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Inflation Model Selection meets Dark Radiation

Thomas Tram, Robert Vallance, Vincent Vennin

TL;DR

The paper assesses how the presence of dark radiation, parameterized by $N_ ext{eff}$, alters inflation-model selection by performing a full Bayesian analysis that jointly varies inflation and cosmological parameters with self-consistent reheating. Using Planck lowP+TT+lensing data and optionally BK14+BAO and local $H_0$ measurements, the authors compare a set of representative single-field potentials (and curvaton scenarios) in both standard and dark-radiation cosmologies, computing Bayesian evidences relative to a Starobinsky reference. They find that most plateau-type models remain favored, while large-field models like power-law inflation can become favoured when $N_ ext{eff}$ is allowed to vary, with $\ tex{Delta}N_ ext{eff}^ ext{PLI} \approx 0.62$. However, incorporating BK14+BAO data typically disfavors PLI again, and including local $H_0$ measurements can restore PLI’s standing, illustrating a deep link between the dark radiation solution to the $H_0$ tension and inflation-model selection. The study also shows curvaton-type non-single-field models can gain footing in extended cosmologies, and robust constraints on the reheating parameter persist across cosmological extensions, underscoring the interplay between early-Universe dynamics and late-time cosmological inferences.

Abstract

We investigate how inflation model selection is affected by the presence of additional free-streaming relativistic degrees of freedom, i.e. dark radiation. We perform a full Bayesian analysis of both inflation parameters and cosmological parameters taking reheating into account self-consistently. We compute the Bayesian evidence for a few representative inflation scenarios in both the standard $Λ$CDM model and an extension including dark radiation parametrised by its effective number of relativistic species $N_\mathrm{eff}$. Using a minimal dataset (Planck low-$\ell$ polarisation, temperature power spectrum and lensing reconstruction), we find that the observational status of most inflationary models is unchanged. The exceptions are potentials such as power-law inflation that predict large values for the scalar spectral index that can only be realised when $N_\mathrm{eff}$ is allowed to vary. Adding baryon acoustic oscillations data and the B-mode data from BICEP2/Keck makes power-law inflation disfavoured, while adding local measurements of the Hubble constant $H_0$ makes power-law inflation slightly favoured compared to the best single-field plateau potentials. This illustrates how the dark radiation solution to the $H_0$ tension would have deep consequences for inflation model selection.

Inflation Model Selection meets Dark Radiation

TL;DR

The paper assesses how the presence of dark radiation, parameterized by , alters inflation-model selection by performing a full Bayesian analysis that jointly varies inflation and cosmological parameters with self-consistent reheating. Using Planck lowP+TT+lensing data and optionally BK14+BAO and local measurements, the authors compare a set of representative single-field potentials (and curvaton scenarios) in both standard and dark-radiation cosmologies, computing Bayesian evidences relative to a Starobinsky reference. They find that most plateau-type models remain favored, while large-field models like power-law inflation can become favoured when is allowed to vary, with . However, incorporating BK14+BAO data typically disfavors PLI again, and including local measurements can restore PLI’s standing, illustrating a deep link between the dark radiation solution to the tension and inflation-model selection. The study also shows curvaton-type non-single-field models can gain footing in extended cosmologies, and robust constraints on the reheating parameter persist across cosmological extensions, underscoring the interplay between early-Universe dynamics and late-time cosmological inferences.

Abstract

We investigate how inflation model selection is affected by the presence of additional free-streaming relativistic degrees of freedom, i.e. dark radiation. We perform a full Bayesian analysis of both inflation parameters and cosmological parameters taking reheating into account self-consistently. We compute the Bayesian evidence for a few representative inflation scenarios in both the standard CDM model and an extension including dark radiation parametrised by its effective number of relativistic species . Using a minimal dataset (Planck low- polarisation, temperature power spectrum and lensing reconstruction), we find that the observational status of most inflationary models is unchanged. The exceptions are potentials such as power-law inflation that predict large values for the scalar spectral index that can only be realised when is allowed to vary. Adding baryon acoustic oscillations data and the B-mode data from BICEP2/Keck makes power-law inflation disfavoured, while adding local measurements of the Hubble constant makes power-law inflation slightly favoured compared to the best single-field plateau potentials. This illustrates how the dark radiation solution to the tension would have deep consequences for inflation model selection.

