Energizing gamma ray bursts via $Z^\prime$ mediated neutrino heating

The pair annihilation of neutrinos $(\nu\overline{\nu}\rightarrow e^+e^-)$ can energize violent stellar explosions such as gamma ray bursts (GRBs). The energy in this neutrino heating mechanism can be further enhanced by modifying the background spacetime over that of Newtonian spacetime. However, one cannot attain the maximum GRB energy $(\sim 10^{52}~\rm{erg})$ in either the Newtonian background or Schwarzschild and Hartle-Thorne background. On the other hand, using modified gravity theories or the Quintessence field as background geometries, the maximum GRB energy can be reached. In this paper, we consider extending the standard model by an extra $U(1)_{\rm{B-L}}$ gauge group and augmenting the energy deposition by neutrino pair annihilation process including contributions mediated by the $Z^\prime$ gauge boson belonging to this model. From the observed energy of GRB, we obtain constraints on $U(1)_{\rm{B-L}}$ gauge coupling in different background spacetimes. We find that the bounds on gauge coupling in modified gravity theories and quintessence background are stronger than those coming from the neutrino-electron scattering experiments in the limit of small gauge boson masses. Future GRB observations with better accuracy can further strengthen these bounds.


I. INTRODUCTION
It has long been realized that the neutrino pair annihilation process plays a significant role in depositing energy to violent stellar processes such as type II supernovae [1][2][3], merging neutron stars [4], binary neutron stars in the last stable orbit [5], Gamma Ray Bursts (GRBs) [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20], etc.It is believed that most of the energy in such stellar explosions is carried away by the three flavors of neutrinos.The emission of a huge number of neutrinos with luminosity L ν ∼ 10 53 erg/s makes the stellar objects cool.A fraction of such huge neutrino flux can also deposit energy into the stellar envelope through neutrino pair annihilation (ν i ν i → e + e − , i = e, µ, τ ), neutrino lepton scattering, and neutrino baryon capture [17].This mechanism is termed neutrino heating.The process ν i ν i → e + e − is important for collapsing neutron stars, binary neutron stars in their last stable orbit, and r-process nucleosynthesis [21].This annihilation process can also continuously give energy to the radiation bubble and is a possible source of powering GRBs [6].In fireball model [22], the subsequent annihilation of electron-positron pair into photons ultimately power the GRB.Here, we assume that all the energy released into electrons is converted into photons to get such high energy in GRBs [6].These GRBs are high energy explosions that have been observed from cosmological distances [13,[23][24][25].They are the most energetic phenomenon represented by intense and prompt γ-ray emissions.We consider the maximum energy emission associated with the short GRBs to be ∼ 10 52 erg [26].However, in a Newtonian background, the neutrino pair annihilation process cannot provide the maximum energy.
The effect of including different background spacetimes near such strong gravity regimes has also been underscored in the literature.This is also apparent from the values of 2GM R which for instance is ∼ 0.7 for collapsing neutron stars, and ∼ 0.4 for supernovae calculations and hence, one cannot neglect the effect of General Relativity (GR) in the strong gravity regime [5].Here, G denotes Newton's gravitational constant, M denotes the mass of the neutron star, and R denotes the length scale.Since the neutron stars are rotating, the rotation parameter in the background metric also has to be included.It has been shown that in the Schwarzschild background, the neutrino heating is enhanced up to a factor 4 for type II supernova, and by up to a factor 30 for collapsing neutron stars relative to the Newtonian result [6].The rotation on the other hand reduces the energy deposition by 38% compared to the non rotating case [10].However, in all three backgrounds (Newtonian, Schwarzschild, and Hartle-Thorne) one cannot attain the maximum GRB energy.The energy deposition due to neutrino heating can also be calculated in different modified gravity backgrounds such as Born-Infeld Reissner-Nordstrom gravity [27], charged Galileon gravity [28], Eddington inspired Born-Infeld gravity [29], Einstein dilaton Gauss-Bonnet gravity [30], Brans-Dicke gravity [31], higher derivative gravity [32], etc as it is done in [17].In some of the modified gravity theories such as the Reissner-Nordstrom solution in the Born-Infeld model [27], one can attain the maximum GRB energy.The quintessence background can also enhance the energy deposition to its maximum value as discussed in [18].The radial variation of the temperature for black hole accretion disk in modified gravity theories can enhance the energy deposition by one order magnitude with respect to GR as is recently discussed in [33].The energy deposition can be enhanced as well due to the presence of topological defects such as global monopole [34].All these studies have considered only the Standard Model (SM) contribution to the neutrino pair annihilation process.
In this paper, we consider for the first time, the possibility of additional contribution from BSM physics to the neutrino pair annihilation process in the context of explaining the observed energy budget of GRBs.In particular, we focus on the scenario where the SM is extended by a general U (1) X gauge group.Such a scenario contains an additional neutral gauge boson (Z ) associated with the U (1) X symmetry.The special case of U (1) X is the U (1) B−L model which has been extensively studied in the literature [35][36][37].The phenomenology of extra Z in these models is particularly interesting and constraints have been obtained on the mass and the coupling strength of Z from several experiments.This includes electroweak precession data [38] , collider searches [39][40][41][42][43][44][45][46] , neutrino-electron scattering experiments [47,48], beam dump experiments [49][50][51], SN1987A [52][53][54][55] etc.If the Z gauge boson has a coupling with neutrinos and electrons then the Z mediated neutrino pair annihilation process can also contribute to the neutrino pair annihilation process.From the reported energy of GRB (10 52 erg) [26,56], we obtain constraints on gauge coupling in Newtonian, Schwarzschild, Hartle-Thorne, Born-Infeld Reissner-Nordstrom gravity, and Quintessence backgrounds.
The paper is organized as follows.In Section II, we obtain the total cross section and the energy deposition rate due to Z mediated process in neutrino pair annihilation.In Section III we compute the angular integration in Newtonian, Schwarzschild, Hartle-Thorne, Born-Infeld Reissner-Nordstrom gravity, and Quintessence spacetime backgrounds for the neutrino heating process.The contribution of Z mediated process to the neutrino heating mechanism in different spacetime backgrounds is calculated in Section IV.In Section V, we obtain the amount of energy enhancement due to Z contribution via neutrino heating process for the above mentioned spacetime backgrounds.In Section VI, we obtain constraints on Z from the GRB observation.Finally, in Section VII, we conclude and discuss our results.
In the following, we have used natural units (c = 1, = 1), and G = 1 throughout the paper.

