Anisotropic ultra-compact object in Serrano–Liska gravity model

We implement a recent model proposed by Serrano–Liska (SL) (Alonso-Serrano and Liška, JHEP, 12:196, 2020) to study the ultra-compact star properties. The matter in the interior star is modeled by the quark model, deducted from QCD theory equipped with anisotropic pressure. Anisotropy is used to increase the compactness of the stars. We intend to see the signature of the “quantum gravity” effect through the SL model in ultra-compact stars. The SL model was motivated by quantum correction appearing in the black hole by adding a logarithmic term in gravity entropy used to derive the effective Einstein field equation. We expect this correction term in the SL model also affects ultra-compact star properties. The SL model with coupling constant c~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{c}}$$\end{document} in logarithmic term equipped with spherically symmetric metric yields correction terms O(c~)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {O}({\tilde{c}})$$\end{document} that can be expressed by a function Ξ(r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varXi (r)$$\end{document}. The Ξ(r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varXi (r)$$\end{document} function vanishes at the star’s exterior. We found that the mass-radius relation prediction by the SL model with anisotropic matter deviates from the one predicted by the standard Tolman–Oppenheimer–Volkoff (TOV) equation for c~≥107\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{c}}\ge 10^7$$\end{document} m2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^2$$\end{document}. We also have a sufficiently deep enough effective potential to produce a quasi-normal mode. We obtain the echo frequency of 15.2 kHz using maximum anisotropic pressure contribution and c~=107\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{c}}= 10^7$$\end{document} m2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^2$$\end{document}. Because the corresponding effective potential is almost indistinguishable from that of GR, this echo frequency value can be indistinguishable to one of GR, but not comparable to the result from GW170817 data analysis, i.e., 72 Hz. To circumvent this problem, we can decrease the value of echo frequency by increasing the magnitude of c~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{c}}$$\end{document} to orders of magnitude than c~=107\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{c}}= 10^7$$\end{document} m2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^2$$\end{document}. On the other hand, too large a strength from the logarithmic correction term is not physically favored because we learn from the black hole case that the logarithmic term is expected to be smaller than that of the Bekenstein term. Therefore, more precise gravitation echo measurements are crucial to understand this issue.


Introduction
One of the exciting discoveries of the 20'th century was the realization that a black hole behaves as a thermodynamical object.They start with the seminal work of Bekenstein and Hawking [1,2] which shows striking relation between the area of black hole's event horizon and entropy.According to Hawking's area theorem, the horizon area cannot be decreased by any classical process [2].However, by implementing the quantum field in the curved space-time, the black hole is behaving like black body radiation [3] which are subtracting the energy from the black hole's mass as it is evaporating [4].Furthermore, the mechanical laws of black holes have resemblance with the laws of thermodynamics [5].
In recent years, many studies show that the connection between the laws of thermodynamics with gravity is broader than before.The first step towards these was the (classical) Einstein field equation can be derived from the proportionality of entropy and black-hole horizon area S = η A, which is the property of entanglement entropy, with the fundamental Clausius relation, d S = δ Q/T .Here T is interpreted as the Unruh temperature and δ Q as the heat flux across the local Rindler horizon [6].By identifying the proportionality constant with the coefficient of the Bekenstein entropy, the classical Einstein equations are satisfied in every space-time point.The discussions for the thermodynamics origin of several modified gravity theories can be found in Refs.[7][8][9][10].The semi-classical Einstein equation can also be derived, with the entropy of the matter is in the form of entanglement entropy (contrary to the previous derivation, which is using the Clausius entropy), and with the local causal diamonds substituting the Rindler horizon [11].A causal diamond essentially is the closed causal geometry bounded by the past and future null geodesics path, puncturing from the boundary of a spacelike ball to a past vertex p and future vertex q.In this derivation, both matter and gravitational entropy were on the same ground.One might worry about the consistencies of the derivation of the classical Einstein equation through thermodynamics in Ref. [6] since two distinct entropy are summed together, namely the entanglement and the Clausius entropies.However, by implementing the formulation of Clausius entropy flux across any bifurcate null surface [12], it is recently shown that both entropies are proportional to each other [13].Interestingly, in Ref. [13] the resulting effective Einstein field equation is in the form of the unimodular gravity rather than the usual Einstein gravity.This fact is due to the restriction that they imposed to fix the local curvature function of the maximally symmetric