The moment of inertia $I$ and the quadrupole moment $Q$ are among the leading
order multipole moments in the asymptotic expansion of the metric functions at
infinity. There are infinitely many other higher order multipole moments,
though, and it is interesting to explore whether similar universal relations
hold for them. This was first addressed in GR [256, 363] where such universal
relations were derived for the higher multipole moments when proper
normalization is applied. Pappas and Sotiriou [258] and Pappas  _et al._ [257]
extended these results to scalarized NSs. This required the generalization of
the multipole moment formalism of Geroch [145], Hansen [149] to scalar-tensor
gravity. Pappas  _et al._ [257] showed that future observations of QPOs of
low-mass x-ray binaries can, in principle, be used can be measure different NS
properties and tell apart different gravity.

Another class of universal relations studied in the context of the DEF model
connects the NS moment of inertia and compactness. This is motivated by
Lattimer and Schutz [205] (see also Breu and Rezzolla [59]), that used this
relation to argue that pulsar timing observations could lead to a measurement
of the moment of inertia to within $10\%$. These relations were generalized to
scalarized NSs with massless scalar field potential by Altaha Motahar  _et
al._ [12], and later to massive self-interacting scalar fields by Popchev  _et
al._ [263]. Similar to the I-Love-Q relations, only the massive scalar field
case leads to large deviations from GR when observational constraints are
taken into account.

Ofengeim [248] proposed to connect the physical parameters of static NSs, such
as mass, radius, central energy density, pressure, and sound speed, at the
maximum-mass point for a given EOS. This is based on the observations that the
nuclear matter EOS (without phase transitions) can be well parameterized by
only two parameters [211]. As a result, Ofengeim [248] derived multiple
constraints on the nuclear matter EOS. Danchev and Doneva [93] generalized
these relations to scalarized NSs. They showed how sensitive these relations
can to the underlying theory of gravity. Thus, all the EOS restrictions
derived in GR should be taken with care, since the possibility for GR
modifications are rarely taken into account when interpreting observational
data.

###  III.4 Extended scalar-tensor theories beyond the Damour-Esposito-Farèse
model

Until now we discussed scalarization in the DEF model or other DEF-inspired
models. However, s we saw in Sec. II.2, spontaneous scalarization is also
possible for other modified theories of gravity. Here, we will discuss such
theories where scalarized NS solutions were obtained.

##### Scalar-tensor theories with disformal coupling

Minamitsuji and Silva [230] studied NS scalarization in scalar-tensor theories
with disformal coupling of the form (17). More specifically, instead of the
standard conformal transformation between Einstein frame metric ($g_{\mu\nu}$)
and Jordan frame metric (${\tilde{g}}_{\mu\nu}$)

$g_{\mu\nu}={\cal A}^{2}(\varphi){\tilde{g}}_{\mu\nu},$ (33)

where ${\mathcal{A}}(\varphi)$ is a function of the scalar field related to
the Jordan frame coupling between the scalar field and the Ricci scalar, we
have a more general and complicated transformation that also involves scalar
field derivatives:

$g_{\mu\nu}=A^{2}(\varphi)\left[{\tilde{g}}_{\mu\nu}+\Lambda
B^{2}(\varphi)\nabla_{\mu}\varphi\nabla_{\nu}\varphi\right],$ (34)

where $B(\varphi)$ is a function of the scalar field and $\Lambda$ is a
parameter of dimensions of [length]2. The motivation behind this modification
of the conformal factor comes from Bekenstein [42] that aimed to find the most
general coupling constructed from the metric $g_{\mu\nu}$ and the scalar field
$\varphi$ that respected causality and the WEP. More recently, Andreou  _et
al._ [17] showed that such disformal coupling is actually equivalent to a
kinetic coupling with the Ricci scalar and thus is contained in the minimal
action (II.2.1) (see Sec. II.2.1 for a detailed discussion).

Minamitsuji and Silva [230] showed that for negative values of the disformal
coupling parameter $\Lambda$, scalarization can be suppressed, while for large
positive values of $\Lambda$ stellar models cannot be obtained. This study
also explored the universal relations between the moment of inertia and the
compactness of NSs, and they determined the range of parameters where these
relations can deviate significantly from GR.

##### Scalar-Gauss-Bonnet theory:

Another well-studied modified gravity theory that allows for NS scalarization
is scalar-Gauss-Bonnet gravity discussed in Sec. II.2. Interestingly,
spacetime curvature itself as well as matter can be the source of the scalar
field in this theory. Scalarized NS solutions in scalar-Gauss-Bonnet gravity
were studied for the first time in Doneva and Yazadjiev [112], Silva  _et al._
[312]. They exhibit quite different features compared to the DEF model while
being qualitatively similar to NS solutions in other classes of Gauss-Bonnet
theories where scalarization is not possible [254, 186, 289]. For example, for
a given central energy density the stellar mass is always smaller compared to
GR. Moreover, normally there is only one bifurcation point at small central
energy densities and afterwards the branches of solutions are terminated
because of violation of the regularity conditions.

