PHYSICAL REVIEW D 93, 064054 (2016) Relativistic stars in bigravity theory
† ‡ Katsuki Aoki,* Kei-ichi Maeda, and Makoto Tanabe Department of Physics, Waseda University, Shinjuku, Tokyo 169-8555, Japan (Received 15 February 2016; published 22 March 2016) Assuming static and spherically symmetric spacetimes in the ghost-free bigravity theory, we find a relativistic star solution, which is very close to that in general relativity. The coupling constants are classified into two classes: Class [I] and Class [II]. Although the Vainshtein screening mechanism is found in the weak gravitational field for both classes, we find that there is no regular solution beyond the critical value of the compactness in Class [I]. This implies that the maximum mass of a neutron star in Class [I] becomes much smaller than that in general relativity (GR). On the other hand, for the solution in Class [II], the Vainshtein screening mechanism works well even in a relativistic star and the result in GR is recovered.
DOI: 10.1103/PhysRevD.93.064054
I. INTRODUCTION [25] as well as by the Vainshtein mechanism. However, in these analysis, gravitational fields are assumed to be weak. Although recent observation has confirmed the big bang It has not been cleared whether the Vainshtein screening scenario, one of the biggest mysterious problems in modern mechanism holds even in the strong gravitational field (e.g., cosmology is the origin of the current accelerating expan- relativistic star and black hole). sion of the Universe [1]. One possibility to explain the The black hole geometry in bigravity has been con- accelerating expansion is a modification of gravitational cerned, which are classified into nondiagonal ansatz [26], interaction on a very large scale. In general relativity (GR), and bidiagonal ansatz [22,24,27–29]. In the former type of it is well known that the graviton is a massless spin-2 field. ansatz, there are only trivial solutions, which are the same The theories with a massive spin-2 field are one of the as those in GR.1 Additionally, the perturbation around the most natural extensions of GR. The nonlinear ghost-free nondiagonal black hole is also identical to GR [32–34]. massive gravity, often dubbed de-Rham–Gabadadze– Hence, the massive graviton does not appear in the non- Tolley (dRGT) theory, was proposed by de Rham et al. diagonal black hole. To find a nontrivial solution, if it [2]. Then the dRGT massive gravity theory has been exists, we should assume both metrics can be simulta- generalized to the bigravity theory by Hassan and Rosen neously diagonal in the same coordinate system. [3], which contains a massless spin-2 field as well as a There exists some special case of the bidiagonal ansatz massive spin-2 field. The present accelerating expansion of such that two metrics are proportional, which we call a the Universe can be explained in the bigravity theory homothetic spacetime. The solutions are also given by [4–18]. those in GR. However, in this case, the massive graviton Since the massive gravity and the bigravity theory are appears in the perturbation around the solutions. As a modified by adding the mass to the graviton, GR should be result, the homothetic Schwarzschild black hole becomes recovered by taking the massless limit. However, the linear unstable against the radial perturbations if the graviton massive gravity, so-called Fierz-Pauli theory [19], cannot mass is sufficiently small [35–37]. The instability of this be restored to the linearized GR even in the massless limit, black hole implies that there would be a hairy black hole which is called the van-Dam-Veltman-Zakharov (vDVZ) solution as well, and that the homothetic Schwarzschild discontinuity [20]. Vainshtein proposed that the vDVZ black hole may transit to the hairy black hole. However, the discontinuity can be evaded by taking into account non- paper [29] showed numerically that such a hairy black hole linear mass terms [21]. Therefore, the nonlinear bigravity does not exist unless the coupling constants satisfy a special theory may have no discontinuity in the massless limit. In condition. One may wonder what we will find in the final fact, GR can be recovered within the Vainshtein radius in stage of gravitational collapse of a compact relativistic star. the weak-field approximation if the coupling constants One may also ask whether there exists a maximum mass of satisfy an appropriate condition (see later) [22–24]. neutron star, beyond which no neutron star cannot exist. Furthermore their analysis can be generalized into the The standard picture in GR is that a star collapses to a cosmological background [17] in which GR is recovered by black hole when the mass exceeds the maximum value. the mechanism similar to that in the ghost condensation However, in bigravity, although there exists a Newtonian
*katsuki‑[email protected] † [email protected] 1There can be a hairy black hole solution in the scalar extended ‡ [email protected] massive gravity [30,31].
2470-0010=2016=93(6)=064054(21) 064054-1 © 2016 American Physical Society KATSUKI AOKI, KEI-ICHI MAEDA, and MAKOTO TANABE PHYSICAL REVIEW D 93, 064054 (2016) κ2 8πG star solution in the weak gravitational field, no stable black f ¼ are the corresponding gravitational constants, hole solution has been found for generic coupling constants. 2 2 2 while κ is defined by κ ¼ κg þ κ . We assume that the In order to investigate what happens when a star is compact f matter action S½m is divided into two parts: and relativistic and then the gravitational interaction becomes very strong, we study a relativistic star in the bigravity theory. ½m ½m S½m ðg; f; ψ ; ψ Þ¼S ðg; ψ ÞþS ðf; ψ Þ; ð2:2Þ A little attention has so far been paid to a relativistic star in the g f g g f f bigravity. Hence, as a first step, we analyze a star solution i.e., matter fields ψ g and ψ f are coupled only to the g-metric with a relativistic effect, and discuss how such a relativistic and to the f-metric, respectively. We call ψ and ψ twin star behaves in the limit of strong gravity. g f matter fluids [38]. In the text, we assume that only a g-matter field exists The ghost-free interaction term between the two metrics and spacetime is asymptotically flat. We then classify the is given by coupling constants into two classes: Class [I] and Class [II]. For Class [I], we find an example of breaking Vainshtein X4 screening mechanism due to the relativistic effect. The Uðg; fÞ¼ bkUkðγÞ; ð2:3Þ static star solution is found when the pressure of the star is k¼0 sufficiently small, while the star solution disappears when where fbkgðk ¼ 0–4Þ are coupling constants and the 4 × 4 the pressure is larger than a critical value. Therefore, in μ matrix γ γ ν is defined by Class [I], the maximum mass of the neutron star in ¼ð Þ bigravity is constrained stronger than one in GR. On the μ ρ μρ γ ργ ν ¼ g fρν; ð2:4Þ other hand, there is no critical value of the pressure for Class [II]. The result of GR is reproduced even in the strong while Uk are the elementary symmetric polynomials of the gravitational field. eigenvalues of the matrix γ, defined explicitly in [11,12]. The paper is organized as follows. The Hassan-Rosen Taking the variation of the action with respect to gμν and bigravity model is introduced in Sec. II. In Sec. III,we fμν, we find two sets of the Einstein equations: derive the basic equations in bidiagonal ansatz of the static μ 2 ½γ μ ½m μ and spherically symmetric spacetime. Taking the limit of G ν ¼ κgðT ν þ T νÞ; ð2:5Þ massless graviton, we discuss behaviors of the solutions Gμ κ2 T ½γ μ T ½m μ deep inside the Vainshtein radius in Sec. IV. We find that ν ¼ fð ν þ νÞ; ð2:6Þ the existence of a neutron star solution is restricted depend- μ Gμ ing on the coupling constants. In Sec. V, we numerically where G ν and ν are the Einstein tensors for gμν and fμν, ½γ μ solve the basic equations without taking the massless limit, respectively. The γ-“energy-momentum” tensors T ν and ½γ μ and confirm that the previous solutions with massless limit T ν are obtained by the variation of the interaction term approximation are valid if the Compton wavelength of the with respect to gμν and fμν, respectively, taking the form graviton mass is sufficiently large compared to the typical [11,12] radius of the star. We summarize our results and give some pffiffiffiffiffiffi 2 −g 2 remarks in Sec. VI. In Appendix A, we summarize the ½γ μ m μ μ ½γ μ m μ T ν ¼ ðτ ν − Uδ νÞ; T ν ¼ − pffiffiffiffiffiffi τ ν; parameter constraint from the existence of a Newtonian κ2 −f κ2 star. In Appendix B, introducing a cosmological constant ð2:7Þ and f-matter field, we discuss solutions with asymptoti- cally nonflat geometry. In Appendix C, we detail the case where beyond the critical value of the pressure for Class [I], in 4 4− which we find a singular behavior. X Xn τμ ≡ −1 nþ1 γn μ U ν ð Þ ð Þ ν bnþk k: ð2:8Þ n¼1 k¼0 II. HASSAN-ROSEN BIGRAVITY MODEL ½m μ ½m μ The matter energy-momentum tensors T ν and T ν We focus on the ghost-free bigravity theory proposed by are given by the variation of matter actions. They are Hassan and Rosen [3], whose action is given by assumed to be conserved individually as Z Z 1 pffiffiffiffiffiffi 1 pffiffiffiffiffiffi S d4x −gR g d4x −fR f ðgÞ ðfÞ ¼ 2κ2 ð Þþ2κ2 ð Þ ∇ ½m μ 0 ∇ T ½m μ 0 g f μT ν ¼ ; μ ν ¼ ; ð2:9Þ 2 Z m pffiffiffiffiffiffi ðgÞ ðfÞ − d4x −gUðg; fÞþS½m ðg; f; ψ ; ψ Þ; ð2:1Þ κ2 g f where ∇μ and ∇μ are covariant derivatives with respect to gμν and fμν. From the contracted Bianchi identities for (2.5) where gμν and fμν are two dynamical metrics, and RðgÞ and and (2.6), the conservation of the γ-energy-momenta is also 2 RðfÞ are their Ricci scalars. The parameters κg ¼ 8πG and guaranteed as