space λ(x), which is by taking the trace of the final field equation rather than imposing matter conservation [6,11].We note that the Einstein field equation's derivation by thermodynamics has also been reported through different routes.For instance, [14] successfully derived Newton and Einstein gravity by using bulk/boundary correspondence, at which the entropy of the boundary follows the Bekenstein-Hawking relation.In Ref. [15], the relativistic hydrostatic equation for radiation in spherically symmetric space-time is derived by using two assumptions, namely, matter obeying Gibbs-Duhem relation and the general maximum entropy principle holds.The generalization of this procedure to an arbitrary self-gravitating fluid is done in Ref. [16].Furthermore, a similar generalization to any static space-time without spherical symmetry restriction and adding Maxwell field coupling also has been made in Refs.[17,18].These consistencies imply that gravity could emerge naturally from thermodynamics.Recently, the work in [19] extended their previous work [13] by taking into account the quantum gravity corrections to the Einstein field equations.They utilize the maximal entropy vacuum hypothesis [11] and incorporate the logarithmic term in the Bekenstein-Hawking relation.This can be considered as the low energy quantum gravity corrections to the entropy-area relation.Several approaches had predicted that this logarithmic term should be present in the quantum gravity, such as loop quantum gravity, string theory, AdS/CFT correspondence, and generalized uncertainty principle phenomenology.It is worth noting that the logarithmic correction in the Serrano-Liska model [19] with c as the SL model coupling constant related to the modified unimodular gravity equation (9), is relevant only for very high energy density.In the case of a simple cosmological model, a modified Raychaudhuri equation exists, although this cannot be found by using the perturbative method [19].
In the case of ρ = 0, the equation can be solved analytically.However, the authors in [19] have shown that this solution do not seem to be physically relevant.Meanwhile, other solutions, that can be found using the perturbative method, yield interesting features, i.e., the classical singularity is resolved.
The Serrano-Liska model [19] has not yet been previously discussed in the context of compact stars.Note that compact stars such as quark stars, neutron stars, or other exotic compact stars are relatively very dense.The signature of quantum gravity corrections might be imprinted in these objects.Furthermore, the matter content in cosmology theories are very dilute.On the other hand, the description inside the black hole, as the densest object in the universe, is unknown.Note, black hole thermodynamics within the Serrano-Liska model and its relation with generalized uncertainty principle (GUP) formalism was discussed in Ref. [20].We think that discussing the compact star within the Serrano-Liska model will be fruitful as a bridge to explore the model's ability to explain gravity on all scales, i.e., from the size of the cosmology to the size of a black-hole.We expect that the strength of the logarithmic term in Eq. ( 1) can be constrained from ultra-compact object properties.Note that the sign of this logarithmic term entropic gravity models (Please see the detail in [19] and the references therein.)depends on the theory, whereas the positive sign is related to thermal fluctuation, and negative sign is related to microcanonical correction [19][20][21][22][23][24][25].Therefore, from phenomenological view the sign of c is determined by combining both theories' contributions.
Much evidence reported recently stated that the compact stars could have pressure anisotropy.Please see Refs.[26,27] and the references therein for a review of the role of anisotropic pressure in compact objects.Note that the role of anisotropic pressure is increasing or decreasing the compactness of the compact object.For example, it has been known that the scalar field, which is the building block of boson stars, has this feature.It is also known that the heaviest neutron stars has a mass around 2.3M [28].However, the theoretical upper bound of the TOV limit is near 2.9M [29].By adding pressure anisotropy, it turns out that the maximum mass can be increased [26,27].By using the ani-sotropic model proposed by the authors of Refs.[30,31], it is reported [26,27] that the issues of hyperon puzzle and large neutron star's maximum mass can be solved.In general, the appearance of anisotropic pressure in compact objects is physically due to many related factors merged in complex ways.Therefore, the anisotropic terms are usually modeled phenomenologically using certain plausible assumptions as of the physical basis.Recently, an EoS model for quark (strange) stars matter had been proposed by Bercerra-Vergara et al. [32], wherein this EoS model the pressures of matter explicitly anisotropic.The EoS actually was calculated earlier by Asbell and Jaikumar [33].This model is an extension of the MIT bag model, which characterizes a degenerate Fermi gas, consisting of up, down, and strange quarks, by adding quark interaction, but the color flavor lock-ed phase is not taken into account in the model.The EoS of Bercerra-Vergara et al. [32], in the limit of zero strange quark mass, will go to the usual MIT bag model.Furthermore, it is also expected that quark stars could be more compact than neutron stars.Therefore, the quark star can have a maximum mass of around 2.9M .Note, the possible mechanisms for the quark stars formation are discussed in Refs.[34][35][36][37].