In Fig. 12, we show the branches of scalarized NS solutions for a coupling
function $f(\varphi)=-\lambda^{2}(2\beta)^{-1}[1-\exp(-\beta\varphi^{2}/4)]$.
Here, $\lambda$ is a parameter with dimensions of [length]-1 that controls the
coupling strength in the action (II.2.3). Note that the sign of $f(\varphi)$
is opposite to the one discussed in Sec. II.2.3 and thus can not lead to BH
spontaneous scalarization since in vacuum $\mu_{{\rm eff}}^{2}>0$ as defined
in (27). However, due to the presence of matter NSs can scalarize. Scalarized
NS solutions were found also for the more “conventional” sign of $f(\varphi)$,
namely $f(\varphi)=\lambda^{2}(2\beta)^{-1}[1-\exp(-\beta\varphi^{2}/4)]$
[112].

Figure 12: Scalarized NSs in scalar-Gauss-Bonnet gravity. We show the mass as
a function of the radius for GR (trivial branch with $\varphi=0$) as well as
several scalarized branches with different values of the scalar field coupling
parameters $\lambda$ and $\beta$. Credit: Doneva and Yazadjiev [112].

Little is currently known about the astrophysical implications of these stars.
Kuan  _et al._ [198] considered the spherically symmetric core collapse of a
non-compact star either to a protoneutron star or to a BH in Gauss-Bonnet
theory. They also proposed a realistic physical mechanism for the formation of
isolated scalarized BHs and NSs The complexity of the problem is greatly
increased with respect to GR, though, and there are fundamental difficulties
such as the loss of hyperbolicity of the evolution equations for certain
ranges of parameters, effectively limiting the maximum possible scalar field
to relatively low values (see also East and Ripley [125, 126], Ripley and
Pretorius [283] for the case of BHs). Core-collapse simulations show that the
remnant in scalar-Gauss-Bonnet theory can be very rich with (de)scalarization
happening at intermediate or final stages of the collapse depending on the
properties of the progenitor and the theory parameters. Since breathing modes
are absent in this theory, the effect on GW emission can be estimated only if
one drops the assumption of spherical symmetry. This is something that has not
been done yet in any modified gravity theory due to problem’s complexity.

##### The Ricci-Gauss-Bonnet model:

One can further modify the action in scalar-Gauss-Bonnet gravity to include
additional terms. This is the case of Ricci-Gauss-Bonnet model discussed in
Sec. II.2.4. This model has interesting advantages such as the possibility to
reconcile scalarization with cosmology. Here we will focus on the NS solutions
this theory that were considered by Ventagli  _et al._ [348]. This work made
thorough exploration of the theory parameter space in order to find the
sectors where NS solutions exist. Since their scalar charge is nonzero, in
these sectors one can put severe constraints on the theory based on the binary
pulsar observations, similar to the DEF model (see Sec. III.1.1).

The free parameters in the theory, as evident from the action (II.2.4), are
$\beta$ and $\alpha$ that control the coupling strength to the Ricci scalar
and the Gauss-Bonnet invariant respectively. In Fig. 13 the existence of NS
solutions is shown in a two-dimensional plot spanning the parameters $\alpha$
and $\beta$ in a broad range, for a fixed central energy density and the MPA1
EOS [237]. The white area corresponds to the region of the parameter space
where the GR solution is stable against scalar perturbations. Throughout the
parameter space that is spanned by $\alpha$ and $\beta$, a new unstable mode
appears every time one crosses a black line. These unstable modes can be
labeled by the number of scalar field nodes (denoted by $n=0,1,2,3$ in the
figure). Any point in the parameter space that lies within some grey region
corresponds to a configuration where the GR neutron star solution is unstable.
The red (respectively blue) area corresponds to the region where scalarized
solutions with $n=0$ (respectively $n=1$) nodes exist.

As one can see, scalarized solutions exist in only part of the parameter
space. In the $\beta>0$ and small $\alpha$ range (in white in the figure) the
GR neutron starts are stable. This is exactly the region most interesting from
the point of view of BH scalarization and connection to cosmology Antoniou
_et al._ [24, 25], see also Sec. IV.1.2. Thus, including a Ricci coupling to
the scalar-Gauss-Bonnet action seems to allow for BH scalarization while
evading the binary pulsar constraints.

Fig. 13 is only for one EOS and a specific central energy density. Different
EOSs and central densities can lead to significant deformations of the
existence and stability regions. The general point made in Ventagli  _et al._
[348], though, is that a stability (white) region always survives.

