Neutrino emission, Equation of State and the role of strong gravity

Neutron-star mergers are interesting for several reasons: they are proposed as the progenitors of short gamma-ray bursts, they have been speculated to be a site for the synthesis of heavy elements, and they emit gravitational waves possibly detectable at terrestrial facilities. The understanding of the merger process, from the pre-merger stage to the final compact object-accreting system involves detailed knowledge of numerical relativity and nuclear physics. In particular, key ingredients for the evolution of the merger are neutrino physics and the matter equation of state. We present some aspects of neutrino emission from binary neutron star mergers showing the impact that the equation of state has on neutrinos and discuss some spectral quantities relevant to their detection such as energies and luminosities far from the source. INTRODUCTION Together with gravity neutrinos drive the evolution of several stellar phenomena such as supernovae, binary mergers, and black hole accretion disks. They are also a key ingredient in the synthesis of heavy elements [1, 2], the production of gamma ray bursts [3, 4] and kilonova [5, 6]. Despite the numerous efforts to fully understand these phenomena there are still puzzles to address. Unrevealing the missing pieces requires complex and computationally expensive simulations of the system under study that bring together details of nuclear physics and gravity. Furthermore, due to the complexity of the problem, it is necessary to post-process the results to shed light on other derived aspects such as the element abundances and neutrino detection. The strong gravitational field generated by a binary neutron star merger changes the spectral properties of the emerging neutrinos. This will affect any physical quantity related to them. For example, the emitted neutrinos can interact with matter outflowing the merger. Via weak interactions neutrinos will change the neutron to proton ratio (electron fraction) of the outflow setting the path that a nuclear reaction chain follows. However, strong gravity effects, such as redshifts and bending of neutrino trajectories, affect the neutrino fluxes altering the electron fraction. As a consequence the final abundances differ from those obtained in the absence of the gravitational field [7]. On another hand, if the merger occurs in the Milky way or in the local galaxy group chances are that we will able to detect those neutrinos with current and future facilities [8, 9, 10]. It is therefore interesting to study the behavior of neutrinos emitted from binary systems. In this proceedings we present post-processed neutrino emission results from state of the art 3D fully relativistic binary neutron star simulations with magnetic fields, neutrino cooling and different equations of state. The original simulation results of the coalescence have been presented in reference [11]. There, the evolution of the matter in connection to the Equation of State (EoS) is discussed, as well as the gravitational wave emission and neutrino detection rates. Here, we expand our discussion of neutrino emission focusing on the evolution of the neutrino surface and its relation to the EoS. EQUATION OF STATE The EoS connects the macroscopic thermodynamical variables to the microphysics of the neutron star. Observables such as maximum masses and radii carry important information about the behavior of nuclear matter at high densiar X iv :1 60 3. 02 75 5v 1 [ nu cl -t h] 9 M ar 2 01 6 ties. Similarly, our theoretical models of nuclear interactions should correctly account for these observables. Among other macroscopic variables, the EoS predicts the temperature, density and matter composition. The neutrino absorption and emission rates are correlated to these same quantities. A microphysical EoS is therefore fundamental to track the merger dynamics and the neutrino emission. We consider three different EoS (tables publically available at www.stellarcollapse.org), NL3, DD2, SFHo, all of them in the frame of the statistical model of Hempel and SchaffnerBielich [12]. They differ in the relativistic mean field model used to describe the nuclear interaction. The NL3 EoS is based on the interaction model of reference [13], the DD2 on [14] and the SFHo on [15]. These EoS predict different radii and maximum masses. The EoS is said to be soft if results in a small radius for a given mass and is stiff if for the same mass the predicted radius is larger. Our simulations consider the coalescence of two neutron stars each with a mass of 1.35M . The SFHo predicts for that mass a neutron star radius of ≈ 12 km. The DD2 predicts a close to 13 km radius while for the NL3 the radius is ≈ 15 km. In that order of ideas the SFHo is the softest of the EoS considered, the DD2 is an intermediate EoS and the NL3 is the stiffest one. NEUTRINO EMISSION Post-merger neutrino luminosities can be as large as 1054 erg/s. This huge amount of neutrinos can reach detectors on Earth and tell us valuable information about the merger and its evolution. At high matter densities (≈ 1014 g/cm3) neutrinos are trapped: the neutrino mean free path λν is shorter than the physical dimensions of the merger, i. e. matter is neutrino opaque. Once the density decreases the neutrino mean free path increases and neutrinos escape carrying energy away and contributing to the cooling of the system. The point of last scattering, the place where neutrinos are free, is