064054-2 RELATIVISTIC STARS IN BIGRAVITY THEORY PHYSICAL REVIEW D 93, 064054 (2016) ðgÞ ðfÞ Since two metrics are proportional, the Einstein equations ∇ ½γ μ 0 ∇ T ½γ μ 0 μT ν ¼ ; μ ν ¼ : ð2:10Þ lead
These equations give nontrivial constraints on solutions, Λ ðKÞ¼K2Λ ðKÞ; ð3:4Þ which are absent in GR. g f which determines K as one of the real roots of this quartic III. STATIC AND SPHERICALLY equation. In the text, we focus on asymptotically homo- SYMMETRIC SPACETIMES thetic solutions, i.e., we assume the boundary condition To find a nontrivial static and spherically symmetric → 1 regular solution, we assume two metrics are bidiagonal in Ng=Nf;Fg=Fg;rg=rf : ð3:5Þ the same coordinate system. Thus, we consider the follow- ing metric forms: Solutions with other asymptotic geometrical structure will be discussed in Appendix B. r02 We introduce new variable μ defined by 2 − 2 2 g 2 2 Ω2 dsg ¼ Ngdt þ 2 dr þ rgd ; ð3:1Þ Fg r μ ≔ f − 1 ð3:6Þ r02 rg 2 2 − 2 2 f 2 2 Ω2 dsf ¼ K Nfdt þ 2 dr þ rfd ; ð3:2Þ Ff with μ > −1, which determines the relation between two radial coordinates rg and rf. From the boundary condition, where the variables fNg;Fg;rg;Nf;Ff;rfg are functions of μ should approach zero at infinity. a radial coordinate r, and a prime denotes the derivative with Introducing new parameters as respect to r. The ansatz has two residual gauge freedoms: → ~ One is a rescaling of time coordinate (t t ¼ ct with c m2κ2 2 ≔ g 2 2 3 being a constant), and the other is redefinition of the radial mg κ2 ðb1K þ b2K þ b3K Þ; ð3:7Þ coordinate (r → r~ðrÞ). The proportional constant factor K is introduced just for convenience. When Ng=Nf ¼ 2κ2 2 m f 2 3 F =F ¼ r =r ¼ 1, g- and f-spacetimes are homothetic m ≔ b1K 2b2K b3K ; : g f g f f 2κ2 ð þ þ Þ ð3 8Þ and the γ energy-momentum tensors turn to be just K “effective” cosmological terms, where the g- and f- 2 3 β ≔ b2K þ b3K cosmological constants are given by 2 2 3 ; ð3:9Þ b1K þ 2b2K þ b3K κ2 Λ 2 g 3 3 2 3 3 gðKÞ¼m 2 ðb0 þ b1K þ b2K þ b3K Þ; b3K κ β3 ≔ ; : 2 2 3 ð3 10Þ κ2 b1K þ b2K þ b3K 2 f −1 −2 −3 Λ ðKÞ¼m ðb4 þ 3b3K þ 3b2K þ b1K Þ: ð3:3Þ f κ2 the Einstein equations are reduced to
2F F0 F2 − 1 r0 F g g g −κ2ρ − Λ 2 1 2 β − 1 μ β − β μ2 − 1 2β μ β μ2 f g 0 þ 2 ¼ g g g þ mg þ ð 2 Þ þð 3 2Þ ð þ 2 þ 3 Þ 0 ; ð3:11Þ rgrg rg rgFf 2F2N0 F2 − 1 N g g g κ2 − Λ 2 1 2 β − 1 μ β − β μ2 − 1 2β μ β μ2 f 0 þ 2 ¼ gPg g þ mg þ ð 2 Þ þð 3 2Þ ð þ 2 þ 3 Þ ; ð3:12Þ rgrgNg rg Ng 2 0 2 −1 2 0 FfFf Ff 2 2 mf 2 2 rgFf þ ¼ −K κ ρ −Λ þ 1þ2ð1þβ2Þμþð1þβ2 þβ3Þμ −ð1þ2β2μþβ3μ Þ ; ð3:13Þ 0 2 f f g 1 μ 2 0 rfrf rf ð þ Þ rf Fg 2 2 0 2 −1 2 FfNf Ff 2 2 mf 2 2 Ng þ ¼ K κ P −Λ þ 1þ2ð1þβ2Þμþð1þβ2 þβ3Þμ −ð1þ2β2μþβ3μ Þ : ð3:14Þ 0 2 f f g 1 μ 2 rfrf Nf rf ð þ Þ Nf
We have two more Einstein equations, which are automatically satisfied since we have two Bianchi identities for gμν and fμν.
064054-3 KATSUKI AOKI, KEI-ICHI MAEDA, and MAKOTO TANABE PHYSICAL REVIEW D 93, 064054 (2016) In the original Lagrangian, we have six unfixed coupling Note that the proportional factor K is not necessary to be κ 2 2 constants f f;big, where m is not independent because it is unity. Since K appears only in the form of K ρf and K Pf, just a normalization factor of bi. In this paper, we use six however, unless f matter exists, the basic equations are free different combinations of those constants; fmg;mf; Λg; from the value of K. In what follows, we assume that there K; β2; β3g, instead of fκf;big, because the behaviors of the is no f-matter just for simplicity. The f-matter effect on the solutions within the Vainshtein radius are characterized by solution will be discussed in Appendix B3. β2 and β3 as we will see later. The original coupling κ Λ β β constants f f;big are found from fmg;mf; g;K; 2; 3g. IV. REGULAR COMPACT OBJECTS: The energy-momentum conservation laws of twin mat- MASSLESS LIMIT ters give Before we present our numerical solutions, we shall 0 Ng discuss some analytic features of a compact object. 0 ρ 0 6 Pg þ ð g þ PgÞ¼ ; ð3:15Þ The radius of a neutron star is about 10 cm, while the Ng Vainshtein radius is given typically by 1020 cm when the 0 Compton wavelength of the graviton mass is the cosmo- Nf P0 þ ðρ þ P Þ¼0; ð3:16Þ logical scale (m−1 ∼ 1028 cm). The magnitude of the f N f f eff f interaction term, which is proportional to the graviton mass squared, is much smaller than the density of a neutron where we assume that twin matters are perfect fluids. The star. Hence, the interaction term seems not to affect the energy-momentum conservation laws of the interaction structure of a neutron star. If we ignore the interaction terms terms, which are equivalent to the Bianchi identities, reduce in the Einstein equations (2.5) and (2.6) [or Eqs. (3.11)– to one constraint equation: (3.14)], we just find two independent Einstein equations in GR. Then both spacetimes are given approximately by GR 2ðF − F ÞðN ð1 − β2 þðβ2 − β3ÞμÞþN ðβ2 þ β3μÞÞ g f g f solutions, which we can solve easily. In bigravity theory, 0 0 2 FgNg FfNf however, we have one additional nontrivial constraint r 1 2β2μ β3μ − 0: 3:17 þ gð þ þ Þ 0 0 ¼ ð Þ equation (2.10) [or (3.18) for a static and spherically rg rf symmetric case] even in the massless limit. This constraint Substituting the Einstein equations (3.12) and (3.14) into will restrict the existence of the solutions. In this section, Eq. (3.17), we obtain one algebraic equation: we consider a compact object in this massless limit. Note that, in this massless limit, the effective action to determine the Stückelberg variable μ is given by C½Ng;Nf;Fg;Ff; μ;Pg;Pf ¼0: ð3:18Þ μ ρ ρ Now we have nine variables Ng;Nf;Fg;Ff; ; g;Pg; f Z and P , and six ordinary differential equations (3.11)– pffiffiffiffiffiffi f S ¼ −Λ4 d4x −gUðμ; g ;f Þ; ð4:1Þ (3.14), (3.15), (3.16) and one algebraic equation (3.18) with eff 2 GR GR two equations of state Pg ¼ PgðρgÞ and Pf ¼ PfðρfÞ.In order to solve those equations numerically, we first take the pffiffiffiffiffiffiffiffiffi derivative of (3.18), and then find seven first-order ordinary Λ κ where 2 ¼ m= , and gGR and fGR are solutions in GR differential equations: which act as external forces to the Stückelberg field.2 This effective action is indeed the same as the noncompact dX ¼ F X½Ng;Nf;Fg;Ff; μ; ρg;Pg; ρf;Pf;r ; ð3:19Þ nonlinear sigma model proposed by [39]. As we will see, dr the massless limit approximation is valid deep inside the dr dμ Vainshtein radius. It implies that, inside the Vainshtain f ¼ r þ μ þ 1 radius, the noncompact nonlinear sigma model with a dr dr curved metric is obtained as the effective theory for the μ ρ ρ ¼ J½Ng;Nf;Fg;Ff; ; g;Pg; f;Pf;r ; ð3:20Þ Stückelberg field. We analyze two models: one is a simple toy model of a F where X ¼fNg;Nf;Fg;Ff;Pg;Pfg, and X and J do not relativistic star, i.e., a uniform-density star, and the other is contain any derivatives. Here we have fixed the radial a more realistic polytropic star with an appropriate equation coordinate as rg ¼ r by use of the gauge freedom. We solve of state for a neutron star. these differential equations from the center of a star (r ¼ 0). In order to guarantee that the above setup gives a correct 2If both metrics are Minkowski ones, this action becomes a solution of our system, we have to impose the constraint total divergence term. Hence it is necessary that one of them is at (3.18) on the variables at the center. least a curved metric.
