White dwarfs as a probe of dark energy
WWhite dwarfs as a probe of dark energy
S. Smerechynskyi , ∗ M. Tsizh , and B. Novosyadlyj , Ivan Franko National University of Lviv,Kyryla and Methodia St., 8, Lviv, 79005, Ukraine and College of Physics and International Center of Future Science of Jilin University,Qianjin St. 2699, Changchun, 130012, R.China (Dated: December 5, 2019)We investigate the radial density distribution of the dynamical dark energy inside the white dwarfs(WDs) and its possible impact on their intrinsic structure. The minimally-coupled dark energy withbarotropic equation of state which has three free parameters (density, equation of state and effectivesound speed) is used. We analyse how such dark energy affects the mass-radius relation for theWDs because of its contribution to the joint gravitational potential of the system. For this weuse Chandrasekhar model of the WDs, where model parameters are the parameter of the chemicalcomposition and the relativistic parameter. To evaluate the dark energy distribution inside a WDwe solve the conservation equation in the spherical static metric. Obtained distribution is used tofind the parameters of dark energy for which the deviation from the Chandrasekhar model mass-radius relation become non-negligible. We conclude also, that the absence of observational evidencefor existence of WDs with untypical intrinsic structure (mass-radius relation) gives us lower limitfor the value of effective sound speed of dark energy c s (cid:38) − (in units of speed of light). PACS numbers: 95.36.+x, 98.80.-kKeywords: cosmology: dark energy–Chandrasekhar model–white dwarfs
I. INTRODUCTION
The nature of dark energy, a substance which causes the observable accelerated expansion of Universe, has becomehighly studied subject in cosmology in the last two decades. A significant part of models that explain it are (usually,scalar) field models of the dark energy [1–8]. Unlike cosmological constant, scalar field dark energy is assumed to bedynamical and perturbable, changing its density across the time and space, and having as a result restrained impacton the evolution of the large scale structures [9]. Such dark energy can be modelled as the perfect or imperfect fluid,which is effectively described by hydrodynamical parameters: the density ρ de , the equation of state parameter w de and the effective speed of sound c s . The most conservative models assume that only density of the dark energy varieswith cosmological time, while models with more degrees of freedom assume that all three parameters are dynamical.Depending on its properties dark energy is referred to as the quintessence ( w de > − w de < −
1) or thequintom, in which w de changes its sing during the evolution. Current observable data doesn’t give strong preferenceto any of these types of dark energy [8–11].It was first noticed by Babichev and co-authors [12], that scalar field dark energy can influence the compact objectsthrough accretion. They analysed how infall of the phantom dark energy ”screens” a black hole’s (BH) gravitationalfield, eventually leading even to its disappearing.The idea, that hidden components of the Universe can influence the compact objects, like BHs or white dwarfs(WDs), through gravity has developed further. In the last decade, a number of works appeared, in which alternativetheories of gravity, also are capable to explain the accelerated expansion of the Universe, were tested on deviation fromthe general relativity at small scales through impact on the observable features of compact objects. For example,Vainshtein mechanism [13, 14], which restores equivalence of Einstein’s general relativity and some of alternativegravity theories at small (Solar System) scales, can be broken inside matter for some cases, such as beyond Horndeskimodels [15–17]. With the purpose of probing this scenario different kind of compact astrophysical objects were chosen.For instance, in work [18] the red and brown dwarfs were used as probes for the modified gravity theories throughimpact on the mass-radius relation, the Chandrasekhar mass limit and the mass-radius relation for the WDs were usedin works [19, 20] in order to obtain independent constraints on the Vainshtein breaking parameter. Similar work [17]was devoted to the study of relativistic objects such as WDs and neutron stars in which it was shown the importanceof post-Newtonian corrections in equilibrium equation for WDs while calculating macroscopic characteristics in theframe of the theory of modified gravity. In