II. NEUTRINO HEATING THROUGH Z
We extend the SM gauge group (SU (3) c × SU (2) L × U (1) Y ) by an additional U (1) X gauge symmetry.Such a scenario includes an extra neutral gauge boson (Z ) associated with the U (1) X symmetry.The latter is broken by an extra singlet scalar field Φ and the gauge boson acquires mass.Three right handed neutrinos are needed in the model to cancel the gauge and gauge-gravity anomalies.The U (1) X charges of the quarks and the leptons can be expressed in terms of that of the Φ and the SM Higgs (H).The charges for the scalars are chosen as 2x Φ and x H 2 [35].The corresponding U (1) X charge of lepton doublet is Q l X = (− 1 2 x H − x Φ ) and the charges of right handed electron and neutrino are The particular choice of x H = 0 and x Φ = 1 leads to the U (1) B−L model [57][58][59][60][61].The general interaction Lagrangian of Z gauge bosons with the leptons is The electron neutrino contributes to the neutrino annihilation process ν e ν e → e + e − via charge current (W ), neutral current (Z), and Z mediated interactions whereas ν µ and ν τ have only Z and Z mediated interactions for the process ν µ,τ ν µ,τ → e + e − .Since all three flavors of neutrinos are there in a hot neutron star, they will all contribute to the energy deposition rate via the Feynman diagrams Fig. 1.The energy deposition rate per unit volume near a hot neutron star is given as [2] q where E ν denotes the energy of neutrino and f ν = 2 (2π) 3 (e Eν /kT + 1) −1 corresponds to their thermal energy distribution function in the phase space which is of Fermi-Dirac type.Here k denotes the Boltzmann constant and T denotes the neutrino temperature.The neutrino velocity is denoted as v ν and σ denotes the cross section in the rest frame.Eq.2 is true for any flavor of neutrinos.Since the term in the first bracket of Eq.2 is a Lorentz invariant quantity, we can calculate its value in the centre of mass frame for ν e ν e → e + e − process.In the U (1) X model, it can be expressed as where for the region of interest we neglect the mass of the electron, since, the energy of neutrino is greater than 10 MeV; G F = 1.166×10 −5 GeV −2 is the Fermi constant and the θ W is the Weinberg angle whose value is sin 2 θ W = 0.23 [62].The first term in Eq.3 corresponds to the W and Z mediated SM processes, the second term is due to the Z contribution only.
The third term arises because of the interference between Z and Z mediated diagrams, while the fourth term stems from the interference between the W and Z mediated processes.For muon and tau types of neutrinos, only Z and Z mediated processes will contribute.Hence, for the scattering ν µ,τ ν µ,τ → e + e − , we obtain In what follows, we focus on the U (1) B−L model which has fixed values of x H and x Φ as x H = 0 and x Φ = 1.The results can easily be generalized to the U (1) X case by using suitable values of x H and x Φ .
Putting P ν = E ν Ω ν , and d 3 p ν = E 2 ν dE ν dΩ ν in the direction of Ω ν , where dΩ ν denotes the solid angle, and assuming T ν = T ν = T we find Hence, from Eq.2 the energy deposition rate due to the electron neutrino pair annihilation (1 + 4 sin 2 θ W + 8 sin 4 θ W ) + 4g where the angular integration Θ(r) is defined as Similarly the energy deposition due to muon and tau type neutrino annihilations in U (1) B−L model is given by qνµ,τ (r) = 21 2(2π) 6 π 4 (kT νµ,τ (r)) In the SM limit ( g M Z → 0), we get back the earlier result [20,63] where the + sign is for ν e ν e pair and the − sign is for ν µ ν µ and ν τ ν τ pairs.