Figure 13:  Regions of existence of scalarized solutions in the
$(\alpha,\beta)$ plane for the MPA1 EOS with central energy density
$\varepsilon_{c}=6.3\times 10^{17}\leavevmode\nobreak\
\text{kg}/\text{m}^{3}$. In GR, a star with this choice of $\varepsilon_{c}$
and EOS is light, with $M_{\text{GR}}=1.12\leavevmode\nobreak\ {\rm
M}_{\odot}$. The red (respectively blue) region is the region where scalarized
solutions with $0$ (respectively $1$) scalar field node exist. The grey
contours obtained by Ventagli  _et al._ [349] are superimposed, which
represent the lines beyond which GR solutions with the same density are
unstable to scalar perturbations with $0,1,2,$ etc nodes. We see that the
region where there exist scalarized solutions with $n$ nodes is included in
the region where the GR solutions are unstable to scalar perturbations with
$n$ nodes, but much smaller. The dashed boundary for the blue region
corresponds to a breakdown of the integration inside the star. Credit:
Ventagli  _et al._ [348].

##### Tensor-multi-scalar theories:

Another class of gravity theories where NS scalarization was considered, is
the tensor-multiscalar theories [86], whose basics we discussed in Sec. II.2.
In this theory the gravitational interaction is mediated by the spacetime
metric $g_{\mu\nu}$ and $N$ scalar fields $\varphi^{a}$ which take values in a
coordinate patch of an $N$-dimensional Riemannian (target) manifold ${\cal
E}_{N}$ with a positive definite metric $\gamma_{ab}(\varphi)$ defined on it
[86, 169].

The main features of tensor-multiscalar theories is the inclusion of more than
one scalar field and a structure called the target space metric. In that
sense, there are two directions to go in order to obtain scalarized NSs. The
first one is to consider a mixture of several, more or less equivalent, scalar
fields that was the approach in Horbatsch  _et al._ [169]. There, the authors
examined the case of two scalar fields in the form of a complex scalar with a
maximally symmetric target space metric $\gamma_{ab}$. Doneva and Yazadjiev
[114] studied scalarization in tensor-multi-scalar gravity in a more
complicated setup, namely when $\gamma_{ab}$ is a three-dimensional maximally
symmetric space together with a nontrivial map $\varphi:$ spacetime
$\rightarrow$ target space. While the solutions in Horbatsch  _et al._ [169]
can be viewed as generalization of the DEF model to multiple scalar fields,
the scalarized NSs in Doneva and Yazadjiev [114] have some quite distinct
properties and they can be considered more as a limiting case of the
topological NSs discovered by Doneva and Yazadjiev [115]. More specifically,
the scalar field is zero in the stellar center, while only its first
derivative is zero in the DEF model. The scalar charge for the scalarized
stars in tensor-multi-scalar theories is zero as well. This automatically
reconciles this theory with the binary pulsar observations (due to the absence
of scalar dipole radiation), while still allowing for large deviations from
GR. Another interesting property is the fact that there is non-uniqueness of
the scalarized solutions with respect to the central energy density –
something that was never observed before at least for scalarized NSs. Fig. 14
illustrates this property. Interestingly, Kuan  _et al._ [199] showed that all
these solutions are stable until the maximum NS mass regardless of the non-
uniqueness.

Figure 14: Tensor-multi-scalar theories: the mass as a function of the central
energy density (left panel) and as a function of the stellar radius (right
panel) for scalarized NSs in tensor-multi-scalar theories where the target
space metric represents a three-dimensional maximally symmetric space and we
have a nontrivial map $\varphi:$ spacetime $\rightarrow$ target space.
Sequences of scalarized solutions for different combinations of parameters are
given. Observe the appearance of non-uniqueness of the solutions with nonzero
scalar field for certain combinations of the parameters (e.g., the small
central energy density region for the blue dotted line). Credit: Doneva and
Yazadjiev [114].

##### Other scalarization models:

Apart from models of scalarized NSs already discussed, other works have
attempted to consider scalarization in other more exotic theories of gravity
or even different types of scalarization.

For example, Azri and Nasri [30] considered NS scalarization in what is called
scalar-connection gravity, where gravity is mediated by a scalar field and the
connection. Other types of scalarization in standard scalar-tensor theories,
but if we allow for different scalar field potentials. Minamitsuji and
Tsujikawa [231] showed this by considering the potential
$V(\varphi)=m_{\phi}^{2}f_{B}^{2}[1+\cos(\phi/f_{B})]$ of a pseudo-Nambu-
Goldstone boson. Here $(\mu_{\phi}$, $f_{B})$ are constants with dimensions of
length and the scalar field has negative mass, $\mu^{2}=-m_{\phi}^{2}$.
Minamitsuji and Tsujikawa [231] showed that in this case the scalar field sits
at its vacuum expectation value far away from the source, while inside a NS, a
symmetry restoration can take place resulting in a new type of scalarization.
This theory also has the advantage that it avoids cosmological instabilities
present in some other scalarization models.