the work [21] its authors attempted to explain existence of sub- and ∗ Electronic address: [email protected] a r X i v : . [ a s t r o - ph . C O ] D ec super-Chandrasekhar WDs as possible progenitors of peculiar Super Novae of Ia type with help of modified gravitytheory. BHs and relativistic stars in scalar-tensor theories of gravity are studied in [22, 23]. In both works authorsfind constraints from observing compact objects on theories. WDs are used for similar purpose in papers [24] and[25]: the corrections to equations that describe them are evaluated and constraints on the possible modifications ofgravity are given.The goal of this paper is to investigate the radial density distribution of the dynamical dark energy inside the WDsand estimate its possible impact on their intrinsic structure. It will give the possibility to estimate the lower limit forthe value of the effective sound speed c s of the dark component. We do it by considering static solution of dark energydistribution in spherically symmetrical metric. We take it into account in the equation of Chandrasekhar model,the dark energy changes joint gravitational potential of the system ”white dwarf + dark energy”, hence changingmass–radius relation for WDs, depending on its parameters, in particular on the effective speed of sound.Though local behaviour of dark energy clustering is an object of study in a lot of works lately (see [26, 27] and [28]for example), and there are even examples of ”mixed star” (dark energy + baryon matter) solutions [29, 30], our work,as of our knowledge, is the first attempt to constrain dark energy parameters through observable WDs properties.The paper is organized as follows. In Section II we briefly remind the Chandrasekhar model of WD and its mainresults. Section III contains the equation of state for the dark energy and the calculation of its radial distributioninside WD. In Section IV we discuss the possibility of setting the constraint on the effective speed of sound of thedark energy using the WDs and in Section V we present our conclusions. II. CHANDRASEKHAR MODEL OF WHITE DWARFS
Typical WD is a spherical object with mass of one half that of the Sun and radius of the order of Earth’s one. In suchextreme dense objects hydrostatic equilibrium is maintained by pressure of relativistic degenerated electron gas [31–33]. High density of matter causes the equation of state of electron gas to be almost independent on temperature, thatis why mechanical and thermal structure of WDs can be treated separately. Such zero-temperature approximation isespecially applicable to massive WDs where finite temperature effects are negligible [34]. In approximation of completedegeneration the equation of state of non-interacting relativistic electron gas can be written in the parametric formas follows P e ( r ) = πm e c h f ( x ( r )) ,f ( x ) = 8 x (cid:90) y dy (cid:112) y = x (2 x − (cid:112) x + 3 ln ( x + (cid:112) x ); (1) ρ ( r ) = µ e m u π ( m e c ) h x ( r ) . Here m e is the electron rest mass, m u stands for atomic mass unit and dimensionless chemical composition parameter µ e determines the number of nucleons per free electron for an averaged nucleus in a star (we assumed here µ e = 2). Thedimensionless Fermi momentum of electrons x = p F /m e c , called relativistic parameter, plays the role of parameter inabove mentioned equation of state.Assuming the hydrostatic equilibrium of a non-rotating gaseous sphere Chandrasekhar obtained the model of theWDs with two parameters [33]: relativistic parameter in stellar centre x and chemical composition parameter µ e (which is close to 2 for all elements except hydrogen). Within this model two important outcomes became famous:peculiar mass-radius relation – radius of WD decreases with increasing mass, on the contrary to normal stars (left panelof Fig. 1); and existence of maximum mass of WD ∼ . M (cid:12) , known as the Chandrasekhar mass limit. This is formallimit for WD with the central density approaching infinity (right panel of Fig. 1). The latter one played the crucialrole in the discovery of the accelerating expansion of the Universe through observations of the distant supernovae ofIa type. This kind of superluminous events are believed to be explosions of WDs exceeding Chandrasekhar limit dueto the accretion of matter from another component of the binary system. M / M s un R/R sun M / M s un x FIG. 1: Characteristics of white dwarfs in Chandrasekhar model: the mass-radius relation (left panel), the Chandrasekharlimit (right).