known as the neutrino surface. A common approach to determine this surface is to find the places where the optical depth τ = 2/3 (see e.g. [16]). In terms of the opacity κν, and the mean free path lν, the optical depth τ is given by τν = ∫ ∞ sν κν(s)ds = ∫ ∞ sν 1 lν(s′) ds′, (1) along the propagation direction ŝ. In this post-processing we chose to study the neutrino emission as seen from an observer located at infinity in the z-axis. Then, the neutrino surfaces are defined by the two-dimensional hypersurfaces where the optical depth of the merger is perpendicular to the equatorial plane. This turns the optical depth into τν(x, y) = ∫ ∞ zν 1 lν(x, y, z′) dz′. (2) We integrate Equation 2 changing the lower limit zν until the value τ = 2/3 is reached. The themodynamical conditions of the neutrino surface are therefore the conditions of the matter at zν. In Equation 2 the mean free path depends of the cross sections σk, lν(x, y, z) = 1 ∑ k nk〈σk(Eν)〉 . (3) The sum above goes over all the relevant scattering and absorption processes k that neutrinos undergo as they diffuse through matter, with nk the number density of the target. 〈σk(Eν)〉 is the thermally averaged (“weighted”) cross section, 〈σk(Eν)〉 = ∫ ∞ 0 σk(Eν)φ(Eν)dEν ∫ ∞ 0 φ(Eν)dEν , (4) where φ(Eν) is the neutrinos Fermi-Dirac flux φ(E) = c 2π2(~c)3 E2 fFD (5) and fFD is the neutrino occupation number. We assume the temperature is equal to the local temperature and zero chemical potential. This procedure removes the energy dependence of the neutrino surface. Note, that the scattering neutrino surfaces introduced above differ from the effective neutrino surfaces (see e.g. [17]). The first ones correspond to the last scattering surfaces, where absorption and scattering neutrino processes are treated equally while in the calculation of the effective neutrino surfaces more emphasis is given to absorption processes. The effective surfaces would be however diffuse in area but will give a better estimate of the neutrino temperature, while the scattering surfaces will give a better estimate of the effective area of decoupling. As our calculation for average energies (explained below) is thermally weighted over the Fermi-Dirac distribution we expect that they will be good estimates. In this analysis we consider neutrino scattering from protons, neutrons and electrons, as is a good approximation to assume that matter is dissociated at the typical values of T=10 MeV (and above) found in simulations. In this way we have the charged current processes νe + n→ p + e−, (6) and ν̄e + p→ e + n. (7) We also consider neutral current processes, elastic scattering from electrons and neutrino-antineutrino annihilation, which affect on equal footing all neutrino flavors. We find proton and neutron number densities assuming charge neutrality Ye = Yp, and the electron number density assuming equilibrium of thermal electrons and positrons with radiation. Details on the cross sections of the above reactions can be found in ref. [10]. As the heavy lepton neutrinos, tau and muon, do not undergo the process of equations 6 and 7 they are the first to decouple form matter. Also as the medium is neutron-rich, due to the original neutron richness of the neutron stars, the last ones to decouple are electron neutrinos given the predominance of the reaction of equation 6. Using our results for the neutrino surfaces we can make estimates of the number of neutrinos leaving the merger per second dN/dt and average energies 〈Eν〉. For the first one we have dN dt = c 2π2(~c)3 ∫ dAdEE2 fFD = ∫ dAdEφ(E), (8) and for the energy rate (energy per sec) dE dt = c 2π2(~c)3 ∫ dAdEE3 fFD = ∫ dAdEEφ(E). (9) In the equations above the integral over dA corresponds to an integral over the neutrino surface. Our estimate for the average neutrino energy is then given by 〈Eν〉 = dE/dt dN/dt . (10) In order to transform the average energy to another reference frame we make use of the fact that the quantity I/c3, with I the specific intensity is an invariant, I E3 = 1 c2 dN d3xd3 p = fFD h3c2 . (11) Noting the quantities measured by an observer with “tilde”, so the observed energy is Ẽ, and dà is the observed area differential, and without tilde analogous emitted quantities E and dA, then we have 1 c2 dN d3 x̃d3 p̃ = 1 c2 dN d3xd3 p , (12) which leads to dN dt̃ = ∫ fFD h3c2 Ẽ2dẼdÃd̃Ω. (13) We can rewrite this in terms of the emitted quantities given that Ẽ = √ g00E, with g00 the redshift (in the Schwarzchild metric for simplicity), assuming that the emission is isotropical dΩ̃ = 4π, and that distances are stretched by a factor of (1 − rs/r) = g−1/2 00 , dN dt̃ = ∫ g 00 φ(E)dEdA. (14) We then get ẼdN/dt̃ = dẼ/dt̃, dẼ dt̃ = ∫ g00φ(E)EdEdA, (15) and the observed average neutrino energy is just the ratio of the above expressions, 〈Ẽν〉 = dẼ/dt̃ dN/dt̃ . Evolution of the neutrino surface: the DD2 EoS Following the methodology of the previous section we find the neutrino surfaces for the three different neutrino flavors. The electron antineutrino surface at t = 7.4 ms after the merger for the DD2 EoS is presented in Figure 2. The height represents the distance, measured from the equatorial plane, at which neutrinos decouple from matter, whereas the color scale shows the matter temperature at these last points of neutrino scattering. With these temperatures we determine the average neutrino energies as described above. -20 -15 -10 -5 0 5 10 15 20 -20 -15 -10 -5 0 5 10 15 20 2 3 4 5 6 7 8 9 10 11 12 z[km] electron antineutrinos t=4500