If we look at the mass-radius plane describing compact objects, it can be classified into three regions with different compactness ranges.If compactness is defined as C ≡ G M/R, the black hole region is limited by condition C = 1/2, while compact objects like white dwarfs, neutron stars, or ordinary quark stars are in the region with 1/3 < C < 4/9.The third region is for the exotic compact objects known in the literature as ultra-compact or black hole mimickers with 4/9 < C < 1/2.The second and third regions do not violate the Buchdahl limit, which describes the maximum amount of mass in fluid stars that can exist in a sphere before its central pressure goes to infinity.However, different from the second region, the third region possesses a photon sphere.The photon sphere is the unstable circular orbit from the external Schwarchild space-time metric.If the object that possesses a photon sphere manifested from a compact binary merger process, the post-merger ringdown waveform could be initially identical to that of a black-hole but with additional subsequent pulses of gravitational wave known as gravitational echos [38].See a review of ultracompact objects and their observational status in Ref. [39] and the references therein.As far as we know, the first work that examines the possibility that the ultra-compact object produced in the GW170817 event be an isotropic quark stars with MIT bag model was done by Mannarelli and Tonelli [40].A systematic investigation on gravitational echos from isotropic ultra-compact stars with MIT bag-like EoS was done by Urbano and Veermae [38].While the investigation of anisotropic stars as ultra-compact objects in general relativity using polytropic EoS has been done in Ref. [41].The authors of Ref. [38] concluded that isotropic ultra-compact objects supported by physical EoS like MIT bag model type could not generate gravitational echos like those obtained from black hole mimickers, even by including rotation.Note that echo can be generated only with the unphysical condition, i.e., the speed of sound violates the causality limit.The anisotropic ultra-compact EoS constructed in Ref. [41] with a large tangential pressure can generate a gravitational echo.However, this object violates dρ/dr ≤ 0 and (P r − P t ) ≥ 0 conditions.Therefore, understanding the underlying micro-scopic responsible for anisotropic pressure proposed in Ref. [41] remains an interesting and open question.
In this paper, we investigate the impact of both the Serrano-Liska (SL) gravity model and anisotropic pressure on an ultracompact object's properties.We use the anisotropic pressure EoS model proposed by the authors in Ref. [32].The motivation to revisit this issue is to know the possible role of quantum gravity, seen in the logarithmic term in the black-hole entropy formula (1), and anisotropic pressures on both the ultracompact object's compactness and its gravitational echo.
The paper is organized as follows.In Sect.2, we discuss the matter content inside the quark star that we use in the Serrano-Liska model.In Sect.3, we discuss the derivations of the modified TOV equation from the Serrano-Liska model.In Sect.4, we discuss the numerical results.Finally, in Sect.5, we conclude our discussions.

Matter equation of state and anisotropy
Here, we use anisotropic quark matter EoS proposed in Ref.