III. DARK ENERGY IN WHITE DWARFSA. Dark energy model
In this work we analyse the scalar field model of dark energy with barotropic equation of state p de = ω ( ρ de ) ρ de c , (2)where p de and ρ de are pressure and density of dark energy, respectively. We consider model for which relation betweenthe equation of state parameter w and the effective speed of sound c s (in the units of speed of light c ) is as follows w = c s − ( c s − w ∞ ) ρ ∞ ρ de . (3)Here ρ ∞ is the background density of dark energy (at r → ∞ ), for which we adopted the value 10 − kg/m [35].Also, we considered two types of dark energy, quintessence with ω ∞ = − . ω ∞ = − . L ( X, U ) with kinetic term X and potential U , the connection withphenomenological quantities is as follows ρ de = 2 X L ,X − L , p de = L , w de = p de c ρ de = L X L ,X , c s = δp de c δρ de = L ,X X L ,XX − L ,X One can obtain considered here linear equation of state for stationary Minkowski or Schwarzschild world and thescalar field dark energy with conditions c s = const > w de < U and a density-dependent kinetic term X [38]. B. Dark energy distribution inside white dwarf
In order to analyse the behaviour of dark energy inside a compact astrophysical object we consider the simplestmodel of WD without rotation and neglect effects of magnetic field, finite temperature and Coulomb interactions onthe mechanical structure. Consequently, we expect spherically symmetric distribution of dark energy inside a star.Also, in this work we do not aim to describe the dynamical evolution of dark energy in the gravitational field ofcompact object, but instead focus on static configuration of system, which consists of two components: matter of WDand dark energy.The space-time metric for spherically symmetric case can be written in the form ds = e ν ( r ) c dτ − e λ ( r ) dr − r (cid:0) dθ + sin θdϕ (cid:1) . (4)In our case the components of metric do not depend on time and can be obtained from Einstein equations withboundary condition λ ( r = 0) = 0 e − λ ( r ) = 1 − πGc r r (cid:90) [ ρ m ( r (cid:48) ) + ρ de ( r (cid:48) )] r (cid:48) dr (cid:48) , (5) ν ( r ) + λ ( r ) = − πGc R (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) . Here ρ m , p m are local density and pressure of stellar matter and ρ de , p de denote corresponding characteristics of darkenergy.In the case when the equilibrium of the gravitational force and pressure gradient is fulfilled, the following equationshold for both components of considered system dp m dr + 12 ( ρ m c + p m ) dνdr = 0 , (6) dp de dr + 12 ( ρ de c + p de ) dνdr = 0 . If the density of dark energy is essentially lower than the matter density and metric function ν is defined by distributionof matter mainly then the last equation gives the radial distribution of dark energy inside a star [35] ρ de ( r ) = ρ ∞ (cid:32) c s − ω ∞ c s + 1 + ω ∞ c s (cid:104) e ν ( r ) (cid:105) − c s c s (cid:33) . (7)In order to solve the system of equations (5)-(6) in the general case we have to know the value for ρ de (0) or thevalue of ν (0) in stellar centre. For this we applied the iterative procedure: in zero approximation we assumed noinfluence of dark energy on WD and having the results of Chandrasekhar model (given in (eq. 1)) we calculated secondequation in (5) at the point r = 0. Following this, we use found potential ν (0) to evaluate the density of the darkenergy at the center with formula (7). At the next step this value was used to solve the system (5)-(6), where bothbaryon matter and dark energy are taken into account when calculating the potential, and recalculate a new value of ρ de (0). Such procedure was repeated until the convergence was reached or iteration limit exceeded. The algorithmstops when relative change of the potential at consecutive iterations is less than 10 − . Usually it takes less then 10iterations to reach the convergence.The system ceases to converge when the effective sound speed c s approaches to zero, meaning that dark energy withvery small c s doesn’t allow static solutions. The specific value of c s when convergence is lost depends on relativisticparameter in the center of the star x and this value of c s increases with growth of x . Technically divergence ismanifested through very rapid growth of density and thus mass of dark energy in the system up to the infinity (ornegative infinity in the case of phantom dark energy).Fig. 2 illustrates the calculated relative deviation of density of dark energy from background for different values ofeffective speed of sound c s and fixed central density of the matter (or relativistic parameter x ). It can be seen thatwith decreasing value of c s the amount of dark energy inside WD increases for quintessential dark energy (decreasesfor phantom one) by orders and becomes concentrated towards stellar centre.Similar behaviour of dark energy one can see when we fixed value of c s but varied central density (or x ) of a star(see Fig. 3). Relative change of ρ de is very sensitive to relativistic parameter in stellar centre x and is even moreabrupt with growing central density of the stellar matter.The radial dependence of WD mass, as well as mass of dark energy inside a star, are shown in figure 4. It showsthat the presence of quintessential dark energy reduces the mass of WD in comparison to result of Chandrasekharmodel, whereas phantom dark energy causes its increase. Also, the stellar radius changes pettily, it grows in the caseof quintessential dark energy and shrinks in the case of phantom one.The obtained negative values of density and mass for phantom dark energy needs some comments. The possibility ofnegative density for this model was already mentioned in [35] (see also [39] for cosmological consequences of phantommodels). It was shown in [40] and [41], that presence of such dark energy can cause UV quantum instability of -0.04-0.02 0 0.02 0.04 0 0.2 0.4 0.6 0.8 1 δ de r/R a) -1-0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1 δ de r/R b) -4x10 -2x10
0 0.2 0.4 0.6 0.8 1 δ de r/Rqph c)FIG. 2: The relative deviation δ de ( r ) = ( ρ de ( r ) − ρ ∞ ) /ρ ∞ as a function of radial coordinate r inside a WD with radius R forfixed value x = 1 and different values of c s : a) c s = 10 − ; b) c s = 10 − ; c) c s = 10 − . Solid lines correspond to quintessentialdark energy with ω ∞ = − .
