Impact of dynamical dark energy on the neutron star equilibrium
Prepared for submission to JCAP
Impact of dynamical dark energyon the neutron star equilibrium
S. Smerechynskyi M. Tsizh B. Novosyadlyj , Ivan Franko National University of Lviv, Kyryla and Methodia Street, 8, Lviv, 79005,Ukraine College of Physics and International Center of Future Science of Jilin University, QianjinStreet 2699, Changchun, 130012, People’s Republic of China
Abstract.
We study the density distribution of the minimally-coupled scalar field dark energyinside a neutron star. The dark energy is considered in the hydrodynamical representationas a perfect fluid with three parameters (background density, equation of state, and effectivesound speed). The neutron star matter is modeled with three unified equations of state,developed by the Brussels-Montreal group. With the calculated density distribution of thedark energy inside a neutron star (and its dependence on the dark energy parameters) weinvestigate how its presence impacts the macroscopic characteristics and the value of the masslimit for neutron stars. From this impact we derive the possible constrains on the effectivespeed of sound of dark energy with the help of maximal known masses of observed neutronstars. In this approach, we have found, that the squared effective speed of sound can not besmaller than ∼ − in units of squared speed of light. Keywords: cosmology: dark energy; stars: neutron stars & pulsars a r X i v : . [ a s t r o - ph . H E ] S e p ontents The local behavior of the perturbed dark energy has become an object of study in a seriesof works of the last decade. The impact of perturbation and clusterization of dark energy onthe matter dynamics was studied on clusters of galaxies and galaxy scales (for example, in[1–4]) and on the astrophysical scale ([5–14]). In general, studies have shown, that the localimpact of minimally coupled perturbed dark energy is mostly negligible, except some specificvalue regions of its parameters space, which can be used for ruling out these values.In particular, the dark energy models as well as the modified gravity theories are testedas an altering factor of the compact objects’ properties. Usually, in these works the authorsinvestigate how the gravitational potential is altered due to hidden component or alternativegravitation theory and how it changes the characteristics of a compact object. The compar-ison of the theoretical predictions with the corresponding observational data can give someconstraint on the value of parameters of the theory. There are also a number of papers wherethe probing objects are, in particular, neutron stars (NS). For example, the gravitationalaether theory is tested with NSs in [9]. In [10] the effect of a logarithmic f(R) theory onrelativistic stars is studied. Similarly, the alternative theories of gravity are tested by NSsin [11] (Einstein-Dilaton-Gauss-Bonnet gravity), [12, 13] (R-squared gravity) and [14] (f(T)gravity).The stationary accretion of dark energy onto the Schwarzschild black hole was studiedin [5]. We studied the static solutions of dark energy dynamical equations in the vicinityof compact objects and found that only relativistic objects with the lowest ratio “radius togravitational radius” disturb the density of dynamical dark energy noticeably [6]. In thepaper [7] we have investigated how the dynamical dark energy inside white dwarfs can changetheir mass-radius relation and have found that the squared effective speed of sound c s of darkenergy must be larger than ∼ · − in order to satisfy the observed mass-radius relation forwhite dwarfs. In the paper [8], we have used the uncertainty of determination of gravitatingmass in the Solar system as an upper limit on the amount of clustered dark energy andobtained the similar constraints for the value of c s ≥ · − . There are plenty of workswhich constrain the dark matter parameters and theories of gravity based on the reduction ofNS maximal mass caused by the accumulation of