[32], where the corresponding EoS formulation is motivated by an EoS from quantum chromodynamics (QCD) as a perturbative series in strange quark mass m s expansion up to O(m 4 s ).The radial pressure P of this is calculated by Asbell and Jaikumar [32,33] ρ is the energy density.B is the Bag constant whose value range is 1.02 × 10 14 g/cm 3 ≤ B ≤ 1.64 × 10 14 g/cm 3 .This value range is taken from [32] which states that strange quark matter is absolutely stable for this range of energy densities.This value range is also validated by a larger value range from Ref. [42].m s is the quark strange mass whose value is 1.78 × 10 −28 kg. a 4 is the strong coupling constant that comes from the QCD corrections on the pressure of the quark free Fermi sea [32].From Ref. [43], it has been shown that a 4 ∼ 0.7 corresponds to maximum mass M ≈ 2.14M and its radius is R ≈ 12 km.Because we want the quark star's mass be as massive as possible while also keeping its radius as small as the model allows, we fix a 4 = 0.7 in this paper.At the limit m s → 0, w is equal to the speed of sound squared for the standard MIT bag model.Usually, in standard MIT bag model calculation, the value used is w = 1/3 [32].Note that there are two theoretical upper bounds of the speed of sound of dense matter proposed and discussed in literature i.e., the stability and causality bound [44] w max = 1 and the conformal bound [45] w max = 1/3.Note that the conformal bound has been demonstrated is satisfied in several classes strongly coupled theories with gravity duals such as QCD and it is saturated only in conformal theory.However, the existence of massive NS is in strong tension with the conformal bound [45].Here we consider w as a parameter because we intend to see the impact of varying w on compact star properties.The tangential pressure P t is proposed by Becerra-Vergara et al. [32] as where P c = P(r = 0).Notice that for 0 < r < R, if we set both a 4⊥ = a 4 and B ⊥ = B, then α ⊥ = α, β ⊥ = β, γ ⊥ = γ , and κ ⊥ = κ, therefore This means no anisotropy if we set both a 4⊥ = a 4 and B ⊥ = B. Again, we replace w = 1/3 [32] with w.Lastly, the anisotropy σ is defined as σ = P − P t .The tangential pressure from [32] is interesting because it is motivated from the EoS calculated by considering QCD correction [33].The parameters B ⊥ and a 4⊥ inside the tangential pressure act like B and a 4 inside the radial pressure, respectively.Therefore the numerical range values for B ⊥ is similar to B, i.e., 1.02 × 10 14 g/cm 3 ≤ B ⊥ ≤ 1.64 × 10 14 g/cm 3 .On the other hand, the value range of a 4⊥ is around the value of a 4 = 0.7 following the choices in Ref.
[32], i.e., 0.07 ≤ a 4⊥ ≤ 1.0.This definition of tangential pressure seems ad hoc.However, the parameters B ⊥ and a 4⊥ have similar physical meaning analogous to the parameters B and a 4 which is not present from three other models in [26].More-over, the three forms of P t in [26] are designed to satisfy the boundary conditions on the center P t (r = 0) = P c and on the surface P t (r = R) = 0.However, the P t from [32] satisfies only the boundary condition on the center P t (r = 0) = P c because all terms on the right hand side of Eq. (3a) vanishes except for P c .
Since inverting Eq. (2a) to obtain the EoS ρ = ρ(P) is analytically hard due to the logarithmic term, we define Then, the energy density can be obtained as Now, there is a circular problem because ρ needs an input from P, which needs an input from ρ.To solve this, we assume that O(m 4 s ) is negligible so that we can replace P with P in Eq. ( 5) so we obtain In practice, to obtain the energy density, we calculate in this order: Eqs. ( 8), ( 7), ( 6), and ( 5).Here the value of quark mass m s is 1.78 × 10 −28 kg and the Bag constant B is between 1.02 and 1.64 × 10 14 g/cm 3 .The speed of sound squared w will be set equal to either 1/3 (constraint from QCD) or 1 (constraint from causality).Numerically, both Δρ and the terms multiplied by β in P are small, so ρ ∼ P/w + 4B.Notice the positive-negative sign in Δρ and Δ 0 ρ.These gave two possibilities of approximation for the energy density depending on the positive-negative sign in both Δρ and Δ 0 ρ.
To show what choice that gives the original EoS (Eq.(2a)), we show the plot from each sign in Fig. 1 and compare them to Eq. (2a) and the usual MIT bag model ρ = P/w + 4B.One can see that the one with positive sign almost coincides with the one from Eq. (2a), hence we shall use the positive sign in Δρ and Δ 0 ρ in the following discussions.