064054-4 RELATIVISTIC STARS IN BIGRAVITY THEORY PHYSICAL REVIEW D 93, 064054 (2016) 1 2 A. The boundary condition at “infinity” 2GM⋆ = F ¼ 1 − ; ð4:7Þ in the massless limit g r The boundary condition at spatial infinity, which is 2 0 outside of the Vainshtein radius, is given by Eq. (3.5). Ngð Þ Ng ¼ FgðrÞ; ð4:8Þ Since the radius of a neutron star is much smaller than the 3FgðR⋆Þ − 1 Vainshtein radius, there exists the weak gravity region even inside of the Vainshtein radius. We then introduce an F ¼ 1;N¼ N ð0Þ; ð4:9Þ ≪ ≪ f f f intermediate scale RI with R⋆ RI RV, where R⋆ and RV are the radius of a star and the Vainshtein radius, where R⋆ and respectively. The space inside the Vainshtein radius can be divided into two regions: the region deep inside the 4π 3 M⋆ ≔ ρ R⋆ ð4:10Þ Vainshtein radius (r
064054-5 KATSUKI AOKI, KEI-ICHI MAEDA, and MAKOTO TANABE PHYSICAL REVIEW D 93, 064054 (2016)
On the other hand, for cases (2) and (3), the real root μ0 always exists for any value of Pgð0Þ=ρg. Furthermore, when we take into account the finiteness of the graviton mass, even if it is very small, we find an additional constraint on the coupling constants fβ2; β3g from the existence of nonrelativistic star with asymptoti- cally homothetic spacetime [24] (see also Appendix A). Since case (2) is completely excluded, we find two classes of the coupling parameters, which provide a relativistic star with asymptotically homothetic spacetime, 3 as follows : ffiffiffiffiffi β −pβ β 0 Class [I]: 2 p<ffiffiffiffiffi 3 andpdffiffiffiffiffi1 þ d2 2 < , Class [II]: − β3 ≤ β2 ≲ β3 and d1 þ d2β2 < 0, where d1 and d2 are some complicated functions of β3, which are defined by (A10) and (A12) in Appendix A, respectively. Assuming β3 > 1, which is necessary for the existence of asymptotically homothetic solution, we show the ranges of Class [I] and Class [II] with this constraint by the shaded light-red region and the hatched light-blue region in Fig. 1, FIG. 1. The constraint on the coupling constants from the existence of the static spherically symmetric solution in bigravity respectively. For its outside (the white region), there exists where we assume m ¼ m and Λ ¼ 0. The parameters are neither nonrelativistic star nor relativistic one. g f g μ classified into two classes: Class [I] (the light-red region) and Even if a real 0 exists, we may not find a regular Class [II] (the hatched light-blue region). Although there is a μ 0 ≤ ∞ solution of ðrÞ in the whole coordinate range ( r< ) regular solution with Vainshtein screening mechanism in the because the real root of (3.18) may disappear at some finite weak gravitational approximation for both classes, the difference radius. In Figs. 2 and 3, we present some examples of Class appears in the case of relativistic star. For Class [I], the maximum [I] and Class [II], respectively. As the example of Class [I], mass of a neutron star is constrained stronger than the case of GR, we choose while the star exists as in the case of GR for Class [II]. The contour lines of maximum mass are presented in the figure, where pffiffiffiffiffi the maximum mass increases as fβ2; β3g are close to β2 ¼ − β3. mg ¼ mf; β2 ¼ −3; β3 ¼ 3; ð4:14Þ while for Class [II], the parameters are chosen as homothetic spacetime, while branch B solutions are not asymptotically flat. mg ¼ mf; β2 ¼ 1; β3 ¼ 3; ð4:15Þ For 1=15
064054-6 RELATIVISTIC STARS IN BIGRAVITY THEORY PHYSICAL REVIEW D 93, 064054 (2016)
FIG. 2. We set mg ¼ mf; β2 ¼ −3; β3 ¼ 3. The top and bottom figures denote the cases of Pgð0Þ=ρg < 1=15 and of Pgð0Þ=ρg > 1=3, respectively. When 1=3 >Pgð0Þ=ρg > 1=15, μ there is no real root of 0. Although there exists a real root of FIG. 3. We set mg ¼ mf, and (a) β2 ¼ 1, β3 ¼ 3, and μ0 for Pgð0Þ=ρg > 1=3, the solutions are disconnected between (b) β2 ¼ −2, β3 ¼ 4. We choose the central pressure as ≲ ≫ 2 the region of r R⋆ and that of r R⋆. As a result, there exists a Pgð0Þ=ρg ¼ 0.01, 0.1, 1, 10. In (a) (β2 − β3 < 0), there are 0 ρ 1 15 ≈ 0 06667 relativistic star for Pgð Þ= g < = . . two branches, and these do not connect for a large pressure of 2 the star. For (b) (β − β3 ¼ 0), there are a nontrivial root (branch 2 pffiffiffiffiffi A) as well as a trivial root μ 1= β3 (branch B). Although these When Pgð0Þ=ρg becomes larger, i.e., if Pgð0Þ=ρg > 1=3, ¼ two roots intersect beyond a critical pressure, there is a regular we again find a real μ0, but there exists no regular μðrÞ for star beyond it in branch A. the whole range of r. μðrÞ exists in two separated regions; one is smaller than some finite radius ( 064054-7 KATSUKI AOKI, KEI-ICHI MAEDA, and MAKOTO TANABE PHYSICAL REVIEW D 93, 064054 (2016) GM⋆ GM⋆ ≔ sup ≈ 0.23; ð4:19Þ R⋆ ub Class ½I R⋆ max which is realized for β2 ≃ −1.48, β3 ≃ 2.19. While in Class [II], any coupling constants give the same maximum value of the compactness, that is, GM⋆ 4 ¼ 9 ; ð4:20Þ R⋆ max which is obtained from the existence condition for a regular interior solution in GR because there is no additional constraint in this class. The upper bound of the compactness in Class [I] is almost the same as the observed value (e.g., the compact- ness is about 0.3 when a radius of a two solar mass neutron star is 10 km, while it is about 0.21 for a two solar mass star with a radius of 14 km [40–42].). In order to give a stringent constraint on the theory by observations, we have to analyze a more realistic star, which will be discussed in the next subsection. C. Polytropic star FIG. 4. ρc-M⋆ and R⋆-M⋆ relations for neutron stars with the polytropic equation of state (4.21). The black solid lines are Giving a more realistic equation of state, we present a obtained in GR or in Class [II]. The maximum mass in Class [I], neutron star solution in the bigravity theory. We then which is shown by the red dots with β2 ¼ −3; β3 ¼ 3 and discuss its mass and radius in order to give a constraint β2 ¼ −1.48; β3 ¼ 2.19, depends on the coupling constants. on the theory or the coupling constants by comparing them with observed values. We assume a simple polytropic-type equation of state, maximum mass of neutron star in GR (see Fig. 4). Hence this limit of ρc provides the maximum mass of a P ¼ Kρ2; ð4:21Þ neutron star in Class [I], which is much smaller than that in GR (or in Class [II]). In Fig. 1, the maximum masses in where we set K ¼ 1.5 × 105 [cgs]. In the massless limit of Class [I] are shown by the contour lines. The maximum β β the graviton, we have two decoupled Einstein equations. mass is largerffiffiffiffiffi as the parameters f 2; 3g come close to β −pβ Then the f-metric is given by the Minkowski spacetime 2 ¼ 3. The upper bound of the maximum mass in 1 72 β ≃ because there is no f-matter, For g-spacetime, we have the Class [I] is at most . M⊙, which is realized at 2 −1 48 β ≃ 2 19 same neutron star solution as that in GR. We present ρ -M⋆ . and 3 . . Hence the maximum mass in Class c 2 and R⋆-M⋆ relations in Fig. 4, where ρ ¼ ρ ð0Þ is the [I] does not reach M⊙, which may be inconsistent with the c g 2 – central density. We find that the maximum mass of a existence of the M⊙ neutron star [43 46]. One might find a 2M⊙ neutron star in Class [I] if we modify the equation of neutron star is about 2M⊙, where M⊙ is the solar mass, for the above equation of state. This result is obtained in GR state, but it will give a strong constraint on the coupling but also it is the case for Class [II] in bigravity because we constants in the theory. always find the regular solution for μðrÞ in the whole As for the compactness, we find coordinate range (0 ≤ r<∞). We show some examples for the same coupling constants (4.15) with several values ρ GM⋆ of the central density c in Fig. 5. ¼ 0.18 for Class½1 ; R⋆ However, for Class [I], we find the additional constraint ub to find the regular μ r as we expect from the result in the ð Þ GM⋆ 0 31 previous subsection. We also present some examples of ¼ . for Class½II : ð4:22Þ R⋆ max μðrÞ for the same coupling constants (4.14) with several values of ρc in Fig. 5. This figure shows there is no regular solution of μðrÞ in the whole region if the density ρc is Although both values are so far consistent with observa- larger than 2.8 × 1014 g=cm3. This upper limit of the tions, the coupling constants in Class [I] may be restricted density does not reach the central density with the again because the above value is just the upper bound. 