8, dash-dotted – phantom dark energy with ω ∞ = − . -1-0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1 δ de r/R a) -2x10
0 0.2 0.4 0.6 0.8 1 δ de r/R b) -2x10 -1x10
0 0.2 0.4 0.6 0.8 1 δ de r/Rqph c)FIG. 3: The same as in Fig. 2 but for fixed value c s = 10 − and different values of x : a) x = 1; b) x = 5; c) x = 10. vacuum through producing a pair of phantom particles and one non-phantom, having energy conserved. The briefdiscussion of the problem of the existence of physical essence with negative density or mass of particles from thegeneral relativity point of view can be found in the recent paper [42]. Though we consider here classical behaviourof the dark energy and field behind it, this means, that one should be careful when trying to obtain real observableconstraints from the solutions for phantom dark energy, remembering there are also quantum limits of such models.In our case, constraints for quintessence and phantom models are almost the same, as divergence occurs almost forthe same values of x and c s . IV. CONSTRAINT ON c s BY WD’S MASS-RADIUS RELATIONA. Influence on mass-radius relation
We calculated the WD masses and radii for various values of relativistic parameter in stellar centre x and effectivespeed of sound c s . Because we are interested in masses of WDs which can be obtained from observations, our resultsshown in figure 5 are represented in the form of object’s mass as function of x .As we can see from the figure, the dark energy inside WD does not reveal itself unless some critical value of x (dependent on c s ) is reached. In the vicinity of this value the dark energy accumulated inside a star causes abruptdeviations from the results of Chandrasekhar model (depicted with dotted line). The lower is the value of effectivespeed of sound, the smaller is critical value of x at which the dark energy begins to make a significant contributioninto the hydrostatic equilibrium of the WD by lowering its mass in the case of quintessence dark energy and increasingit in the case of phantom one. The reason of such behaviour is following. Higher x corresponds to higher density ρ m of matter in the center of WD and hence, deeper potential well of the system. Deeper potential well causes growthof ρ de . In the case of quintessential dark energy, given equation of state of dark energy makes its behaviour close tomatter one, which means that for satisfying the hydrostatic equilibrium the lower mass of matter is necessary. Indeed,when ρ de (cid:29) ¯ ρ de then w → c s , p de → c s ρ de c and dark energy contributes into the metric functions, as it follows from M / M s un qphNo DE 1.3 1.35 1.4 0.8 1-0.3 0 0.3 0 0.2 0.4 0.6 0.8 1 M de / M s un r/R FIG. 4: Radial dependence of white dwarf mass (top) and mass of dark energy inside a star (bottom). Also, the zoomed-inpart of the dependence near stellar surface is shown on the top panel. Solid lines correspond to quintessential dark energy with ω ∞ = − .
8, dash-dotted – phantom one with ω ∞ = − .