Modified TOV equation from Serrano-Liska model
We start the discussion in this section from the modified unimodular gravity equation from Ref. [19]: where S ab = R ab − (1/4)g ab R, t ab = T ab − (1/4)g ab T , R = R ab g ab , T = T ab g ab , and κ N = 8π G.Here we use the natural units c = h = 1.The constant is related to the Planck scale by c = Dl 2 p with D a dimensionless constant.We use Fig. 1 The EoS from our approximation has two possible solutions, i.e., from choosing either positive or negative sign in both Δρ and Δ 0 ρ.
The "approx plus" ("approx minus") sign refers to positive (negative) sign in Eqs. ( 6) and (8).The "original" sign refers to Eq. (2a).The "MIT" sign refers to ρ = P/w+4B.In this plot, we use w = 1/3, a 4 = a 4⊥ = 0.7, B = B ⊥ = 1.02 × 10 14 g/cm 3   the spherically symmetric metric as where dΩ the infinitesimal element of a 2-sphere.We use here the ideal fluid stress tensor form with anisotropic pressure as follows [31,[46][47][48]] with u a = δ a 0 (−g 00 ) −1/2 is the fluid 4-velocity and k a = δ a r (g rr ) −1/2 is the unit radial vector.Notice that g ab u a k b = 0.The tangential pressure is related to anisotropy σ by σ = P − P t and g ab + u a u b − k a k b is the projection operator onto the 2-surface orthogonal to both u a and k a .By this definition, we can obtain the non-zero components of t ab as The requirement that the model sould be satisfy the Bianchi identity T ab ;b = 0 leads to the following expression where the semi-colon symbol denotes covariant derivative and the comma symbol denotes partial derivative.
We inspect the conservation constraint in Eq. ( 13), and we found that the Ricci tensor and stress tensor can be Taylor expanded as (1) , (14b) T = T (0) + cT (1) , ( where we only kept the first order.Hence S ab = S (0) ab + cS ab .Substituting these expressions into Eq.( 13), we obtain for the O(1) and O( c) terms, respectively, ,b , (15a) From the non-perurbative part in Eq. (15a) the boundary condition of asymptotically flat space-time, we obtain To calculate the perturbative part in Eq. (15a), we assume the components from the tensors and scalars with subscript (1) are negligible.Therefore by using the metric components explicitly, we have the following non-zero components of the Ricci tensor as where the prime symbol denotes differentiation with respect to r .These make S ab = 0 if a = b.Now, since the O(1) terms in Eq. ( 9) is just the usual unimodular gravity equation ab , we substitute this into Eq.(15b), which becomes obtain R (1) + κ N T (1)   ,b where the Einstein summation convention is not used.Using b = r and the assumption that the EoS ρ satisfies the thermodynamic constraint, which implies ρ = ρ(P), then we obtain with Ξ(r ) an additional degrees of freedom that contains the higher order curvature corrections from O( c) in Eq. ( 9), and it satisfies We obtain this equation by assuming that the contributions of g (1)  ab , t (1)  ab , etc. are very small in this perturbative part.Then the r component in the Bianchi identity becomes which is used in Eq. ( 20).The components of Eq. ( 9) can then be rewritten as where, again, the Einstein summation convention is not used.
Here, G ab = R ab − (1/2)g ab R. Setting a = 0 and assuming we obtain Setting a = r , we obtain We numerically integrate Eqs. ( 20), ( 21), (24), and (25) from the center of the star r = r c ∼ 0 to its surface r = R.At the center, the boundary conditions are At the surface, we demand the solutions satisfy and Ξ(R) = 0.The central pressure P c is chosen arbitrarily.Then α(r c ) can be determined from the shooting method, i.e., form an arbitrary value of α(r c ) = α c,old we use and recalculate the equations again from the center.Recalculating α c stops when |α c,new − α c,old | < ε α , where ε α > 0 is arbitrarily small.Using similar method to calculate α(r c ), Ξ(r c ) can be found by first starting the calculation from α(r c ) = Ξ c (Eq. (26a)) at the center and recalculate again by and stop recalculating when |Ξ c,new − Ξ c,old | < ε Ξ , where ε Ξ > 0 is arbitrarily small.It is interesting to note that in the limit of c → 0, Eqs. ( 21), (24), and (25) become the usual TOV equations in GR (it is denoted as TOV GR in the next discussion).In the numerical results, we need to note that c has units of m 2 .