064054-8 RELATIVISTIC STARS IN BIGRAVITY THEORY PHYSICAL REVIEW D 93, 064054 (2016) Here, by use of the freedom of time coordinate rescaling, 4 we choose Ngð0Þ¼1 without loss of generality. We determine the values of variables at r ¼ δr by using up to second order of (5.1). In this section, we focus only on the branch A solution since we are interested in an asymptotically flat spacetime. We will give some remarks for the branch B, which gives an asymptotically AdS spacetime, in Appendix B. A. A uniform density star We first discuss a uniform density star, i.e., ρg ¼ constant. The dimensionless parameters characteriz- ing the star are κ2ρ 2 ð0Þ 0 ρ g g=meff;Pg ð Þ= g; ð5:4Þ where we have defined 2 2 2 meff ¼ mg þ mf; ð5:5Þ which gives the effective graviton mass on the homothetic spacetime. The first parameter in (5.4) is evaluated as κ2ρ −1 2 β −3 β 3 g g 6GM⋆ m FIG. 5. We set mg ¼ mf, and (a) 2 ¼ ; 3 ¼ (Class [I]) ¼ eff ; ð5:6Þ β −1 β 3 ρ 1 71 2 and (b) 2 ¼ ; 3 ¼ (Class [II]). We choose c ¼ . × meff R⋆ R⋆ 1014 g=cm3 (red solid curves), 3.35 × 1014 g=cm3 (blue dashed 14 3 curves) and 18.9 × 10 g=cm (green dotted curves), whose star which is much larger than unity because m−1 is the 0 6 1 0 2 0 eff masses are . M⊙, . M⊙ and . M⊙, respectively. Compton wavelength of the graviton and then it must be a cosmological scale. V. REGULAR COMPACT OBJECTS: Once the parameters (5.4) are given, the proper value of NUMERICAL RESULTS μð0Þ is determined by a shooting method to adjust the correct boundary condition (3.5) at infinity as well as the In this section, we numerically solve the basic equations asymptotic flatness. Then all coefficients in Eq. (5.1) are under the metric ansatz (3.1) and (3.2) with a g-matter field. fixed by this μð0Þ from the expanded basic equations order We find a relativistic star solution and confirm the previous by order, results obtained in the massless limit when the graviton We use μ0 as the center value of μð0Þ in the case of mass is sufficiently small. massless limit. When the value of the graviton mass is We numerically integrate Eqs. (3.15), (3.19) and (3.20) sufficiently small, the proper value of μð0Þ is close to μ0. outwards from the center r ¼ 0. The constraint equa- Hence, we start to search for μð0Þ near μ0 to find a regular tion (3.18) is used to evaluate the boundary values at the solution with the correct boundary condition. center. Since it must be satisfied in the region of r>0 too, we use this constraint to check the accuracy of our 4 ∞ ≠ 1 numerical solutions in r>0. Although it gives Ngð Þ , if we wish to find the boundary condition N ð∞Þ¼1, we redefine new lapse functions as Since the equations are seemingly singular at r ¼ 0,We g start our calculations from r ¼ 0 þ δr with δr ≪ 1.All ~ NgðrÞ ~ NfðrÞ variables are expanded around r ¼ 0 as NgðrÞ¼ ; NfðrÞ¼ ð5:2Þ Ngð∞Þ Ngð∞Þ X 1 and the new time coordinate as X ¼ XðnÞð0Þrn; ð5:1Þ n! n¼0 ~ t ¼ Ngð∞Þt: ð5:3Þ ðnÞ ~ ~ ~ where X ð0Þ is the nth derivative of the variable X New metrics defined by Ng, Nf and t satisfy the boundary ~ ∞ 1 at r ¼ 0. condition Ngð Þ¼ at infinity. 064054-9 KATSUKI AOKI, KEI-ICHI MAEDA, and MAKOTO TANABE PHYSICAL REVIEW D 93, 064054 (2016) To check the boundary conditions at infinity, we evaluate μ the eigenvalues of γ ν, i.e., KNf λ0 ≔ ; ð5:7Þ Ng 0 Krf=Ff λ1 ≔ ; ð5:8Þ 1=Fg Krf λ2 ¼ λ3 ≔ ¼ Kð1 þ μÞ: ð5:9Þ r If all eigenvalues approach the same constant K as r → ∞, the solution is asymptotically homothetic. Then the γ energy-momentum tensor will become a “cosmological” constant (Λg) term at infinity. We find our solution with an asymptotic flatness, if Λg ¼ 0, which we have assumed for our coupling constants. 1. Class [I] As an example in Class [I], we choose the same coupling constants as before, i.e., Λg ¼ 0;mg ¼ mf; β3 ¼ −3; β4 ¼ 3: ð5:10Þ FIG. 6. A typical solution for the branch A. We set The branch A solution approaches an asymptotically flat −2 Pgð0Þ=ρg ¼ 5 × 10 . The shooting parameter is tuned to be homothetic spacetime. In Fig. 6, we show a numerical μð0Þ¼0.03093. The vertical bar represents the star surface solution by setting κ2ρ =m2 ¼ 2.5 × 105,5 for which the −1 0 00141288 g g eff (R⋆=meff ¼ . ). typical value of the Vainshtein radius is given by 2 1=3 R ≔ GM⋆=m ∼ 30R⋆: 5:11 2 V ð effÞ ð Þ If we choose the larger value of the parameter as κgρg= 2 7 m ¼2.5×10 , the solution exists for Pgð0Þ=ρg > 0.0666, GR is recovered within the Vainshtein radius. eff λ which is closer to the value in the massless limit. Hence, we We note 1 is discontinuous at the star surface R⋆.Itis expect that the massless limit approximation is valid for the because the discontinuity of the matter distribution leads realistic value κ2ρ =m2 ∼ 1043. the discontinuity of r0 as seen in Eq. (3.20). This g g eff f If the solution exists, the inner structure of the star as discontinuity disappears when we discuss a continuous well as the gravitational field are restored to the result of matter distribution such as a polytropic star (4.21) as shown GR because of the Vainshtein mechanism. We find the in Fig. 8. differences between our numerical solution and the semi- 0 ρ Changing the central value of the pressure Pgð Þ= g,we analytic solution in the massless limit are very small as find the solution disappears for P 0 =ρ > 0.0665.Itis gð Þ g shown in one example of the pressure Pg in Fig. 7. This fact consistent with the argument in the massless limit, in which also confirms the validity of the massless limit approxi- 0 ρ 1 15 ≈ 0 06667 the critical value is given by Pgð Þ= g ¼ = . . mation if the graviton mass is sufficiently small. We Hence even in the case with a finite graviton mass, there conclude that the bigravity for Class [I] cannot reproduce exists a critical value of the pressure beyond which a the result in GR beyond the critical value of Pgð0Þ=ρg. regular star solution does not exist. 