III. NEUTRINO HEATING IN DIFFERENT BACKGROUND SPACETIMES
In this section, we calculate the effect of geometries in neutrino heating.We mainly focus on Hartle-Thorne (HT) geometry, the Born-Infeld generalization of Reissner-Nordstrom geometry (BIRN), and Quintessence geometry (Quint).We also calculate the neutrino heating in different limiting cases such as Newtonian (Newt) and Schwarzschild (Sch) backgrounds.

A. Hartle-Thorne background
The geodesic outside a slowly rotating neutron star with only a dipole correction on a static star is governed by the HT metric [64] where r denotes the distance from the origin, φ is the longitude, M is the mass of the neutron star, and J is the specific angular momentum.For J = 0, Eq.10 reduces to the Schwarzschild metric and for both J = M = 0, we have the flat or the Newtonian metric.
Since, we have considered the planar motion for the massless particle so we take θ = π 2 and the null geodesic g µν V µ V ν = 0 , where V µ = dX µ dλ and X µ ≡ (t, r, θ, φ).Hence, the geodesic equation of the HT metric for the planar motion becomes We can also derive the generalized momenta as where E denotes the energy per unit mass of the system and L denotes the angular momentum per unit mass.We can solve Eq.12 and obtain the expressions of ṫ and φ in terms of L and E as Using Eq.12, we can write Eq.11 as Putting the expressions of ṫ and φ from Eq.13, we can write Eq.14 as The local Lorentz tetrad for the given metric can be written as [10] e where the upper index denotes the row and the lower index denotes the column.The tangent of the angle (θ r ) between the trajectory and the tangent vector is the ratio between the radial and longitudinal velocity that can be expressed in terms of dr dφ as Comparing Eq.15 and Eq.17 we obtain the impact parameter b If the neutrino is emitted tangentially (θ R = 0) from the neutrinosphere of radius R ν i , then its trajectory defined by an angle θ r with radius r is given by where i = e, µ, τ .In the Newtonian limit, Eq.19 becomes (cos r , and in the non rotating Schwarzschild limit Eq.19 becomes (cos . Puting the expressions of ṫ and φ (Eq.13) in Eq.14 we obtain where the effective potential V eff defining the trajectory of a massless particle for a slowly rotating neutron star system is given as To find the neutrinosphere radius which corresponds to the last stable circular orbit for the neutrinos, we have to impose dV ef f dr = 0. Then from Eq.21 we obtain (Note that our expressions Eq.22 and Eq.23 are slightly different from those given in [10] due to some typographical errors.) If there is no rotation (Schwarzschild solution), then we get the neutrinosphere orbit at a radius = 3M .In that case, the massless neutrino which has a radius < 3M is gravitationally bound.However, if we solve Eq.22 for the rotating case, we can find at least one real root for which the neutrinosphere radius is < 3M .If the neutrinos are emitted tangentially from the neutrinosphere surface, then from Eq.18 we can also write, We can calculate the neutrinosphere radius and the impact parameter by solving Eq.22 and Eq.23 simultaneously for different values of J M 2 .The event horizon is calculated using We are interested in the domain r > R ν for the energy deposition rate.
In TABLE I we have summarized the event horizons, neutrinosphere radius, and impact parameter for different values of J M 2 .Now we compute the angular integration factor Θ(r) which appears in Eq.7.Choosing dΩ = dµdφ, where µ = sin θ and Ω = (µ, 1 − µ 2 cos φ, 1 − µ 2 sin φ), we can write The solution of Eq.24 in HT background is found as where, The temperature of the free streaming neutrinos at a radius r is related to the temperature of the neutrino at the neutrinosphere by the gravitational redshift.The neutrino temperature varies linearly with the redshift as whereas the observable quantity, the luminosity varies quadratically with the redshift as The neutrino luminosity for each ν i species for a blackbody neutrino gas can also be written as where a = 0.663 is the radiation constant in natural units.
Using Eq.25, Eq.27, Eq.28, Eq.29 we can calcualte The energy deposition rate is enhanced with increasing T 9 ν i (r)Θ ν i (r) which has distinct values in different background spacetimes.

