum measurements.
  • ...and 1 more figures