Pulsars are rotating neutron stars that are observed to be slowing down implying a loss of their kinetic energy. There can be several different physical mechanisms involved in their spindown process. The properties of fastrotating pulsars depend on the nature of the neutron star matter which can also affect the spindown mechanisms. In this work we examine three different physical phenomena contributing to the spindown magnetic dipole radiation gravitational mass quadrupole radiation due to the mountain formation gravitational mass current quadrupole radiation or the rmodes and calculate the expressions for the braking indices due to all of them. We have also considered jointly the implications of the uncertainties of the equation of the state of neutron star matter and rapid rotation on the braking indices corresponding to the aforementioned processes and their combinations. In all cases the rapid rotation results in a departure from the standard values in the literature for the braking index when the rotational effects are ignored. If generated with a saturation amplitude within the range of 104  101 the rmode oscillations dominate the spindown of millisecond pulsars. We also explore the braking index in the context of millisecond magnetars. This study examines two braking index measurements in the context of newly born millisecond magnetars from two observed short gammaray bursts. The measured braking indices for these objects are consistent with our estimation which allows us to conclude that the spin frequency of the remnants is within the range of sim 550850 Hz.
['Astrophysics - High Energy Astrophysical Phenomena']
Structural response of neutron stars to rapid rotation and its impact on the braking index
{'Structural response of neutron stars to rapid rotation and its impact on the braking index': "Avishek Basu, 1, ∗ Prasanta Char, 2, 3, † and Rana Nandi 4, ‡  \n1 Jodrell Bank Centre for Astrophysics, School of Physics and Astronomy, University of Manchester, Manchester, M13 9PL, UK 2 Departamento de F'ısica Fundamental, Universidad de Salamanca, Plaza de la Merced S/N, 37008 Salamanca, Spain 3 Space Sciences, Technologies and Astrophysics Research (STAR) Institute, Universit'e de Li'ege, Bˆat. B5a, 4000 Li'ege, Belgium 4 Department of Physics, School of Natural Sciences, Shiv Nadar  \nInstitution of Eminence, Greater Noida 201314, Uttar Pradesh, India  \nPulsars are rotating neutron stars that are observed to be slowing down, implying a loss of their kinetic energy. There can be several different physical mechanisms involved in their spin-down process. The properties of fast-rotating pulsars depend on the nature of the neutron star matter, which can also affect the spin-down mechanisms. In this work, we examine three different physical phenomena contributing to the spin-down: magnetic dipole radiation, gravitational mass quadrupole radiation due to the 'mountain' formation, gravitational mass current quadrupole radiation or the r-modes, and calculate the expressions for the braking indices due to all of them. We have also considered jointly the implications of the uncertainties of the equation of the state of neutron star matter and rapid rotation on the braking indices corresponding to the aforementioned processes and their combinations. In all cases, the rapid rotation results in a departure from the standard values in the literature for the braking index when the rotational effects are ignored. If generated with a saturation amplitude within the range of 10 -4 -10 -1 , the r-mode oscillations dominate the spin-down of millisecond pulsars. We also explore the braking index in the context of millisecond magnetars. This study examines two braking index measurements in the context of newly born millisecond magnetars from two observed short γ -ray bursts. The measured braking indices for these objects are consistent with our estimation, which allows us to conclude that the spin frequency of the remnants is within the range of ∼ 550-850 Hz.", 'I. INTRODUCTION': "The majority of neutron stars (NS) are observed as pulsars, which are rapidly rotating and observed to spin down over time, indicating a loss of rotational kinetic energy. The spin-down rate is given as ˙ Ω = -k Ω n [1], where n is the braking index and k is a positive constant. The braking index captures information about the mechanism of energy loss. Depending on the emission loss mechanism k can be a function of radius, moment of inertia, magnetic field strength etc (discussed later in the text). In most cases, the pulsar's rotational energy loss is attributed to the magnetic dipolar radiation (MDR) from the pulsar magnetosphere for which the braking index is 3 [2]. However, pulsars can lose energy via gravitational wave (GW) radiation if there is non-axisymmetric deformation or r-mode instability, for which the braking index is 5 and 7 respectively [3]. Therefore, it is natural to expect a rapidly rotating neutron star to lose energy via all three channels if the conditions are favourable. Apart from these three channels of energy loss, pulsars also lose energy due to the particle wind [4, 5].  \nThe spin-period of pulsars spans a large range. The slowest pulsar has a rotation period of 76 s [6] to the most rapidly rotating being 1.39 ms [7]. The total population is typically categorized as normal (or 'slow') pulsars  \nand millisecond pulsars, with the demarcation around 16 ms [8]. The braking index has been measured for a few younger pulsars and found be less than 3 [9-15], which is often attributed to the evolution of the magnetic field, modified magnetosphere or even the interaction of supernova fallback disk [4, 16, 17]. A similar sub-3 braking index can be achieved in rapidly rotating pulsars, which is purely due to structural change in the star due to rapid rotation. In such a scenario, the k cannot be assumed independent of the spin frequency [18], which has been discussed in detail in Sec. IV.  \nThe effect of structural evolution is prominent over a spin frequency of 200 Hz [18]. Currently, the ATNF Pulsar Catalogue 1 [19] records 392 pulsars with spinfrequency more than 200 Hz, which corresponds to ∼ 11% of the total pulsar population discovered till date. These rapidly rotating pulsars are important in probing the physics of dense nuclear matter. The physics of the nuclear interaction and the constituents of the matter determines the dense matter equation of state (EOS), which in turn governs the stellar structure and its response to rapid rotation. To study the frequency evolution of the stellar structure and the braking index, we have adopted a semi-agnostic approach to construct the dense matter equation of state following the formalism of Gandolfi et al. [20], which is summarized in Sec. II.  \nThere are various types of constraints that can be applied to the EOS model. Astrophysical observations  \nprovide several constraints relevant to the high density part of the EOS. In particular, the radio observations of the most massive pulsar, PSR J0740+6620 [21, 22] provide the most stringent constraint on the NS EOS. The tidal deformability measurements from the multimessenger observations of the binary neutron star merger event GW170817 put additional constraints on the EOS [23-25]. In recent years, the Neutron star Interior Composition Explorer (NICER) collaboration has reported a few simultaneous mass-radius measurements of PSR J0030+0451, PSR J0740+6620, and PSR J0437-4715, respectively, that also help us to constraint the Mass-radius plane and consequently the EOS parameter space [2632]. On the other hand, the latest constraint on the lowdensity part of the EOS comes from the advancements of the chiral effective field theory ( χ -EFT) calculations [33-37].  \nWe have employed a Bayesian analysis framework, supplemented by the measurements from the astrophysical observations to constrain the EOS model parameters and present the results of the analysis in Sec. II. In Sec. III, we used the constrained EOS to investigate their effects on stellar structure under rapid rotation. In Sec. IV, we have presented an ab initio derivation of the braking index under the assumption of energy loss through MDR and GWs due to finite deformation and r-mode oscillation. The analytical expressions have been later combined to study the frequency-dependent evolution of the braking index in context to millisecond pulsars and nascent millisecond magnetars born from NS-NS mergers or other formation scenarios. Finally, we summarise our findings of this work in Sec. V.", 'II. DESCRIPTION OF NEUTRON STAR MATTER': 'In this section, we describe the EOS of dense nuclear matter. We follow the work by Gandolfi et al. [20] to construct a double polytrope EOS for our purpose. With this particular form of the function, Gandolfi et al. [20] could parametrize the results of the ab-initio microscopic calculations using Quantum Monte Carlo methods. We use this parametrization to create the EOS directly from nuclear empirical parameters (NMPs). We summarize the necessary equations in the following. The energy per particle for the pure neutron matter (PNM) is expressed as a function of density,  \nE PNM ( ρ ) = a ( ρ ρ 0 ) α + b ( ρ ρ 0 ) β , (1)  \nwhere, ρ 0 is the nuclear saturation density.  \nThe symmetry energy is defined as, E sym ( ρ ) = E PNM ( ρ ) -E SNM ( ρ ), where E SNM ( ρ ) is the energy per particle of the symmetric nuclear matter (SNM). Then, the slope of symmetry energy becomes, L ( ρ ) =  \n3 ρ 0 ∂E sym ( ρ ) /∂ρ . At saturation, they can be expressed as,  \nJ = E sym ( ρ 0 ) = a + b -E SNM , L ( ρ 0 ) = 3( aα + bβ ) . (2)  \nThe density dependence of the symmetry energy can be parametrized by,  \nE sym ( ρ ) = C ( ρ ρ 0 ) γ . (3)  \nThus, we get, C = a + b -E SNM from equation 2 at ρ = ρ 0 . Finally, from the condition, P = ρ 2 ∂E SNM ∂ρ | ρ = ρ 0 , we get,  \nγ = aα + bβ a + b -E SNM . (4)  \nUsing the relations above, the general expression of the EOS as a function of baryon number density ( ρ ) and proton fraction ( x = ρ p /ρ , where ρ p is the proton density) becomes,  \nE ( ρ, x ) = E PNM ( ρ ) + C ( ρ ρ 0 ) γ [ (1 -2 x ) 2 -1 ] . (5)  \nTo describe the NS matter, we also need to include the contribution from the lepton. In a cold NS, the lepton density is regulated by the β -equilibrium condition, µ n = µ p + µ e . Beyond 2 ρ 0 , we construct the EOS using the piecewise constant speed of sound EOS model [38] up to the density of 12 ρ 0 . We randomly sample the density points within this interval of the c s -ρ plane and the corresponding speed of sound points complying with the constraints 0 ≤ c s < 1. For the low-density part of the EOS, we have used the standard Baym-PethickSutherland (BPS) EOS ( < ∼ 0 . 001 fm -3 ) for the outer crust and Negele-Vautherin (NV) EOS (up to 0 . 08 fm -3 ) for the inner crust[39, 40]. The crust EOS is smoothly joined with the core EOS given by equation 5.  \nAfter setting up our EOS model, we continue to explore the parameter space of our model using various theoretical, experimental, and observational constraints within a Bayesian analysis. We describe briefly the constraints and their implementation in the following.  \nFirst, we consider constraints around saturation density. The EOS generated within our model must satisfy the constraints coming from the chiral effective field theory ( χ -EFT) calculations. We used the results from Drischler et al. [36] to implement the ( χ -EFT) constraints for the EOSs in β -equilibrium in the range (0 . 5 , 1 . 1) ρ 0 . The priors of our model parameters are given in table I. The ranges of ρ 0 and E SNM are chosen according to the experimental knowledge. Another condition is on the calculated values of E sym and L at saturation. We have imposed the E sym to be in the range of 25 to 40 MeV and L to be within 30 and 80 MeV.  \nFIG. 1. Shows the posterior distribution of the model parameters along with the symmetry energy at the saturation density ( J ) and the slope of symmetry energy at the saturation density ( L ).  \n<!-- image -->  \nWe have used the mass-measurements of massive pulsars, and combined tidal deformability ( ˜ Λ) from GW170817 as our observational constraints on the EOS. In particular, we have used the mass-measurement of J0740+6620 as 2 . 08 ± 0 . 07 M ⊙ reported by [22] as a Gaus-  \nlikelihood,  \nP (data M max | X ) = 1 2 [ 1 + erf ( M max ( X ) /M ⊙ -2 . 08 0 . 07 √ 2 )] , (6)  \nwhere, ( X = a, α, b, β, E SNM , ρ 0 ) represents our model parameters. Then, we applied the improved constraints on ˜ Λ from Abbott et al. [25]. The value of ˜ Λ de-  \nFIG. 2. The brown lines indicate the constrained EOSs ( ∼ 6K) corresponding to the posterior distributions of figure 1 and the blue lines indicate the EOS which falls within the 68% confidence interval of the constrained posterior distribution of the EOS parameters.  \n<!-- image -->  \nn the mass ratio q = m 1 /m 2 and the chirp mass ( M chirp = ( m 1 m 2 ) 3 / 5 ( m 1 + m 2 ) 1 / 5 ), where m 1 , m 2 are the masses of primary and secondary objects of the binary system, respectively. The chirp mass has been determined rather precisely to be M chirp = 1 . 186 ± 0 . 001 M ⊙ for GW170817. We use the publicly available data from LIGO-VirgoKagra (LVK) collaboration 2 assuming the low-spin prior and construct the likelihood as,  \nP (data LVK | X ) = ∫ dm 1 ∫ dm 2 P ( m 1 , m 2 | X ) × P (data LVK | m 1 , m 2 , Λ 1 ( m 1 , X ) , Λ 2 ( m 2 , X )) , (7)  \nwhere, P ( m 1 , m 2 | X ) is the prior distribution for the component masses of the binary. For simplicity, we have chosen a uniform prior for m 1 and m 2 . We use a Gaussian kernel density estimator to construct the GW likelihood from the discreet data. We have fixed the chirp mass to its median value because of the high precision of the measurement. Then, we construct binaries corresponding to that chirp mass by varying m 1 and determine the corresponding m 2 . For each EOS, we compute the tidal deformabilities (Λ 1 , Λ 2 ) for the pairs of ( m 1 , m 2 ), and find the probability using equation 7.  \nThe final likelihood function with all the astrophysical  \nFIG. 3. The M-R diagrams for all ∼ 6K EOSs from figure2 have been shown in the brown coloured lines and the blue lines indicate the M-R relations corresponding to the 68% confidence interval of the posterior distribution of the constrained EOS parameters.  \n<!-- image -->  \nTABLE I. Summary of the prior range used for our EOS model parameter.  \nconstraints is simply the product,  \nP (data astro | X ) = P (data M max | X ) × P (data LVK | X ) . (8)  \nFinally, we sample the posterior using a nested sampling algorithm [41], in the dynesty software package [42]. The posterior distributions of the EOS parameters for nuclear matter are shown in Fig. 1. We have also included the derived values for E sym and L at ρ 0 . We have found E sym = 31 . 49 +3 . 43 -3 . 11 MeV and L = 58 . 82 +10 . 24 -8 . 44 MeV at 68% CI. These values are well within the agreed ranges found in literature [43, 44]. This affirms the robustness of our EOS samples used to study the rotational properties of pulsars in this work. We have shown the corresponding EOSs and their mass-radius curves in Fig. 2 and 3, respectively. We can see from Fig. 3 that the EOSs generated in our exercise span a large range of radii for a 1 . 4 M ⊙ star. Therefore, these curves are able to represent the parameter space of neutron star matter. Note  \nthat, we have not used the data from NICER measurements in our analysis as we see that our 68% mass-radius curves in Fig. 3 are consistent with 68% of those sources.', 'III. ROTATIONAL EFFECTS ON THE STRUCTURE OF NEUTRON STARS': 'To study the effect of rotation on the macroscopic quantities of the star we have used the RNS code 3 [45, 46]. For every EOS computed in the Sec II, we calculate the gravitational mass ( M ), the equatorial radius ( R e ), the ratio of the polar to the equatorial radius ( R p /R e ) and the moment of inertia ( I ) as a function of rotation frequency ( f = Ω / 2 π , where Ω is the angular velocity of the star), keeping the baryon mass ( M b ) fixed at 2 M ⊙ .  \nA rapidly rotating star can sustain more mass than a non-rotating star with the same central density. To obtain the sequence of the above-mentioned macroscopic quantities with rotation frequency, we treat central density as the parameter and search for the frequency which a star of baryon mass of 2 M ⊙ can sustain. However, since a low central density leads to a low-mass star, the frequency needed to meet the M b = 2 M ⊙ condition may be more than the Kepler frequency (defined as the maximum frequency of rotation that a star can support before mass-sheading) if the central density is too low. Hence, the RNS code does not converge. As we increase the central density, the M b = 2 M ⊙ criteria is fulfilled at increasingly lower frequencies. When the central density is very high, the condition of M b = 2 M ⊙ is achieved at a frequency ( < ∼ 200 Hz) that is too low for the RNS code to handle. Therefore, we identify the densities corresponding to the highest and the sequence between those two densities for all ∼ 6000 EOSs in an automated fashion in our pipeline built around the RNS code.  \nThe rapid rotation generates a large amount of centrifugal force, which leads to the departure from the spherical shape of the star that is characterised by the ratio of the polar to the equatorial radius R p /R e . The dependence of this ratio on frequency is shown in the bottom left panel of Fig. 4 for all EOS.lowest frequencies for which the RNS code converges. Subsequently, we generate the Due to the deformation in the shape of the star, the definition of the radius becomes ambiguous in such a case, yet, in many scenarios, for estimating various physical quantities, the radius of the star is required as discussed in Sec. IV. Hence, in this work, we define a frequency-dependent radius by averaging over the circumferential radius of a rotating star given as  \nR ≡ ⟨ R (Ω) ⟩ = ∫ 2 π 0 R ( θ, Ω) dθ ∫ 2 π 0 dθ . (9)  \nThe expression of circumferential radius R ( θ, Ω) = R e [1 -¯ Ω(0 . 788 -1 . 03 x ) cos 2 θ ] is taken from the previous work by AlGendy and Morsink [47] and Suleimanov et al. [48], where θ is the polar angle, ¯ Ω = Ω( R 3 e /GM ) 1 / 2 and x = GM/c 2 R e . The EOS and rotation frequency-dependent average radius of the star is shown in the lower right panel of Fig. 4, which can be seen to decrease strongly at higher frequencies ( > ∼ 850Hz). Also at the higher frequency, the R p can be ∼ 25% smaller than the R e , therefore our method of computing a unique value of a radius and attributing a spherical shape to the star may have some shortcomings even if it mathematically renders a valid solution. In the upper left and right panels of Fig. 4 we show the variation of the gravitational mass and moment of inertia as a function of the rotation frequency for all EOS. It is evident from Fig. 4 that all these physical quantities have significant frequency dependence and will contribute to the braking index of the NS which we will discuss in the next section. We have confined our studies to the rotational frequency above > ∼ 200 Hz following Hamil et al. [18], where the frequency dependence of the macroscopic quantities has been shown to be prominent above > ∼ 200 Hz only.  \nThe high-frequency behaviour of the macroscopic quantities observed in Fig. 4 is controlled by the stiffness of the EOSs. At a given density, a stiffer EOS produces more pressure than a softer one and, therefore, can defy gravity more easily. Hence, if there is no rotation, the stiffer EOS will make a star of a larger radius and a lower central density if a certain number of baryons (equivalent to a fixed baryon mass) are put together. Consequently, the star with the stiffer EOS will be less bound, resulting in a higher gravitational mass. Note the gravitational binding energy is defined as M b -M . This behaviour remains even if the stars rotate. Therefore, the upper curves in M -vsf , R -vsf , and I -vsf plots in Fig. 4 correspond to stiffer EOSs. With increasing frequency, the equatorial radii of all the stars (irrespective of the underlying EOS) grow, making them less bound and having higher gravitational mass. Furthermore, since at a given rotational frequency, the star with the stiffer EOS is less bound than the softer EOS, it becomes unstable at a lower frequency. In other words, the Kepler frequency is smaller in the case of stiffer EOSs. It explains the behaviour seen in the M -f plot. The R p /R e -vsf curve captures the evolution of the stellar deformation as the function of rotation frequency. It is clear from the above discussion that for the stiffer EOSs, the equatorial radius increases much faster with the small change to the rotation frequency compared to the softer EOSs. Consequently, the ratio R p /R e falls faster in the case of stiffer EOSs. Therefore, the left curves in the R p /R e -vsf plot belong to stiffer EOSs.  \n45  \nFIG. 4. This figure shows the variation of four structural quantities of neutron stars as a function of the rotation frequency for all EOS obtained in Sec II. The blue lines indicate the relation for all ∼ 6K EOS and the purple lines corresponds to the solutions of EOS constructed from the 68% C.I. of the posterior distribution of parameters shown in Fig. 1. The upper left panel shows the variation of the gravitational mass as the function of the rotation frequncy, the upper right panel shows the variation of the moment of inertia, the lower left panel shows the variation of the ratio of the polar to equitorial radius and the lower right panel shows the variation of average radius of the star as the function of rotation frequency. For every EOS the baryon mass of the star is kept fixed at 2 M ⊙ , further details can be found in the Sec. III.  \n<!-- image -->', 'IV. BRAKING INDEX': 'In this section, we present the derivation of the braking index and its dependence on the spin frequency. Later in this section, we discuss the results in the context of millisecond pulsars and newly born millisecond magnetars in detail.', 'A. Calculation of braking index': "Pulsars spin down at the cost of their rotational kinetic energy. If the spin-down energy ˙ E through a mechanism is proportional to -Ω n +1 , the spin-down relation is obtained as  \nd dt ( 1 2 I Ω 2 ) = I Ω ˙ Ω ∝ -Ω n +1 ⇒ ˙ Ω = -k Ω n , (10)  \nwhere k is a constant which depends on various physical parameters of the neutron stars depending on the mechanism of the emission loss discussed later in this section. The braking index n depends both on the first and the second derivative of the spin frequencies as  \nn = Ω ¨ Ω ˙ Ω 2 . (11)  \nThe numeric value of the braking index is the signature of the mechanism which dominates the spin-down of a pulsar.  \nThe energy loss by the MDR is given as  \n| ˙ E EM | = 2 3 µ 2 Ω 4 sin 2 α ≡ E EM R 6 Ω 4 , (12)  \nwhere the magnetic moment | µ | 2 = B 2 p R 6 / 4, with the surface magnetic field strength B p , stellar radius R , α is the angle of inclination between the rotation and the magnetic axis and E EM = B 2 p sin 2 α/ 6. The corresponding spin-down rate using Eq. 10 leads to  \n˙ Ω MDR = -2 B 2 p R 6 sin 2 α 3 I Ω 3 ⇒-k MDR Ω 3 , (13)  \n̸  \nindicating the MDR leads to braking index 3. However, in Sec. III we have noticed at higher spin-frequency the R and depends on f making k MDR dependent on f and hence the braking index. On the other hand, the f dependence of I , will not allow us to arrive at the expression 13, as dI/d Ω = 0. Hence, for a rapidly rotating star, the modification to the spin-down expression is essential.  \nThe expression given in Eq. 12 is valid if the external magnetic field is purely dipole. However, studies show the presence of complex structures of higher order magnetic multipole near the stellar surface [49]. But, at a larger distance from the surface, the approximation of the dipolar field line is still valid. Therefore in this work, we have assumed that the pulsar magnetosphere is purely dipolar. Apart, from the geometrical consideration of the magnetic field, we also ignore any possible evolution of the dipolar field with time. Otherwise, it would contribute by an amount 2Ω ˙ B/ ˙ Ω B to the braking index. Similarly, the evolution of α would allow the departure of the braking index by an amount 4Ω ˙ α/ ˙ Ωtan α , which has been also ignored for the current study. As we do not consider any evolution in B and α we use them as a parameter in our model.  \nA long-lived, non-axisymmetric deformation of neutron stars would result in energy loss via the continuous GW (details can be found in a recent review by Gittins [50]). These deformations are referred to as mountains and generate a certain amount of ellipticity ( ϵ ), whose magnitude is still uncertain, depending on the process leading to the formation of the mountains and the crustal EOS. However, a typical value of maximum ellipticity that a neutron star crust can sustain is ∼ 10 -6 [51-53]. The strong magnetic field can lead to the deformation of a star generating ϵ ∼ 10 -12 and the misalignment between the magnetic axis and the rotational  \naxis leads to the GW emission [54-56]. Similarly, mountains are generated from the accretion process, they can be either thermal mountains [57] or magnetic mountains. The thermal mountains produced from the temperaturesensitive nuclear reactions on the surface of the neutron stars survive for a much shorter duration ∼ 0 . 2 yrs (this a typical value for more details refer to Gittins [50] and the references therein) between the accretion-mediated outburst phase of neutron stars. However, the magnetic field-supported mountains of the accreted material may survive over much longer time scales > ∼ 10 8 yr [58] giving rise to ϵ ∼ 10 -7 to 10 -8 . Implies, that the mountain formed at the end of the accretion phase (even after the companion's disappearance) can survive for a decent fraction of a neutron star's lifetime. The amount of energy loss via the GW emission is proportional to the sixth power of the spin frequency and the ellipticity of the star, and is given as  \n| ˙ E Q GW | = 32 5 G c 5 I 2 ϵ 2 Ω 6 ≡ E Q I 2 ϵ 2 Ω 6 , (14)  \nwhere E Q = 32 G/ 5 c 5 . The corresponding spin-down rate, following the Eq. 10 is  \n˙ Ω Q GW = -E Q Iϵ 2 Ω 5 ⇒ = -k Q GW Ω 5 , (15)  \nindicating the braking index in this case is 5. However, a similar argument like in the case of MDR spin-down rate given in Eq.13, is valid here, where rotational effects on I would not lead to the expression we have arrived at Eq. 15.  \nThe all-sky search for the continuous gravitational waves did not lead to the discovery of a signal but has enabled to establish an upper limit of ϵ < ∼ 10 -6 (depending on the distance and the frequency) [59]. Hence, in our case, we use ϵ = 10 -7 which is an order of magnitude lower than the established upper limit.  \nThe r-modes are quasi-toroidal oscillations in neutron stars where the Coriolis force acts as the restoring force [60]. Counterrotating r-modes can become unstable to gravitational radiation reaction via the ChandrasekharFriedman-Schutz (CFS) mechanism [61, 62]. It has been shown that in the absence of any fluid dissipation, CFS instability can arise at any rotational frequency of the star [63-65]. In realistic neutron stars, however, several damping mechanisms are present, such as due to the bulk and shear viscosities, which lead to the r-modes amplitude achieving a saturation value ( α S ), as shown in Owen et al. [60]. The microscopic origin of the damping mechanisms relates to the nature and the temperature of the neutron star matter. In the angular velocitytemperature (Ω -T ) plane, the interplay between the gravitational wave emission timescale and viscous dissipation timescales curve out an instability region. When the star resides within the unstable region, the gravitational wave emission contributes significantly to the rotational energy loss becoming another mechanism for the spindown of the star. The frequency of the emitted GW  \nis 4/3 the rotation frequency of the star [3]. Considering previous studies, the r-mode saturation amplitude can attain a value between 10 -4 to 10 -1 . For these values of α S , the energy loss due to r-mode is given by [60, 66],  \n| ˙ E R GW | = ( 4 3 ) 8 4 πG 25 c 7 α 2 S M 2 R 6 Ω 8 J 2 ≡ E R α 2 S M 2 R 6 Ω 8 J 2  \n(16)  \nwhere, E R = ( 4 3 ) 8 4 πG 25 c 7 , and J is given by,  \nJ = 1 MR 4 ∫ R 0 ρ ( r ) r 6 dr. (17)  \nThe expression 10 gives the spin-down rate as  \n˙ Ω R GW = -E R α 2 S M 2 R 6 J 2 I Ω 7 , (18)  \nindicating the spin-down due to r-mode leads to a braking index of 7. The above expression 18 is under the approximation of the no rotational effect on the stellar structure. However, we have shown that for rapid rotation there exists significant structural evolution, hence a revision to the above expressions is important.  \nWhen all the above three mechanisms of energy loss are active, the rotational energy loss is given by  \nd dt ( 1 2 I Ω 2 ) = -˙ E EM -˙ E Q GW -˙ E R GW , (19)  \nwhere ˙ E EM , ˙ E Q GW and ˙ E R GW are the rate of loss of energy via the electromagnetic radiation, gravitational quadrupolar radiation and the r-mode. The relation between the spin-down rate and the other macroscopic properties of the neutron star's interior and magnetosphere can be obtained by expanding the left-hand side derivative of the Equation 19, where we consider a nonzero time evolution of the moment of inertia.  \n˙ I +2 ˙ Ω I = -2 E EM R 6 Ω 3 -2 E Q I 2 ϵ 2 Ω 5 -2 E R α 2 S M 2 R 6 Ω 7 J 2 (20)  \nIn Equation 20, we use the chain rule to write ˙ I = dI d Ω d Ω dt = I ' ˙ Ω and further simply the equation to obtain ˙ Ω as  \n˙ Ω = -2 E EM R 6 Ω 3 +2 E Q I 2 ϵ 2 Ω 5 +2 E R α 2 S M 2 R 6 Ω 7 J 2 I ' Ω+2 I (21)  \nSimilarly by taking the derivative of Equation 20 and applying the chain rule to write I = I '' ˙ Ω 2 + I ' ¨ Ω, we obtain  \n¨ Ω = [ -6 E EM Ω 2 ˙ Ω R 5 (2 R ' Ω+ R ) -2 E Q Iϵ ˙ ΩΩ 4 (2 I ' ϵ Ω+5 Iϵ ) -2 E R α 2 S MR 5 Ω 6 ˙ Ω J × (2 M ' R Ω+6 MR ' Ω+2 MR Ω J ' +7 MR ) -3 I ' ˙ Ω 2 -I '' ˙ Ω 2 Ω] / [2 I + I ' Ω] . (22)  \nAll the quantities with ( ' ) denote the derivative with respect to the angular velocity Ω. Combining the ˙ Ω and ¨ Ω from the expressions in the given in the Equation 21 and 22, we obtain the braking index using the relation 11 as a function of the spin angular velocity Ω.  \nn (Ω) = n EM + n Q GW + n R GW + n I , where n EM = 6 E EM R 5 Ω 3 (2 R ' Ω+ R ) 2 E EM R 6 Ω 3 +2 E Q I 2 ϵ 2 Ω 5 +2 E R α 2 S M 2 R 6 Ω 7 J 2 , n Q GW = 2 E Q I Ω 5 ϵ 2 (2 I ' Ω+5 I ) 2 E EM R 6 Ω 3 +2 E Q I 2 ϵ 2 Ω 5 +2 E R α 2 S M 2 R 6 Ω 7 J 2 , n R GW = 2 E R α 2 S R 5 M Ω 7 J × (2 M ' R Ω J +6 MR ' Ω J +2 MR J ' Ω+7 MR J ) 2 E EM R 6 Ω 3 +2 E Q I 2 ϵ 2 Ω 5 +2 E R α 2 S M 2 R 6 Ω 7 J 2 , n I = 3 I ' Ω+ I '' Ω 2 . (23)  \n-2 I + I ' Ω  \nThe expression of n (Ω) above shows the evolution of the braking index with the spin-frequency, which is dependent on the EOS through the frequency derivative of the macroscopic quantities like the M , R and I . The n EM term is the contribution of MDR loss, n Q GW is the contribution from the GW radiation due to long-lived deformations, n R GW is the contribution from the emission loss via the r-mode oscillations. The term n I depends  \npurely on the moment of the inertia and its higher order derivatives, which has been previously derived in Glendenning et al. [67]. The quantities E EM , ϵ and α S can be considered as the switches which by setting to zero help in analysing the braking index in the absence of a particular channel of energy loss. For example, setting α S = 0 will help in understanding the evolution of the braking index purely due to the electric dipolar emission  \nloss and the GW emission loss due to finite ϵ . In the limit of slow rotation where all the derivatives w.r.t Ω tend to zero, by setting ϵ and α S to zero we obtain n = 3, by setting E EM and α S = 0, we obtain braking index 5 and finally by setting E EM = ϵ = 0, we obtain n = 7, which are in agreement with the limiting cases known in the literature.", 'B. Effect of rapid rotation on n EM , n Q GW and n R GW': "In Sec IV A, we have shown how the expressions in Eq. 23 reduce to the known values of the braking index in the literature when the rotational effects are ignored by setting all derivatives w.r.t Ω to zero. However, when the effect of rotation is taken into account the departure of n EM , n Q GW and n R GW from 3, 5 and 7 occurs, which depends on the EOS through the variation of the macroscopic quantities and their derivatives as the function of the Ω.  \nIf the energy loss is considered through the magnetic dipolar radiation only, then  \nn EM = 3 ( 1 + 2 R ' Ω R ) + n I , (24)  \nwhich can be obtained by setting ϵ and α S = 0. Similarly by assuming the loss of energy is via the GW emission due to the non-axisymmetric long-lived deformation (that is by setting B p = α S = 0) the braking index is given as  \nn Q GW = 5 ( 1 + 2 I ' Ω 5 I ) + n I . (25)  \nLikewise, for the emission to happen purely from the rmode oscillation in the absence of the electromagnetic dipolar radiation and the GW emission from deformations the braking index takes the form  \nn R GW = 7 ( 1 + 2 M ' Ω 7 M + 6 R ' Ω 7 R + 2 J ' Ω 7 J ) + n I , (26)  \nwhich is obtained by setting B p = ϵ = 0. In Fig. 5 we have shown the dependence of these three braking indices n EM , n Q GW and n R GW separately as a function of f , for three different EOS. The solid yellow line corresponds to one of the EOS described in Sec. II with the highest weight (the parameters of the EOS are: α = 0.699, β = 2.151, a = 9 . 423 MeV, b = 5 . 662 MeV, ρ 0 = 0 . 152 fm -3 and E SNM = -17 . 934 MeV), the blue dashed line is for the density-dependent relativistic mean field (RMF) EOS, DDME2 [68] and another RMF EOS with nonlinear meson field couplings, S271v6 [69] is shown by the black solid line. The departure of n EM , n Q GW and n R GW from the known constant value of 3, 5 and 7, respectively is due to the rotational effect. We have used B p = 10 8 G and α = 45 · to obtain the dependence of n EM as a function of Ω, for n Q GW and n R GW we use ϵ = 10 -7 and α S = 10 -4 respectively.  \nThe response to the rapid rotation depends on the stiffness of the EOS, which is different for different EOSs (refer to Sec. III for detailed discussion), giving rise to different braking index curves. However, the difference in response to the n EM and n R GW at frequencies < ∼ 550 Hz is negligible for at least these three representative EOS shown in the Fig. 5, unlike in the case of n Q GW . Both n EM and n R GW systematically decrease with the increase in the spin frequency, which indicates at frequencies > ∼ 200 Hz the contribution of n I is significant. The importance of n I can be argued from Fig. 4, where the R is almost constant at f ∼ 200 -750 Hz, indicating R ' ∼ 0 which would lead to n EM ≃ 3 of Eq. 24 at this frequency range, hence, the deviation seen in n EM from 3 in Fig. 5 can be attributed to n I . A similar trend in the variation of n R GW can be seen in Fig. 5, which is also attributed to the n I . The finite (but small positive) value of M ' /M and finite (but small negative) value of J ' / J leads to a different slope than the n EM in the initial fall of n R GW from the value 7.  \nThe spin frequency evolution of n Q GW depends only on I and its derivative which changes appreciably even at f > ∼ 200 Hz, making n Q GW always > 5 (the first term of Eq. 25) and an increasing function of f in the absence of n I . Whereas n I is always < 0, and the nature of its fall depends on the EOS. Therefore, there may exist a range of frequencies for which the n Q GW > 5 in the presence of the n I term as shown in Fig. 5. However, unlike n EM and n R GW (at least with the example for these EOS) we find the EOS-dependent variation in n Q GW is distinguishable even at the frequencies ∼ 250 -300 Hz. But as the spin frequency increases the variation of all three braking indices for different EOS can be distinguished at the f ∼ 716 Hz, the spin frequency of PSR J1748 -2446ad (maximum spin frequency of a pulsar known till date), an eclipsing binary millisecond pulsar in the globular cluster Terzan 5 [7]. However, at much larger frequencies ( f > ∼ 850 Hz), we find the large change in radius (see Fig. 5) indicating the size decreases with frequency, which could be an artefact because of forceful fitting of a sphere to a largely deformed star. Hence, the results beyond f > ∼ 850 Hz must be interpreted with care.", 'C. Effect of B p , ϵ and α S on the braking index.': "In the previous Sec. IV B, we have discussed the rotational effect on all three separate components of the braking index individually in the absence of the other two channels of energy loss. Here we present the result of Ω dependent braking index taking all three energy loss channels into account for the same set of three representative EOSs used in Sec. IV B. We further investigate the dependence of the braking index for various physically motivated combinations of parameters of B p , ϵ and α S . The results are shown in Fig. 6, where the green colour indicates the results for the EOS obtained from our agnostic EOS model (see Sec. II for details). The same  \nFIG. 5. This figure shows the effect of rotation on the braking index of pulsars (see Sec. III for detailed discussions and references). The left panel indicates the braking index purely due to magnetic dipolar radiation of magnetic field strength B p = 10 8 G and α = 45 · . The middle panel shows the variation of the braking index if the energy loss happens only through the GW emission in the presence of long-lived deformations generating ϵ = 10 -7 (r-mode and magnetic dipole radiation ignored). The right panel shows the variation of the braking index as the function of rotation frequency when the energy loss happens through the r-mode only with the amplitude α S = 10 -4 . The solid yellow line represents one of the EOS from our sample (see Sec. IV B for the details), the blue dashed line indicates the DDME2 EOS and the black solid line indicates S271v6 EOS. The vertical magenta line indicates f = 716 Hz line which is the maximum spin-frequency of a pulsar discovered to date.  \n<!-- image -->  \nsample EOS has been used in Sec. IV B and has been shown with the solid yellow line in Fig. 5. The blue and red colour corresponds to S271v6 and DDME2 EOS, and different line style indicates the different combinations of B p , ϵ and α S .  \nThe upper left panel of the Fig. 6 shows the variation of n (Ω) for neutron stars with a typical value B p = 10 8 G, ϵ = 10 -7 and for two values of α S = 10 -1 and 10 -4 shown using the solid and dashed line respectively. The justification for using such value of ϵ and α S has been presented in Sec. IV A. To arrive at the typical value of B p , we find the median value of B p of pulsars with f > 200 Hz from the ATNF pulsar catalogue [19]. The overlap of the dashed and solid lines for all the EOSs indicates that the effect of two different parameter sets can not be distinguished. The braking index of ∼ 7 around f ∼ 200 -400 Hz indicates that the evolution is mostly dominated by the GW emission due to r-mode instability. The MDR loss and GW radiation from the mass quadropolar moment do not have any significant contribution towards the braking index. However, the departure from the value of 7 has a contribution from n I and other frequency-dependent structural quantities (as discussed in Sec. IV B) which makes n < 7. Therefore, this  \nfigure indicates if the r-mode instability is generated in a typical millisecond pulsar (which is dependent on the spin frequency and the redshifted stellar temperature, for details see Kraav et al. [70]), with α S ranging between 10 -4 -10 -1 , then the spin-down would be dominated by the GW radiation through the r-mode instability.  \nAt higher frequency f ∼ 716 Hz the contribution of n I and rotational effects contributing to n R GW leads to 6 < n < 7. At a much higher frequency of f > ∼ 850 Hz, the rapid decline in braking index is due to a sharp fall in the radius as discussed previously in Sec. IV B and III.  \nIn the upper right-hand side panel of Fig. 6, we explore the evolution of the braking index in millisecond pulsars in the absence of radiation in GW through r-mode instability. We use B p = 10 8 G, and two separate values of ϵ = 10 -7 and 10 -10 shown in the solid and dashed line respectively. Both lines overlap with braking index ∼ 3, the deviation from n = 3 is due to the rotational effect. This also indicates that the emission loss through GWis insignificant even with a deformation of ϵ = 10 -7 . Our findings are consistent with the current observation limit [59]. We find that a large deformation which can generate ϵ > ∼ 10 -3 can result in the emission of GW large enough to impact the rotational evolution of the neutron  \nFIG. 6. This figure captures the dependence of braking index as the function of the rotation frequency on the magnetic field strength B p , the ellipticity due to long-lived deformation ϵ and the r-mode saturation amplitude α S . However, we use a constant inclination angle α = 45 · . The variation has been studied using three representative EOSs, the red colour lines indicate DDME2 EOS, the blue colour line indicates S271v6 EOS and the green line indicates an EOS from our sample (see Sec. IV C) for details. The results for different combinations of B p , ϵ and α S have been shown with different line styles. Refer to Sec. IV C for detailed discussion and implications.  \n<!-- image -->  \nstars. However, such a high magnitude of ellipticity is unlikely for normal millisecond pulsars [71].  \nThe lower left panel of Fig. 6 shows the evolution of the braking index if a neutron star has an ultra-strong magnetic field B p = 10 15 G, making them fall in the class of magnetars. The main motivation for using such a strong magnetic field is to study the frequency-dependence on the braking index of a newly born millisecond magnetar [72], which may form due to the accretion-induced collapse of white dwarfs [73] or a merger of two neutron stars [74, 75]. The formation of these objects is believed to manifest as the Gamma Ray bursts (GRBs) [76]. In the lower-left panel of Fig. 6, we find if α S = 10 -4 and ϵ = 10 -7 at lower frequency f ∼ 200 -600 Hz the braking index is ∼ 3, indicating the magnetic dipolar radiation dominates the spin-down rate of the star. At higher frequencies, f > ∼ 600 Hz the departure from n ∼ 3 is due to  \nthe rotational effects and is sensitive to the EOS.  \nWhen the saturation amplitude of the r-mode is large, that is, α S = 10 -1 (shown in the lower left panel of Fig. 6), we find at low rotation frequency ( f ∼ 200Hz) the braking index is ∼ 3, but finally rises to ∼ 6 close to f = 716 Hz. The increase in the braking index from ∼ 3.5 to ∼ 6 is the manifestation of Ω 8 dependence on the energy loss via the r-mode. The braking index cannot reach the value of 7 because of the rotational effects where n I is also dominant along with the contribution from R ' /R and J ' / J , where all of them are negative quantities. The behavior of the braking index found here illustrates the importance of incorporating the rotational effects to interpret the spin-down evolution of the remnants produced from the astrophysical transient events discussed above. For example, if the rotational effect is not considered, one can misinterpret a measurement of  \nFIG. 7. Evolution of the braking index as the function of rotation frequency. The blue band and the red band show the 90 percentile and 68 percentile of the braking index respectively, which originates from the uncertainties in the EOS computed in Sec. II. The over plot of the yellow and grey band is the braking index of GRB 140903A and GRB 130603B with their 1 σ uncertainty measured from their light curves.  \n<!-- image -->  \nn = 5 for a magnetar as the spin-down dominated by the GW emission due to a finite mass quadrupolar moment.  \nWe have further investigated the evolution of the braking index in the parlance of a newborn millisecond magnetar with a much higher value of ellipticity ( ϵ = 10 -3 ), based on the measurement of Xie et al. [77]. The results are shown in the lower right panel of Fig. 6, we use the same colour convention of EOSs mentioned earlier in the section. The dashed line style indicates the braking index evolution with B p = 10 15 G, ϵ = 10 -3 and α S = 10 -1 , this is the combination with the highest magnetic field, ellipticity and r-mode saturation amplitude. We find the evolution of the braking index is exactly identical to the parameter set B p = 10 15 G, ϵ = 10 -7 and α S = 10 -1 (shown by the solid lines in lower left panel of Fig. 6), which implies that even with 4 orders of magnitude larger ellipticity the evolution is still dominated by the r-mode instability if the saturation amplitude is 10 -1 . Whereas, if the α S ≤ 10 -4 , we find the evolution of the braking index is dominated by the magnetic dipolar radiation, which is shown by the solid line with the B p = 10 15 G, ϵ = 10 -3 and α S = 10 -4 . The dashed-dotted line indicates the evolution of braking index in absence of r-mode but finite deformation of ϵ = 10 -3 and B p = 10 15 G, which indicates again the magnetic dipole radiation is the dominant mode of the energy loss from the neutron star.", 'D. EOS uncertainties and the braking index of millisecond magnetar': 'The long and short GRBs are observed to show an X-ray plateau phase after the prompt emission, which lasts for ∼ 10 -1000 s indicating an ongoing energy injection from the central engine [74, 75, 77-80]. There exists debate on the nature of the central engine but millisecond magnetar is one of the plausible models often used in literature. The spin-down of the magnetar is believed to power the X-ray plateau region of the light curve [81, 82], which is modelled to obtain the braking index [77, 83]. We have shown above that the braking index depends on the structure of the neutron stars which is pronounced at higher frequencies. Given the remnant is a newly born millisecond magnetar with spin period P ∼ 1 -3 ms [77] it is expected to have the dependence from EOS on its braking index. Therefore, we compute the range of possible braking index at every given rotational frequency, which gets translated from the uncertainties in the EOS obtained in Sec. II. We have used Eq. 23 to compute the braking index for all the EOS constructed in Sec. II, which has been supplemented with all the frequency-dependent profiles of M,I,R and J (as discussed in the Sec. III and shown in Fig. 4) and their derivatives. We have used B p = 10 15 G, ϵ = 10 -7 and α S = 10 -4 , where the values of ϵ and α S is conservative estimates based on our results of Sec. IV C. The result of our analysis is shown in Fig. 7. The 90 percentile and the 68 percentile in the braking index as the function of frequency are shown by the blue and red band respectively which have been computed from the weights on the EOS parameter obtained from the Bayesian analysis in Sec. II. The range of the braking index gets wider at higher frequencies due to the contribution of the rotational effects from different EOS of various stiffness. The yellow and the grey horizontal bands are the braking index of GRB 140903A and GRB 130603B with the 1 σ uncertainty measured from their lightcurve by Lasky et al. [83]. The measured braking index passes through our estimated band of braking index indicating if the millisecond magnetar is formed with a typical value of B p = 10 15 G, ϵ = 10 -7 and α S = 10 -4 or less, the spin frequency of the central engine can be between ∼ 550 -850 Hz for GRB 140903A and f < ∼ 650 Hz for GRB 130603B.', 'V. CONCLUSION': "In this paper, we have investigated the effect of stellar rotation on the braking index and its dependence on the EOS. We have generated a large number of EOS by employing a semi-agnostic model and constrained them using the latest astrophysical observations from the gravitational and radio wave bands within a Bayesian framework. We find our constrained parameter space is also consistent with the observational findings of neutron star's mass and radius from X-ray bands. The con-  \nstrained nuclear EOSs have been used to study the effect on the macroscopic quantities of rapidly rotating neutron stars like mass, radius, moment of inertia and the ratio of polar to equatorial radius. Our analysis shows neutron stars with stiffer EOS experience larger deformations in the shape at a given frequency, in comparison to the softer EOS. Consequently, neutron stars with stiffer EOS have smaller Kepler frequencies.  \nNext, we study the effect of rotation on the braking index of neutron stars. We present the first calculation of the braking index accounting for the rotational effects in the presence of energy loss by magnetic dipole radiation and GWs due to non-axisymmetric deformations and r-mode oscillations. We have shown that the rotation of the star can impact its braking index significantly. The deviation of the braking index from 3 to smaller values in the case of magnetic dipole radiation is due to the rotational effects. This rotational effect is prominent only when the spin frequency > ∼ 200 Hz. However, the low braking index in normal pulsars has been explained through various other mechanisms, such as, the evolution of magnetic inclination angle, which has not been considered here. Similarly, we have shown if the neutron stars spin-down is purely due to GW radiation from non-asymmetric deformations or r-mode oscillations, the effect of rapid rotation leads to the significant departure of braking index from their classical value of 5 and 7 respectively.  \nWe further explore the dependence of the braking index on the rotation frequency for various physically motivated combinations of surface magnetic field strength, ellipticity of the star and the amplitude of r-mode oscillations. We find if there is r-mode instability generated in a typical millisecond pulsar with α S between 10 -4 -10 -1 , then angular momentum loss through GW r-mode instability will dominate the spin-down of the pulsar. Whereas, extremely large deformation of ϵ > ∼ 10 -3 is required for the spin-down of a typical millisecond pulsar to be dominated by GW emission due to time-dependent mass quadrupolar moment, which has been ruled out under various physical conditions. Our analysis shows that if the ϵ < ∼ 10 -7 the spin-down is dominated by the magnetic dipole radiation in the absence of r-mode instability, which is consistent with the current limit of the GW observations.  \nWe extended our study in the light of millisecond magnetars, which are believed to be born from NS-NS mergers, core-collapse of a massive star or accretion-induced  \ncollapse of a white dwarf. The spin-down energy is believed to power the X-ray plateau phase of GRBs. These millisecond magnetars can have large magnetic field strength ∼ 10 15 G. We find at large spin-frequencies > ∼ 600 Hz the presence of large amplitude r-mode oscillations dominates the spin-down rate. However, the effect of rapid rotation suppresses the braking index closer to 6 instead of 7. Whereas, the impact of non-axisymmetric deformation on braking index is less pronounced unless ϵ > 10 -3 . However, if the birth spin frequency is low ∼ 200 Hz or r-mode amplitude is less than 10 -1 , spindown is always dominated by the magnetic dipole radiation.  \nWe have used the constrained EOS to compute the permissible range of the braking index as a function of the rotational frequency. The width of the range of the braking index increases with rotation frequency, indicating a wide range of EOS-dependent structural change of neutron stars in response to large centrifugal force. We find a millisecond magnetar with the magnetic field strength 10 15 G, deformation ϵ = 10 -7 and r-mode amplitude α S = 10 -4 has braking index ∼ 3 at low frequency and n < 3 at higher frequency 4 as shown in Fig. 7. Our band of computed braking is in good agreement with the measured braking index of GRB 140903A and 130603B, from where we find the spin frequency of the central engine of GRB 140903A could be between ∼ 550 -850 Hz, which is subjected to systematics at higher spin-frequency end. Our model of spin-frequency dependent braking index hints towards a potential method to constrain the dense mater EOS by precisely measuring the braking index and the spin frequency from the high-energy GRB light curves.", 'ACKNOWLEDGEMENTS': "Pulsar research at Jodrell Bank Centre for Astrophysics and Jodrell Bank Observatory is supported by a consolidated grant from the UK Science and Technology Facilities Council (STFC). P.C. is supported by European Union's HORIZON MSCA-2022-PF-01-01 Programme under Grant Agreement No. 101109652, project ProMatEx-NS. P.C. also acknowledges past support of the Fonds de la Recherche Scientifique-FNRS, Belgium, under Grant No. 4.4501.19. Part of the computation has been performed in the high-performance computing facility 'Magus,' which is available at the Shiv Nadar Institution of Eminence.  \nof the rigid co-rotating magnetosphere), which naturally lead to n < 3 scenario. However, this modification in the magnetosphere is beyond the scope of this paper and has been confined to the simple dipolar approximation of the magnetosphere.  \n- [1] R. N. Manchester, J. M. Durdin, and L. M. Newton, Nature 313 , 374 (1985).\n- [2] J. E. Gunn and J. P. Ostriker, Nature 221 , 454 (1969).\n- [3] K. Riles, Living Reviews in Relativity 26 , 3 (2023), arXiv:2206.06447 [astro-ph.HE].\n- [4] A. K. Harding, I. Contopoulos, and D. Kazanas, ApJL 525 , L125 (1999), arXiv:astro-ph/9908279 [astro-ph].\n- [5] A. Spitkovsky, ApJL 648 , L51 (2006), arXiv:astroph/0603147 [astro-ph].\n- [6] M. Caleb et al. , Nature Astron. 6 , 828 (2022), arXiv:2206.01346 [astro-ph.HE].\n- [7] J. W. T. Hessels, S. M. Ransom, I. H. Stairs, P. C. C. Freire, V. M. Kaspi, and F. Camilo, Science 311 , 1901 (2006), arXiv:astro-ph/0601337 [astro-ph].\n- [8] P. Halder, S. Goswami, P. Halder, U. Ghosh, and S. Konar, Res. Notes AAS 7 , 213 (2023), arXiv:2310.06230 [astro-ph.HE].\n- [9] M. A. Livingstone, V. M. Kaspi, F. P. Gavriil, R. N. Manchester, E. V. Gotthelf, and L. Kuiper, Astrophys. Space Sci. 308 , 317 (2007), arXiv:astro-ph/0702196.\n- [10] P. Weltevrede, S. Johnston, and C. M. Espinoza, MNRAS 411 , 1917 (2011), arXiv:1010.0857 [astro-ph.SR].\n- [11] M. A. Livingstone and V. M. Kaspi, The Astrophysical Journal 742 , 31 (2011).\n- [12] A. G. Lyne, R. S. Pritchard, and F. Graham Smith, MNRAS 265 , 1003 (1993).\n- [13] P. T. Boyd, G. W. van Citters, J. F. Dolan, K. G. Wolinski, J. W. Percival, R. C. Bless, J. L. Elliot, M. J. Nelson, and M. J. Taylor, ApJ 448 , 365 (1995).\n- [14] J. Roy, Y. Gupta, and W. Lewandowski, MNRAS 424 , 2213 (2012), arXiv:1205.6264 [astro-ph.SR].\n- [15] C. M. Espinoza, A. G. Lyne, M. Kramer, R. N. Manchester, and V. M. Kaspi, The Astrophysical Journal Letters 741 , L13 (2011).\n- [16] K. Chen and M. Ruderman, ApJ 402 , 264 (1993).\n- [17] A. Melatos, MNRAS 288 , 1049 (1997).\n- [18] O. Hamil, J. R. Stone, M. Urbanec, and G. Urbancov'a, Phys. Rev. D 91 , 063007 (2015), arXiv:1608.01383 [astroph.HE].\n- [19] R. N. Manchester, G. B. Hobbs, A. Teoh, and M. Hobbs, Astron. J. 129 , 1993 (2005), arXiv:astro-ph/0412641.\n- [20] S. Gandolfi, J. Lippuner, A. W. Steiner, I. Tews, X. Du, and M. Al-Mamun, J. Phys. G 46 , 103001 (2019), arXiv:1903.06730 [nucl-th].\n- [21] H. T. Cromartie et al. , Nature Astron. 4 , 72 (2019), arXiv:1904.06759 [astro-ph.HE].\n- [22] E. Fonseca et al. , Astrophys. J. Lett. 915 , L12 (2021), arXiv:2104.00880 [astro-ph.HE].\n- [23] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. Lett. 119 , 161101 (2017), arXiv:1710.05832 [gr-qc].\n- [24] B. P. Abbott et al. (LIGO Scientific, Virgo, Fermi GBM, INTEGRAL, IceCube, AstroSat Cadmium Zinc Telluride Imager Team, IPN, Insight-Hxmt, ANTARES, Swift, AGILE Team, 1M2H Team, Dark Energy Camera GW-EM, DES, DLT40, GRAWITA, Fermi-LAT, ATCA, ASKAP, Las Cumbres Observatory Group, OzGrav, DWF (Deeper Wider Faster Program), AST3, CAASTRO, VINROUGE, MASTER, J-GEM, GROWTH, JAGWAR, CaltechNRAO, TTU-NRAO, NuSTAR, PanSTARRS, MAXI Team, TZAC Consortium, KU, Nordic Optical Telescope, ePESSTO, GROND, Texas Tech  \nUniversity, SALT Group, TOROS, BOOTES, MWA, CALET, IKI-GW Follow-up, H.E.S.S., LOFAR, LWA, HAWC, Pierre Auger, ALMA, Euro VLBI Team, Pi of Sky, Chandra Team at McGill University, DFN, ATLAS Telescopes, High Time Resolution Universe Survey, RIMAS, RATIR, SKA South Africa/MeerKAT), Astrophys. J. Lett. 848 , L12 (2017), arXiv:1710.05833 [astroph.HE].  \n- [25] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. X 9 , 011001 (2019), arXiv:1805.11579 [gr-qc].