B. Born-Infeld generalization of Reissner-Nordstrom solution
In the following, we will discuss the energy deposition for the Born-Infeld Reissner-Nordstrom (BIRN) gravity solution.The reason is that in this background the energy deposition matches quite well with the observed maximum GRB energy for certain values of modified gravity parameters.
In the Born-Infeld model, the nonlinear electromagnetic generalization of the Reissner-Nordstrom solution is defined by the metric as [27] where Here, b denotes the Born-Infeld parameter defined as the magnitude of the electric field at r = 0, Q denotes the electric charge, and F denotes the Legendre's elliptic function of the first kind given as  The values of b are chosen in such a way that as r → 0, f ( r M ) → +∞ which corresponds to the soliton like behaviour.For 0.5M Q 0.9M , f ( r M ) has two zeros.For Q = 0.5M , the metric has a soliton like behavoiur for b 4.2  M .In Fig. 2(b) we consider the special case Q = M which corresponds to the extreme case in black hole terminology.In the extreme case with b = 0.5225/M (blue solid line), the metric Eq.31 has one horizon and goes to +∞ near r = 0.For b 0.53/M (red dashed line), the metric has a naked singularity, and for b ≤ 0.516 M (solid purple line), the metric behaves as a Schwarzschild spacetime.
In BIRN background we can write where L BIRN obs in BIRN background is and x ν i is given as In and the quintessence parameter c [66].The black hole solution surrounded by a quintessence field is governed by the metric as given in Eq.31 with different expressions of f (r) as Note that c = Λ 3 and ω = −1 corresponds to the cosmological constant scenario.The For the black hole solution in presence of a quintessence field, Eq.30 becomes (1 where L Quint obs in Quintessence background is and x Quint Note that c = 0 corresponds to the Schwarzschild solution.
IV. Z CONTRIBUTION TO NEUTRINO HEATING IN DIFFERENT SPACE-

TIME BACKGROUND
In this section, we obtain expressions for the rate of energy deposition due to SM+Z mediated neutrino heating in different spacetime backgrounds.

A. Hartle-Thorne background
The total amount of energy deposition in the HT background due to neutrino heating in the region beyond the neutrinosphere is given as The total energy deposition by all the neutrino species is QHT νe + QHT νµ,τ .If we only include the W and Z mediated diagrams i.e, only the SM contribution for the process νν → e + e − and denote D = 1 ± 4 sin 2 θ w + 8 sin 4 θ w , where + sign is for ν e ν e pair, and − sign is for ν µ ν µ and ν τ ν τ pairs then the total rate of energy deposition in HT spacetime background becomes where Q51 = where we assume R νe ≈ R νµ,ντ = R, r = yR, and, x HT ν = x HT is given by Eq 26.Eq.41 is for a single neutrino flavor and thus one needs to calculate the total energy deposition contributed by all three flavors.
Putting J = 0 in Eq.42 gives the function F for the Schwarzschild background with For M → 0 and J → 0, the above function becomes F (0) = 1 corresponding to the Newtonian background with  In TABLE II we present the maximum energy deposited in Newtonian, Schwarzschild, and Hartle-Thorne backgrounds due to the neutrino pair annihilation process in SM.Here we have used Eq.41 for the Hartle-Thorne background and the equivalent expressions for Schwarzschild and Newtonian backgrounds by taking J = 0 and J = M = 0 in Eq.41 respectively.We also consider that the neutrinos are emitted from the neutrinosphere, the value of D is 1.23, the neutrino luminosity at infinity is ∼ 10 53 erg/s, and R = 20 km.
We infer from TABLE II that the observed maximum GRB energy cannot be explained by modifying the background spacetime with Schwarzschild geometry.If we include rotation in the background spacetime (Hartle-Thorne), the rate of energy deposition decreases as compared to the Schwarzschild case.
If the Z mediated process is included then Eq.41 will be modified.However, Eq.42 remains unchanged for any particular spacetime mentioned above, as it does not depend on the BSM physics.In the following, we obtain expressions for the rate of energy deposition in the Hartle-Thorne background including the Z mediated neutrino pair annihilation process.
Using Eq.6, Eq.8, and Eq.30, we can write the energy deposition rate for the electron neutrino in HT metric as where and x HT νe is given by Eq.26 for ν e .Similarly, the energy deposition rates for ν µ and ν τ in the HT background become where x HT νµ,τ is given by Eq.26 for ν µ,τ .From Eq.43 and Eq.44, we can similarly obtain the energy deposition rates for three flavors of neutrinos in the Schwarzschild and the Newtonian background with proper choices of x Sch ν and x Newt ν by putting J = 0 and J = M = 0 in Eq.26 respectively.