2. Class [II] 5This value is too small for a realistic neutron star with a As an example in Class [II], we choose one of the massive graviton responsible for the present accelerating expan- κ2ρ 2 ∼ 1043 previous coupling constants, i.e., sion of the Universe, for which we have g g=meff . However, because of the technical reason for numerical calcu- Λ 0 β 1 β 3 lation, we choose the above value. For the realistic value, we g ¼ ;mg ¼ mf; 3 ¼ ; 3 ¼ ð5:12Þ expect that the solution may be closer to the case of the massless limit. and we set 064054-10 RELATIVISTIC STARS IN BIGRAVITY THEORY PHYSICAL REVIEW D 93, 064054 (2016) ½m¼0 ½m¼0 FIG. 7. jPg − Pg j=Pg where Pg is the numerical solution ½m¼0 with a finite mass and Pg is the solution in massless limit. We 0 ρ 5 10−2 κ2ρ 2 2 5 105 set Pgð Þ= g ¼ × and (5.10) with g g=meff ¼ . × κ2ρ 2 2 5 107 (the red solid curve), g g=meff ¼ . × (the blue dashed κ2ρ 2 2 5 105 curve) and (5.12) with g g=meff ¼ . × (the green dotted ½m¼0 ½m¼0 curve). We note Pg − Pg > 0 for (5.10), while Pg − Pg < 0 for (5.12). κ2ρ 2 2 5 105 g g=meff ¼ . × : ð5:13Þ In this case, we can find a regular star for any values of Pgð0Þ. The solution is almost the same as the massless limit (or GR) as shown in Fig. 7. We conclude that in the bigravity theory in Class [II] the results in GR are recovered ρ 1 71 1014 3 −1 104 and the Vainshtein mechanism holds even in a strong FIG. 8. We set c ¼ . × g=cm and meff ¼ km, 0 601 gravity limit. for which the mass of the neutron star is . M⊙. The shooting parameter is tuned to be μð0Þ¼−0.13334. The vertical bar represents the star surface (R⋆ ¼ 17.7 km). B. Polytropic star For a neutron star with a realistic equation of state, we can also confirm the above results, i.e. the massless limit is confirm that the massless limit solution is a good approxi- valid. Here we again assume the polytropic equation of mation for the sufficiently small graviton mass. state (4.21). One typical example of the solutions in Class [I] is VI. CONCLUDING REMARKS shown in Fig. 8, where we choose the coupling constants as (5.10) and Assuming static and spherically symmetric spacetimes, We have presented a relativistic star solution in the ρ 1 71 1014 3 −1 104 c ¼ . × g=cm ;meff ¼ km: ð5:14Þ bigravity theory. For simplicity, we have considered only g-matter fluid and given only asymptotically flat solutions We find a neutron star solution with in the text. Some solutions with the other conditions are discussed in Appendix B. M⋆ ¼ 0.601M⊙;R⋆ ¼ 17.7 km; ð5:15Þ First we obtain the solutions under the massless limit approximation in Sec. IV. Then, by solving the basic which is the same as those in the massless limit. Our equations numerically without the approximation in numerical calculation shows that increasing the central Sec. V, we confirm such an approximation is valid since density ρc, the solution exists only for M⋆ ≲ 0.882M⊙ for the graviton mass, if it exists, must be sufficiently small. the coupling constants (5.10). We have obtained M⋆ ≲ We find that the coupling constants are classified into 0.886M⊙ in the massless limit. If we choose the larger two classes: Class [I] and Class [II]. For both classes, the value of the Compton wavelength of the graviton as Vainshtein screening is found in the weak gravitational −1 105 meff ¼ km, the mass upper limit increases as field. However, when we take into account a relativistic M⋆ ≲ 0.884M⊙, which is closer to the value in the massless effect, the Vainshtein screening mechanism may not work limit. in some strong gravity regime in Class [I]. In fact, to find a For Class [II], we always find the same solution as that in regular function of μðrÞ in Class [I], the central pressure is GR. As a result, as the case of a uniform-density star, we constrained, and as a result, the maximum mass is much 064054-11 KATSUKI AOKI, KEI-ICHI MAEDA, and MAKOTO TANABE PHYSICAL REVIEW D 93, 064054 (2016) TABLE I. The maximum masses and the maximum compactnesses for Class [I] and Class [II]. For Class [I], the maximum values depend on the coupling constants. Then we only show the upper bounds which are realized for β2 ≃ −1.48, β3 ≃ 2.19. Class Class [I] Class [II] Coupling β3 > 1 & d1 þ d2β2 < 0 pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi Constants β2 < − β3 − β3 < β2 ≲ β3 Equation of state Uniform density Polytrope Uniform density Polytrope Mass Mub ¼ 1.72M⊙ Mmax ¼ 2.03M⊙ Compactness M⋆ 0.23 M⋆ 0.18 M⋆ 0.44 M⋆ 0.31 R⋆ jub ¼ R⋆ jub ¼ R⋆ jmax ¼ R⋆ jmax ¼ 2 smaller than that in GR as shown in Fig. 1. Beyond this 2c3 þ 3c4 > 0; ð6:3Þ maximum mass, the Vainshtein mechanism does not work well since the GR solution is not obtained. which excludes the possibility of (6.1). Hence, Class [II] On the other hand, there is no additional constraint for cannot admit the de Sitter solution as well as the Class [II], and the structure of star as well as the Minkowski solution as vacuum solutions. To obtain a de gravitational field are restored to those in GR for the Sitter solution in Class [II], we should first assume the de expected small graviton mass. The Vainshtein screening Sitter spacetime is a vacuum solution, which is similar to mechanism works well in Class [II]. introducing a cosmological constant by hand in GR. In Table I, we summarize our results. In this paper, we have assumed that both static g- and f- The result suggests that Class [II] is favored from the spacetimes are static with respect to the same time existence condition of a neutron star. As the necessary coordinate t, and the Stückelberg field μ is also static. condition of Class [II], the parameters should satisfy However, there is a possibility such that the existence of the critical value in Class [I] might be caused by the above 2 β2 − β3 ≤ 0; ð6:1Þ simple ansatz. The static ansatz of the Stückelberg field may not be necessary to obtain an (approximate) static spacetime. In fact, in the case of cosmology, a homo- as shown in Fig. 1. However, those parameters should geneous configuration of the Stückelberg field leads to an happen to satisfy instability, while the inclusion of an inhomogeneity in the Stückelberg field gives a stable solution, which describes β2 − β 0 2 3 > ; ð6:2Þ an (approximate) homogeneous spacetime due to the Vainshtein screening [17]. Hence, to draw a final con- from the cosmological point of view, which constraint clusion about the existence of a massive neutron star (and comes from finding a stable solution in the early Universe also a black hole solution), relaxing the static ansatz of the in bigravity [17]. There is no intersection of the parameters Stückelberg field, we should extend our analysis to the because the boundaries of Class [II] and of the cosmo- spacetime with dynamical Stückelberg fields, which we logical constraint coincide exactly. If we take the param- leave for our future work. eters in Class [I] from the cosmological constraint, the equation of state of the star will be strongly constrained to ACKNOWLEDGMENTS find a two solar mass neutron star. Conversely, if we assume Class [II] from the astrophysical point of view, the We would like to thank Kotaro Fujisawa, Takashi problem of ghost or gradient instability may reappear in the Nakamura, and Mikhail S. Volkov for useful discussions and comments. This work was supported in part by Grants- early Universe. in-Aid from the Scientific Research Fund of the Japan There is another problem in Class [II] parameters. An Society for the Promotion of Science (No. 25400276 and advantage of the bigravity theory is, even if we assume the No. 15J05540). Minkowski solution as a vacuum solution, we can find a de Sitter solution as another vacuum solution. The cosmo- APPENDIX A: WEAK GRAVITY logical constant is typically given by an order of magnitude APPROXIMATION of the graviton mass, which implies that the present acceleration of the Universe can be explained by the In this Appendix, we discuss the constraint on the graviton mass. For this case, as discussed in [11], fbig’s coupling constants in order to find a successful are given by two coupling constants