the dark matter in its interior (see [15–18]).In particular, authors of [19] analyzed compact objects that contain dark matter admixed– 1 –ith ordinary matter made of neutron star and white dwarf materials and dependence ofmaximum radius of such objects on dark matter properties. In this work we use a similarapproach towards another part of the dark sector – minimally coupled dark energy.The paper has the following structure. In Section 2 we describe the equations of statefor NS matter and hence present the model which governs the matter distribution inside NS.Section 3 is devoted to the description of the dark energy model and the radial distributionof the dark energy inside NS and its dependence on the dark energy parameters. In Section 4we deduce the constraint on the effective speed of sound for the dark energy using the NSsand in Section 5 we give our conclusions. The NSs exist due to the pressure of the degenerate gas of Fermi-particles, neutrons, muchlike the electrons in the case of white dwarfs, though, nuclear forces play an important role inthe former ones. The state of the matter inside white dwarfs is well known, but we are stilluncertain about the composition and equation of state (EoS) for the matter at the higherdensities that correspond to the NS interiors [20].Based on a given EoS one can yield the maximum mass of NS configuration, the so-calledTolman-Oppenheimer-Volkoff mass limit [21, 22]. The last one is similar to the Chandrasekharmass limit [23–25], but strongly depends on the incorporated physics, resulting in differentstiffness of EoS. The maximum mass limit for NS is approximately . M (cid:12) for soft equationsof state and reaches M (cid:12) in the case of stiff ones [26, 27]. The maximal known mass ofobserved NSs is a crucial value for testing different EoSs (see, for example, [27–31]).For the description of the NS interior we have exploited three unified EoSs developedby the Brussels-Montreal group [32–35]. Despite the matter being under different physicalconditions and states, such an EoS is valid throughout all parts of a neutron star – from theouter envelope to its crust and core. We will use the same denotations BSk19, BSk20 andBSk21 for the considered EoSs as in [35]. They differ by its stiffness, BSk21 is the stiffest oneand BSk19 is the softest one.The analytical representation of the equations of state with variables ξ = lg( ρ m / g · cm − ) and ζ = lg( p m / dyn · cm − ) ( lg denotes log ) is defined by the following parametrization: ζ = a + a ξ + a ξ a ξ a ( ξ − a )] + 1 + a + a ξ exp [ a ( a − ξ )] + 1 + a + a ξ exp [ a ( a − ξ )] + 1+ a + a ξ exp [ a ( a − ξ )] + 1 + a a ( ξ − a )] + a a ( ξ − a )] , (2.1)where p m and ρ m are the local pressure and density of the NS (or baryonic) matter, respec-tively. It is the same for three considered cases, only the values of coefficients a - a aredifferent (see [35]). Such representation of EoS simplifies its usage for the consideration ofthe inner structure of NSs. The fitting procedure introduces the errors of the macroscopic NScharacteristics, but they are far below the observational uncertainties [35].Using these equations of state, one can solve the hydrostatic equilibrium equation toobtain the corresponding mass-radius relations, shown in Fig. 1. As we can see, the stifferthe EoS is, the higher maximal mass of NS corresponds to it. These values (in solar massunits) as well as the corresponding central densities ρ m (0) ≡ ρ c of NSs are given in Table 1.The maximal known masses for observed NSs are: . +0 . − . M (cid:12) for PSR J0740+6620(here and below the confidence interval is 68.3%), obtained from the measurement of the rela-– 2 – M / M s un R, km BSk19BSk20BSk21
Figure 1 . Mass-radius relation for neutron stars with three considered equations of state.
Table 1 . The maximum masses and corresponding central densities of neutron stars for three equationof states BSk19, BSk20, BSk21 [35].
EoS M max /M (cid:12) ρ c , g/cm BSk19 .
86 3 . BSk20 .
16 2 . BSk21 .
