# Neutrino Oscillations in Neutrino-Dominated Accretion Around Rotating Black Holes

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## Abstract

**:**

## 1. Introduction

## 2. Hydrodynamics

#### 2.1. Units, Velocities and Averaging

#### 2.2. Conservation Laws

#### 2.3. Equations of State

^{,}2 and

## 3. Neutrino Oscillations

#### 3.1. Equations of Oscillation

- Due to axial symmetry, the neutrino density is constant along the $\mathbf{z}$ direction. Moreover, since neutrinos follow null geodesics, we can set ${\dot{p}}_{z}\approx {\dot{p}}_{\varphi}=0$.
- Within the thin disk approximation (as represented by Equation (10)) the neutrino and matter densities are constant along the $\mathbf{y}$ direction and the momentum change due to curvature along this direction can be neglected, that is, ${\dot{p}}_{y}\approx 0$.
- In the LRF, the normalized radial momentum of a neutrino can be written as ${p}_{x}=\pm r/\sqrt{{r}^{2}-2Mr+{M}^{2}{a}^{2}}$. Hence, the typical scale of the change of momentum with radius is $\Delta {r}_{{p}_{x},\mathrm{eff}}={\left|dln{p}_{x}/dr\right|}^{-1}=(r/M)\left({r}^{2}-2Mr+{M}^{2}{a}^{2}\right)/\left(M{a}^{2}-r\right)$, which obeys $\Delta {r}_{{p}_{x},\mathrm{eff}}>{r}_{s}$ for $r>2{r}_{\mathrm{in}}$. This means we can assume ${\dot{p}}_{x}\approx 0$ up to regions very close to the inner edge of the disk.
- We define an effective distance $\Delta {r}_{\rho ,\mathrm{eff}}={\left|dln\left({Y}_{e}{n}_{B}\right)/dr\right|}^{-1}$. For all the systems we evaluated, we found that it is comparable to the height of the disk $(\Delta {r}_{\rho ,\mathrm{eff}}\sim 2-5$ ${r}_{s}$). This means that at any point of the disk we can calculate neutrino oscillations in a small regions assuming that both the electron density and neutrino densities are constant.
- We neglect energy and momentum transport between different regions of the disk by neutrinos that are recaptured by the disk due to curvature. This assumption is reasonable except for regions very close to the BH but is consistent with the thin disk model (see, e.g., [128]). We also assume initially that the neutrino content of neighboring regions of the disk (different values of r) do not affect each other. As a consequence of the results discussed above, we assume that at any point inside the disk and at any instant of time an observer can describe both the charged leptons and neutrinos as isotropic gases around small enough regions of the disk. This assumption is considerably restrictive but we will generalize it in Section 5.

## 4. Initial Conditions and Integration

## 5. Results and Analysis

## 6. Discussion

- The equation of vertical hydrostatic equilibrium, Equation (15), can be derived in several ways [124,127,131]. We followed a particular approach consistent with the assumptions in [127], in which we took the vertical average of a hydrostatic Euler equation in polar coordinates. The result is an equation that leads to smaller values of the disk pressure when compared with other models. It is expected that the pressure at the center of the disk is smaller than the average density multiplied by the local tidal acceleration at the equatorial plane. Still, the choice between the assortment of pressure relations is tantamount to the fine-tuning of the model. Within the thin disk approximation, all these approaches are equivalent, since they all assume vertical equilibrium and neglect self-gravity.
- Following the BdHN scenario for the explanation of GRBs associated with Type Ic SNe (see Section 2), we considered a gas composed of ${}^{16}$O at the outermost radius of the disk and followed the evolution of the ion content using the Saha equation to fix the local NSE. In [107], only ${}^{4}$He is present, and in [112], ions up to ${}^{56}$Fe are introduced. The affinity between these cases implies that this particular model of disk accretion is insensible to the initial mass fraction distribution. This is explained by the fact that the average binding energy for most ions is very similar; hence, any cooling or heating due to a redistribution of nucleons, given by the NSE, is negligible when compared to the energy consumed by direct photodisintegration of alpha particles. Additionally, once most ions are dissociated, the main cooling mechanism is neutrino emission, which is similar for all models; the modulo includes the supplementary neutrino emission processes included in addition to electron and positron capture. However, during our numerical calculations, we noticed that the inclusion of non-electron neutrino emission processes can reduce the electron fraction by up to $\sim 8\%$. This effect was observed again during the simulation of flavor equipartition alluding to the need for detailed calculations of neutrino emissivities when establishing NSE state. We obtained similar results to [107] (see Figure 3), but by varying the accretion rate and fixing the viscosity parameter. This suggests that a more natural differentiating set of variables in the hydrodynamic equations of an $\alpha $-viscosity disk is the combination of the quotient $\dot{M}/\alpha $ and either $\dot{M}$ or $\alpha $. This result is already evident in, for example, Figures 11 and 12 of [107], but was not mentioned there.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BdHN | Binary-Driven Hypernova |