B. Born-Infeld generalization of Reissner-Nordstrom solution
If we only include the SM neutrino pair annihilation processes, then the total rate of energy deposition in the BIRN background becomes where is given as where f (r) is defined in Eq.32.Putting all the values of the parameters as given above and choosing R M = 2, Q = M and b = 0.30 M we obtain the total rate of energy deposition QBIRN 51 ∼ 1.5 × 10 52 erg/s.This value matches quite well with the maximum GRB energy.
We consider that the contribution of Z in neutrino heating is limited to be no larger than the experimental uncertainty in the measurement of GRB energy.We can calculate the energy deposition rate for ν e in BIRN background including the Z mediated process as where the expression of x BIRN ν i is given in Eq.35.Similarly, the energy deposition for ν µ and ν τ in BIRN background is same as Eq.47 with y νe , R νe and x νe will be replaced by y νµ,τ , R νµ,τ and x νµ,τ respectively.In addition, there will be a negative sign in front of 4 sin 2 θ W and the last term in the square bracket should be omitted due to only contributions from Z and Z mediated diagrams.

C. Quintessence background
The total rate of energy deposition for the SM neutrino pair annihilation process in presence of a Quintessence background becomes QQuint where is given as where f (r) is defined in Eq.36.For R M = 3, ω = − 2 3 , and c = 2.487×10 −2 we obtain the total rate of energy deposition QQuint 51 ∼ 9.99 × 10 51 erg/s which matches well with the maximum GRB energy.
We consider that in quintessence background the contribution of Z is limited to be no larger than the measurement uncertainty.The energy deposition due to electron neutrino pair annihilation in quintessence background including the Z mediated process is where x Quint νµ,τ is given by Eq.39 for ν µ,τ .

V. QUANTITATIVE ESTIMATES OF NEURINO HEATING DUE TO Z CON-TRIBUTION IN DIFFFERENT SPACETIME BACKGROUNDS
In this section, we study the effect of Z mediated contribution to the pair annihilation of neutrinos (νν) and calculate the energy deposited in GRB for Newtonian, Schwarzschild, Hartle-Thorne, Born-Infeld Reissner-Nordstrom, and Quintessence backgrounds.

A. Newtonian Background
In Newtonian background, the SM contribution to νν → e + e − process gives the value of the energy deposition as ∼ 1.5 × 10 50 erg whereas the maximum energy in a GRB is 10 52 erg.
We investigate to what extent the inclusion of Z mediated process can energize a GRB.
In Fig. 4, the red line denotes the variation of the ratio of the rate of energy depositions in BSM to that in SM with respect to The figure shows that the ratio decreases with increasing → ∞ (here it happens at M Z g 10 3 GeV) the BSM effect goes away as expected and the ratio becomes unity.For smaller values of there can be a greater enhancement in the energy deposition compared to the SM contribution.The black dashed line denotes the enhancement in energy required with respect to that of the SM+Newtonian background to explain the maximum energy in GRB.