27 2 . tivistic Shapiro delay [36]; and . +0 . − . M (cid:12) for PSR J2215+5135, yielded by the simultaneousfitting of radial velocity curves and three-band light curves [37, 38]. It is worth mentioningthat there are objects which are reported to have masses even larger. For instance, pulsarPSR B1957+20 with mass ∼ . M (cid:12) [39], but systematic uncertainties of the mass determina-tion are large. It is argued, that the accuracy of this measurement is partly limited by opticalflares and variable emission lines of companion’s stellar wind [38, 40]. There are evidencesindicating significantly lower mass of that pulsar ( ∼ . M (cid:12) [41, 42]). Thus, to date, themaximum mass of observed NSs, in general, agrees with estimates on the upper bound of NSmasses made within different approaches [27, 43–49].One can infer from the comparison of theoretical maximum masses from Table 1 andobserved ones, that the maximum mass values predicted by the equations of state BSk20 andBSk21 are inside the range inferred from observations and BSk19 does not allow the existenceof such massive NSs. Probably, there is a discrepancy for BSk20 EoS and PSR J2215+5135mass, if no evidence of significant uncertainty of its mass determination is found. In thefollowing sections we will use all three mentioned equations of state, bearing in mind thatBSk19 can not explain the existence of the most massive among known NSs and we considerit only for the sake of consistency and analysis.– 3 – Dark energy inside a neutron star
It was shown that the minimally-coupled scalar field model of dark energy with barotropicEoS p de = w ( ρ de ) c ρ de , (3.1)where p de and ρ de are pressure and density of dark energy, respectively, can agglomerate insideand in vicinity of a compact object when the EoS parameter w and the squared effective speedof sound c s ( c s in the units of speed of light) are related as [5, 6] w = c s − ( c s − w ∞ ) ρ ∞ ρ de . (3.2)Here ρ ∞ is the background density of dark energy (at r → ∞ ), which in our case is equal to − g/cm [50]. The value of background dark energy density was chosen to be higher thanthe cosmological value in Λ CDM model. The reason for this is that dark energy is assumed toundergo clusterization process along with dark matter during initial perturbation growth. Weconsidered only the quintessence type of dark energy with w ∞ > − . The model, describedby equation (3.2), implies the constant effective speed of sound of dark energy.The hydrodynamical representation of the scalar field dark energy as a perfect or im-perfect fluid with barotropic EoS is usually used in cosmology. The Lagrangian of the field L ( X, U ) with kinetic term X and potential U , is connected to phenomenological hydrody-namical quantities as follows [51] c ρ de = 2 X L ,X − L , p de = L , w = p de c ρ de = L X L ,X , c s = δp de c δρ de = L ,X X L ,XX − L ,X . The scalar field dark energy with conditions c s = const > and w < in stationaryMinkowski or Schwarzschild world is governed by the Klein-Gordon or hydrodynamical con-tinuity equations. In [6] we have shown how the eq. (3) is deduced in the framework of theseconditions and in [52] how the scalar field variables are related with hydrodynamical ones forthis dark energy model. In order to estimate the influence of the dynamical dark energy on the equilibrium condition ina NS we suppose that it is the non-rotational non-magnetic star which is in static equilibrium:the pressure gradient of baryon matter balances the gravitational attraction of the total massin a given sphere in the star, as well as the pressure gradient of the dark energy balances thegravitational attraction of the same mass. The Einstein and conservation law equations forminimally coupled baryonic and dark energy are used. Therefore, we considered a sphericallysymmetric object for which the space-time metric can be written in the form ds = e ν ( r ) c dτ − e λ ( r ) dr − r (cid:0) dθ + sin θdϕ (cid:1) . (3.3)If we limit ourselves to the case of static configuration of dark energy inside a NS,the components of metric