##  IV Black hole scalarization

In this section, we discuss spontaneous scalarization of BHs in three parts.
First, in Sec. IV.1 we consider models in which BHs can scalarize in vacuum
due to couplings between scalar field and curvature scalars. Next, in Sec.
IV.2 we discuss models in which scalarization is induced by the presence of
extra fields (e.g., gauge or matter fields) in the BH spacetime. Finally, in
Sec. IV.3 we review a selection of other models of BH scalarization.

###  IV.1 Black hole scalarization: vacuum spacetimes

As we saw in Sec. II, some gravity theories admit the same vacuum BH solutions
as GR, and yet can give rise to new branches of solutions, with scalar hair,
once certain conditions are met. The prototypical example is described by
scalar-Gauss-Bonnet theories, whose action, in the absence of matter, is given
by Eq. (II.2.3), namely,

$\displaystyle S$ $\displaystyle=\frac{1}{16\pi G}\int{\rm
d}^{4}x\,\sqrt{-g}\,\left[R-2g^{\mu\nu}\partial_{\nu}\varphi\partial_{\mu}\varphi+f(\varphi)\mathscr{G}\right].$

As we discussed in Sec. I, the first models exhibiting BH scalarization in
vacuum were proposed in Doneva and Yazadjiev [113] and Silva  _et al._ [312].
These models remain to date the most studied BH scalarization models in the
literature and for this reason, they will be our main focus in this section.

As explained in Sec. II.2.3, in theories described by the action (II.2.3), a
no-hair theorem guarantees that the BH solutions of GR are unique to the
theory, as long as $({\rm d}^{2}f/{\rm
d}\varphi^{2})_{\varphi=\varphi_{0}}\mathscr{G}<0$, for some constant
$\varphi_{0}$ [312]. If this condition is violated, scalar field perturbations
can become tachyonic unstable and the endpoint in expected to be a nonlinear,
scalarized BH [283]. Scalarized BHs have been shown to form dynamically, as
outcomes from the core collapse of an initially unscalarized star [198].

In what follows, we first review our understanding of isolated scalarized BHs
in scalar-Gauss-Bonnet theories, focusing on their properties and their
stability. We then review what is understood about when they are found in
binaries. Finally, we give an overview of vacuum BH solutions in models that
generalize the action (II.2.3).

####  IV.1.1 Spherically symmetric and rotating scalarized black holes

Spontaneously scalarized BH solutions were first found in Doneva and Yazadjiev
[113], Silva  _et al._ [312]. What distinguishes these works is the choice of
the coupling function $f(\varphi)$ that couples the scalar field to the Gauss-
Bonnet invariant [cf. Eq. (II.2.3)], respectively,

$\displaystyle f$
$\displaystyle=(\lambda^{2}/12)\,[1-\varepsilon\exp(-6\varphi^{2})],$ (35a)
$\displaystyle f$ $\displaystyle=(\eta/8)\,\varepsilon\,\varphi^{2},$ (35b)

where $\lambda^{2}$ and $\eta$ are coupling constants with dimensions of
[length]2 and $\varepsilon=\pm 1$, which should not to be confused with the
energy density of Sec. III. Both coupling functions agree in the
small-$\varphi$ limit (i.e., when $\varphi\ll 1$) and therefore make the same
prediction for the onset of scalarization of GR BHs. This threshold can be
found by searching for bound state solutions, i.e., time-independent solutions
of the linearized field equation for $\varphi$ around a GR BH, which are
regular at the event horizon $r_{\rm h}$ and that vanish at spatial infinity.
Hence, the determination of the scalarization threshold reduces to a boundary
value problem. In the simplest case of a Schwarzschild BH of mass $M$, the
dimensionless quantity $\eta/M^{2}$ plays the role of the eigenvalue and can
be determined numerically with standard techniques, such as the shooting
method. The smallest eigenvalue gives the threshold for the formation of the
“fundamental” scalarized BH solution (in the sense that the radial profiles of
the scalar field has $n=0$ nodes). The other eigenvalues give the thresholds
for the formation of “excited states”, that is, solutions with one or more
nodes. In Fig. 15 we show the results of such a calculation in the spacetime
of a Schwarzschild BH and $\varepsilon=1$, as done in Silva  _et al._ [312].
We remark that no bound states can form for $\varepsilon=-1$, because the
effective mass of scalar field perturbations is positive and therefore the
effective potential is positive definite. See the discussion in Sec. II.1.2.

Figure 15: Results of the numerical integration of the equation of a monopole
($\ell=m=0$) scalar perturbation $\delta\varphi$ on a Schwarzschild BH
background. Top panel: the asymptotic value of $|{\rm d}\sigma/{\rm d}r|$
(with $\sigma\equiv r\,\delta\varphi$), evaluated at a point $r_{\rm max}\gg
r_{\rm h}$, far away from the event horizon $r_{\rm h}=2M$, as a function of
$\eta/M^{2}$. Cusps mark when bound state solutions form. They signal the
transition between stable Schwarzschild BHs and scalarized solutions. Bottom-
left panel: effective potential $V_{\rm eff}$ (see Sec. II.1.2) for
$(\eta/M)^{2}=0$ and $5$. In the latter case, $V_{\rm eff}$ develops a
negative region and can support bound states. Bottom-right panel: radial
profiles of $\delta\varphi$ for the first three bound states, corresponding to
$\eta/M^{2}=2.902$, $19.50$ and $50.93$. These profiles have $0$, $1$ and $2$
nodes, respectively. The properties of these solutions are shared between the
models of Doneva and Yazadjiev [113], Silva  _et al._ [312]. Credit: Silva
_et al._ [312].