Paper Structure

This paper contains 13 sections, 9 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Method
  3. Dark radiation and the primordial power spectrum
  4. Primordial power spectrum from inflation
  5. Bayesian model selection
  6. Results
  7. Inflationary model selection
  8. An example of non single-field slow-roll models: curvaton scenarios
  9. Combining different data sets
  10. Constraining the reheating parameter
  11. Constraining the effective number of relativistic species
  12. Including local measurements of the present expansion rate
  13. Conclusion

Figures (7)

  • Figure 1: Scalar spectral index $n_{{\mathrm{S}}}$ and tensor-to-scalar ratio $r$ predicted by some of the models considered in this work. For natural inflation ($\mathrm{NI}$) and power-law inflation ($\mathrm{PLI}$), the arrow indicates in which direction $f$ [cf. equation. (\ref{['eq:pot:ni']})] and $\alpha$ [cf. equation. (\ref{['eq:pot:pli']})] respectively vary. For $\mathrm{MC}_5\mathrm{LFI}_2$ (see section \ref{['sec:curvaton']}), the arrow denotes how $n_{{\mathrm{S}}}$ and $r$ change when the contribution from the curvaton field $\sigma$ increases. The two remaining models are quadratic large field inflation ($\mathrm{LFI}_2$) and Higgs inflation (the Starobinsky model, $\mathrm{HI}$). In this figure, the inflaton is assumed to oscillate at the quadratic minimum of its potential once inflation ends (this assumption is dropped in the rest of the paper and is used here for display convenience only) and the colour denotes the reheating temperature $T_\mathrm{reh}$, see section \ref{['sec:reheating']} (for power-law inflation, the predictions are independent of $T_\mathrm{reh}$). The pink shaded surfaces are the one- and two-sigma contours of the Planck 2015 lowP+TT+lensing data when the standard $\Lambda\mathrm{CDM}$ cosmological model is assumed, while for the grey shaded surfaces, the effective number of relativistic degrees of freedom $N_\mathrm{eff}$ is allowed to vary.
  • Figure 2: Bayesian evidence of the inflationary models considered in this work when $N_\mathrm{eff} = 3.046$ is fixed to its standard value (left column, pink) and is allowed to vary in the interval $\Delta N_\mathrm{eff}\in[-2,3]$ (right column, grey). In both cases, the Bayesian evidence is normalised to Higgs inflation (HI, the Starobinsky model), taken as a reference model. The analysis is performed with the Planck 2015 lowP + TT + lensing data. The typical numerical sampling error is around $\ln\mathcal{E}\sim 0.3$. The vertical grey lines denote Jeffreys' scale, and the ${}^\vee_\wedge$ symbols display the best-fit values normalised to HI.
  • Figure 3: Same as in figure \ref{['fig:nsr']} with observational constraints derived when adding the Bicep2/Keck (BK14) and BAO data in the left panel, and the Bicep2/Keck + BAO + Planck high-$\ell$ polarisation data (TTTEEE) in the right panel. The green shaded surfaces are the one and two sigma contours when the standard $\Lambda\mathrm{CDM}$ cosmological model is assumed, while for the blue shaded surfaces, the effective number of relativistic degrees of freedom $N_\mathrm{eff}$ is allowed to vary. In both panels, the constraints from the base Planck 2015 lowP+TT+lensing data displayed in figure \ref{['fig:nsr']} have been recalled (pink and grey shaded areas).
  • Figure 4: Posterior distributions on the reheating parameter $R_\mathrm{reh}$ for the single-field models considered in this paper, in the standard cosmological model (solid lines) and when $N_\mathrm{eff}$ is allowed to vary (dashed lines). The analysis is performed with the Planck 2015 lowP + TT + lensing data. Power-law inflation is not displayed since this potential is conformally invariant. This means that its predictions do not depend on $R_\mathrm{reh}$ and flat posterior distributions would be obtained in both cases.
  • Figure 5: Posterior distributions on the effective number of neutrino species $\Delta N_\mathrm{eff} = N_\mathrm{eff}-3.046$ for the single-field models discussed in this paper, from the Planck 2015 lowP + TT + lensing data.
  • ...and 2 more figures