will not depend on time and can be obtained from the Einstein– 4 –quations with the boundary condition λ ( r = 0) = 0 [6] e − λ ( r ) = 1 − πGc r r (cid:90) (cid:2) ρ m ( r (cid:48) ) + ρ de ( r (cid:48) ) (cid:3) r (cid:48) dr (cid:48) ,ν ( r ) + λ ( r ) = − πGc ∞ (cid:90) r (cid:20) ρ m ( r (cid:48) ) + ρ de ( r (cid:48) ) + p m ( r (cid:48) ) + p de ( r (cid:48) ) c (cid:21) e λ ( r (cid:48) ) r (cid:48) dr (cid:48) . (3.4)Here ρ m , p m are the local density and pressure of baryonic matter and ρ de , p de denote thecorresponding characteristics of dark energy. Other boundary conditions are the following: ν ( ∞ ) = − λ ( ∞ ) , ρ m ( R + ) = 0 , ρ de ( ∞ ) = ρ ∞ .With the metric functions given in (3.4) we numerically solved the equilibrium equationsfor both baryonic matter and dark energy dp m dr + 12 ( ρ m c + p m ) dνdr = 0 ,dp de dr + 12 ( ρ de c + p de ) dνdr = 0 , (3.5)applying the iterative procedure. On the initial step we evaluated the gravitational potentialwithout the dark energy influence, and, thereafter, found the distribution of the dark energy insuch potential. Then we solved the system of equations for baryonic matter and dark energydensities and their joint gravitational potential, and compared the results with ones obtainedin the previous step. Then we re-evaluated the distribution of dark energy and baryonicmatter in the new potential. Such procedure was repeated until the solutions converged orthe iteration limit exceeded (for more details see [7]). δ de r / R ρ m / ρ n -0.80-0.85-0.90-0.95-0.99 (a) δ de r / R ρ m / ρ n -0.80-0.85-0.90-0.95-0.99 (b) δ de r / R ρ m / ρ n -0.80-0.85-0.90-0.95-0.99 (c) Figure 2 . The relative deviation δ de ( r ) = ( ρ de ( r ) − ρ ∞ ) /ρ ∞ of dark energy density as a functionof radial coordinate r inside a neutron star with radius R for the case of BSk20 EoS: (a) ρ c = 2 ρ n , c s = 0 . ; (b) ρ c = 2 ρ n , c s = 0 . ; (c) ρ c = 5 ρ n , c s = 0 . . Line types correspond to different values ofthe parameter w ∞ : from − . for top curve to − . for bottom curve. The upper x-axis correspondsto matter density at given radial coordinates. The solutions of the system of equations (3.4–3.5) for the dark energy component areshown in Fig. 2 in the form of radial dependence of the relative deviation of the dark energydensity δ de ( r ) inside a star. The results correspond to BSk20 EoS for two different values ofeffective speed of sound c s = 0 . and . , and central density of the NS matter ρ c = 2 ρ n and ρ n , where ρ n = 2 . · g/cm is the so called normal nuclear density [20]. The line types– 5 –orrespond to the different values of the parameter w ∞ given in the figure. The upper x-axiscorresponds to the matter density at the given radial coordinates.One can see, that the relative deviation of the dark energy density in NS is very sensitiveto the value of c s and is increasing as the latter one is decreasing. It follows from thecomparison of Figs. 2a and 2b, corresponding to the same value of ρ c . δ de ( r ) increases alsowith increasing central density of baryonic matter ρ c at constant effective speed of sound ofdark energy c s (Figs. 2b and 2c).Also, one can infer that lowering w ∞ causes a smaller deviation of the dark energydensity from the background one. This makes no surprise: it is well known that the darkenergy with w ∞ = − is not perturbed at all, so one would expect the deviation to be smalleras w ∞ approaches -1 and vanishes at w ∞ = − .The solutions for the case of the stiffest of considered EoSs, namely BSk21, are illus-trated in Fig. 3. Similar dependencies of the relative deviation of the dark energy density onparameters w ∞ , c s and ρ c can be inferred from Figs. 3a–3c. However, it should be noted thatthe amount of dark energy inside a NS is larger for the case of stiffer EoS assuming the samevalues of other parameters. δ de r / R ρ m / ρ n -0.80-0.85-0.90-0.95-0.99 (a) δ de r / R ρ m / ρ n -0.80-0.85-0.90-0.95-0.99 (b) δ de r / R ρ m / ρ n -0.80-0.85-0.90-0.95-0.99 (c) Figure 3 . The same as in Fig. 2, but for the case of BSk21 equation of state.