2. Short-dashed line corresponds to the result of Chandrasekhar modelwithout taking into account dark energy. Here we assumed x = 10 . c s = 4 · − . s2 =1e-45e-54e-53e-52e-51e-5 M / M s un x M de / M s un x c s2 =5e-4c s2 =1e-4c s2 =5e-5c s2 =4e-5c s2 =3e-5c s2 =2e-5c s2 =1e-5 FIG. 5: Left panel: The dependence of WD mass on x for a set of values of effective speed of sound of quintessential darkenergy c s (10 − , 2 · − , 3 · − , 4 · − , 5 · − , 10 − ). Zoomed-in part of the dependence near x = 10 for both types ofdark energy with c s = 4 · − is shown in the central part of the figure. Downward deviation corresponds to quintessentialdark energy, upward – phantom one. Short-dashed line corresponds to the result of Chandrasekhar model. Right panel: Thedependence of mass of quintessential type dark energy on parameter x for the same set of values for c s . TABLE I: The critical values of relativistic parameter in stellar centre x for different values of effective speed of sound fordark energy of both considered types. c s x q x ph − . . · − . . · − . . · − . . · − . . − . . equations (3) and (5). This breaks hydrostatic equilibrium and causes gravitational collapse of system. In the caseof phantom dark energy ρ de changes its sign and becomes negative. As a consequence, for satisfying the hydrostaticequilibrium the larger mass of matter is necessary. In both cases the process is rapid, as sort of positive feedbackloops are created, changing the properties of dark energy inside and in the nearest vicinities of WD.In the left panel of Fig. 5 the depeneces M ( x ) are presented for WDs in the models without dark energy (short-dashed line) and with quintessence dark energy with different values of c s (main part) and with phantom andquintessence dark energy in the insert which is zoom-in of central part of figure. One can see that deviations forboth quintessential and phantom dark energy take place approximately at the same values of x .The corresponding masses of dark energy of quintessential type (in log-scale) as a functions of relativistic parameterin stellar centre x are shown in the right panel of figure 5. The amount of dark energy inside a dwarf steeply increaseswith x and strongly depends on the effective speed of sound c s . Our solutions yield infinite values of densities when c s = 0. This is consequence of considered equation of state of the dark energy and, correspondingly, solution (7),where c s is in the denominator of exponent. One can conclude, that dark energy with c s = 0 is excluded from possiblemodels. B. Constraints
As we saw in Fig. 5, there are some critical values of x depending on c s at which the dark energy changes the M − x relation for WD noticable. These values are given in Table I for both considered types of dark energy.In the papers [43, 44] authors employed the Chandrasekhar model of WDs to solve the inverse problem for largesample ( ∼ x (cid:46) . x can be sufficiently high – up to x max ≈ . x for real WDs.Also, WDs in binary systems can be used to estimate the maximal value of x . As can be seen in the right panelof Fig. 6, both tails of the distribution by mass (or by x ) are more populated than for field (single) dwarfs. Thereason is a mass transfer between the components of binary systems. As was mentioned above, it is believed thatprogenitors of Ia type Super Novae (SN) events are WDs in close binaries with masses near the Chandrasekhar limit.Formally, in the frame of Chandrasekhar model, they occur at x → ∞ . However, it was shown first in [47] that massaccretion onto WD can cause the instability before reaching the Chandrasekhar limit due to effects of general theoryof relativity and/or neutronization. The critical values of central density in the case of carbon WD were found to beof the same order for both effects (2 . · and 3 . · kg/m , respectively) [48]. The recent values for generaltheory of relativity effects are very similar (see, for example [49]). The corresponding values of relativistic parameter x max are 23 . .