Numerical results
First, we shall discuss the impacts from varying parameters of anisotropic EoS of modified MIT bag model proposed in Ref. [32] in mass and radius relation of quark stars modified MIT bag model with w = 1/3 within Serrano-Liska gravity model using ( c = 1 m 2 ).The impacts of B and B ⊥ parameters variation are shown in Fig. 2 and the impact of a 4 and a 4⊥ parameters variation are shown in Fig. 3.We obtain similar behaviors of quark stars mass-radius within GR as discussed by the authors of Ref. [32].Decreasing (increasing) B can increase (decrease) both maximum mass and its corresponding radius.Note that this trend also has found if we used the standard isotropic MIT bag EoS.At the same time, B ⊥ value variation yields an insignificant effect in mass and radius of Fig. 2 In this plot we vary B and B ⊥ .Increasing B will decrease the maximum mass, but varying B ⊥ does not produce very different results.The shaded regions, from top to bottom, are taken from the upper limit mass of neutron stars 2.01M < M < 2.16M [49], the mass constraint of J0348+0432 pulsar M = (2.01 ± 0.04)M [50], and the one of two neutron stars mass range from GW170817 1.17M < M < 1.6M [51] the stars.However, the mass and radius of quark stars are sensitive to a 4 and a 4⊥ parameter values variations.To increase the maximum mass, one can choose either to decrease a 4⊥ value or to increase a 4 value.It means the anisotropic term of this model impacts the compactness of the stars.However, it can be seen that as long as we use w = 1/3, the mass-radius relation predicted by this model never crosses the photon sphere line.However, if we increase the w value, the compactness increases significantly.It is evident in Fig. 4 that the maximum mass and its corresponding radius using w = 1 is higher than the one using w = 1/3, and the massradius relation with w = 1 crosses the photon-sphere line in the region near the maximum mass.This result is compatible with previous results [40] using the standard MIT bag anisotropic EoS.However, different to the finding in Ref. [41] by using a covariant anisotropic model with polytropic EoS, the maximum impact of the anisotropic of pressure based on the modified MIT bag model with w = 1/3 is still not sufficient to provide mass-radius relation that can cross the photon-sphere line.It means that quark stars within the modified MIT bag model proposed in Ref. [32] with w < 1 is still not sufficiently compact to be classified as ultra-compact objects.Now, we discuss the impact from the c variation of the Serrano-Liska model.We have found that the result is still indistinguishable from TOV GR case if c < 10 7 m 2 not only for w = 1/3 but also for w = 1. Figure 4 has shown the results for c = 10 7 m 2 .For comparison, we also show the results for c = 1 m 2 , which coincides with TOV GR case.It is also evident from Fig. 4 that for the significant large value of c i.e., c = 10 7 m 2 , if w = 1/3, the mass-radius relation Fig. 3 In this plot we vary a 4 and a 4⊥ .Decreasing a 4⊥ will increase the maximum mass.On the contrary, decreasing a 4 will decrease the maximum mass Fig. 4 In this plot we vary both w and c.Increasing either w or c will increase the maximum mass.We obtain the results from c = 1 m 2 coincides with TOV GR results still does not cross the photon-sphere line.Furthermore, if we take the value of c ≥ 10 7 m 2 , the situation becomes problematic.Since small c is needed to make the magnitude of the additional logarithmic term in the black hole entropy formula be less than the Beckenstein term.It means that the quantum gravity effect predicted by Serrano-Liska model within modified MIT bag anisotropic fluid EoS model can not be seen not only from a compact object with w = 1/3 like quark stars but also from an ultra-compact object with w = 1.Therefore, the choice of EoS matter is essential to obtain significant compactness.One cannot depend entirely on the corrections due to the modified gravity theories.
