[Paper Review] Low reheating temperatures in monomial and binomial inflationary potentials

Thomas Rehagen, Graciela B. Gelmini|arXiv (Cornell University)|Apr 15, 2015

Cosmology and Gravitation Theories34 references9 citations

TL;DR

This paper investigates reheating temperatures in monomial and binomial inflationary models using Planck 2015 and BICEP2/Keck Array CMB data. It finds that a $φ^1$ potential with canonical reheating ($w_{\text{re}}=0$) yields $T_{\text{re}} \lesssim 6 \times 10^{10}\,\text{GeV}$, excluding instantaneous reheating, and opens the possibility of dark matter production during reheating, leading to distinct relic abundances and momentum distributions for WIMPs, sterile neutrinos, and axions.

ABSTRACT

We investigate the allowed range of reheating temperature values in light of the Planck 2015 results and the recent joint analysis of Cosmic Microwave Background (CMB) data from the BICEP2/Keck Array and Planck experiments, using monomial and binomial inflationary potentials. While the well studied $ϕ^2$ inflationary potential is no longer favored by current CMB data, as well as $ϕ^p$ with $p>2$, a $ϕ^1$ potential and canonical reheating ($w_{re}=0$) provide a good fit to the CMB measurements. In this last case, we find that the Planck 2015 $68\%$ confidence limit upper bound on the spectral index, $n_s$, implies an upper bound on the reheating temperature of $T_{re}\lesssim 6 imes 10^{10}\,{ m GeV}$, and excludes instantaneous reheating. The low reheating temperatures allowed by this model open the possiblity that dark matter could be produced during the reheating period instead of when the Universe is radiation dominated, which could lead to very different predictions for the relic density and momentum distribution of WIMPs, sterile neutrinos, and axions. We also study binomial inflationary potentials and show the effects of a small departure from a $ϕ^1$ potential. We find that as a subdominant $ϕ^2$ term in the potential increases, first instantaneous reheating becomes allowed, and then the lowest possible reheating temperature of $T_{re}=4\,{ m MeV}$ is excluded by the Planck 2015 $68\%$ confidence limit.

Motivation & Objective

• To determine the allowed range of reheating temperatures in monomial and binomial inflationary models using the latest CMB data.
• To assess whether low reheating temperatures—down to the BBN limit of 4 MeV—are compatible with current Planck and BICEP2/Keck Array constraints.
• To explore the implications of low reheating temperatures for dark matter production mechanisms, particularly non-thermal production during reheating.
• To investigate how small deviations from a $φ^1$ potential (via a subdominant $φ^2$ term) affect the allowed reheating temperature range.
• To evaluate the viability of instantaneous reheating and canonical reheating in these models under observational constraints.

Proposed method

• Used monomial potentials $V \propto \phi^p$ and binomial potentials $V \propto \phi^p + b\phi^q$ with $b\phi^{q-p} \ll 1$ to model inflationary dynamics.
• Applied the slow-roll approximation to compute the number of e-folds $N_k$ corresponding to the pivot scale $k=0.05\,\text{Mpc}^{-1}$.
• Calculated the spectral index $n_s$ and tensor-to-scalar ratio $r$ as functions of $N_k$, which depends on the reheating temperature $T_{\text{re}}$.
• Incorporated an effective equation of state $w_{\text{re}}$ for reheating to model non-instantaneous reheating, with $\rho \propto a^{-3(1+w_{\text{re}})}$.
• Used Planck 2015 $68\%$ confidence limit on $n_s = 0.9683 \pm 0.0059$ and the BICEP2/Keck Array/Planck upper bound $r < 0.12$ at 95% CL to constrain $T_{\text{re}}$.
• Evaluated the impact of varying the subdominant term $b$ in binomial potentials on the allowed $T_{\text{re}}$ range, particularly for $p=1$, $q=2$.

Experimental results

Research questions

• RQ1 What is the maximum allowed reheating temperature for a $φ^1$ inflationary potential under canonical reheating, given Planck 2015 and BICEP2/Keck Array constraints?
• RQ2 Can instantaneous reheating be consistent with current CMB data for a $φ^1$ potential, or is it excluded?
• RQ3 How does the inclusion of a small subdominant $φ^2$ term in a binomial potential affect the allowed reheating temperature range?
• RQ4 At what value of the subdominant coupling $b$ does instantaneous reheating first become allowed in a $φ^1 + b\phi^2$ potential?
• RQ5 When is the minimum allowed reheating temperature of $4\,\text{MeV}$ excluded by the $68\%$ confidence limit on $n_s$ in the binomial model?

Key findings

• For a $φ^1$ potential with canonical reheating ($w_{\text{re}} = 0$), the Planck 2015 $68\%$ confidence limit on $n_s$ implies an upper bound of $T_{\text{re}} \lesssim 6 \times 10^{10}\,\text{GeV}$.
• Instantaneous reheating is excluded by the CMB data for the $φ^1$ model, as it would require $n_s$ to fall outside the $68\%$ confidence interval.
• For $w_{\text{re}} = -1/3$, the allowed reheating temperature range is $10^7\,\text{GeV} \lesssim T_{\text{re}} \lesssim 2 \times 10^{14}\,\text{GeV}$, but $T_{\text{re}} \lesssim 6 \times 10^{10}\,\text{GeV}$ remains the upper bound from $n_s$.
• In the binomial model with $p=1$, $q=2$, instantaneous reheating becomes allowed when $b = 5.3 \times 10^{-3}$, as the predicted $n_s$ enters the $68\%$ confidence range.
• The minimum reheating temperature of $4\,\text{MeV}$ is excluded when $b = 1.4 \times 10^{-2}$, as the predicted $n_s$ at that temperature falls outside the $68\%$ confidence interval.
• For $5.3 \times 10^{-3} < b < 1.4 \times 10^{-2}$, all reheating temperatures from $4\,\text{MeV}$ to the upper bound are consistent with both $n_s$ and $r$ constraints under canonical reheating.