1, meaning x can not exceed it as the SN explosion occures.Thus, supposing that there exist field WDs with such high masses that correspond up to x max ≈
10 and/or that Iatype Super Nova events are explosions of WDs in the binaries with relativistic parameter in their centres x max ≈ x , one can constrain c s . For each value of the former one there is a corresponding value of c s for which the solution for hydrostaticalequilibrium equation ceases to exist, i.e. DE with lower values of c s would distroy the WD. Within this assumptionwe have found the lower limit for the value of effective speed of sound when the deviation from the Chandrasekharmodel becomes non-negligible: for field (single) WDs c s (cid:38) · − and for WDs in binary systems c s (cid:38) − . It is N x N M/M sun
SDSS DR4WDMS
FIG. 6: Left panel: Distribution of white dwarfs of spectral class DA from SDSS DR4 [45] by x . Right panel: Comparison ofdistributions by mass of white dwarfs from SDSS DR4 and white dwarfs in binary systems with main sequence stars (WDMS)[46]. interesting to point out, that these constrains are close to the ones, obtained in work [50], where we’ve used currentprecision of the measuring gravitating mass in the Solar System to constrain the value of the speed of sound of thedark energy. V. CONCLUSIONS
In this work we have considered a dark energy with the barotropic equation of state in static gravitational fieldof WDs. We have obtained the distribution of dark energy in a WD using Chandrasekhar model and calculated itsimpact on object’s characteristics in self-consistent way. Investigation of “mass-radius” relation for WDs with darkenergy inside has shown that deviation of the WD mass from one in the model without dark energy appears to betiny, unless some critical value of relativistic parameter x is reached, though deviation of density of dark energy inthe center of star from background dark energy density can be noticeable. Deviation of mass of WD in comparisonto Chandrasekhar model is negative for quintessential type of dark energy and positive for phantom one. The criticalvalue of x decreases with decreasing value of dark energy effective speed of sound c s .Using this, we have compared the critical values of relativistic parameter when the concentration of dark energy istoo high to maintain the equilibrium of WD with maximum value of x obtained in Chandrasekhar model for observed(single) WDs x max (cid:46)
10 and with value at which another effects affect stellar structure such as neutronization oreffects of general theory of relativity for massive WDs in binary systems, x max (cid:46)
25. Supposing that the dark energyhas no or has negligible influence on WD structure, which allows WD with such relativistic parameters to exist, wecan conclude that minimal value of squared effective speed of sound is c s ≈ · − in the case of field WDs and c s ≈ − for dwarfs near the Chandrasekhar limit in the binary systems. Acknowledgements
This work was supported by the projects of Ministry of Education and Science of Ukraine ΦΦ-63Hp (No.0117U007190) and Formation and characteristics of elements of the structure of the multicomponent Universe, gammaradiation of supernova remnants and observations of variable stars (No. 0119U001544). [1] B. Ratra and J.E. Peebles, Cosmological consequences of a rolling homogeneous scalar field, Phys. Rev. D , 3406 (1988)[2] M.S. Turner, D. Huterer, Cosmic Acceleration, Dark Energy, and Fundamental Physics, J. Phys. Soc.Jap. , 111015(2007) [3] Special issue on dark energy, Eds. G. Ellis, H. Nicolai, R. Durrer, R. Maartens, Gen. Relat. Gravit. (2008)[4] R.R. Caldwell, M. Kamionkowski, The Physics of Cosmic Acceleration, Ann. Rev. Nucl. Part. Sc. , 397 (2009)[5] L. Amendola and S. Tsujikawa, Dark Energy: theory and observations, Cambridge University Press, 507 p. (2010)[6] A. Blanchard, Astron. Astroph. Rev. , 595 (2010)[7] Lectures on Cosmology: Accelerated expansion