For completeness, in Fig. 5 we show the Ξ profiles from a single value of central pressure (upper panel) and the values Fig. 5 These are the result of Ξ and Ξ c = Ξ(r = 0).In this figure, all values of Ξ and Ξ c have negative value, except for some Ξ at exactly r = R. a Shows the profiles of Ξ with respect to r from a single value of central density corresponding to central pressure P c = 0.89 × 10 15 g/cm 3 .b Shows Ξ c the value of Ξ at the center with respect to various values of central density of Ξ at the center from the various value of central pressure (lower panel).From the profiles in the upper panel, we can see that R (1) + κ N T (1) is negatively valued inside the star.From the lower panel, we can see that the value of Ξ c = Ξ(r = 0) increases (to a larger negative value) nonlinearly to central density.This behavior leads to more recalculation from the shooting method, making a significant time increase in running the numerical calculation.Interestingly, the value of Ξ c in the lower panel does not change if we change the c value.This is actually not surprising since the equation governing Ξ (Eq.( 20)) does not depend on c.On the other hand, Ξ c is sensitive to the EoS.This is evident from the fact that the ρ c − Ξ c curve from w = 1 case is more chaotic than from w = 1/3 case, and the former shows a smaller Ξ c magnitude than in the latter.However, the trend of increasing Ξ c as ρ c increases happens to both w = 1 and w = 1/3 cases.
Since the model can produce ultracompact object i.e., the case with w = 1, we calculate its echo frequency and its effective potential because we expect it could be a possible quantum gravity signature within this model in quasi normal modes.The results are shown in Fig. 6.To obtain maximum compactness, we increase w, decrease a 4⊥ , and increase c.We obtain compactness C = 0.362 by the formula C = G M/Rc 2 = 1.47667 × (M/M ) × (km/R).(For clarity, we show the definition of compactness in SI units and the number 1.47667 came from using the SI unit.This gives us 2.95 km as the Schwarzschild radius of the Sun.Note that ultra-compact objects have compactness C > 1/3.)This object has large mass and small radius (M = 3.53M and R = 14.4 km).The upper panel shows r tortoise r * which is useful to calculate τ echo by [38,40,52] which in this case, gives us f echo = π/τ echo =15.2 kHz.Our result for c ≤ 10 7 m 2 is compatible with results from Ref. [40], where the strange star's EoS produces frequencies on the order of tens of kHz.However, this result is much larger than the result from GW170817 data analysis, as reported in Ref. [53].The GW170817 event corresponds to gravitational echo with a frequency of 72 Hz.Ref. [52] shows that this 72 Hz is compatible with an ultra-compact object in the form of a constant density star with compactness very near Buchdahl limit (C = 4/9) and mass 2-3 M .This clearly shows the difference of matter EoS, i.e., the EoS in Ref. [52] has infinite speed of sound w = ∞, but ours has finite speed of sound.The lower panel shows the effective potential with and without correction term Δ from the Serrano-Liska model This Δ correction term arises from employing Eqs. ( 24) and ( 25) in deriving the effective potential.In the limit of c → 0, the correction term Δ vanishes and the potential became the usual Regge-Wheeler potential inside a star [54].It is surprising to see that there is no significant shift for the effective potential even though c is already large.The reason is as follows.We show the profiles of pressure, energy density, Ξ(r ), the metric functions α and β, and the ratio between the correction term Δ with respect to ρ − P in Fig. 7.We see that the Δ/(ρ − P) magnitude is getting larger as r → 0 but the mass term and the l term dominates in this region.When the mass term and the l term do not dominate as r → R, Δ/(ρ − P) magnitude is small.These results provide an EoS candidate with w = 1 that can produce a static star with compactness C > 1/3.This is an improvement from the result by Urbano and Veermäe in Ref. [38], which says that in the case of MIT bag model C > 1/3 can be satisfied if the causality limit w ≤ 1 is violated.The reason lies in the anisotropic pressure which can stiffen the EoS thus making the star more compact [26].More Fig. 6 These are the r tortoise and effective potential result from a single central pressure P c = 0.89 × 10 15 g/cm 3 , which leads to compactness of an ultra-compact object G M/R = 0.362 recently, Zhang [55] had shown that an EoS from interacting quark matter without anisotropic pressure but with w = 1/3 can obtain C > 1/3.This is interesting because