Stability of relativistic stars with scalar hairs

The dawn of gravitational-wave (GW) astronomies Abbott2016 shed new light on the physics around compact objects such as black holes and neutron stars (NSs). For example, the GW170817 event GW170817 arising from binary NSs placed constraints on the mass-radius relation of NSs by their tidal deformations. The accumulation of GW events in the future will allow us to probe the accuracy of General Relativity (GR) in the strong gravitational regime Berti ; Barack . In particular, whether or not some extra degrees of freedom are present around compact objects are a great concern, along with the problem of dark energy and dark matter.

The simplest candidate for such an extra degree of freedom is a scalar field $\phi$. Theories in which the scalar field is coupled to the gravitational sector are called scalar-tensor theories Fujii . The nonminimal coupling $F(\phi)R$, where $F$ is a function of $\phi$ and $R$ is the Ricci scalar, is a typical example of the direct interaction between the scalar field and gravity. In the presence of baryonic fluids, the matter sector indirectly feels an interaction with the scalar field through the coupling to gravity. This matter coupling manifests itself after performing a so-called conformal transformation to the Einstein frame in which the Ricci scalar does not have a direct coupling to $\phi$ DeFelice:2010aj . In scalar-tensor theories, such matter couplings can modify the internal structure of relativistic stars.

In scalar-tensor theories with the nonminimal coupling $F(\phi)R$, it is known that there are some NS solutions with scalar hairs on a spherically symmetric and static background. For the theories in which the coupling $F(\phi)$ contains an even power-law function $\phi^{n}$, there exists a field profile of nonvanishing $\phi$, while satisfying $F_{,\phi}(0)=0$ as in GR (where $F_{,\phi}={\rm d}F/{\rm d}\phi$) Damour ; Damour2 . If the second derivative $F_{,\phi\phi}$ is positive at $\phi=0$, then the GR branch ($\phi=0$ everywhere) can be unstable due to a negative mass squared proportional to $-F_{,\phi\phi}(0)$. This allows a possibility for triggering tachyonic growth of the scalar field toward a scalarized branch with nonvanishing $\phi$, whose phenomenon is dubbed spontaneous scalarization.

A typical example of the nonminimal coupling triggering spontaneous scalarization is of the form $F(\phi)=e^{-\beta\phi^{2}/(2M_{\rm pl}^{2})}$ Damour ; Damour2 , where $\beta$ is a constant and $M_{\rm pl}$ is the reduced Planck mass. Provided that $\beta<0$, the two conditions $F_{,\phi}(0)=0$ and $F_{,\phi\phi}(0)>0$ are satisfied. More precisely, spontaneous scalarization can occur for the coupling $\beta<-4.35$ Harada:1998ge ; Novak:1998rk ; Silva:2014fca ; Freire:2012mg , whose upper bound is insensitive to the change of equations of state inside the Ns. The properties of hairy solutions and its observational consequences were investigated in several contexts, e.g., GW asteroseismology Sotani1 ; Sotani2 , rotating NSs Sotani:2012eb ; Doneva:2013qva ; Doneva:2014faa ; Pani:2014jra , and the influence on particle geodesics around NSs Doneva:2014uma ; Pappas:2015npa . Recent studies showed that black holes can also exhibit spontaneous scalarization, in the presence of couplings to a Gauss-Bonnet term Kleihaus:2015aje ; Doneva:2017bvd ; Silva:2017uqg ; Antoniou:2017acq ; Antoniou:2017hxj ; Minamitsuji:2018xde ; Cunha and to an electromagnetic field Stefanov ; Herdeiro1 ; Herdeiro2 ; Herdeiro3 ; Ikeda .

There are also other nonminimally coupled theories admitting hairy NS solutions, e.g., Brans-Dicke (BD) theories Brans with a scalar potential. The nonminimal coupling in BD theories can be expressed in the form $F(\phi)=e^{-2Q\phi/M_{\rm pl}}$, where $Q$ is a constant related to the so-called BD parameter $\omega_{\rm BD}$ as $2Q^{2}=1/(3+2\omega_{\rm BD})$ Yoko . Since this coupling does not satisfy conditions for the occurrence of spontaneous scalarization, there exist only hairy solutions with nonvanishing scalar-field profiles Cooney:2009rr ; Arapoglu:2010rz ; Orellana:2013gn ; Astashenok:2013vza ; Yazadjiev:2014cza ; Resco:2016upv . For example, $f(R)$ gravity belongs to a class of BD theories with $\omega_{\rm BD}=0$ Ohanlon ; Chiba03 , in the presence of a scalar potential arising from the deviation from GR. For the Starobinsky model $f(R)=R+R^{2}/(6m^{2})$ Staro , the existence of a positive constant mass $m$ gives rise to an exponential growing mode of $\phi$ outside the star Ganguly ; Kase:2019dqc . This is not the case for the models $f(R)=R+aR^{p}$ with $1<p<2$ Kase:2019dqc ; Dohi:2020bfs , in which the effective mass of $\phi$ can approach 0 toward spatial infinity. In the latter case, there are hairy NS solutions with the mass-radius relation modified from that in GR.