The density distributions of dark energy obtained in the previous section give us the possibilityto calculate its total Lagrangian mass inside a star. Fig. 4 illustrates the dark energy mass M de (in solar mass units) as a function of central density ρ c of baryonic matter for the valuesof c s ranging from . to . (depicted with different colors). The dependence on theparameter w ∞ is shown by shadowed regions between solid lines corresponding to w ∞ = − . and dash-dotted lines ( w ∞ = − . ). The dark energy mass rises steeply with central matterdensity with an exception of the region near ρ c /ρ n ≈ and then reaches saturation levelwhich is in the range of one percent of total mass for all considered values of c s and w ∞ (seezoomed-in part of the figure in the upper left corner). The amount of dark energy is higherfor smaller values of c s and larger values of w ∞ . Moreover, the results for M de are moresensitive to the change of c s .Aiming to constrain the parameters of dark energy we have studied its influence onthe NS mass. The dark energy does not reveal itself until a certain value of central matterdensity is reached because of very strong dependence on the ratio of gravitational radius– 6 – de / M s un ρ c / ρ n c s2 =0.008c s2 =0.009c s2 =0.010c s2 =0.011c s2 =0.012c s2 =0.0131e-401e-351e-301e-251e-201e-151e-101e-051 1 2 3 4 5 6 7 8 9 100.010.02 Figure 4 . The dark energy mass (in solar mass units and logarithmic scale) as a function of theneutron star central density for the case of BSk21 equation of state. The colors correspond to differentvalues of c s : from . (top) to . (bottom). The results for w ∞ = − . are depicted with solidlines and ones for w ∞ = − . are depicted with dash-dotted lines, shadowed area between theselines corresponds to the results for in-between values of w ∞ . Zoomed-in region of M de /M sun ∼ − is shown in the upper left corner of the figure in the linear scale. to stellar surface one . This value depends on the chosen EoS for the NS matter and theparameters of dark energy. Thus, in order to analyze the impact of each parameter, we haveconsecutively fixed all of them except one. As was mentioned above, in our calculations weadopted ρ ∞ = 10 − g/cm , and this parameter was not changed at all.The total mass of a NS configuration (including the dark energy inside) as a function ofits central matter density is shown in Fig. 5a for three considered equations of state (labeledrespectively) and the same values of squared effective speed of sound c s as in Fig. 4, herethe EoS parameter remained fixed ( w ∞ = − . ). Similarly to white dwarfs, the quintessencetype of dark energy reduces the NS mass, acting matter-like, and contributing to the jointgravitational potential. The corresponding mass-radius relations for two considered EoSs forNS matter are shown in Fig. 5b.As we can see from both figures, the influence of dark energy becomes crucial at somevalue of ρ c which depends on c s . At the central matter densities higher than this “turn-off” point, the dark energy content causes abrupt deviation from the model without thelatter one (black solid lines). The amount of dark energy accumulated inside a neutron starwith central matter density near the turn-off point can be seen in Fig. 6 for the model with ρ c = 8 . ρ n , c s = 0 . and BSk21 EoS as an example. On the left panel (Fig. 6a) the ratiobetween dark energy mass m de ( r ) and mass of baryonic matter m m ( r ) is shown as functions ofdimensionless radial coordinate for given values of the parameter w ∞ (depicted with differentline styles and colors). Because of the concentration of dark energy towards the center (seeFig. 