# Incorporating a Radiative Hydrodynamics Scheme in the Numerical-Relativity Code BAM

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## Abstract

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## 1. Introduction

## 2. Fundamental Equations

#### 2.1. 3 + 1 Decomposition and Spacetime Evolution

#### 2.2. General Relativistic Radiative Hydrodynamics

#### 2.3. Neutrino Leakage

#### Underlying Hypotheses of the Neutrino Leakage Scheme

- For simplicity, we consider only electrons and positrons as representative leptons within the fluid.
- The considered neutrino flavors are electron neutrinos ${\nu}_{e}$, electron antineutrinos ${\overline{\nu}}_{e}$, and heavy lepton neutrinos/antineutrinos ${\nu}_{\mu ,\tau},\phantom{\rule{3.33333pt}{0ex}}{\overline{\nu}}_{\mu ,\tau}$, collectively grouped as a single species ${\nu}_{x}$ with statistical weight 4.
- Neutrinos obey the ultra-relativistic Fermi–Dirac distribution in local $\beta $-equilibrium and have the same temperature as the matter. Hence, the relativistic chemical potentials (i.e., including rest-masses of protons, neutrons, and electrons) for electron-flavored neutrinos read$${\mu}_{{\nu}_{e}}=-{\mu}_{{\overline{\nu}}_{e}}={\mu}_{p}+{\mu}_{e}-{\mu}_{n},$$

- 4.
- The emission of neutrinos is isotropic in the fluid rest-frame and is given by$${\Psi}^{\mu}=-{n}_{b}\mathcal{Q}{u}^{\mu},$$$$\mathcal{Q}\equiv Q\left({\nu}_{e}\right)+Q\left({\overline{\nu}}_{e}\right)+Q\left({\nu}_{x}\right).$$To see that Equation (18) corresponds to an isotropic emission, note that the projection of ${\Psi}^{\mu}$ onto the hypersurface orthogonal to the fluid worldlines via the projector ${h}_{\mu \nu}={g}_{\mu \nu}+{u}_{\mu}{u}_{\nu}$ vanishes (i.e., ${h}_{\mu \nu}{\Psi}^{\mu}=0$). Hence, neutrinos are emitted such that no net momentum flux is perceived in the fluid comoving frame.
- 5.
- The source term $\mathcal{R}$ is given by$$\mathcal{R}\equiv R\left({\overline{\nu}}_{e}\right)-R\left({\nu}_{e}\right),$$
- 6.
- Neutrinos are treated as a `test’ fluid. Hence, the projections of ${T}_{\mathrm{rad}}^{\mu \nu}$, which act as sources of spacetime curvature, are neglected.

#### 2.4. Emissivities and Production Rates

- (i)
- Direct Urca process, comprised of positron capture by neutrons$$\begin{array}{c}\hfill {e}^{+}+n\to p+{\overline{\nu}}_{e},\end{array}$$$${e}^{-}+p\to n+{\nu}_{e}.$$
- (ii)
- Electron–positron pair annihilation$${e}^{-}+{e}^{+}\to {\nu}_{e}+{\overline{\nu}}_{e}\phantom{\rule{4pt}{0ex}},\phantom{\rule{2.em}{0ex}}{e}^{-}+{e}^{+}\to {\nu}_{\mu}+{\overline{\nu}}_{\mu}\phantom{\rule{4pt}{0ex}},\phantom{\rule{2.em}{0ex}}{e}^{-}+{e}^{+}\to {\nu}_{\tau}+{\overline{\nu}}_{\tau}\phantom{\rule{4pt}{0ex}}.$$
- (iii)
- Transversal plasmon decay$$\gamma \to {\nu}_{e}+{\overline{\nu}}_{e}\phantom{\rule{4pt}{0ex}},\phantom{\rule{2.em}{0ex}}\gamma \to {\nu}_{\mu}+{\overline{\nu}}_{\mu}\phantom{\rule{4pt}{0ex}},\phantom{\rule{2.em}{0ex}}\gamma \to {\nu}_{\tau}+{\overline{\nu}}_{\tau}.$$