The hairy NS solutions in theories of spontaneous scalarization and BD theories have an additional pressure induced by a matter coupling with the scalar field. Under the occurrence of spontaneous scalarization, for example, the additional pressure works to be repulsive against gravity, in which case the radius and mass of star tend to be increased relative to those in GR (see Refs. Maselli ; Minami ; Chagoya ; Kase:2017egk ; Chagoya2 ; Ogawa ; Kase:2020yhw for related papers). Naively, one might expect that such hairy solutions can be unstable against perturbations. To study the stability of relativistic stars with scalar hairs, it is necessary to properly incorporate perturbations of the matter sector besides those of gravity and the scalar field. In this paper, we will address this issue by dealing with baryonic matter as a perfect fluid.

For a k-essence scalar field with the Lagrangian $P(X)$ Scherrer , where $P$ is a function of the field kinetic energy $X=-\partial_{\mu}\phi\partial^{\mu}\phi/2$, it is known that the corresponding energy-momentum tensor reduces to that of a perfect fluid for a time-like scalar field, i.e., $X>0$ Hu05 ; Arroja . This is the case for a time-dependent cosmological background, so that the k-essence Lagrangian was extensively used to describe the perfect-fluid dynamics especially in the context of late-time cosmic acceleration DMT10 ; KT14b ; Heisenberg:2016eld ; Gum . On the spherically symmetric and static background, however, the scalar field is space-like ($X<0$) and hence the perfect fluid cannot be described by the k-essence Lagrangian. Instead, we will employ the matter action advocated by Schutz and Sorkin Sorkin , which allows one to describe the perfect fluid in any curved background (see also Refs. Brown ; DGS ) .

To accommodate both theories of spontaneous scalarization and BD theories presented in Sec. II, we will consider scalar-tensor theories given by the Lagrangian ${\cal L}=G_{4}(\phi)R+G_{2}(\phi,X)$ in the presence of a perfect fluid. This belongs to a subclass of Horndeski theories with second-order field equations of motion Horndeski ; Horn1 ; Horn2 ; Horn3 . We carry out all the analysis in the Jordan frame, in which the matter sector is minimally coupled to gravity. After deriving covariant equations of motion and applying them to the spherically symmetric and static background in Sec. III, we proceed to the discussion about the separation of perturbations into odd- and even-parity modes in Sec. IV. In Horndeski theories without matter, the stabilities of hairy black hole solutions against odd- and even-parity perturbations were studied in Refs. DeFelice:2011ka ; Kobayashi:2012kh ; Kobayashi:2014wsa ; Kobayashi:2014eva ; Ogawa:2015pea ; Babichev:2016rlq ; Khoury:2020aya .

In Sec. V, we expand the full action up to second order in odd-parity perturbations with the multipoles $l\geq 2$ and show that the ghost is absent for $G_{4}>0$ with the propagation speed of gravity equivalent to that of light. In Sec. VI, we derive conditions for the absence of ghosts and Laplacian instabilities in the even-parity sector. The propagating degrees of freedom are different depending on the multipoles $l$, so we separately discuss the three cases $l\geq 2$, $l=1$, and $l=0$. The analysis of odd- and even-parity perturbations was also performed in Ref. Sotani:2005qx for the nonminimal coupling of Refs. Damour ; Damour2 by using an energy-momentum tensor of the perfect fluid. Since this approach is based on the equations of motion rather than the action principle, it is not straightforward to identify the dynamical degrees of freedom and their stability conditions. In this paper, we address this problem by dealing with the perfect fluid as the Schutz-Sorkin action and integrate out all the nondynamical perturbations from the action. Indeed, the dynamical degree of freedom in the matter sector is of a nontrivial form, which affects the propagation of scalar GWs.

In Sec. VII, we apply our general stability conditions to hairy relativistic stars present in theories of spontaneous scalarization and BD theories. We show that these hairy solutions are stable under the conditions $G_{4}>0$, $\rho+P>0$, and $c_{m}^{2}>0$, where $\rho$, $P$, and $c_{m}^{2}$ are the density, pressure and sound speed squared of matter respectively. In particular, as long as $0<c_{m}^{2}\leq 1$, all the speeds of propagation associated with even-parity perturbations in the gravity sector are subluminal inside the star, while they are equivalent to the speed of light outside the star. This fact is consistent with the speed of gravity constrained from the GW170817 event GW170817 .

Throughout the paper, we adopt the natural units for which the speed of light $c$, the reduced Planck constant $\hbar$, and the Boltzmann constant $k_{B}$ are set to unity.

