Leptogenesis and Low Reheating Temperatures

Technical Summary: Leptogenesis and Low Reheating Temperatures

Problem Statement
Standard cosmological models often assume high reheating temperatures ($T_{RH} \gtrsim 10^9$ GeV) to facilitate thermal leptogenesis, where right-handed neutrinos (RHNs) are produced from the thermal bath. However, observational constraints on Big Bang Nucleosynthesis (BBN) only require $T_{RH} > \sim 4$ MeV, leaving open the possibility of cosmological histories with very low reheating temperatures. In such low-$T_{RH}$ scenarios, standard leptogenesis faces a severe challenge: if $T_{RH}$ falls below the sphaleron decoupling temperature ($T_{sph} \sim 130$ GeV), any lepton asymmetry generated after this epoch cannot be converted into a baryon asymmetry. Furthermore, in standard matter-like reheating ($w_\phi = 0$), the entropy production during the reheating phase dilutes any pre-existing asymmetry by a factor proportional to $(T_{RH}/T_{sph})^5$. For $T_{RH} \sim 1$ GeV, this dilution factor exceeds $10^{10}$, rendering the observed baryon asymmetry unattainable via standard non-instantaneous reheating mechanisms.

Methodology
The authors investigate leptogenesis within the canonical Type-I seesaw framework, focusing on non-thermal RHN production via inflaton decays ($\phi \to NN$) during a non-instantaneous reheating phase. They employ a set of coupled Boltzmann equations to track the evolution of:

1. The inflaton energy density ($\rho_\phi$), including the effects of condensate fragmentation.
2. The radiation energy density ($\rho_R$).
3. The RHN number density ($n_N$) and energy density ($\rho_N$).
4. The $B-L$ asymmetry ($n_{B-L}$).

The study compares two classes of inflationary potentials:

• Standard Starobinsky-like ($k=2$): $V(\phi) \propto (1 - e^{-\sqrt{2/3}\phi/M_P})^2$, leading to matter-like oscillations ($w_\phi = 0$).
• Generalized Starobinsky-like ($k \ge 4$): $V(\phi) \propto (1 - e^{-\sqrt{2/3}\phi/M_P})^k$, leading to radiation-like ($k=4, w_\phi=1/3$) or stiffer equations of state.

The analysis incorporates the fragmentation of the inflaton condensate for $k \ge 4$, where self-interactions convert condensate energy into a bath of low-mass inflaton quanta. The authors also account for gravitational production of radiation and RHNs, demonstrating their negligibility in the relevant parameter space.

Key Contributions and Results

1. Failure of Matter-Like Reheating ($k=2$):
   For standard Starobinsky inflation ($k=2$), the inflaton mass is constant. Consequently, RHNs are produced continuously throughout the entire reheating epoch. If $T_{RH} < T_{sph}$, the continuous production of RHNs leads to a lepton asymmetry generated after sphaleron decoupling, which is never converted to baryons. Additionally, the entropy generated during the matter-dominated reheating phase dilutes the asymmetry by $(T_{RH}/T_{sph})^5$. Numerical results confirm that even with optimized branching ratios, the final baryon asymmetry is suppressed by orders of magnitude ($Y_B < 10^{-26}$ for $T_{RH} = 1$ GeV), making successful leptogenesis impossible in this regime.

2. Success of Radiation-Like Reheating ($k \ge 4$):
   For generalized potentials with $k \ge 4$, the inflaton is massless in the vacuum but possesses a time-dependent effective mass ($m_{eff}$) during oscillations. Crucially, $m_{eff}$ decreases as the scale factor $a$ increases. This leads to a kinematic shutoff of the $\phi \to NN$ channel at a specific scale factor $a^*$, determined by $m_{eff}(a^*) = 2M_N$.

   • Early Production: RHNs are produced efficiently only in the early stage of reheating ($a < a^*$), creating a fixed initial population.
   • Decoupling: Once $m_{eff} < 2M_N$, non-thermal production ceases. The RHN population then simply redshifts and decays.
   • Insensitivity to $T_{RH}$: Because the RHN population is fixed early (at $a^*$) and the subsequent evolution of the asymmetry depends on the ratio of scale factors rather than temperature-dependent dilution factors in the same way as the $k=2$ case, the final baryon asymmetry becomes largely independent of the reheating temperature $T_{RH}$.

3. Analytic and Numerical Verification:
   The authors derive an analytic expression for the final baryon asymmetry $Y_B$ for $k=4$:
   $$Y_B(a_{RH}) \propto |\epsilon| y_{\phi NN}^2 \left( \frac{M_P}{\rho_{end}^{1/4}} \right) \left( \frac{\hat{m}}{2M_N} \right)$$
   This result demonstrates that $Y_B$ is independent of $T_{RH}$ (up to mild logarithmic dependencies via $g_*$) and the RHN mass $M_N$. Numerical simulations confirm that the observed baryon asymmetry ($Y_B \approx 8.7 \times 10^{-11}$) can be achieved for arbitrarily low reheating temperatures, down to the BBN bound of $\sim 4$ MeV.

4. Parameter Space:
   The viable parameter space is primarily constrained by the inflaton-RHN coupling ($y_{\phi NN}$) and the CP-violating parameter ($|\epsilon|$ or $\delta_{eff}$). The RHN decay rate ($y_N$) must satisfy specific bounds to ensure RHNs decay before they dominate the energy density but after the kinematic shutoff. The results hold for both fermionic and bosonic reheating scenarios.

Significance
The paper establishes that the incompatibility between low reheating temperatures and standard leptogenesis is not a fundamental limitation but a consequence of the specific equation of state ($w_\phi = 0$) associated with quadratic potentials. By considering generalized Starobinsky potentials ($k \ge 4$) where the inflaton condensate behaves like radiation ($w_\phi \ge 1/3$), the mechanism of leptogenesis is robust against low $T_{RH}$. The kinematic shutoff of RHN production prevents the dilution of the asymmetry by late-time entropy production. This finding significantly expands the viable cosmological histories for the early universe, allowing for successful baryogenesis even in scenarios where reheating temperatures are as low as the BBN limit, without requiring exotic new physics beyond the Type-I seesaw framework.