- (i)
- Neutrino-elastic scattering on a representative heavy nucleus X and atomic mass number A.$$\begin{array}{ccc}& & {\nu}_{e}+A\to {\nu}_{e}+A,\phantom{\rule{14.22636pt}{0ex}}{\overline{\nu}}_{e}+A\to {\overline{\nu}}_{e}+A,\phantom{\rule{14.22636pt}{0ex}}{\nu}_{x}+A\to {\nu}_{x}+A.\hfill \end{array}$$
- (ii)
- Neutrino-elastic scattering on free nucleons$$\begin{array}{ccc}& & {\nu}_{e}+[n,p]\to {\nu}_{e}+[n,p],\phantom{\rule{14.22636pt}{0ex}}{\overline{\nu}}_{e}+[n,p]\to {\overline{\nu}}_{e}+[n,p],\phantom{\rule{14.22636pt}{0ex}}{\nu}_{x}+[n,p]\to {\nu}_{x}+[n,p].\hfill \end{array}$$
- (iii)
- Electron-flavor neutrino absorption on free nucleons$${\nu}_{e}+n\to p+{e}^{-}\phantom{\rule{4pt}{0ex}},\phantom{\rule{2.em}{0ex}}{\overline{\nu}}_{e}+p\to n+{e}^{+}.$$

## 3. Numerical Implementation

#### 3.1. Code Updates

- In our previous studies using the BAM code, we used mainly one-parameter piecewise polytropes EoSs together with an ideal-gas thermal contribution. Now, we have extended this infrastructure to enable the use of three-dimensional tables. In general, these tables have a finite range of validity defined as a domain $\mathcal{D}$ with$$\begin{array}{c}\hfill \mathcal{D}=\{(\rho ,T,{Y}_{e}):{\rho}^{min}\le \rho \le {\rho}^{max},{T}^{min}\le T\le {T}^{max},{Y}_{e}^{min}\le {Y}_{e}\le {Y}_{e}^{max}\}.\end{array}$$For this purpose, EoS evaluations should only be performed within this domain (i.e., additional checks have to be incorporated into BAM).
- Previously adopted EoSs allowed us to use a simple and fast converging root-finding procedure for the conservative-to-primitive conversion. This is not the case for a three-parameter tabulated EoS, since numerical derivatives computed by trilinear interpolations are noisy. In our case, we use the methods outlined in [73,81] to ensure a robust conservative-to-primitive conversion.
- Once we employ three-parameter EoSs, we also have to solve Equation (24).
- We make use of a static and cold atmosphere to model vacuum, that is, grid points with $\rho \le {\rho}_{\mathrm{fac}}\times {\rho}_{\mathrm{atm}}$ (here, we use ${\rho}_{\mathrm{fac}}=10$ and ${\rho}_{\mathrm{atm}}=10\times {\rho}_{min}$) are set to$$\begin{array}{c}\hfill \rho ={\rho}_{\mathrm{atm}},\phantom{\rule{28.45274pt}{0ex}}{v}^{i}=0,\phantom{\rule{28.45274pt}{0ex}}T={T}^{min}=0.1\phantom{\rule{3.33333pt}{0ex}}\mathrm{MeV},\phantom{\rule{28.45274pt}{0ex}}{Y}_{e}={Y}_{e,\mathrm{atm}}.\end{array}$$We use ${Y}_{e,\mathrm{atm}}=0.4$ in our TOV simulations to reproduce the conditions of the testbeds reported in the literature, and ${Y}_{e,\mathrm{atm}}={Y}^{min}=0.01$ in our BNS runs so that the pressure of the atmosphere ${p}_{\mathrm{atm}}=p({\rho}_{\mathrm{atm}},{T}_{\mathrm{atm}},{Y}_{e,\mathrm{atm}})$ is lowest.

#### 3.2. Free Emission Rates and Optical Depth Estimates

## 4. Neutrino-Induced Collapse of Single TOV Stars

## 5. BNS Simulations

#### 5.1. Initial Data

#### 5.2. Short Inspiral Simulations

#### 5.2.1. Post-Merger Stage

#### 5.2.2. Spectrograms

#### 5.2.3. Neutrinos Emission

#### 5.3. Long Inspiral Simulations

#### 5.3.1. Post-Merger Stage

#### 5.3.2. Neutrino Emissions

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Convergence of the Code

**Figure A1.**Constraint evolution and conserved quantities.

**Left**panel: BHB$\Lambda \varphi $.

**Right**panel: BHB$\Lambda \varphi $-$\nu $. We show the $L2$ norm of the Hamiltonian constraint ${\left|\right|\mathcal{H}\left|\right|}_{2}$, the baryonic mass variation $\Delta {M}_{b}/{M}_{b}^{0}$, and the electrons number variation $\Delta {N}_{e}/{N}_{e}^{0}$ for the BHB$\Lambda \varphi $ (left panel) for all four resolutions of Table 2. We observe good behavior of the Hamiltonian constraint and conserved quantities with increasing resolution in both cases. Physical quantities were extracted from the grid level $l=1$, and the vertical lines mark the merger for each resolution.