of the Univese. Lect. Notes in Physics 800, Ed. G. Wolschin, Springer,BerlinHeidelberg, 188 p. (2010)[8] B. Novosyadlyj, V. Pelykh, Yu. Shtanov, and A. Zhuk, Dark Energy: Observational Evidence and Theoretical Modelsedited by V. Shulga, Academperiodyka, Ukraine, 381 p. (2013)[9] M. Tsizh, B. Novosyadlyj, Advanc. Astron. Space Phys. , 51 (2015), arXiv:1508.05518[10] B. Novosyadlyj, O. Sergijenko, R. Durrer, V. Pelykh, J. Cosmol. Astropart. Phys. , 30 (2014) arXiv:1312.6579[11] Planck Collaboration: Y. Akrami, F. Arroja, M. Ashdown et al. (2018), arXiv:1807.06205[12] E. Babichev, V. Dokuchaev, Yu. Eroshenko, Phys. Rev, Lett. , 021102 (2004), astro-ph/0505618[13] A. I. Vainshtein, Phys. Lett. B , 393 (1972)[14] E. Babichev and C. Deffayet, Clas. Quant. Grav. , 184001 (2013), arXiv:1304.7240[15] G. Horndeski, Intern. J. Theor. Phys. , 363 (1974)[16] T. Kobayashi, Y. Watanabe and D. Yamauchi, Phys. Rev. D , 064013 (2015), arXiv:1411.4130[17] E. Babichev, K. Koyama, D. Langlois, R. Saito, Clas. Quant. Grav. , 235014 (2016), arXiv:1606.06627[18] J. Sakstein, Phys. Rev. D , 124045 (2015), arXiv:1511.01685[19] R. K. Jain, C. Kouvaris, N. G. Nielsen, Phys. Rev. Lett. , 151103 (2016), arXiv:1512.05946[20] S. Banerjee, S. Shankar and T. Singh, J. Cosmol. Astropart. Phys. , 004 (2017), arXiv:1705.01048[21] U. Das and B. Mukhopadhyay, J. Cosmol. Astropart. Phys. , 045 (2015), arXiv:1411.1515[22] J. Chagoya, G. Tasinato, J. Cosmol. Astropart. Phys. , 006 (2018), arXiv:1803.07476[23] A. Lehebel Compact astrophysical objects in modified gravity , doctoral thesis, arXiv:1810.04434[24] I. D. Saltas, I. Sawicki, I. Lopes, J. Cosmol. Astropart. Phys. , 028 (2018), arXiv:1803.00541[25] N. Nari, M. Roshan, Phys. Rev. D , 024031 (2018),arXiv:1802.02399[26] M. Donnari, M. Merafina, M. Arca-Sedda, 14th Marcel Grossmann meeting, arXiv:1602.00889[27] S. Dhawan, A. Goobar, E. M¨ortsell, J. Cosmol. Astropart. Phys. , 024 (2018), arXiv:1710.02374[28] B. Novosyadlyj, M. Tsizh, Yu. Kulinich, Gen. Relat. Grav., , 30 (2016), arXiv:1602.08050[29] R. Chan, M.F.A. da Silva, J.F. Villas da Rocha, Gen. Relat. Grav. , 1835 (2009), arXiv:0803.3064[30] S. S. Yazadjiev, Phys. Rev. D , 127501, (2011), arXiv:1104.1865[31] S. Chandrasekhar, Mon. Not. Roy. Astron. Soc. , 456 (1931)[32] S. Chandrasekhar, Mon. Not. Roy. Astron. Soc. , 207 (1935)[33] S. Chandrasekhar, ”An Introduction to the Study of Stellar Structure”, Chicago, Illinois, University of Chicago Press,1939, LCCN: 39-8320 (PREM); CALL NUMBER: QB461 .C45[34] M. Vavrukh and S. Smerechinskii, Astron. Rep. , 913 (2013)[35] B. Novosyadlyj, Yu. Kulinich, M. Tsizh, Phys. Rev. D , 063004 (2014), arXiv:1404.0276[36] O. Sergijenko, B. Novosyadlyj, Phys. Rev. D , 083007 (2015), arXiv:1407.2230[37] E. Babichev, V. Dokuchaev, and Yu. Eroshenko, Usp. Fiz. Nauk , 1287 (2013).[38] B. Novosyadlyj, Ukr. J. Phys. , 998 (2019)[39] B. Novosyadlyj, O. Sergijenko, R. Durrer, V. Pelykh, Phys. Rev. D , 083008 (2012)[40] S. M. Carroll, M. Hoffman, M. Trodden, Phys. Rev. D , 023509 (2003)[41] J. M. Cline, S. Jeon, and G. D. Moore, Phys. Rev. D , 043543 (2004)[42] H. Socas-Navarro, Astron. Astrophys. , A5 (2019)[43] M. Vavrukh, S. Smerechynskyi, N. L. Tyshko, Astron. Rep. , 505 (2011)[44] M. Vavrukh, S. Smerechynskyi, Astron. Rep., , 363 (2012)[45] P.-E. Tremblay, Bergeron, and A. Gianninas, Astrophys. J. , Issue 2, 128 (2011), arXiv:1102.0056[46] A. Rebassa-Mansergas, B. T. G¨ansicke, M. R. Schreiber, D. Koester, Rodr´ıguez-Gil, Mon. Not. Roy. Astron. Soc. ,Issue 1, 620 (2010), arXiv:0910.4406[47] Ya. B. Zel’dovich, I. D. Novikov, ”Relativistic astrophysics. Vol.1: Stars and relativity”, Chicago, University of ChicagoPress, 1971, ISBN 0-486-69424-0[48] S. L. Shapiro, S. A. Teukolsky, ”Black Holes, White Dwarfs, and Neutron Stars. The Physics of Compact Objects”, CornellUniversity, Ithaca, New York, Wiley (1983), ISBN 0471873160[49] A. Mathew and M. K. Nandy, Research Astron. Astrophys.17