the EoS came from a similar theory as the EoS used in this paper.However, the model in Ref. [55] predicts echo frequency in kHz similar to our result.On the other hand, Alho et al. [56] had shown that elastic matter EoS can produce static anisotropic stars with compactness exceeding the Buchdahl bound (C = 4/9) if superluminal wave propagation in the material exists.However, this is considered unphysical.They found that the compactness limit will be smaller than the Buchdahl limit (C 0.376) if physically admissible solutions and radial stability conditions are considered.Thus Alho et al. conjecture that true black hole mimickers require beyond-GR effects.Our results agree with this conjecture because reaching compactness larger than Buchdahl bound is tricky even when the model used here is an extension of GR and includes a model of anisotropic pressure.According to Cardoso and Pani [39], very anisotropic stars can reach compactness beyond the Buchdahl limit even arbitrarily close to the black hole limit.However, this is not the only possibility.Very large compactness can be reached by other exotic stars, Fig. 7 These are the profiles the produces the bottom panel in Fig. 6 e.g., quasiblack holes, wormholes, dark stars, and gravastars.As implied by Fig. 4, the effect of c within the range used may not be too helpful to increase the compactness.The type of matter used and the introduction of anisotropic pressure play a more significant role in increasing compactness.On the other hand, If we significantly increase the values of c beyond c = 10 7 m 2 , the strength of the logarithmic term in the entropy formula in Eq. ( 1) becomes significantly increased.On the other hand, a significant value of c is not physically favored because we learn from the black hole case that the logarithmic term is expected to be smaller than that of the Bekenstein term.

Conclusions
In this paper, we use the SL gravity model [13] to calculate the properties of ultra-compact stars.We use the modified MIT bag model with anisotropy pressure proposed in Ref. [32] to describe the star's matter.We intend to see the possible signature of quantum gravity reflected in compact objects predicted by the SL model.The logarithmic term in the black hole entropy motivates the SL model.Here, we conclude several points as follows.By equipping the SL model with spherically symmetric metric, the correction terms O( c) in the modified unimodular gravity equation can be contained in a new function Ξ(r ).We also found from w = 1/3 and w = 1 cases that the SL model's prediction for mass-radius relation deviates from standard TOV GR if we set c > 10 7 m 2 .We also see that even with a sufficiently large value of c, the ultra-compact stars from w < 1 are still not sufficiently compact to be in the family of ultra-compact objects.The effective potential is almost indistinguishable from GR's if we still use c ≤ 10 7 m 2 .To make it distinct, we should increase c until c 10 7 m 2 .(This is because by increasing c, we increase the compactness, so the width of the well in the effective potential will be larger, hence echo frequency will be smaller.To see this, for instance, see Fig. 6 in Ref. [57].)However, when we set c = 10 7 m 2 and have the maximum anisotropic pressure, we obtain C = 0.362, and its echo frequency is 15.2 kHz.This value is not comparable to 72 Hz from GW170817 reported in Ref. [53].This 72 Hz value is obtainable by employing a non-rotating star with constant density [52], i.e. infinite speed of sound.Note the difference to our EoS that has finite speed of sound.Therefore, to lessen the echo frequency value, we need to either use EoS with larger speed of sound or increase c again.However, the latter will increase the strength of the logarithmic correction term in the black hole entropy formula.To this end, more precise gravitation echo measurements are crucial to understanding this issue.
Table 1 This is the data of maximum mass and its corresponding radius shown graphically in Fig. 2 B (g/cm 3 )  3 This is the data of maximum mass and its corresponding radius shown graphically in Fig. 4. The "TOV-GR" sign denotes calculation using the usual TOV equation in GR framework g/cm 3  .Subscript "ori" refers to Eq. (2a), "positive" and "negative" refer to the sign in Eqs. ( 6) and ( 8  , respectively.The step size for r is actually 0.01 km, but due to the large data size, here we show only some of them.We stop the shooting method for finding

Table 2
This is the data of maximum mass and its corresponding radius shown graphically in Fig.3

Table 4
EOS data shown in Fig.1.All data are in unit 10 15