\n- [26] M. C. Miller, F. K. Lamb, A. J. Dittmann, S. Bogdanov, Z. Arzoumanian, K. C. Gendreau, S. Guillot, A. K. Harding, W. C. G. Ho, J. M. Lattimer, R. M. Ludlam, S. Mahmoodifar, S. M. Morsink, P. S. Ray, T. E. Strohmayer, K. S. Wood, T. Enoto, R. Foster, T. Okajima, G. Prigozhin, and Y. Soong, ApJL 887 , L24 (2019), arXiv:1912.05705 [astro-ph.HE].\n- [27] T. E. Riley, A. L. Watts, S. Bogdanov, P. S. Ray, R. M. Ludlam, S. Guillot, Z. Arzoumanian, C. L. Baker, A. V. Bilous, D. Chakrabarty, K. C. Gendreau, A. K. Harding, W. C. G. Ho, J. M. Lattimer, S. M. Morsink, and T. E. Strohmayer, ApJL 887 , L21 (2019), arXiv:1912.05702 [astro-ph.HE].\n- [28] M. C. Miller, F. K. Lamb, A. J. Dittmann, S. Bogdanov, Z. Arzoumanian, K. C. Gendreau, S. Guillot, W. C. G. Ho, J. M. Lattimer, M. Loewenstein, S. M. Morsink, P. S. Ray, M. T. Wolff, C. L. Baker, T. Cazeau, S. Manthripragada, C. B. Markwardt, T. Okajima, S. Pollard, I. Cognard, H. T. Cromartie, E. Fonseca, L. Guillemot, M. Kerr, A. Parthasarathy, T. T. Pennucci, S. Ransom, and I. Stairs, ApJL 918 , L28 (2021), arXiv:2105.06979 [astro-ph.HE].\n- [29] T. E. Riley, A. L. Watts, P. S. Ray, S. Bogdanov, S. Guillot, S. M. Morsink, A. V. Bilous, Z. Arzoumanian, D. Choudhury, J. S. Deneva, K. C. Gendreau, A. K. Harding, W. C. G. Ho, J. M. Lattimer, M. Loewenstein, R. M. Ludlam, C. B. Markwardt, T. Okajima, C. Prescod-Weinstein, R. A. Remillard, M. T. Wolff, E. Fonseca, H. T. Cromartie, M. Kerr, T. T. Pennucci, A. Parthasarathy, S. Ransom, I. Stairs, L. Guillemot, and I. Cognard, ApJL 918 , L27 (2021), arXiv:2105.06980 [astro-ph.HE].\n- [30] T. Salmi, S. Vinciguerra, D. Choudhury, T. E. Riley, A. L. Watts, R. A. Remillard, P. S. Ray, S. Bogdanov, S. Guillot, Z. Arzoumanian, C. Chirenti, A. J. Dittmann, K. C. Gendreau, W. C. G. Ho, M. C. Miller, S. M. Morsink, Z. Wadiasingh, and M. T. Wolff, ApJ 941 , 150 (2022), arXiv:2209.12840 [astro-ph.HE].\n- [31] S. Vinciguerra, T. Salmi, A. L. Watts, D. Choudhury, T. E. Riley, P. S. Ray, S. Bogdanov, Y. Kini, S. Guillot, D. Chakrabarty, W. C. G. Ho, D. Huppenkothen, S. M. Morsink, Z. Wadiasingh, and M. T. Wolff, ApJ 961 , 62 (2024), arXiv:2308.09469 [astro-ph.HE].\n- [32] D. Choudhury et al. , Astrophys. J. Lett. 971 , L20 (2024), arXiv:2407.06789 [astro-ph.HE].\n- [33] I. Tews, T. Kruger, K. Hebeler, and A. Schwenk, Phys. Rev. Lett. 110 , 032504 (2013), arXiv:1206.0025 [nucl-th].\n- [34] K. Hebeler, J. M. Lattimer, C. J. Pethick, and A. Schwenk, Astrophys. J. 773 , 11 (2013), arXiv:1303.4662 [astro-ph.SR].\n- [35] J. E. Lynn, I. Tews, J. Carlson, S. Gandolfi, A. Gezerlis, K. E. Schmidt, and A. Schwenk, Phys. Rev. Lett. 116 , 062501 (2016), arXiv:1509.03470 [nucl-th].\n- [36] C. Drischler, K. Hebeler, and A. Schwenk, Phys. Rev. Lett. 122 , 042501 (2019), arXiv:1710.08220 [nucl-th].\n- [37] S. Huth, C. Wellenhofer, and A. Schwenk, Phys. Rev. C 103 , 025803 (2021), arXiv:2009.08885 [nucl-th].\n- [38] I. Tews, J. Carlson, S. Gandolfi, and S. Reddy, Astrophys. J. 860 , 149 (2018), arXiv:1801.01923 [nucl-th].\n- [39] G. Baym, C. Pethick, and P. Sutherland, Astrophys. J. 170 , 299 (1971).\n- [40] J. W. Negele and D. Vautherin, Nucl. Phys. A 207 , 298 (1973).\n- [41] J. Skilling, Bayesian Analysis 1 , 833 (2006).\n- [42] J. S. Speagle, Mon. Not. Roy. Astron. Soc. 493 , 3132 (2020), arXiv:1904.02180 [astro-ph.IM].\n- [43] M. Oertel, M. Hempel, T. Klahn, and S. Typel, Rev. Mod. Phys. 89 , 015007 (2017), arXiv:1610.03361 [astroph.HE].\n- [44] J. Margueron, R. Hoffmann Casali, and F. Gulminelli, Phys. Rev. C97 , 025805 (2018), arXiv:1708.06894 [nuclth].\n- [45] N. Stergioulas and J. L. Friedman, Astrophys. J. 444 , 306 (1995), arXiv:astro-ph/9411032.\n- [46] T. Nozawa, N. Stergioulas, E. Gourgoulhon, and Y. Eriguchi, Astron. Astrophys. Suppl. Ser. 132 , 431 (1998), arXiv:gr-qc/9804048.\n- [47] M. AlGendy and S. M. Morsink, ApJ 791 , 78 (2014), arXiv:1404.0609 [astro-ph.HE].\n- [48] V. F. Suleimanov, J. Poutanen, and K. Werner, A&A 639 , A33 (2020), arXiv:2005.09759 [astro-ph.HE].\n- [49] C. Kalapotharakos, Z. Wadiasingh, A. K. Harding, and D. Kazanas, ApJ 907 , 63 (2021), arXiv:2009.08567 [astro-ph.HE].\n- [50] F. Gittins, Classical and Quantum Gravity 41 , 043001 (2024), arXiv:2401.01670 [gr-qc].\n- [51] G. Ushomirsky, C. Cutler, and L. Bildsten, MNRAS 319 , 902 (2000), arXiv:astro-ph/0001136 [astro-ph].\n- [52] B. Haskell, D. I. Jones, and N. Andersson, MNRAS 373 , 1423 (2006), arXiv:astro-ph/0609438 [astro-ph].\n- [53] F. Gittins, N. Andersson, and D. I. Jones, MNRAS 500 , 5570 (2021), arXiv:2009.12794 [astro-ph.HE].\n- [54] B. Haskell, L. Samuelsson, K. Glampedakis, and N. Andersson, MNRAS 385 , 531 (2008), arXiv:0705.1780 [astro-ph].\n- [55] S. Kalita and B. Mukhopadhyay, MNRAS 491 , 4396 (2020).\n- [56] J. Soldateschi, N. Bucciantini, and L. Del Zanna, A&A 654 , A162 (2021), arXiv:2106.00603 [astro-ph.HE].\n- [57] L. Bildsten, ApJL 501 , L89 (1998), arXiv:astroph/9804325 [astro-ph].\n- [58] M. Vigelius and A. Melatos, MNRAS 395 , 1985 (2009), arXiv:0902.4484 [astro-ph.HE].\n- [59] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. D 100 , 024004 (2019), arXiv:1903.01901 [astro-ph.HE].\n- [60] B. J. Owen, L. Lindblom, C. Cutler, B. F. Schutz, A. Vecchio, and N. Andersson, Phys. Rev. D 58 , 084020 (1998), arXiv:gr-qc/9804044.\n- [61] S. Chandrasekhar, Phys. Rev. Lett. 24 , 611 (1970).\n- [62] J. L. Friedman and B. F. Schutz, Astrophys. J. 222 , 281 (1978).  \n- [63] N. Andersson, Astrophys. J. 502 , 708 (1998), arXiv:grqc/9706075.\n- [64] J. L. Friedman and S. M. Morsink, Astrophys. J. 502 , 714 (1998), arXiv:gr-qc/9706073.\n- [65] L. Bildsten, Astrophys. J. Lett. 501 , L89 (1998), arXiv:astro-ph/9804325.\n- [66] K. S. Thorne, Rev. Mod. Phys. 52 , 299 (1980).\n- [67] N. K. Glendenning, S. Pei, and F. Weber, Phys. Rev. Lett. 79 , 1603 (1997).\n- [68] G. A. Lalazissis, T. Niksic, D. Vretenar, and P. Ring, Phys. Rev. C 71 , 024312 (2005).\n- [69] C. J. Horowitz and J. Piekarewicz, Phys. Rev. C 66 , 055803 (2002), arXiv:nucl-th/0207067.\n- [70] K. Y. Kraav, M. E. Gusakov, and E. M. Kantor, Phys. Rev. D 109 , 043012 (2024).\n- [71] N. K. Johnson-McDaniel and B. J. Owen, Phys. Rev. D 88 , 044004 (2013), arXiv:1208.5227 [astro-ph.SR].\n- [72] Z. G. Dai and T. Lu, A&A 333 , L87 (1998), arXiv:astroph/9810402 [astro-ph].\n- [73] V. V. Usov, Nature 357 , 472 (1992).\n- [74] A. Rowlinson, P. T. O'Brien, B. D. Metzger, N. R. Tanvir, and A. J. Levan, MNRAS 430 , 1061 (2013), arXiv:1301.0629 [astro-ph.HE].\n- [75] H.-J. Lu and B. Zhang, ApJ 785 , 74 (2014).\n- [76] H.-J. Lu, L. Lan, and E.-W. Liang, ApJ 871 , 54 (2019), arXiv:1812.03465 [astro-ph.HE].\n- [77] L. Xie, D.-M. Wei, Y. Wang, and Z.-P. Jin, The Astrophysical Journal 934 , 125 (2022).\n- [78] J. A. Nousek, C. Kouveliotou, D. Grupe, K. L. Page, J. Granot, E. Ramirez-Ruiz, S. K. Patel, D. N. Burrows, V. Mangano, S. Barthelmy, A. P. Beardmore, S. Campana, M. Capalbi, G. Chincarini, G. Cusumano, A. D. Falcone, N. Gehrels, P. Giommi, M. R. Goad, O. Godet, C. P. Hurkett, J. A. Kennea, A. Moretti, P. T. O'Brien, J. P. Osborne, P. Romano, G. Tagliaferri, and A. A. Wells, ApJ 642 , 389 (2006), arXiv:astroph/0508332 [astro-ph].\n- [79] P. T. O'Brien, R. Willingale, J. Osborne, M. R. Goad, K. L. Page, S. Vaughan, E. Rol, A. Beardmore, O. Godet, C. P. Hurkett, A. Wells, B. Zhang, S. Kobayashi, D. N. Burrows, J. A. Nousek, J. A. Kennea, A. Falcone, D. Grupe, N. Gehrels, S. Barthelmy, J. Cannizzo, J. Cummings, J. E. Hill, H. Krimm, G. Chincarini, G. Tagliaferri, S. Campana, A. Moretti, P. Giommi, M. Perri, V. Mangano, and V. LaParola, ApJ 647 , 1213 (2006), arXiv:astro-ph/0601125 [astro-ph].\n- [80] B. Zhang, Y. Z. Fan, J. Dyks, S. Kobayashi, P. M'esz'aros, D. N. Burrows, J. A. Nousek, and N. Gehrels, ApJ 642 , 354 (2006), arXiv:astro-ph/0508321 [astro-ph].\n- [81] B. Zhang and P. M'esz'aros, ApJL 552 , L35 (2001), arXiv:astro-ph/0011133 [astro-ph].\n- [82] B. D. Metzger, D. Giannios, T. A. Thompson, N. Bucciantini, and E. Quataert, MNRAS 413 , 2031 (2011), arXiv:1012.0001 [astro-ph.HE].\n- [83] P. D. Lasky, C. Leris, A. Rowlinson, and K. Glampedakis, ApJL 843 , L1 (2017), arXiv:1705.10005 [astro-ph.HE].\n- [84] I. Contopoulos and A. Spitkovsky, Astrophys. J. 643 , 1139 (2006), arXiv:astro-ph/0512002."}