c3 and c4. The Vainshtein screening mechanism in a weak gravity system. existence condition of the de Sitter solution as well as We assumed only g-matter field for simplicity, and chose the Minkowski solution yields the gauge rg ¼ r. We define new variables by 064054-12 RELATIVISTIC STARS IN BIGRAVITY THEORY PHYSICAL REVIEW D 93, 064054 (2016) Φ −Ψ ffiffiffiffi ffiffiffiffi N ¼ e g ;F¼ e g ; ðA1Þ C 2 j p þ CΛj p g g m μ¼−1= β3 μ¼−1= β3 pffiffiffiffiffi pffiffiffiffiffi 12GM⋆ Φ −Ψ 2 f f ≈− ð1 − 1= β3Þ β3δμ: ðA8Þ Nf ¼ e ;Ff ¼ e ; ðA2Þ r3 with the following conditions: Since the right-hand side is negative, the necessary con- dition is given by jΦ j; jΨ j; jΦ j; jΨ j ≪ 1; ðA3Þ g g f f ffiffiffiffi ffiffiffiffi − C 2 j p þ CΛj p m μ¼−1= β3 μ¼−1= β3 jrΦ0 j; jrΨ0 j; jrΦ0 j; jrΨ0 j ≪ 1; ðA4Þ 2 pffiffiffiffiffi g g f f β − β β 0 ¼ 5=2 ð 2 3Þðd1 þ 2d2Þ > ; ðA9Þ β3 where a prime denotes the derivative with respect to r. μ From the basic equations, we find a septic equation for as where pffiffiffiffiffi pffiffiffiffiffi C 2 μ C μ C μ 0 2 2 m ð Þþ Λð Þþ matterð Þ¼ ; ðA5Þ d1 ≔−6m β3ð1 − β3Þ g pffiffiffiffiffi 2 1 − 6 β 13β − 6β3=2 C 2 C C þ m ð 3 þ 3 3 ÞðA10Þ where m ; Λ and matter are explicitly defined in [17]. f These terms have typical magnitudes given by pffiffiffiffiffi pffiffiffiffiffi 2 þð1 − β3Þ ð−1 þ 4 β3ÞΛg; ðA11Þ 2 7 C 2 ∼ O μ m meff × ð Þ; pffiffiffiffiffi pffiffiffiffiffi 2 2 2 5 d2 ≔ 3mgð1 − β3Þ þ m ð1 − 6 β3 þ 3β3Þ: ðA12Þ CΛ ∼ Λg × Oðμ Þ; f However the constraint (A9) is not sufficient, because it and the last term is given by does not guarantee that the function μðrÞ is a single-valued function in the domain RI 064054-13 KATSUKI AOKI, KEI-ICHI MAEDA, and MAKOTO TANABE PHYSICAL REVIEW D 93, 064054 (2016) APPENDIX B: ASYMPTOTICALLY NONFLAT SOLUTIONS In this Appendix, we analyze asymptotically nonflat solutions. We consider only the case such that the γ energy- momentum tensor approaches to a cosmological constant at infinity. There are two types of nonasymptotically flat spacetimes: One is an asymptotically homothetic space- time, and the other is an asymptotically nonhomothetic one. We find de Sitter, Minkowski or anti–de Sitter spacetime at infinity. Here we have also assumed that a regular solution exists in the whole range of the radial coordinate r (0 ≤ r<∞), which excludes a spacetime approaching the Nariai solution asymptotically. When there exists a positive cosmologicalpffiffiffiffiffiffiffiffiffiffi constant, the cosmological horizon may appear at r ≃ 3=Λg. From the regularity condition at the horizon, the metric variables should satisfy FIG. 10. The parameter space for a successful Vainshtein Ng ¼ Fg ¼ Nf ¼ Ff ¼ 0 ðB1Þ screening. We set m ¼ m ; Λ ¼ 0. The colored (the light-blue g f pgffiffiffiffiffi and red) regions satisfy β2 − β3 < 0, d1 þ β2d2 < 0.However, at the horizon. We regard the solution as satisfying this only the hatched light-blue region satisfies the condition such that boundary condition when we obtain N2;N2;F2; μðrÞ is a single-valued function. g f g 2 10−4 Ff < . pffiffiffiffiffi β2 − β3 > 0;d1 þ β2d2 > 0: ðA14Þ 1. Effective cosmological constants First, we summarize when the γ energy-momentum ffiffiffiffiffi p tensor is reduced to just a cosmological constant at infinity. For both (a) and (b), the branch of μ ≃ −1= β3 in r ≪ RV connects the branch of μ ¼ 0 at r ¼ ∞. However, case For the ansatz (3.1) and (3.2), we find the eigenvalues λ λ λ λ γμ (a) gives the single-valued function μðrÞ, while case (b) is f 0; 1; 2; 3g of ν as not. It indicates that the ratio of two radial coordinates are 6 not single-valued function. For case (c), there are two t KNf λ0 ≔ γ t ¼ curves (c-1) and (c-2) and these are disconnected. Note that, Ng the branch (c-2) can be extended to infinity. This branch is Kr0 F λ ≔ γr f g not an asymptotically Minkowski solution, but an asymp- 1 r ¼ 0 totically AdS solution similarly to the branch C which will rgFf be discussed in Appendix B3. θ Krf λ2 ≔ γ θ ¼ ¼ Kðμ þ 1Þ As a result, the parameter constraint is approximately rg given by φ Krf λ3 ≔ γ φ ¼ ¼ Kðμ þ 1Þ: ðB2Þ pffiffiffiffiffi rg β2 − β3 ≲ 0;d1 þ β2d2 < 0; ðA15Þ Then, the γ energy-momentum tensor is given by 2 as shown in Fig. 10. The hatched light-blue region gives a κ ½γ t 2 2 T ¼ −ðb0 þ 2b1λ2 þ b2λ Þ − λ1ðb1 þ 2b2λ2 þ b3λ Þ; successful Vainshtein screening solution. We can show m2 g t 2 2 numerically that there is no regular asymptotically homo- pffiffiffiffiffi ðB3Þ thetic solution in the narrow region along β2 ¼ β3 (the red region), in which μðrÞ is not a single-valued function such as (b) in Fig. 9, and should then be excluded. 2 κ ½γ r 2 2 6 T ¼ −ðb0 þ 2b1λ2 þ b2λ Þ − λ0ðb1 þ 2b2λ2 þ b3λ Þ; A similar behavior is found in the context of cosmology, for m2 g r 2 2 which the ratio of cosmic times is not a single-valued function [11]. ðB4Þ 064054-14 RELATIVISTIC STARS IN BIGRAVITY THEORY PHYSICAL REVIEW D 93, 064054 (2016) κ2 asymptotic structures of branch A when we introduce a ½γ θ − ½γ r 2 ðTg θ Tg rÞ nonzero cosmological constant. m For branch A, the results are the same both in Class [I] λ − λ λ λ λ λ ¼ð 2 0Þðb1 þ b2ð 1 þ 2Þþb3 1 2Þ and in Class [II]. When we introduce a negative cosmo- ¼ðλ2 − λ1Þðb1 þ b2ðλ0 þ λ2Þþb3λ0λ2Þ logical constant, the solution approaches the homothetic – 2 anti de Sitter spacetime at infinity as shown in Fig. 11 þðλ1 − λ0Þðb1 þ 2b2λ2 þ b3λ2Þ; ðB5Þ (Ng=Nf;Fg=Ff;rg=rf → 1). For a positive cosmological 2Λ ≲ 3 2 ½γ θ ½γ φ constant, when g meff (the Higuchi bound) is sat- Tg θ ¼ Tg φ: ðB6Þ isfied, the solution seems to approach a homothetic de Sitter spacetime. Since we cannot solve the basic equations γ We then find in the following three cases that the beyond the cosmological horizon, we cannot conclude energy-momentum tensor turns to be a cosmological definitely that the solution is asymptotically homothetic, constant: but as shown in Fig. 12, the solution seems to approach a Case (i) homothetic spacetime because the eigenvalues coincide ≈ −1 2Λ ≳ λ λ λ around r meff before the horizon. However, if g 0 ¼ 1 ¼ 2 ¼ constant: ðB7Þ 3 2 meff, a regular solution disappears as discussed in the Case (ii) Appendix of [17]. As a result, branch A always approaches a homothetic spacetime if the cosmological constant 2 2Λ ≲ 3m2 λ0 ¼ λ2;b1 þ 2b2λ2 þ b3λ2 ¼ 0: ðB8Þ satisfies g eff. Case (iii) b. Branch B 2 For branch B, there is no regular solution in Class [II] for λ1 ¼ λ2;b1 þ 2b2λ2 þ b3λ ¼ 0: ðB9Þ 2 any cosmological constant. On the other hand, in Class [I], 2 We note that the equation b1 þ 2b2λ2 þ b3λ2 ¼ 0 is equivalent to 2 1 þ 2β2μ þ β3μ ¼ 0; ðB10Þ where we use λ2 ¼ Kð1 þ μÞ. Case (i) gives an asymptotic homothetic spacetime, i.e., an asymptotic de Sitter or anti–de Sitter spacetime as well as an asymptotic Minkowski spacetime. In addition, as we will show in the next subsection, we also find a solution with a cosmological constant given by Case (ii). 