**Figure A2.**Phase difference of the $(2,2)$ mode of the GW signal between different resolutions with respect to the retarded time u. The BHB$\Lambda \varphi $ case is presented on the

**left**panel and the BHB$\Lambda \varphi $-$\nu $ case is presented on the

**right**panel. Solid vertical lines mark the merger for each resolution.

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**Figure 1.**TOV solutions for the SHT-NL3 EoS in neutrino-less $\beta $-equilibrium and constant $T=30\phantom{\rule{3.33333pt}{0ex}}\mathrm{MeV}$. The marked points refer to the configurations of Table 1: maximal (green diamond), A (black circle), B (blue circle), and C (red circle).

**Left**panel: mass-radius curve.

**Right**panel: mass-central rest-mass density curve. The solid line represents the stable branch, while the dashed line represents the unstable branch.

**Figure 2.**Time evolution of quantities of interest.

**Upper**panels: central rest-mass density for simulations A, B, and C.

**Left**panel: without NLS, where the density stably evolves around an equilibrium state for each run.

**Right**panel: with NLS, where the wobbly evolution results from the coupling between the matter motion and the neutrinos emission. A final density growth marks the formation of an apparent event horizon.

**Lower**panel: total luminosity evolution for simulations A-$\nu $, B-$\nu $, and C-$\nu $. We notice a rapid burst at the beginning of the simulations. Likewise, the oscillating pattern of the luminosity is a consequence of the coupling between neutrinos and matter. After the formation of the apparent event horizon, the luminosity abruptly decreases.

**Figure 3.**Evolution of the temperature profile inside the star along the coordinate x direction at the innermost level $l=2$. Run A (

**left**panel): we present temperatures up to the level $l=2$ boundary to show that in this case, within our simulations timespan, the NS has increased radius with respect to $t=0$. We note that at $t=1.08\phantom{\rule{3.33333pt}{0ex}}\mathrm{ms}$ and $t=2.66\phantom{\rule{3.33333pt}{0ex}}\mathrm{ms}$, the NS is still expanding and ejecting material. Run A-$\nu $ (

**right**panel): the vertical lines mark the position of the electron neutrino neutrinosphere. In general, outside the neutrinosphere, the material has lower temperatures and the cooling becomes less effective towards the core due to the dominance of diffusive processes. By the end of the first expansion cycle (∼$t=1.08\phantom{\rule{3.33333pt}{0ex}}\mathrm{ms}$), the internal temperatures are nearly unchanged, whereas a large portion outside the neutrinosphere is cooler. Likewise, on the verge of the gravitational collapse ($t=2.66\phantom{\rule{3.33333pt}{0ex}}\mathrm{ms}$), the layers outside of the neutrinosphere are cold, while the interior is hotter due to compression.

**Figure 4.**Snapshots of the A-$\nu $ simulation at $t=0.00,\phantom{\rule{3.33333pt}{0ex}}1.08,\phantom{\rule{3.33333pt}{0ex}}2.66\phantom{\rule{3.33333pt}{0ex}}\mathrm{ms}$ from left to right.

**Upper**panels: logarithm of the electron neutrinos emissivity.

**Lower**panels: electron fraction. The solid black lines are contours of constant rest-mass density with ${log}_{10}\left(\rho \phantom{\rule{3.33333pt}{0ex}}\left[\mathrm{g}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{cm}}^{-3}\right]\right)=7,12,14$. The outermost line marks the interface between the star and the atmosphere.

**Figure 5.**Post-merger evolution of the maximum rest-mass density ${\rho}_{max}$ reached by the different EoSs considered in this article. Solid lines refer to the runs without NLS, while dashed lines represent the NLS runs. Our results for the SFHo and DD2 cases are in agreement with those reported in [58]. For the BHB$\Lambda \varphi $ EoS, we use resolution R4 (see Table 2).