While both models (35) agree in their prediction of the threshold for
scalarization, they differ significantly in their prediction of the properties
of the nonlinear, scalarized solution, as already discussed in Sec. II.1. In
Fig. 16 we show the branches of scalarized BHs, in a parameter space spanned
by the dimensionless scalar charge and BH mass. The solutions were obtained by
solving the full system of field equations (see e.g. Julié  _et al._ [180] for
detailed discussion). The results are for the Gaussian coupling (35a) with
$\varepsilon=1$. We show the fundamental, zero-node scalar field solution
(dashed, red curves) and also the excited scalarized BHs (solid, green and
blue curves). We see that the branches of excited BHs can terminate at finite
masses $M$. This is due to a violation of an inequality at the BH horizon
which must be satisfied for the scalar field to be real-valued and is typical
of scalar-Gauss-Bonnet theories; see e.g., Kanti  _et al._ [182] and Antoniou
_et al._ [23]. The parameters in the coupling function (35a) are chosen in
such a way that this disappearance of BH solutions does not affect the
fundamental branch. For other couplings, such as Eq. (35b), even this branch
can violate the regularity condition shortly after bifurcation [see e.g.,
Doneva  _et al._ [106], Silva  _et al._ [311]]. Hence, the domain of existence
of scalarized solutions in the Gaussian model (35a) is larger with respect to
that of the “quadratic” model (35b).

Figure 16: The scalar charge $D$ normalized to the coupling parameter
$\lambda$ as a function of the normalized BH mass $M/\lambda$ for sequences of
scalarized BH solutions. The branches with different colors corresponds to
solutions with different number of scalar field nodes, where only the nodeless
scalar field solutions can be stable [49]. The GR solutions have zero scalar
field and thus zero $D$, and that is why they lie on the $x$-axis, depicted in
magenta. The scalarized branches are symmetric with respect to the $y$-axis
due to the reflection symmetry ($\varphi\to-\varphi$) in the theory. Figure
adopted from Doneva and Yazadjiev [113].  Figure 17: Rotating scalarized BHs
in the Gaussian model (35a). The shaded regions represent the domain of
existence of scalarized rotating BHs in the plane spanned by the dimensionless
scalar charge ($Q_{\rm s}/\lambda$) and spin ($j=a/M$). These regions are
bound by the Kerr family of solutions $Q_{\rm s}/\lambda=0$ (solid blue line)
and, in the nonrotating limit, by the Schwarzschild solution (black dot-dashed
line), and also when the regularity condition for the scalar field is violated
(dotted curve). The inset shows the case for positive coupling constant
studied in Cunha  _et al._ [84], while the main panel shows nonlinear spin-
induced scalarized BHs. Credit: Herdeiro  _et al._ [164].

Asymptotically flat, spinning scalarized BHs in the Gaussian models (with
$\varepsilon=1$) and quadratic theories were explored respectively by Cunha
_et al._ [84] and Collodel  _et al._ [82]. In discussing these solutions, it
is convenient to consider a plane spanned by the dimensionless scalar charge
$Q_{s}/\lambda$ and dimensionless spin parameter $j\equiv J/M^{2}$, where $J$
is the angular momentum of the BH. Figure 17 shows the domain of existence of
scalarized rotating BH solutions. In particular, focus on the inset which
corresponds to $\varepsilon=1$ in Eq. (35a). We see that as the spin
increases, so does the existence domain (shaded region) of the solutions. This
result can be understood in terms of the Gauss-Bonnet invariant not being
positive-definite for the Kerr metric [74]. More specifically, a Kerr BH
develops increasingly larger regions where the Gauss-Bonnet invariant is
negative as the spin increases and hence suppressing scalarization. As a
consequence, the deviations relative to GR predictions on physical observables
(e.g., the location of the innermost stable circular orbit (ISCO) of massive
particles or the BH shadow) are smaller for rapidly rotating BHs. See Fig. 18
for an example. Collodel  _et al._ [82] studied the quadratic coupling (35b)
and reached similar conclusions. They also obtained radially excited solutions
(i.e., scalar field profiles with nodes in the radial direction) and also the
existence of “angularly” excited rotating BHs (i.e., scalar field profiles
with nodes in the angular directions).

Figure 18: Gravitational lensing and shadow produced by rotating BHs under
similar observation conditions. The left panel corresponds to a scalarized BH
with mass $M/\lambda\approx 0.237$ and spin $j=0.24$. The right panel
corresponds to a comparable Kerr BH. Credit: Cunha  _et al._ [84].

