Neutrino flavor transformation in the lepton-asymmetric universe

We investigate neutrino flavor transformation in the early Universe in the presence of a lepton asymmetry, focusing on a two-flavor system with 1–3 mixing parameters. We identify five distinct regimes that emerge in an approximate treatment neglecting collisions as the initial lepton asymmetry at high temperature is varied from values comparable to current constraints on the lepton number down to values at which the neutrino-neutrino forward-scattering potential is negligible. The characteristic phenomena occurring in these regimes are (1) large synchronized oscillations, (2) minimal flavor transformation, (3) asymmetric ($\nu$- or $\overline{\nu }$-only) MSW, (4) partial MSW, and (5) symmetric MSW. We examine our numerical results in the framework of adiabaticity, and we illustrate how they are modified by collisional damping. Finally, we point out the existence of matter-neutrino resonances in the early Universe and show that they suffer from nonadiabaticity.

Figure 1

Schematic illustration of the landscape of coherent flavor evolution in the inverted hierarchy as a function of lepton asymmetry $\mathcal{L}=L_{{\nu }_e}-L_{{\nu }_x}$. The black swath at the top of the figure indicates the realm of lepton asymmetries that are currently excluded by ${}^4\mathrm{He}$ measurements. The five regimes at subconstraint values of $\mathcal{L}$ are labeled by their most prominent characteristics.

Figure 2

Magnitudes of the individual diagonal potentials for $\epsilon =3$: $|{\mathcal{H}}_{\text{vac},\mathrm{z}}|$ (red, upper solid curve at $T=1\mathrm{MeV}$), $|{\mathcal{H}}_{e,z}|$ (blue, other solid curve), and $|{\mathcal{H}}_{\nu ,z}|$ (black, dashed). The last of these was computed assuming no flavor transformation. From top to bottom the dashed curves correspond to degeneracy parameters ${\eta }_{{\nu }_e}=5\times {10}^{-3}$, $5\times {10}^{-5}$, $2\times {10}^{-6}$, $1.5\times {10}^{-7}$, and $5\times {10}^{-8}$ [Eq. (25)]; chemical potentials in ${\nu }_{\mu }$ were taken to be 0.

Figure 3

Symmetric MSW: $P_{z,\mathrm{int}}$ (blue, upper curve at $T=20\mathrm{MeV}$) and ${\overline{P}}_{z,\mathrm{int}}$ (red) in the IH with initial degeneracy parameters ${\eta }_{{\nu }_e}=5\times {10}^{-8}$, ${\eta }_{{\nu }_x}=0$.

Figure 4

Partial MSW: $P_{z,\mathrm{int}}$ (blue, upper curve at $T=20\mathrm{MeV}$) and ${\overline{P}}_{z,\mathrm{int}}$ (red) in the IH with initial degeneracy parameters ${\eta }_{{\nu }_e}=1.5\times {10}^{-7}$, ${\eta }_{{\nu }_x}=0$.

Figure 5

Partial MSW: $P_{z,\epsilon }$ for $\epsilon =1.15$ (blue, bottommost curve at $T=2\mathrm{MeV}$), 2.36 (red, topmost curve at $T=2\mathrm{MeV}$), and 4.78 (purple), with the same parameters as in Fig. 4. Note that here and in subsequent plots $P_{z,\epsilon }$ has been normalized to an initial value of unity for each $\epsilon$; this choice puts all modes on equal footing for the purpose of comparing flavor evolution.

Figure 6

Partial MSW: ${\mathcal{H}}_z$ (blue, nearly vertical curve) and $|{\mathcal{H}}_T|$ (red) as functions of $T$ for the $\epsilon =4.78$ mode shown in Fig. 5.

Figure 7

Asymmetric MSW: $P_{z,\mathrm{int}}$ (blue, upper curve at $T=20\mathrm{MeV}$) and ${\overline{P}}_{z,\mathrm{int}}$ (red) in the IH with initial degeneracy parameters ${\eta }_{{\nu }_e}=2\times {10}^{-6}$, ${\eta }_{{\nu }_x}=0$.

Figure 8

Minimal transformation: $P_{z,\mathrm{int}}$ (blue, upper curve at $T=10\mathrm{MeV}$) and ${\overline{P}}_{z,\mathrm{int}}$ (red) in the IH with initial degeneracy parameters ${\eta }_{{\nu }_e}=5\times {10}^{-5}$, ${\eta }_{{\nu }_x}=0$.

Figure 9

Large synchronized oscillations: $P_{z,\mathrm{int}}$ (blue, upper curve at $T=10\mathrm{MeV}$) and ${\overline{P}}_{z,\mathrm{int}}$ (red) in the IH with initial degeneracy parameters ${\eta }_{{\nu }_e}=5\times {10}^{-3}$, ${\eta }_{{\nu }_x}=0$.

Figure 10

Large synchronized oscillations: ${\overline{P}}_{z,\epsilon }$ for $\epsilon =1.15$ (upper blue curve at $T=0.2\mathrm{MeV}$), 2.36 (upper red curve at $T=0.2\mathrm{MeV}$), 3.57 (purple), 4.78 (lower blue curve at $T=0.2\mathrm{MeV}$), and 5.99 (lower red curve at $T=0.2\mathrm{MeV}$), computed with the same parameters as in Fig. 9. A spectral swap—wherein modes below the threshold ${\epsilon }_{\text{th}}\approx 3.5$ change flavor and those above do not—is evident.

Figure 11

Top end of the synchronized-oscillation regime: $P_{z,\mathrm{int}}$ (blue, upper curve at $T=10\mathrm{MeV}$) and ${\overline{P}}_{z,\mathrm{int}}$ (red) in the IH with initial degeneracy parameters ${\eta }_{{\nu }_e}=5\times {10}^{-2}$, ${\eta }_{{\nu }_x}=0$.

Figure 12

Resonant comoving energy ${\epsilon }_{\text{res}}$ (red, solid) as a function of T, with ${\mathcal{H}}_{\nu }=0$. Also plotted is the Fermi-Dirac average-energy mode ${\epsilon }_{\text{avg}}\approx 3.15$ (black, dashed).

Figure 13

Minimal transformation (damped): $P_{j,\mathrm{int}}$ for $j=z$ (blue, topmost curve at $T=1\mathrm{MeV}$), $j=x$ (red, bottommost curve at $T=1\mathrm{MeV}$), and $j=y$ (purple), with the parameters of the minimal-transformation scenario in Fig. 8 (initial degeneracy parameters ${\eta }_{{\nu }_e}=5\times {10}^{-5}$, ${\eta }_{{\nu }_x}=0$), in the presence of collisional damping. Antineutrinos undergo qualitatively similar evolution.

Figure 14

Asymmetric MSW (damped): $P_{z,\mathrm{int}}$ for neutrinos (blue, upper curve at $T=20\mathrm{MeV}$) and ${\overline{P}}_{z,\mathrm{int}}$ (red), with the parameters of the asymmetric-MSW scenario in Fig. 7 (initial degeneracy parameters ${\eta }_{{\nu }_e}=2\times {10}^{-6}$, ${\eta }_{{\nu }_x}=0$), in the presence of collisional damping.

Figure 15

Asymmetric MSW (damped): $P_{z,\epsilon }$ in the collisionally damped scenario depicted in Fig. 14, for $\epsilon =1.15$ (blue, topmost curve at $T=2\mathrm{MeV}$), $\epsilon =3.57$ (red), and $\epsilon =5.99$ (purple, bottommost curve at $T=2\mathrm{MeV}$).