3), it dominates in the central part of the object (except the model with w ∞ = − . for which m de < m m even in stellar center). But with growing radial coordinate r , the ratiodecreases and reaches the values less than 0.01 on the surface for all considered w ∞ . On the The ratio M de /M (cid:28) for objects with R (cid:28) r g for c s > − [6], where R is radius of object, r g is itsgravitational radius. So, the dynamical dark energy practically does not influence the gravitational field inthe normal stars. – 7 – s2 =0.0080.009 0.0100.0110.012 0.013 M / M s un ρ c / ρ n no dark energy 0 0.5 1 1.5 2 2.5 2 4 6 8 10 12 14BSk19BSk19BSk20BSk20BSk21BSk21J0740+6620J2215+5135 (a) M / M s un R, km c s2 =0.008c s2 =0.009c s2 =0.010c s2 =0.011c s2 =0.012c s2 =0.013no dark energy 0 0.5 1 1.5 2 2.5 10 11 12 13 14 15BSk20BSk20 BSk21BSk21 (b) Figure 5 . (a) The total mass of neutron star configuration (including the dark energy inside) asa function of its central matter density for three considered equations of state for NS matter anddifferent values of c s and w ∞ = − . for dark energy (see text for details); (b) Mass-radius relationfor two equations of state of NS matter in models with dark energy with different values of c s (depictedwith different colors) and without it (black solid lines). right panel (Fig. 6b) we can see the radial dependencies of dark energy mass (dash-dottedlines), mass of NS matter (solid lines) and total mass, which is the sum of the previous two(dotted lines), in units of solar mass for two values of w ∞ (depicted with different colors). Thebaryonic masses are only less than 1 percent lower at r = R than total ones (see zoomed-inregion in the upper right corner of the figure) and both baryonic and total masses are slightlylower for the model with w ∞ = − . , while the amount of dark energy is higher in this case.Therefore, more than 99 percent of the total mass of stable NS consists of baryonic mass.Returning to Fig. 5, at the higher central matter densities the amount of dark energyis so high, that the mass of the matter drops and in this region of ρ c there are no stableequilibrium configurations. The pressure of baryonic matter can no more resist gravitationforce from potential strengthened by dark energy. Therefore, for a given value of c s we obtaina corresponding existence region for central matter densities of NSs, roughly constrained bythis turn-off point.This fact can be used to find a lower bound on the parameter c s , but for that purposeone needs the observational data on NS masses. With grey color in Fig. 5a we depicted theupper bound range for maximum NS mass given in papers [46–48], the results of which arebased on the observations of the binary NS merger GW170817 [49]. The masses of the mostmassive NSs, PSR J0740+6620 and PSR J2215+5135, are indicated with labeled straightlines. In addition, 90% credible region for maximum mass, obtained in paper [27] with help ofBayesian model selection analysis for NS mass distribution, is shown as the green filled area.One can immediately conclude from Fig. 5a, that the accuracy of NS mass determinationor/and the increase of the number of NSs with known masses are crucial for setting a tightconstraint on the effective sound speed of dark energy. Considering the upper bounds formaximum NS mass (grey region) we found that c s should be larger than . in the caseof BSk21 EoS, and . for BSk20. Assuming the mass estimation of PSR J0740+6620 isreliable, we can constrain c s (cid:38) . for BSk21. It should be mentioned, that these constraintsdepend on the considered EoS and the accuracy of the mass determination. In general, for afixed NS model, the higher the maximum mass of NSs is found, the higher is the lower limit– 8 – de (r) / m m (r) r / R -0.8-0.85-0.9-0.95-0.99 (a) m a ss / M s un r / R totalmatterdark energy-0.8-0.99 (b) Figure 6 . The dark energy to baryonic mass ratio m de ( r ) /m m ( r ) as functions of dimensionless radialcoordinate for given values of w ∞ (depicted with different colors and line types); (b) The radialdependencies of dark energy mass (dash-dotted lines), mass of baryonic matter (solid lines) and totalmass, which is the sum of previous two (dotted lines), in units of solar mass for two values of w ∞ (depicted with different colors). Zoomed-in surface region (0 . − r/R is shown in the upper rightcorner. The results correspond to the choice of BSk21 EoS. Other parameters have values ρ c = 8 . ρ n , c s = 0 . . for c s value. On the other hand, such high maximum masses make possible and at some pointeven require stiffer equations of state (to allow their