B. Schwarzschild Background
As discussed earlier, including the GR effects can enhance the rate of energy deposition via neutrino annihilation [6] compared to the Newtonian calculation.In this section, we explore the effect of BSM physics if the background spacetime is Schwarzschild.
In Fig. 5 we have shown the variation of the ratio of energy deposition in BSM+Schwarzschild to SM+Newtonian cases with respect to

FIG. 6. Variation of the ratio of rate of energy depositions in BSM+Schwarzschild and
SM+Newtonian processes with respect to R M .We have shown the variations for = 300 GeV (blue), and

C. Hartle-Thorne Background
The Hartle-Thorne metric which includes the rotation can also enhance the energy deposition in neutrino heating as compared to the Newtonian background.
In Fig. 7 we have shown the variation of the ratio of energy depositions in merging neutron stars in Hartle-Thorne and Newtonian background with respect to energy in GRB.
In Fig. 9 we obtain the variation of the ratio of energy deposition rate in BIRN and Newtonian background with respect to Future observations with better accuracy can improve our results for modified gravity backgrounds up to one order of magnitude.Moreover, it is important to mention here that we have not taken the effects of any trapping of neutrinos [74] and nonlinear magnetic fields [75,76] that can significantly change the energy deposition rate.In conclusion, our study highlights the importance of the inclusion of BSM contribution to the neutrino annihilation process in constraining new physics.This can open up new avenues in understanding the physics of GRBs.

5 M 5 M
BIRN background Q=0.5 M, b = 4.Q= 0.6 M, b= 2.Q= 0.8 M, b= 1.03 M Q= M, b = 0.5225 f ( r M ) vs. r M , with different Q and b. f ( r M ) vs. r M , with fixed Q and different b.

FIG. 2 .
FIG. 2. (a)Variation of f ( r M ) with respect to r M for different values of charge and Born-Infeld parameter in BIRN background.(b)Variation of f ( r M ) with respect to r M for different values of Born-Infeld parameter with Q = M in BIRN background.
the limit, b → ∞, the metric turns into the linear Einstein-Maxwell Reissner-Nordstrom solution.If we further put Q → 0, we obtain the Schwarzschild solution.C. Quintessence backgroundQuintessence is a dynamical, time dependent, and spatially inhomogeneous scalar field which was first introduced as an alternative of cosmological constant (Λ) to explain the accelerated expansion of the universe[65].The quintessence field around a massive gravitating object can deform the spacetime.The spacetime geometry in presence of quintessence is parametrized by two quantities, the equation of state ω which can take values −1 < ω < − 1 3

FIG. 3 .
FIG. 3. Variation of f ( r M ) with respect to r M for different values of ω and c for a black hole solution surrounded by a quintessence field.

FIG. 4 .
FIG. 4. Variation of the ratio of rate of energy depositions in BSM and SM processes with respect to M Z g in Newtonian background.

MFIG. 5 .
FIG. 5. Variation of the ratio of rate of energy depositions in BSM+Schwarzschild andSM+Newtonian processes with respect to M Z g .We have shown the variations for R M = 3 (red) and negligible and the enhancement in energy deposition obtained is ∼ 4 − 30 (depending on the values of R M ) because of changing the spacetime geometry from Newtonian to Schwarzschild.The black dashed line denotes the enhancement in energy required with respect to that of the SM+Newtonian background to explain the maximum energy in GRB.In Fig.6we have shown the variation of the ratio of the rate of energy depositions in BSM+Schwarzschild and SM+Newtonian cases with respect to R M .We have varied R

2 FIG. 7 .
FIG. 7. (a)Variation of the ratio of energy depositions in merging neutron stars in Hartle-Thorne and Newtonian background with respect to M Z g for fixed R M .(b)Variation of the ratio of energy depositions in merging neutron stars in Hartle-Thorne and Newtonian background with respect to

8 FIG. 8 .
FIG. 8. (a)Variation of the ratio of energy depositions in merging neutron stars in Hartle-Thorne and Newtonian background with respect to R M for J M 2 = 0.1.(b)Variation of the ratio of energy depositions in merging neutron stars in Hartle-Thorne and Newtonian background with respect to R M for J M 2 = 0.8.

M = 3 , ω = − 2 5 (FIG. 12 . 5 .FIG. 13 .
FIG. 10. (a)Variation of the ratio of energy depositions in Born-Infeld Reissner-Nordstrom background with respect to R M for fixed Q = M, b = 0.10 M .(b)Variation of the ratio of energy depositions Born-Infeld Reissner-Nordstrom background with respect to R M for fixed Q = M, b = 0.30 M .

TABLE II
. Rate of energy emission due to neutrino pair annihilation for short GRB in Newtonian, Schwarzschild, and Hartle-Thorne background.Here we only consider the SM processes.