**Figure 6.**Snapshots of the remnant $10\phantom{\rule{3.33333pt}{0ex}}\mathrm{ms}$ after the merger in the x-y plane. From left to right: DD2, DD2-$\nu $, SFHo, and SFHo-$\nu $. From top to bottom: rest-mass density, temperature, and electron fraction. Overall, we note the formation of bar-like structures (see the extended $\rho \ge {10}^{13}\phantom{\rule{3.33333pt}{0ex}}\mathrm{g}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{cm}}^{-3}$ central regions and the $\rho \ge {10}^{12}\phantom{\rule{3.33333pt}{0ex}}\mathrm{g}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{cm}}^{-3}$ arms in the top panels) surrounded by dense disks. The temperature profiles exhibit a hot interface between the bar and the disk. Finally, the effects of the NLS are perceivable on the electron fraction, where the disk becomes more neutron-rich, as opposed to the cores and spiral arms.

**Figure 7.**Spectrograms of the GWs for optimally oriented binaries and extracted at ${r}_{\mathrm{ext}}=600\phantom{\rule{3.33333pt}{0ex}}{M}_{\odot}$. The countour lines represent the simulations without NLS.

**Left**panel: SFHo-$\nu $ simulation, where the dominant post-merger frequency is 2.95 kHz.

**Right**panel: DD2-$\nu $ simulation, where post-merger peak frequency is ≈2.4 kHz. Both panels share the same properties as presented in [58].

**Figure 8.**Neutrinos source luminosity evolution for the SFHo-$\nu $ simulation (

**left**panel) and the DD2-$\nu $ simulation (

**right**panel). The electron antineutrinos have higher luminosity until ∼$5\phantom{\rule{3.33333pt}{0ex}}\mathrm{ms}$ after the merger.

**Figure 9.**Effective emissivities $10\phantom{\rule{3.33333pt}{0ex}}\mathrm{ms}$ after the merger in the x-y plane of electron neutrinos (

**left**panels) and electron antineutrinos (

**right**panels) for the SFHo-$\nu $ (

**upper**panels) and DD2-$\nu $ (

**lower**panels) runs. Here, we notice that the emissions are small at the densest portion of the core, concentrate at the hot parts of the disk/spiral arms, and decrease towards the outer regions of the disk.

**Figure 10.**Snapshots of hydrodynamical quantities of the simulations using BHB$\Lambda \varphi $ and BHB$\Lambda \varphi $-$\nu $ $10\phantom{\rule{3.33333pt}{0ex}}\mathrm{ms}$ after the merger on the x-y plane. From top to bottom, we have the rest-mass density, temperature, and electron fraction.

**Figure 11.**Neutrino luminosity evolution for the BHB$\Lambda \varphi $-$\nu $ simulation. Similarly to the short inspiral simulations, we have peak luminosities at ∼2–3 ms after the merger, followed by a decrease towards the end of the simulation.

**Figure 12.**Effective emissivities $10\phantom{\rule{3.33333pt}{0ex}}\mathrm{ms}$ after the merger in the x-y plane of electron neutrinos (

**left**panels) and electron antineutrinos (

**right**panels) for the BHB$\Lambda \varphi $-$\nu $ run. Differently than for the cold ID case, we notice that the emissivities at the core are greater as a consequence of the isentropic thermal profile, which produces temperatures of tens of MeV within the NSs. The more potent emissions are found in the outer core and along the spiral arms, mostly in the inner disk region.

**Table 1.**Properties of the TOV stars. From left to right, the columns read: model name, central rest-mass density, gravitational mass, baryonic mass, emitted neutrino energy (up to the collapse for those evolved with NLS), grid resolution, and distance between grid points on the finest level. The central densities for this test were chosen to meet the same initial conditions of [73]. The top row, `Maximal’, refers to the model at the onset of instability.

Model | ${\mathit{\rho}}_{\mathit{c}}\phantom{\rule{3.33333pt}{0ex}}\left({10}^{15}\phantom{\rule{3.33333pt}{0ex}}\mathbf{g}/{\mathbf{cm}}^{3}\right)$ | $\mathit{M}\phantom{\rule{3.33333pt}{0ex}}\left({\mathit{M}}_{\odot}\right)$ | ${\mathit{M}}_{\mathit{b}}\phantom{\rule{3.33333pt}{0ex}}\left({\mathit{M}}_{\odot}\right)$ | $\mathit{E}\phantom{\rule{3.33333pt}{0ex}}\left({10}^{51}\phantom{\rule{3.33333pt}{0ex}}\mathbf{erg}\right)$ | n | ${\mathit{h}}_{2}\phantom{\rule{3.33333pt}{0ex}}\left(\mathbf{m}\right)$ |
---|---|---|---|---|---|---|

