Gravitational leptogenesis, reheating, and models of neutrino mass

Gravitational leptogenesis refers to a class of baryogenesis models in which the matter-antimatter asymmetry of the Universe arises through the standard model lepton-number gravitational anomaly. In these models chiral gravitational waves source a lepton asymmetry in standard model neutrinos during the inflationary epoch. We point out that gravitational leptogenesis can be successful in either the Dirac or Majorana neutrino mass scenario. In the Dirac mass scenario, gravitational leptogenesis predicts a relic abundance of sterile neutrinos that remain out of equilibrium, and the lepton asymmetry carried by the standard model sector is unchanged. In the Majorana mass scenario, the neutrinos participate in lepton-number-violating interactions that threaten to wash out the lepton asymmetry during postinflationary reheating. However, we show that a complete (exponential) washout of the lepton asymmetry is prevented if the lepton-number-violating interactions go out of equilibrium before all of the standard model Yukawa interactions come into equilibrium. The baryon and lepton asymmetries carried by right-chiral quarks and leptons are sequestered from the lepton-number violation, and the washout processes only suppress the predicted baryon asymmetry by a factor of ${ϵ}_{\mathrm{w}.\mathrm{o}.}=\pm O(0.1)$. The sign of ${ϵ}_{\mathrm{w}.\mathrm{o}.}$ depends on the model parameters in such a way that a future measurement of the primordial gravitational wave chirality would constrain the scale of lepton-number violation (heavy Majorana neutrino mass).

Figure 1

The baryon asymmetry $Y_{\mathsf{B}}=n_{\mathsf{B}}/s$ generated from gravitational leptogenesis in a model where the Hubble scale at the end of inflation is $H_e$, the plasma temperature at the end of reheating is $T_{\mathrm{RH}}$, and baryon-minus-lepton number is assumed to be conserved after inflation (Dirac mass scenario). The left panel shows the case where the effective equation of state during reheating is $w=0$ and the right panel shows $w=1/3$. The amplitude of chiral gravitational waves is parametrized by ${\mathcal{H}}_{R-L}^{\mathrm{GW}}$ [see Eq. (5)], which we take to be ${\mathcal{H}}_{R-L}^{\mathrm{GW}}=-{10}^{14}$ in drawing the contours, but more generally $Y_{\mathsf{B}}\propto -{\mathcal{H}}_{R-L}^{\mathrm{GW}}$ as in Eq. (9).

Figure 2

Scattering processes in which a heavy Majorana neutrino $N$ mediates lepton-number-violating interactions among the standard model leptons $L$ and Higgs bosons $\mathrm{\Phi }$.

Figure 3

We show the effective washout factor, ${ϵ}_{\mathrm{w}.\mathrm{o}.}$, which corrects the formula in Eq. (9) to account for lepton-number violation due to heavy right-handed neutrino exchange. We vary the equation of state during reheating in the dashed blue, dotted red, solid black, and dotted-dashed green lines. The thin gray line shows the approximation in Eq. (16) for $w=0$. At high reheat temperature ${ϵ}_{\mathrm{w}.\mathrm{o}.}\simeq 0.09$, but this value has an $O(1)$ uncertainty from our rough estimation of ${\mathrm{\Gamma }}_{\mathrm{w}.\mathrm{o}.}$ in Eq. (10).

Figure 4

The baryon-to-entropy ratio $Y_{\mathsf{B}}=n_{\mathsf{B}}/s$ generated from gravitational leptogenesis in a Majorana-mass model where the Hubble scale at the end of inflation is $H_e$, the plasma temperature at the end of reheating is $T_{\mathrm{RH}}$, and the effective equation of state during reheating is $w=0$ (left panel) and $w=1/3$ (right panel). We take ${\mathcal{H}}_{R-L}^{\mathrm{GW}}=-{10}^{14}$ to draw the contours, but more generally $Y_{\mathsf{B}}\propto -{\mathcal{H}}_{R-L}^{\mathrm{GW}}$. The washout of lepton number by approximately an order of magnitude is apparent for $T_{\mathrm{RH}}\gtrsim {10}^{11}\mathrm{GeV}$.

Figure 5

The effect of varying the mass scale of the heavy Majorana neutrinos, $m_N$, and the reheating temperature (for matter-dominated expansion, $w=0$, during reheating) on the resulting baryon-to-entropy ratio $Y_{\mathsf{B}}=n_{\mathsf{B}}/s$ that is generated from gravitational leptogenesis. In making this figure, we have taken $H_e={10}^{13}\mathrm{GeV}$. Note that the baryon asymmetry changes sign at $m_N\sim {10}^{12}\mathrm{GeV}$. For $m_N>{10}^{12}\mathrm{GeV}$, $\mathrm{sign}(Y_B)=-\mathrm{sign}({\mathcal{H}}_{R-L}^{\mathrm{GW}})$, while $\mathrm{sign}(Y_B)=\mathrm{sign}({\mathcal{H}}_{R-L}^{\mathrm{GW}})$ for $10^6<m_N<{10}^{12}\mathrm{GeV}$.