BH | Black Hole |

CF | Coordinate Frame |

CO${}_{\mathrm{core}}$ | Carbon–Oxygen Star |

CRF | Co-rotating Frame |

GRB | Gamma-ray Burst |

IGC | Induced Gravitational Collapse |

ISCO | Innermost Stable Circular Orbit |

LNRF | Locally Non-Rotating Frame |

MSW | Mikheyev–Smirnov–Wolfenstein |

NDAF | Neutrino-Dominated Accretion Flows |

NS | Neutron Star |

NSE | Nuclear Statistical Equilibrium |

SN | Supernova |

## Appendix A. Transformations and Christoffel Symbols

## Appendix B. Stress–Energy Tensor

## Appendix C. Nuclear Statistical Equilibrium

${\mathit{K}}_{1}$ | ${\mathit{K}}_{2}$ | ${\mathit{K}}_{3}$ | ${\mathit{Z}}_{1}$ | ${\mathit{Z}}_{2}$ | ${\mathit{Z}}_{3}$ | ${\mathit{Z}}_{4}$ |
---|---|---|---|---|---|---|

$-0.907347$ | $0.62849$ | $0.278497$ | $4.50\times {10}^{-3}$ | $170.0$ | $-8.4\times {10}^{-5}$ | $3.70\times {10}^{-3}$ |

## Appendix D. Neutrino Interactions and Cross-Sections

**Table A2.**Constants used throughout this appendix to calculate emissivities and cross-sections. All quantities are reported in Planck units.

Symbol | Value | Name |
---|---|---|

${M}_{w}$ | $6.584\times {10}^{-18}$ | W boson mass |

${g}_{w}$ | 0.653 | Weak coupling constant |

${g}_{a}$ | 1.26 | Axial-vector coupling constant |

${\alpha}^{*}$ | $\frac{1}{137}$ | Fine structure constant |

${sin}^{2}{\theta}_{\mathrm{W}}$ | 0.231 | Weinberg angle |

${cos}^{2}{\theta}_{c}$ | 0.947 | Cabibbo angle |

${G}_{F}$ | $1.738\times {10}^{33}$ | Fermi coupling constant |

${C}_{v,e}$ | $2{sin}^{2}{\theta}_{\mathrm{W}}+1/2$ | Weak interaction vector constant for ${\nu}_{e}$ |

${C}_{a,e}$ | $1/2$ | Weak interaction axial-vector constant for ${\nu}_{e}$ |

${C}_{v,e}$ | ${C}_{v,e}-1$ | Weak interaction vector constant for ${\nu}_{x}$ |

${C}_{a,e}$ | ${C}_{a,e}-1$ | Weak interaction axial-vector constant for ${\nu}_{x}$ |

${\sigma}_{0}$ | $6.546\times {10}^{21}$ | Weak interaction cross-section |

#### Appendix D.1. Neutrino Emissivities

- Pair annihilation: ${e}^{-}\phantom{\rule{-0.166667em}{0ex}}+{e}^{+}\phantom{\rule{-0.166667em}{0ex}}\to \nu +\overline{\nu}$

- Electron capture and positron capture: $p+{e}^{-}\phantom{\rule{-0.166667em}{0ex}}\to n+{\nu}_{e}$, $n+{e}^{+}\phantom{\rule{-0.166667em}{0ex}}\to p+{\overline{\nu}}_{e}$ and $A+{e}^{-}\phantom{\rule{-0.166667em}{0ex}}\to {A}^{\prime}+{\nu}_{e}$

- Plasmon decay: $\tilde{\gamma}\to \nu +\overline{\nu}$.