In this section, we briefly review several theories of relativistic stars with scalar hairs. We consider a scalar field $\phi$ with a nonminimal coupling of the form $F(\phi)R$. The scalar field can have a kinetic term of the form $\omega(\phi)X$, where $\omega$ is a function of $\phi$ and $X=-(1/2)g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$ is the kinetic energy (with metric tensor $g^{\mu\nu}$). We also allow for the existence of a field potential $V(\phi)$. Then, the action of such scalar-tensor theories is given by

where $g$ is a determinant of metric tensor $g_{\mu\nu}$, and $M_{\rm pl}$ is the reduced Planck mass. The action ${\cal S}_{m}$ corresponds to that of the matter field $\Psi_{m}$. We assume that the matter field is minimally coupled to gravity in the Jordan frame given by the metric $g_{\mu\nu}$. In Sec. III, we specify the action ${\cal S}_{m}$ to be of the form of a perfect fluid.

Performing a so-called conformal transformation $(g_{\mu\nu})_{E}=F(\phi)g_{\mu\nu}$ of the metric, we obtain the following Einstein-frame action DeFelice:2010aj ,

where the subscript “$E$” represents quantities in the Einstein frame, and

In the Einstein frame the field $\varphi$ does not have a direct coupling with the Ricci scalar $R_{E}$, but the matter sector has an interaction with the scalar field through the metric $(g_{\mu\nu})_{E}=F(\phi)g_{\mu\nu}$.

In what follows, we will present two classes of theories which belong to the action (1).

In this paper, we focus on scalar-tensor theories given by the action,

where $G_{4}$ is a function of the scalar field $\phi$, and $G_{2}$ depends on both $\phi$ and $X$. The action (11) accommodates scalar-tensor theories with Eq. (1) as a special case. A perfect fluid minimally coupled to gravity can be described by a Schutz-Sorkin action of the form Sorkin ; Brown ; DGS

The matter density $\rho$ depends on the fluid number density $n$ alone. The vector field $J^{\mu}$ corresponds to a current, while the scalar quantity $\ell$ is a Lagrange multiplier. In terms of $J^{\mu}$, the number density can be expressed as

The fluid four-velocity $u_{\mu}$ is related to $J_{\mu}$, as

From Eq. (13), there is the relation $u^{\mu}u_{\mu}=-1$. The quantities ${\cal A}_{i}$ and ${\cal B}^{i}$ are spatial vectors characterizing intrinsic vector modes.

To study the stability of relativistic stars around the spherically symmetric and static space-time (30), we consider metric perturbations $h_{\mu\nu}$ on the background metric $\bar{g}_{\mu\nu}$, such that

There are also perturbations of the scalar field $\phi$ and quantities appearing in the Schutz-Sorkin action (12). In terms of the spherical harmonics $Y_{lm}(\theta,\varphi)$, the scalar field can be expressed in the form,

where $\bar{\phi}(r)$ is the background value, and $\delta\phi_{lm}$ is the perturbed part being a function of $t$ and $r$. In the following, we omit the subscripts $l,m$ from the perturbation $\delta\phi_{lm}(t,r)$ and also apply the same rule to other perturbed quantities appearing below. Under the rotation in two-dimensional plane ($\theta,\varphi$), any scalar perturbation has the parity $(-)^{l}$, which is called the even mode Regge:1957td ; Zerilli:1970se . The odd mode corresponds to a perturbation with the parity $(-)^{l+1}$. The metric components $h_{tt},h_{tr},h_{rr}$ transform as scalars under a two-dimensional rotation in the ($\theta,\varphi$) plane, so they only possess even-parity modes.

The $\theta$ and $\varphi$ components of any vector field $V_{\mu}$ contain both even- and odd-modes, whereas the temporal and radial components $V_{t}$ and $V_{r}$ possess the even mode alone. To accommodate the odd-parity contribution to $V_{\mu}$, we introduce the tensor $E_{ab}=\sqrt{\gamma}\,\varepsilon_{ab}$, where $\gamma$ is the determinant of two dimensional metric $\gamma_{ab}$ (with $a,b$ either $\theta$ or $\varphi$), and $\varepsilon_{ab}$ is the anti-symmetric symbol with $\varepsilon_{\theta\varphi}=1$. The $\theta,\varphi$ components of fluid four velocity $u_{\mu}$, for example, can be expressed as

where, in the second line, we expressed two scalars $v$ and $\tilde{v}$ in terms of the expansion of spherical harmonics. The first and second terms of Eq. (43) correspond to even- and odd-parity perturbations, respectively. The metric components $h_{t\theta},h_{t\varphi},h_{r\theta},h_{r\varphi}$ transform as vectors under the two-dimensional rotation in the ($\theta,\varphi$) plane, so they can be also expressed in terms of the sum of even- and odd-modes analogous to Eq. (43).

The components $T_{ab}$ of any symmetric tensor $T_{\mu\nu}$ contain both even- and odd-modes. For instance, the metric components $h_{ab}$ are written in the form,

where, in the second line, we expressed the three scalars $K$, $G$, and $U$ in terms of the expansion of spherical harmonics.

In summary, metric perturbations $h_{\mu\nu}$ and perturbed quantities present in the action (11) can be decomposed into even- and odd-modes in the following way.