####  IV.1.2 The stability of scalarizated black holes and implications to
model building

Having established the existence of scalarized BHs, the natural next task is
to study their stability. A first indication comes from the study of the
thermodynamical properties of such BHs. Doneva and Yazadjiev [113] found that
the scalarized solutions belonging to the fundamental branch in the Gaussian
model are thermodynamically favored relative to the Kerr solution (35a), in
the sense that they have a smaller Wald entropy [351]. This suggest the
stability of the former, i.e., in a dynamical process BHs in this model would
evolve towards a state of smaller entropy. In contrast, in the quadratic
coupling (35b) model always results in scalarized BHs with higher entropy
relative to their GR counterparts. These observations remain true in the case
of rotating BHs [164, 82].

Figure 19:  Scalar-charge-mass diagram for scalarized BHs with a quadratic
scalar-Gauss-Bonnet model with a scalar field potential
$V(\varphi)=(\mu^{2}/2)\varphi^{2}+(\lambda/2)\varphi^{4}$. The figure uses
dimensionless quantities obtained by factors of $\eta^{1/2}$: BH charge
$\hat{Q}=Q/\eta^{1/2}$ and mass $\hat{M}/\eta^{1/2}$, and scalar field
parameters $\hat{\mu}=\mu\eta^{1/2}$ and $\hat{\lambda}=\lambda\eta^{1/2}$.
The vertical line marks the threshold for scalarization $\hat{M}_{\rm th}$.
BHs with $\hat{M}\geqslant\hat{M}_{\rm th}$ are radially unstable, while the
Schwarzschild solution is stable. For large $\hat{\lambda}$, we can obtain
solutions with $\hat{M}<\hat{M}_{\rm th}$ and form two branches. The ones
which are radially unstable solutions are denoted with a dashed line (upper
branch) and those stable to radial perturbations are denoted with a solid line
(lower branch). The dot marks are the solutions marginally stable, which
correspond to a minimum mass, but maximally scalar charged BH for fixed
$\hat{\mu}$ and $\hat{\lambda}$. Credit: Macedo  _et al._ [212].  Figure 20:
Similar to Fig. 19, but now in “mixed” model, where a quadratic coupling
between scalar field and Ricci tensor is added to the quadratic scalar-Gauss-
Bonnet interaction. In this figure, similar to Fig. 19, we show the charge
$\hat{Q}=Q/\alpha^{1/2}$ and mass $\hat{M}/\alpha^{1/2}$ made dimensionless by
division by $\alpha^{1/2}$, for different values of scalar-Ricci coupling
$\beta$. See Eq. (II.2.4) for the theory’s action. The black lines correspond
to values of $\beta$ for which all scalarized BHs have masses below the GR
instability mass threshold, while the red lines mark values of $\beta$ that
lead to scalarized BH masses that are larger than the GR mass threshold. Past
the turning points (marked with blue dots), the BHs are expected to be
unstable to radial perturbations. Credit: Antoniou  _et al._ [25].

A complementary question is whether scalarized BHs are stable to linear
perturbations. The first step in this direction was carried by Blázquez-
Salcedo  _et al._ [49] which considered radial scalar field and metric
perturbations. They showed that Schwarzschild BHs become unstable at the
scalarization threshold and that the stability of the fundamental (i.e., the
zero-node) scalarized solution depends on which model in Eq. (35) is
considered: in the quadratic case all solutions are unstable, whereas stable
solutions exist in the Gaussian model.

This important difference between the two models can be traced back to the
higher-than-quadratic terms in the scalar field in the theory’s action. More
concretely, Silva  _et al._ [311] and Minamitsuji and Ikeda [228] showed the
inclusion of a quartic-interaction, say
$(\zeta/16)\,\varphi^{4}\,{\mathscr{G}}$, is sufficient to yield stable BH
solutions, as long as, $\zeta$ is negative and satisfies $\zeta/\eta<-0.8$.
These two conditions are satisfied by the Gaussian model. Indeed, expanding
Eq. (35a) to order ${\cal O}(\varphi^{4})$, we find $\zeta/\eta=-3/2$, helping
explain the stability of BHs in this model.

It was later shown by Macedo  _et al._ [212] that the original quadratic model
can be made stable under radial perturbations by making the scalar field
massive and self-interacting
$V(\varphi)\propto\mu^{2}\varphi^{2}+\lambda\varphi^{4}$, as discussed in Fig.
19. From an EFT perspective, these terms are of lower order compared to the
quartic scalar-Gauss-Bonnet interaction and hence should be included. A
similar analysis was also performed for the Ricci-Gauss-Bonnet model by
Antoniou  _et al._ [25] and is show in Fig. 20. See also Doneva  _et al._
[108] for a study of scalarization with massive scalar fields.