- Nucleon-nucleon bremsstrahlung ${n}_{1}+{n}_{2}\to {n}_{3}+{n}_{4}+\nu +\overline{\nu}$.

#### Appendix D.2. Cross-Sections

- Neutrino annihilation: $(\nu +\overline{\nu}\to {e}^{-}\phantom{\rule{-0.166667em}{0ex}}+{e}^{+})$.

- Electron (anti)neutrino absorption by nucleons: $({\nu}_{e}+n\to {e}^{-}+p$ and ${\overline{\nu}}_{e}+p\to {e}^{+}+n)$.

- (anti)neutrino scattering by baryons: $(\nu +{A}_{i}\to \nu +{A}_{i}$ and $\overline{\nu}+{A}_{i}\to \overline{\nu}+{A}_{i})$.

- (anti)neutrino scattering by electrons or positrons: $(\nu +{e}^{\pm}\to \nu +{e}^{\pm}$ and $\overline{\nu}+{e}^{\pm}\to \overline{\nu}+{e}^{\pm})$.

#### Appendix D.3. Neutrino-Antineutrino Pair Annihilation

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1. | |

2. | We will consider accretion rates of up to 1${M}_{\odot}$ s${}^{-1}$. These disks reach densities of ${10}^{13}$ g cm${}^{-3}$. Baryons can be lightly degenerate at these densities but we will still assume that the baryonic mass can be described by an ideal gas. |

**Figure 1.**Schematic representation of the physical system. Due to conditions of high temperature and density, neutrinos are produced in copious amounts inside the disk. Since they have very low cross-sectional areas, neutrinos are free to escape but not before experiencing collective effects due to the several oscillation potentials. The energy deposition rate of the process $\nu +\overline{\nu}\to {e}^{-}\phantom{\rule{-0.166667em}{0ex}}+{e}^{+}$ depends on the local distribution of electronic and non-electronic (anti)neutrinos, which is affected by the flavor oscillation dynamics.

**Figure 2.**Total number emissivity for electron and positron capture ($p+{e}^{-}\phantom{\rule{-0.166667em}{0ex}}\to n+{\nu}_{e}$, $n+{e}^{+}\phantom{\rule{-0.166667em}{0ex}}\to p+{\overline{\nu}}_{e}$) and electron–positron annihilation (${e}^{-}\phantom{\rule{-0.166667em}{0ex}}+{e}^{+}\phantom{\rule{-0.166667em}{0ex}}\to \nu +\overline{\nu}$) for accretion disks with $\dot{M}=0.1{M}_{\odot}$ s${}^{-1}$ between the inner radius and the ignition radius.

**Figure 3.**Properties of accretion disks in the absence of oscillations with $M=3{M}_{\odot}$, $\alpha =0.01$, $a=0.95$. (

**a**,

**b**) The mass fraction inside the disk. We have plotted only the ones that appreciably change. (

**c**) The electron degeneracy parameter. (

**d**) The comparison between the neutrino cooling flux ${F}_{\nu}$ and the viscous heating ${F}_{\mathrm{heat}}$. (

**e**) The baryon density. (

**f**) The temperature. (

**g**,

**h**) The neutrino number density. (

**i**,

**j**) The average neutrino energies.

**Figure 4.**Total optical depth (left scale) and mean free path (right scale) for neutrinos and antineutrinos of both flavors between the inner radius and the ignition radius for accretion disks with (

**a**) $\dot{M}=1{M}_{\odot}$ s${}^{-1}$ and (

**b**) $0.01{M}_{\odot}$ s${}^{-1}$.

**Figure 5.**Oscillation potentials as functions of r with $M=3{M}_{\odot}$, $\alpha =0.01$, $a=0.95$ for accretion rates (

**a**) $\dot{M}=1{M}_{\odot}$ s${}^{-1}$ and (

**b**) $\dot{M}=0.01{M}_{\odot}$ s${}^{-1}$, respectively. The vertical line represents the position of the ignition radius.

**Figure 6.**Survival provability for electron neutrinos and antineutrinos for the accretion disk with $\dot{M}=0.1{M}_{\odot}$ s${}^{-1}$ at $r=10{r}_{s}$. The survival probabilities for neutrinos and antineutrinos in both plots coincide. (

**a**) Inverted hierarchy and (

**b**) normal hierarchy.