2. Relativistic star with g-matter Just for simplicity, we discuss a uniform-density star only with g-matter fluid. We use the parameters (4.14) as an example for Class [I], and parameters (4.15) for Class [II]. We then choose κ2ρ 2 2 5 105 0 ρ 5 10−2 g g=meff ¼ . × ;Pgð Þ= g ¼ × : ðB11Þ Since the central pressure (B11) is lower than the critical value, we find a regular star solution both in Class [I] and in Class [II]. As discussed in Sec. IV, there are two branches A and B. a. Branch A (homothetic spacetime at infinity) FIG. 11. The typical solution with a negative cosmological Λ − 2 2 2 In the text, we consider branch A without a cosmological constant for branch A. We set g ¼ meff and mg ¼ mf; constant, in which case, the branch A solution approaches β2 ¼ −3; β3 ¼ 3. The shooting parameter is tuned to be μð0Þ¼ the Minkowski homothetic spacetime. Here, we discuss 0.0305. The vertical bar represents the surface of the star. 064054-15 KATSUKI AOKI, KEI-ICHI MAEDA, and MAKOTO TANABE PHYSICAL REVIEW D 93, 064054 (2016) FIG. 12. The same as Fig. 11 in the case of a positive FIG. 13. The typical solution with a negative cosmological cosmological constant ðΛ ¼ m2 Þ. The shooting parameter is 2 2 β −3 g eff constant for branch B in Class [I]. We set mg ¼ mf, 2 ¼ , tuned to be μð0Þ¼0.0326. β 3 Λ −75 2 −1 5l 3 ¼ and g ¼ meff (meff ¼ AdS). The shooting param- eter is tuned to be μð0Þ¼0.4936. there exists a regular solution only if we introduce a negative l ≲ −1 cosmological constantpffiffiffiffiffiffiffiffiffiffiffiffiffiffi with AdS meff as shown in Fig. 13, β2 → 1 − β2; β3 → 1 − 2β2 þ β3: ðB13Þ l ≔ −3 Λ where AdS = g is the AdS curvature radius. Note that this solution is not asymptotically homothetic. The One can see that parameters in Class [II] are still in Class eigenvalues λ0 and λ2 approach the same value with [II] after the transformation (B12). Therefore, the case only 2 satisfying 1 þ 2β2μ þ β3μ ¼ 0, for which the interaction with f-matter in Class [II] is equivalent to the case only term becomes just a cosmological constant as discussed in with g-matter in Class [II], which we have already Appendix B1. Although the g- and f-spacetimes are not discussed in the previous subsection. The only nontrivial homothetic at infinity, both spacetimes approach asymp- effect of f-matter exists in Class [I], which we discuss here. totically to some AdS spacetimes. We briefly comment on the case of ρg ∼ ρf. For this case, the asymptotically homothetic branch cannot be extended 3. Relativistic star with f-matter inward similarly to the solution (c-1) in Fig. 9 [17]. Although the result presented in [17] is only the case Here, we discuss the effect of the f-matter field. For of Class [I], we find the same behavior even for Class [II]. ρ ≫ ρ simplicity, we assume f g, for which we regard that One exceptional case is a homothetic solution. If the g-spacetime is almost vacuum. 2 ½m μ 2 2 ½m μ κ T ν ¼ K κ T ν, there exists a homothetic solution, The action of the bigravity is symmetric for g- and f- g f i.e., N ¼ N ;F ¼ F and μ ¼ 0, for which the solution is spacetimes under the transformation g f g f identical to that in GR in the whole space region. g ↔ f; b0 ↔ b4;b1 ↔ b3: ðB12Þ a. Massless limit approximation Then the case only with f-matter is equivalent to the case In the massless limit, the interior solution is given by only with g-matter for corresponding coupling constants 1 under the transformation (B12), i.e., Fg ¼ ; ðB14Þ 064054-16 RELATIVISTIC STARS IN BIGRAVITY THEORY PHYSICAL REVIEW D 93, 064054 (2016) Ng ¼ Ngð0Þ; ðB15Þ 2GM 1=2 1 − ⋆ 2 Ff ¼ 3 rf ; ðB16Þ R⋆ 3FfðR⋆Þ − FfðrfÞ Nf ¼ Nfð0Þ ; ðB17Þ 3FfðR⋆Þ − 1 P ðr Þ F ðr Þ − F ðR⋆Þ f f ¼ f f f ; ðB18Þ ρfð0Þ 3FfðR⋆Þ − FfðrfÞ FIG. 14. The same as Fig. 2 in the case of the f-star. where we assume a uniform density for f matter fluid. The b. Branch C g-spacetime is just a Minkowski solution. The exterior solution is given by We set 2 2 β −3 β 3 mg ¼ mf; 2 ¼ ; 3 ¼ ; ðB26Þ Fg ¼ 1; ðB19Þ κ2ρ 2 2 5 105 0 ρ 5 10−2 f f=meff ¼ . × ;Pfð Þ= f ¼ × ; ðB27Þ Ng ¼ Ngð0Þ; ðB20Þ and ρ ¼ constant. 2GM 1=2 f ⋆ 1 Ff ¼ 1 − ; ðB21Þ For the above parameter setting with K ¼ ,an rf −0 76 2 ≲ asymptotically AdS solution is found for . meff Λ ð1Þ ≲ 0.05m2 . This solution in branch C is asymptoti- 2 0 g eff Nfð Þ cally homothetic because the eigenvalues λ0; λ1; λ2 Ng ¼ FfðrfÞ; ðB22Þ 3FfðR⋆Þ − 1 converge to the same constant although its value is not unity as shown in Fig. 15. where we define the gravitational mass by The reason is as follows: When we fix parameters Λ β β Z fmg;mf; g;K; 2; 3g, the original coupling constants R⋆ κ M 4π 2ρ f f;big are determined. Once the original coupling con- ⋆ ¼ rf fdrf; ðB23Þ 0 stants are given, all homothetic solutions given by ~ 2 and R⋆ is the radius of the f-star measured in f-spacetime. fμν ¼ K gμν ðB28Þ Similarly to the argument in Sec. IV, the ratio must be are characterized by the proportional factor K~ which is one N ð0Þ 2 f ¼ : ðB24Þ of the roots of the quartic equation, Ngð0Þ 3FfðR⋆Þ − 1 ~ ~ 2 ~ ΛgðKÞ¼K ΛfðKÞ: ðB29Þ The center value of μ is given by a root of 2 2 In the range of −0.76m ≲ Λgð1Þ ≲ 0.05m , there are 2 eff eff ð3P ð0Þð2β2 − 3β3Þþρ ð2β2 − β3ÞÞμ0 ~ f f four real roots for K. For instance, when we set Λgð1Þ¼0, þð6Pfð0Þð1 − β2 − β3Þþ2ρfÞμ0 we find þ 3P ð0Þð1 − 2β2Þþρ ¼ 0; ðB25Þ f f K~ ¼ −0.604; 1; 1.44; and 3.83; ðB30Þ thus there are two branches (branches C and D) similar to and find four homothetic solutions (one Minkowski, one de the case of the g-star. Branch C approaches a homothetic Sitter, and two AdS spacetimes). It turns out that the spacetime as we will see later. ~ 1 44 We chose the coupling constants as (5.10) in Class [I]. solution we solved approaches K ¼ . homothetic Λ 1 44 0 The solution in the massless limit is shown in Fig. 14.For spacetime. Since gð . Þ < , it is the asymptotically the case of the f-star, the wormhole geometry is not found. AdS spacetime. Λ 1 ≲ −0 76 2 Now we solve the basic equations for each branch without Note that when we assume gð Þ . meff, there are the massless limit approximation. only two real roots of K~ , e.g, 064054-17 KATSUKI AOKI, KEI-ICHI MAEDA, and MAKOTO TANABE PHYSICAL REVIEW D 93, 064054 (2016) Vainshtein radius, there is a singularity at a radius near the Compton wavelength of the massive graviton. Thus we will not discuss branch D furthermore. APPENDIX C: WORMHOLE-TYPE SOLUTION In Class [I], as shown in Fig. 2(b), we cannot find a regular solution beyond the critical value of the pressure. The solution turns to a closed spacetime or a wormhole- type spacetime beyond the critical value. In this Appendix, we shall discuss what kind of worm- hole-type structure is obtained in the bigravity theory. To find a solution with a wormhole-type structure, we should integrate the basic equations from the wormhole throat. As mentioned in Sec. IV B, a wormhole throat corresponds to the point of J ¼ ∞, where the function J is the Jacobian for the radial coordinate transformation from rg to rf. When we find J ¼ ∞ at some radius, such a coordinate trans- formation is singular. That is, we cannot define the trans- formation rg → rf at the point. Similarly, we cannot define the transformation rf → rg at the point of J ¼ 0. When the coordinate transformation rf ¼ rfðrgÞ is not well defined (i.e., J ¼ ∞) at some point, we cannot integrate beyond such a singular point as a function of rg. However, the inverse function rg ¼ rgðrfÞ is well ∞ FIG. 15. A typical solution for branch C. We set defined at J ¼ . As a result, we can solve the equations −2 and find the solution as a function of r by using the radial Pf=ρf ¼ 5 × 10 . The shooting parameter is tuned to be f μð0Þ¼0.541. coordinate rf, i.e., the basic equations to be solved are dX −1F ~ −0 586 1 ¼ J X; ðC1Þ K ¼ . ; and ; ðB31Þ drf 2 dr r μ 1 for Λ ð1Þ¼−m . In this case, we cannot find a regular g − f d g eff ¼ 2 þ 1 μ Λ 1 ≲ −0 76 2 drf ð1 þ μÞ drf þ solution for branch C in gð Þ . meff. In the case of 3 2 2 ≫ Λ 1 ≳ 0 05 2 −1 meff= gð Þ . meff, there are four homothetic ¼ J : ðC2Þ solutions, e.g., Although the point of J ¼ ∞ is a curvature singularity as K~ ¼ −0.621; 1; 1.11; and 4.85; ðB32Þ we will see, we can continue to solve the equations and find the solution beyond such a singularity. 