The nonlinear stability of BHs in this model have also been studied in the
time domain by Ripley and Pretorius [283]. They found evidence for regions in
mass-coupling parameter space in which the end state of the radial instability
of the Schwarzschild BH is a stable scalarized BH. For larger couplings
however, regions where the time-evolution equations change character from
hyperbolic to elliptic appear outside the BH horizon. This signals a regime in
which the theory does not have a well-posed initial-value problem. (See
Hilditch [166], Ripley [282] for discussions.) These results are in agreement
with earlier findings by Blázquez-Salcedo  _et al._ [49], but in the linear
regime. The radial stability of scalarized BHs in the Ricci-Gauss-Bonnet model
introduced in Sec. II.2.4, was analyzed by Antoniou  _et al._ [26]. They found
that although the Ricci scalar does not affect the scalarization threshold, it
can, at the nonlinear level, stabilize the BHs in the quadratic model.

The problem of nonradial perturbations of scalarized BH solutions, which is
relevant in the context of testing such theories with ringdown GW signals, was
addressed in a series of papers by Blázquez-Salcedo and collaborators [47,
48]. On the other hand, the stability of rotating scalarized BHs remain as an
open problem, even in the slow rotating limit.

####  IV.1.3 Spin-induced scalarization

One property of the Gauss-Bonnet invariant of the Schwarzschild metric is that
it is positive definite everywhere and, as a consequence, spontaneous
scalarization can only occur for positive values of the coupling constant.
Does this always have to be the case? In the case of the Kerr metric, it is
known that regions where $\mathscr{G}<0$ outside the outer horizon are
possible if the BH spins with $a/M\geqslant 0.5$ [74] suggesting the
possibility of a _spin-induced scalarization_ when the coupling constant is
negative, i.e., $\varepsilon=-1$ in Eqs. (35).

Although conceptually simple, the problem is not straightforward to analyse
because (i) the analytic form of $\mathscr{G}$ forbids the separation of
variables as done in Sec. I and thus the determination of the scalarization
threshold, in principle, will require more sophisticated numerical methods
compared to the Schwarzschild case and (ii) the Kerr metric (with
$a/M\geqslant 0.5$) has regions where $\mathscr{G}$ can be either negative or
positive, where in the latter regions, for negative coupling constants, the
effective mass may trigger a superradiant instability [64]. To overcome these
difficulties, Dima  _et al._ [101] reduced the $(2+1)$-dimensional
nonseparable scalar field equation into a $(1+1)$-dimensional system of
equations which are coupled through spherical harmonic multipole indices $l$
and $m$. They then performed time-domain numerical integration of the scalar
field borrowing methods previously developed to study superradiant instability
[103]. Dima  _et al._ [101] showed that the tachyonic instability is the
dominant one and charted the parameter space in which spin-induced
scalarization would occur. This is shown in Fig. 21. Hod [167] worked
analytically in the infinitely-large coupling limit ($\eta/M^{2}\to\infty$)
and confirmed the expectation that $a/M\gtrsim 0.5$ is the minimal necessary
spin value for which spin-induced scalarization occurs. The corrections for
large, but finite, coupling were obtained by Hod [168].

Figure 21: Instability timescale $\tau$ for spin-induced scalarization in the
dimensionless spin $a/M$ and dimensionless coupling constant $-\eta/M^{2}$
plane. The instability threshold for the total reconstructed field is shown by
the solid green line, while the threshold when the $m=0$ modes are excluded is
shown by a blue dotted line. The red dashed line corresponds to the
instability threshold for the $m=0$ odd modes, while the dot-dashed cyan line
marks the instability threshold for the spherical mode $l=m=0$. Credit: Dima
_et al._ [101].

Nonlinear spin-induced scalarized BH solutions were obtained by Herdeiro  _et
al._ [164] and by Berti  _et al._ [44]. They confirmed the existence of
scalarized BHs solutions in the parameter space region in which linear theory
predict the spin-induced tachyonic instability. In Fig. 17 we show the domain
of the spin-induced scalarized BHs in the Gaussian model (35a). We see that
scalarized solutions only exist for $j\gtrsim 0.5$, and that their scalar
charge increases with spin. Moreover, there are solutions that can violate the
Kerr bound $j\leq 1$ on BH spins.

A complementary approach to that of Dima  _et al._ [101], was used by Doneva
_et al._ [104], Doneva and Yazadjiev [116]. In these works the scalar field
equation is evolved in (2+1)-dimensions, i.e., without doing a multipolar
decomposition. The influence of a non-vanishing scalar field mass explored in
Doneva  _et al._ [105]. A similar time-domain study, but using the
hyperboloidal foliation method [376], was performed by Zhang  _et al._ [377]
and a model with both the Pontryagin and Gauss-Bonnet invariants was studied
by Myung and Zou [242]. Annulli  _et al._ [19] studied the effect of strong
magnetic fields on spin-induced scalarization. They found that the magnetic
field suppresses this effect, i.e., it pushes the scalarization threshold to
larger values of the dimensionless spin $j$ of the BH.

