Nucleon-nucleon interaction and the neutrino surface of neutron star mergers

. Neutrinos act as an important cooling mechanism in an array of explosive astrophysical events. Modeling of their ﬂuxes and spectra is key for the interpretation of their detection and the understanding of the synthesis of heavy elements. The neutrino surface is key in the modeling of neutrino spectra. In previous work, the points of last neutrino scattering were found assuming no interaction between nucleons. In this work we incorporate this interaction using a few di ﬀ erent models and observe di ﬀ erences in where the neutrinos decouple from the system.


Introduction
Neutrinos are known to determine the fate of explosive environments such as core-collapse supernova, and mergers involving neutron stars. Neutrinos cool matter and can also provide energy to produce winds, in which heavy elements can be synthesized [1]. Compared to photons, neutrinos decouple from matter at much higher densities providing a window to stellar interiors. Thus, understanding neutrino scattering from highly dense matter is of paramount importance and as such has gotten a lot of attention (see e.g. [2][3][4][5][6] to name a few).
The neutrino surface is defined by the last points of neutrino scattering in a stellar environment. The local temperature at decoupling is determinant in the modeling of neutrino spectra from emitting sources. In previous work, we have studied the neutrino surface, fluxes, and detection, from neutron star mergers and accretion disks around black holes [7][8][9]. In those works, we assumed that matter was dissociated into neutrons, protons, and electrons, and we ignored in-medium effects. In this work, we present our first study of the neutrino surface when nucleonic interactions are introduced in the neutrino scattering off nucleons. We use the formalism of the virial equation of state (EoS) and Skyrme forces in the framework of density functional theory (DFT). Below we present the simulation used, our post-processing approach, and results that motivate future work.

Neutrinos from neutron star mergers
Our results are based on a post-processing study of 3D general relativistic simulations of a neutron star merger. These simulations included strong magnetic fields, neutrino cooling and different microscopic Equations of State (EoS) [8]. For this particular study we used a Figure 1: The left panel shows our density profile in g/cm 3 , and the right panel shows the temperature profile in MeV snapshot at approximately 3.5 ms after merger of a simulation with the DD2 EoS [10]. This EoS predicts a radius of 13 km for an isolated neutron star of mass 1.35 M . In ref. [8] this EoS was classified as intermediate in terms of stiffness. The density and temperature profile as seen in the equatorial plane of the merger are shown in Figure 1.
To determine the neutrino surface we must examine the optical depth which we defined in along a path in the z direction as where n i represents the density of species i, and σ i represents the thermally averaged cross section of species i (we use Einstein summation). A blackbody at optical depth 2/3, will radiate approximately the same energy as our system [11], and thus we find z 0 , the last point of neutrino scattering, such that τ = 2/3. In order to find the set of points z 0 , we must include the cross sections of all interactions undergone by neutrinos in the system. In our previous work cross sections between neutrinos and nucleons did not account for nucleonic interaction [9], which modify the scattering cross sections, and may be significant due to the highly dense nature of the system. As a first pass, here we modified the cross sections of the neutral current reactions ν + p → ν + p and ν + n → ν + n, and will include in-medium effects in the current reactions in future work.

Neutrinos and Nucleon-nucleon interactions
The study of neutrino scattering in a medium of strongly interacting nucleons is done by via the nuclear matter response functions χ. These quantify the change in expectation values in a given observable, in response to some perturbation. The dissipative component of these response functions relates to the structure factors S , that modify the free differential cross section. Using the temperature dependent fluctuation dissipation theorem we have [12] where β = 1/k B T , T is the temperature, q is the momentum and ω the energy loss. In the absence of interactions, these response functions turn into the identity in position space, or delta functions centered around ω = 0 in momentum space.

The Virial EoS
The virial EOS is a model independent method for describing finite-temperature low-density nuclear matter that takes into account interactions between species. Previous development has been done to study the response function of low density warm matter. We refer the reader to Ref. [13] for details. Here we summarize the main points.
In this formalism the pressure of a system of different species i in terms of the fugacity, to second order, can be written as [14,15] P where T is the temperature, z i = e (µ i +E i )/T is the fugacity (with chemical potential µ i and binding energy E i > 0), s i is the spin degeneracy of a nucleon and λ i = √ 2π/m i T is the thermal wavelength, all these for species i. b i j , are the second virial coefficients (note b i = b ii ) and characterize interactions between particles of species i and j. It is possible to determine the second virial coefficients from nuclear scattering phase shifts (see [14]). For a system with one single species (for example neutrons) the vector response is written as [16,17] Horowitz et al [13] found that for a mixture of neutrons and protons which densities are given by n n(p) = 2 λ 3 z n(p) + 2b n z 2 n(p) + 2b pn z n z p , the vector response S V is where C n(p) v are weak charges of neutron and proton. Similarly, the axial or spin response S A was found via the densities of spin polarized nuclear matter where the virial coefficients b + and b − , describe the interaction between two particles of the same species (neutrons or protons) with spins aligned in the same direction and opposite direction, respectively. b + pn and b − pn correspond to the second virial coefficients of a neutron and a proton with spins pointing in the same direction and opposite direction, respectively. z ± i is the fugacity of species i with spin up (+) or down (−). In this way, Finally, the in-medium differential cross section per unit volume V can be written in terms of the structure factors as [13] 1 where G F is the Fermi constant, g A is the axial charge of the nucleon, E v is the neutrino energy, S A and S v are response functions derived from our virial equation of state, and θ is the angle of deflection. Note that by finding our cross section per volume we find our inverse mean free path, nσ.

Skyrme forces
Nuclear interactions have also been studied by using energy density functionals, which provide the energy of the system as a functional for a variety of different densities. In this way interaction energy is included by experimentally tuning different parameters. One common energy density functional is the Skyrme energy density functional, whose standard form looks as follows [18] where the C's represent parameters tuned to experimentally known quantities, T represents an isospin index, where T=0 is the density regardless of isospin, T=1 is the difference in densities between the two isospin densities. The other variables represent different types of densities (eg. Spin, kinetic, etc). We numerically calculate the structure factors from these functionals, following the work of A. Pastore et al. [19]. In terms of the Skyrme functionals the differential cross section is given by where E v (E v ) is the incoming(outgoing) neutrino energy, k is the outgoing neutrino momentum, and S (0,0) , S (1,0) and S (1,1) can be derived from response functions in different spin channels. Finally, we also make use of the fit provided by Horowitz et al [13] that matches the virial EoS at low density and follows Random Phase Approximation results at high density. Figure 2 shows a slice of the neutrino surface corresponding to the simulation discussed in section 2. We have found differences when the neutrino scattering off nucleons includes the correlation between nucleons caused by nucleon-nucleon interactions. The effect is stronger on the edges of the merger, on the equatorial plane, away from the center of gravity where the density has decreased. The height at which neutrinos decouple is higher when the nucleonnucleon interactions are ignored. Therefore, neutrinos decouple at higher temperatures due to nucleon correlations. We also note that surfaces using Skyrme energy density functionals are closer to the non-interacting case. However not much difference was seen between the different parameterizations of the Energy density functional. In future work we will study the impact of these differences in the neutrino detection as seen on neutrino observatories on Earth.