Maximal | 1.068 | 2.797 | 3.506 | - | - | - |

A | 1.079 | 2.797 | 3.310 | 2.433 | 256 | 111 |

B | 1.111 | 2.796 | 3.309 | 2.191 | 256 | 111 |

C | 1.218 | 2.784 | 3.293 | 2.075 | 256 | 111 |

**Table 2.**Binary neutron star simulations. From left to right, the columns read: simulation name, gravitational mass of the stars ($A,\phantom{\rule{3.33333pt}{0ex}}B$) in isolation, baryonic mass of the stars, compactness of the stars, tidal deformability of stars, ADM mass and ADM angular momentum of the BNS at the beginning of the simulation, initial coordinate distance between the stars, number of points per direction on the static levels, number of points per direction on the moving levels, and grid spacing at the finest level.

Model | ${\mathit{M}}^{\mathit{A},\mathit{B}}\phantom{\rule{3.33333pt}{0ex}}\left[{\mathit{M}}_{\odot}\right]$ | ${\mathit{M}}_{\mathit{b}}^{\mathit{A},\mathit{B}}\phantom{\rule{3.33333pt}{0ex}}\left[{\mathit{M}}_{\odot}\right]$ | ${\mathcal{C}}^{\mathit{A},\mathit{B}}$ | ${\mathit{\Lambda}}^{\mathit{A},\mathit{B}}[\times {10}^{3}]$ | ${\mathit{M}}_{\mathbf{ADM}}\phantom{\rule{3.33333pt}{0ex}}\left[{\mathit{M}}_{\odot}\right]$ | ${\mathit{J}}_{\mathbf{ADM}}\phantom{\rule{3.33333pt}{0ex}}\left[{\mathit{M}}_{\odot}^{2}\right]$ | ${\mathit{d}}_{0}\phantom{\rule{3.33333pt}{0ex}}\left[\mathbf{km}\right]$ | n | ${\mathit{n}}_{\mathbf{mv}}$ | ${\mathit{h}}_{6}\phantom{\rule{3.33333pt}{0ex}}\left[\mathbf{km}\right]$ |
---|---|---|---|---|---|---|---|---|---|---|

DD2 | $1.200$ | $1.292$ | $0.134$ | $1.616$ | $2.375$ | $5.612$ | $36.2$ | 256 | 128 | $0.199$ |

SFHo | $1.200$ | $1.300$ | $0.148$ | $0.860$ | $2.376$ | $5.673$ | $38.0$ | 256 | 128 | $0.186$ |

BHB$\Lambda \varphi -\mathrm{R}1$ | $1.350$ | $1.458$ | $0.144$ | $0.944$ | $2.679$ | $8.021$ | $58.8$ | 128 | 64 | $0.417$ |

BHB$\Lambda \varphi -\mathrm{R}2$ | $1.350$ | $1.458$ | $0.144$ | $0.944$ | $2.679$ | $8.021$ | $58.8$ | 192 | 96 | $0.278$ |

BHB$\Lambda \varphi -\mathrm{R}3$ | $1.350$ | $1.458$ | $0.144$ | $0.944$ | $2.679$ | $8.021$ | $58.8$ | 256 | 128 | $0.209$ |

BHB$\Lambda \varphi -\mathrm{R}4$ | $1.350$ | $1.458$ | $0.144$ | $0.944$ | $2.679$ | $8.021$ | $58.8$ | 320 | 160 | $0.167$ |

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Gieg, H.; Schianchi, F.; Dietrich, T.; Ujevic, M.
Incorporating a Radiative Hydrodynamics Scheme in the Numerical-Relativity Code BAM. *Universe* **2022**, *8*, 370.
https://doi.org/10.3390/universe8070370

**AMA Style**

Gieg H, Schianchi F, Dietrich T, Ujevic M.
Incorporating a Radiative Hydrodynamics Scheme in the Numerical-Relativity Code BAM. *Universe*. 2022; 8(7):370.
https://doi.org/10.3390/universe8070370

**Chicago/Turabian Style**

Gieg, Henrique, Federico Schianchi, Tim Dietrich, and Maximiliano Ujevic.
2022. "Incorporating a Radiative Hydrodynamics Scheme in the Numerical-Relativity Code BAM" *Universe* 8, no. 7: 370.
https://doi.org/10.3390/universe8070370