2 For simplicity, we assume vacuum spacetimes, i.e., there for Λ ¼ 0.1m . The solution may approach the K~ ¼ 1.11 g eff is neither g-matter nor f-matter. A wormhole throat of homothetic solution with Λ ð1.11Þ > 0. However, because −1 g g-spacetime is given by J ¼ dr =dr ¼ 0, at which we of a numerical instability, we cannot confirm that there is a g f assume the variables N ;F ;N ;F ; μ are finite. Setting regular solution approaching de Sitter spacetime for g g f f 2 2 the radial coordinate as r ¼ r , we find the derivatives 3m =2 ≳ Λ ð1Þ ≳ 0.05m . f eff g eff of g-variables are finite at J−1 0 because Eqs. (3.11) and Finally, we give a comment for the case of ¼ Λ 1 ≳ 3 2 2 (3.12) yield gð Þ meff= . In this case, the Jacobian J ¼ drf=drg diverges before reaching the cosmological horizon. 2 dFg mgrf 2 Therefore, this solution has the curvature singularity as → − ð1 þ 2β2μ þ β3μ Þ; ðC3Þ dr 2F discussed in Appendix C. f f dN g → 0 c. Branch D ; ðC4Þ drf For branch D, we cannot construct any regular solution with or without a cosmological constant by our numerical for J−1 → 0. Furthermore, Eqs. (3.13) and (3.14) indicate approach. Although the solution is regular below the that the derivatives of f-variables are also finite at J−1 ¼ 0, 064054-18 RELATIVISTIC STARS IN BIGRAVITY THEORY PHYSICAL REVIEW D 93, 064054 (2016) μ 2 10−3 −1 and Eq. (C2) indicates d =drf is finite. Hence, the first setting the graviton mass as meff ¼ × af and by derivatives of all variables are finite even at J−1 ¼ 0.Since choosing the differential equations are first order, we can solve the −1 0 equations numerically beyond J ¼ by use of the rf NgðafÞ¼FgðafÞ¼0.86070: ðC8Þ coordinate. Since two metrics are symmetric in the bigravity theory, the μðafÞ is tuned as μðafÞ¼0.03847, which gives the above argument is also applied to the point of J ¼ 0,whichis asymptotically AdS spacetime. Here we have introduced 0 awormholethroatinf-spacetime, At J ¼ ,thecoordinate a typical length scale of the wormhole rS by transformation rg ¼ rgðrfÞ is not well defined, but the solution is obtained as a function of rg beyond this singularity. rS ≔ 2GMgð∞Þ; ðC9Þ In the case of Λg ¼ 0, the branch B solution contains a singularity at some radial point. To find a regular worm- where we define a mass function MgðrÞ by hole-type solution, we should introduce a negative cosmo- logical constant. Here we set the parameters as 2GM ðrÞ Λ F2ðrÞ¼1 − g − g r2: ðC10Þ g r 3 g 2 2 β −3 β 3 g mg ¼ mf; 2 ¼ ; 3 ¼ ; ðC5Þ and Figure 16 shows the relation between two radial coor- dinates. The top panel gives rf=rg in terms of the rg Λ −75m2 1 l 0.2m−1 : C6 g ¼ eff ð AdS ¼ eff Þ ð Þ coordinate. It shows that rf=rg takes two different values at the same radius rg. One branch ðrf=rg → 1Þ approaches the We first use the g-radial coordinate rg. Suppose that a homothetic AdS spacetimes, while another branch wormhole throat exists in the f-spacetime (which we call ðrf=rg → 1.183Þ approaches the nonhomothetic AdS the f-throat), so J ¼ 0 at a radius rg ¼ af. The value of Ng on the throat is arbitrary by the rescaling freedom of the time coordinate, and the value of Fg gives the gravitational field strength at the throat, which characterizes the property of the wormhole. Since we have two algebraic equations at the f-throat as J 0; C 0; C7 jrg¼af ¼ jrg¼af ¼ ð Þ where C is the constraint equation defined by Eq. (3.18), when we give the values of Fg and Ng at rg ¼ af, the values NfðafÞ;FfðafÞ are determined by Eqs. (C7) as functions of μðafÞ. We first solve variables outward on the rg coordinate system, and find an asymptotically homothetic AdS space- time by tuning the value of μðafÞ. Next, we solve variables inward with respect to the rg coordinate. When we find the −1 point of J ¼ 0 at a radius rg ¼ ag, which is the wormhole throat in g-spacetime (the g-throat),7 we cannot continue to integrate the basic equations numerically on the rg coor- dinate. Then we switch the radial coordinate from rg to rf, and solve variables with respect to the rf coordinate beyond the point of J−1 ¼ 0. Finally we find a global wormhole- type solution (example is given in Figs. 16 and 17), by 7The throat condition (C7) is different from the analysis in [47], where the paper assumed that two wormhole throats are FIG. 16. The relation of two radial coordinates rg and rf. From located at the same spacetime point. However, we assume, → 1 → although both spacetimes show wormhole structures, two worm- the top panel, we find the ratio approaches rf=rg or rf=rg 1 183 → ∞ 0 −1 0 hole throats are located at the different spacetime points as shown . as rg :J¼ and J ¼ correspond to the worm- in Fig. 16. hole throats of f-spacetime and of g-spacetime, respectively. 064054-19 KATSUKI AOKI, KEI-ICHI MAEDA, and MAKOTO TANABE PHYSICAL REVIEW D 93, 064054 (2016) FIG. 17. Ricci scalars of g-spacetime and f-spacetime which FIG. 18. The variation rates of the mass function Mg (red solid) are presented by the red curve and the blue dashed curve, and the ratio Ng=Fg (blue dashed) as functions of rf=rg. −1 respectively. The g-throat (J ¼ 0) corresponds to rf=rg ¼ 1.2563, while the f-throat (J ¼ 0) exists at rf=rg ¼ 1.03847. the g-spacetime Ricci curvature diverges. Inversely, only the f-spacetime Ricci scalar diverges at the f-throat. This spacetime. Two different asymptotic structures are con- behavior is quite similar to the case of the cosmology [11]. nected by the wormhole. Figure 16 shows that the g-throat Finally, we discuss the Vainshtein screening. Since the γ and the f-throat are located at the different points. energy-momentum tensor cannot be ignored at the throat We depict the Ricci curvature scalar of the f-metric as point, the Vainshtein screening mechanism is no longer well as one of the g-metric in Fig. 17, where we have used guaranteed. We may find a deviation from the GR result. In the variable rf=rg to parametrize the radial coordinate, fact, the geometry of the vacuum spacetime turns to a instead of either rg or rf, because either coordinate rg or rf wormhole geometry, which does never appear in GR. To is not a single-valued function near the throats. The g-throat see the differences of the metric functions from GR, we −1 0 1 2563 (J ¼ ) is located at rf=rg ¼ . and the f-throat show the variation rates of the mass function Mg and the 0 1 03847 (J ¼ ) is founded at rf=rg ¼ . . The Ricci curvature ratio Ng=Fg in Fig. 18. In GR, two functions are exactly scalar of the g-metric diverges at the g-throat. It is caused by constant. In the bigravity, although two functions are not the divergence of the γ energy-momentum tensor at the exactly constant, these are almost constant. Hence, the wormhole throat. As shown in Fig. 17, the Ricci scalar goes metric functions are well approximated by the ∞ → 1 2563 − ϵ −∞ to þ as rf=rg . , while it goes to as Schwarzschild-AdS metric (up to their first derivatives) → 1 2563 ϵ 0 ϵ ≪ 1 rf=rg . þ with < . Note that f-space- although the topology of the solution is different from the time curvature is finite even at the g-throat of J−1 ¼ 0. Only Schwarzschild-AdS spacetime. [1] S. 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