existence), and in that case the oppositeis true: the model with stiffer EoS of NS lowers the bound on c s . M / M s un ρ c / ρ n no dark energy-0.80-0.85-0.90-0.95-0.99 (a) M / M s un ρ c / ρ n no dark energy-0.80-0.85-0.90-0.95-0.99 (b) Figure 7 . Similar as in Figure 5a, but for different values of w ∞ and fixed c s (see text for details):(a) c s = 0 . ; (b) c s = 0 . . As a next step, we investigated how the location of a turn-off point depends on parameter w ∞ at fixed value of c s . The results for the total mass of NS as a function of ρ c for c s = 0 . are illustrated in Fig. 7a and for c s = 0 . – in Fig. 7b. As in Fig. 5a, the results of themodel without dark energy are shown with black solid lines. We considered the values of w ∞ ranging from − . to − . . For these values of w ∞ , the central matter density ρ c corresponding to the turn-off point varies approximately on 3.9% and 4.0% for BSk21 andBSk20, respectively, in the case c s = 0 . , while 3.5% and 5.0% are corresponding variations– 9 –or the case c s = 0 . . The values of the turn-off point for different values of dark energyparameters c s and w ∞ and three considered EoSs are given in Table 2. Thus, one can inferfrom Fig. 7 and Table 2 that the results are less sensitive to the choice of w ∞ than c s for allconsidered equations of state. Table 2 . The values of turn-off point – the central matter density of NS ρ c (in units of normal nucleardensity ρ n ) at which the dark energy with different values of c s and w ∞ reveals itself on the mass –central density dependence (see Figs. 5a and 7). The results are given for three considered equationsof state BSk19, BSk20 and BSk21. BSk21 w ∞ c s .
008 0 .
009 0 .
010 0 .
011 0 .
012 0 . − .
80 4 . . . . . . − .
90 4 . . . . . . − .
99 4 . . . . . . BSk20 w ∞ c s .
008 0 .
009 0 .
010 0 .
011 0 .
012 0 . − .
80 5 . . . . . . − .
90 5 . . . . . . − .
99 5 . . . . . . BSk19 w ∞ c s .
008 0 .
009 0 .
010 0 .
011 0 .
012 0 . − .
80 6 . . . . . . − .
90 6 . . . . . . − .
99 7 . . . . . . In this paper we have analyzed the impact of the dynamical scalar field quintessence darkenergy on the NS. The density distribution was found from the numerical solutions of theconservation equation for NS matter and dark energy in joint potential, which correspondsto the static equilibrium of NS. We studied how this distribution depends on the parametersof dark energy (EoS w ∞ and squared effective speed of sound c s ) and also, on the centraldensity of baryonic matter ρ c . We have found, that the relative deviation of the dark energydensity δ de ( r ) inside a neutron star increases as w ∞ and ρ c grow and as c s decreases.We have also established, that there is a turn-off point in the dependence of NS mass onits central matter density determined by the amount of dark energy inside a star. When ex-ceeding this turn-off point, the dark energy with certain set of parameters ( c s and w ∞ ) makesimpossible a stable static solution of equilibrium equation to exist, meaning that consideredcombination of parameters ( M and ρ c or M − R relation) is impossible.Using this, we have established, that current limitation on the maximal mass of NSsallows one to constrain the minimal value of speed of sound c s relying on certain NS model.We have used the Brussels-Montreal EoS for NS matter with the set of parameters BSk20-21– 10 –nd the estimations of maximal NS mass from binary NS merger GW170817 to obtain thelowest possible minimum value for c s (cid:38) − . We have also found that the dependence of theturn-off point on the background EoS parameter of dark energy w ∞ is weak, so we didn’t useit to establish constraints on w ∞ .This constraint is stronger than one obtained from consideration of "white dwarfs + darkenergy" system ( c s (cid:38) − ), which means that NSs are more suitable objects for studying thisdark component of the Universe. On the other hand, large uncertainties in the determinationof masses of NSs, as well as knowledge of the matter state inside them, postpone obtaininginteresting results for further perspective. Acknowledgements
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