The case of spin-induced scalarization has been much less studied in terms on
its stability and on how additional nonlinear interactions affect the scalar
charge.

####  IV.1.4 Scalarized black holes in binary systems

Most work in BH scalarization has focused on isolated BHs, but to be able to
confront these models against GW observations one has to turn to the two-body
problem. What phenomenology would we expect in BH binaries? To answer this
question we will first consider works that explored the nonlinear regime of
the late inspiral and merger of BH binaries, a regime that relies on
_numerical relativity_. Next, we will consider works that focus on the
inspiral alone, a regime which can be modeled with PN theory.

Figure 22: Dynamical descalarization in binary BH head on collisions. Scalar
field and Gauss–Bonnet dynamics on the $xy$–plane of the BHs with initial
mass-ratio $q=m_{1}/m_{2}=1/2$. We show the amplitude of $\log_{10}|\Phi|$
(color map) together with the isocurvature levels of the Gauss–Bonnet
invariant at the beginning of the evolution (top left), during the BHs’
approach (top right), shortly before the collision (bottom left) and shortly
after the merger (bottom right). These levels correspond to $1M^{-4}$ (solid
line), $10^{-1}M^{-4}$ (dashed line), $10^{-2}M^{-4}$ (dot-dashed line) and
$10^{-3}M^{-4}$ (dotted line). Credit: Silva  _et al._ [314].

The first study in this direction was done by Silva  _et al._ [314], which
studied the scalar field dynamics (i.e., in the decoupling limit) in head-on
collisions and nonspinning, quasicircular inspirals of binary BH spacetimes
obtained from numerical relativity simulations. This work adapted the methods
developed in Witek  _et al._ [360] for shift-symmetric scalar-Gauss-Bonnet
theory to theories which allow for spontaneous scalarization. In particular,
because they were interested at phenomenology near the scalarization
threshold, Silva  _et al._ [314] adopted the quadratic model (35b), with a
coupling strength $\eta$ such that both (or one, depending on the mass ratio)
binary components or the remnant BH can carry a scalar field bound state
through the simulation. They showed that BH binaries can either form a
scalarized remnant or dynamically descalarize by shedding off the initial
scalar hair (i.e., the scalar bound state configuration) depending on the
values of the coupling constants and the mass ratio between the two BHs. An
example of dynamical descalarization is shown in Fig. 22. Dynamical
descalarization was also shown to occur in the nonlinear scalarization models
(see Sec. II.3.5) by Doneva  _et al._ [110], that simulated head-on BH
collisions and worked in the decoupling limit.

A natural question that follows is: what would happen when $\varepsilon=-1$,
i.e., the case in which spin-induced scalarization occurs? This is relevant
from an observational point of view, because BH remnant of binary BH
coalescences have typical dimensionless spins of $j\sim 0.7$. This value
comfortably meets the criteria for spin-induced scalarization to happen. This
question was explored in Elley  _et al._ [127], which found through a suite of
numerical relativity simulations that: (i) spin-induced dynamical
descalarization can happen, when the remnant BH spin is smaller than that of
the initially spinning (and scalarized) binary components and (ii) spin-
induced dynamical scalarization in an initially nonspinning BH binary system
can result in a scalarized rotating BH. In the latter case, for values of the
coupling constant near the scalarization threshold for the remnant BH, the
tachyonic instability can be delayed to some time $\approx 100M$ after the
binary merger, and thus, hide the scalar field throughout the binary’s
inspiral.

Figure 23: Snapshots of the scalar field, here labeled $\Phi$, and the
$\mathscr{G}$ invariant in the $xz$-plane llustrating the phenomenon of spin-
induced dynamical scalarization. The color map indicates the amplitude of the
scalar field. The isocurvature contours are the same as Fig. 22. of the
$\mathscr{G}$ invariant correspond to $1M^{-4}$ (solid line), $10^{-1}M^{-4}$
(dashed line), $10^{-2}M^{-4}$ (dot-dashed line), and $10^{-3}M^{-4}$ (dotted
line). Black (red) lines correspond to positive (negative) values of
$\mathscr{G}$. We show the inspiral (top left), half an orbit before merger
(top right), formation of the first common apparent horizon (bottom left) and
about $200M$ after the merger. Credit: Elley  _et al._ [127].

A restriction of the decoupling limit is that it does not capture the
backreaction of the scalar field onto the spacetime metric, thus not allowing
one to derive the modifications to the gravitational-waves in the binary set-
ups mentioned previously. These difficulties can be partially overcome by
employing a formulation of the theory’s field equations in a generalized
harmonic gauge, first developed by Kovács and Reall [192, 191] [see also
Ripley [282]]. This formulation in principle allows one to evolve the complete
scalar field and metric dynamics in scalar-Gauss-Bonnet gravity. However, the
first binary BH coalescence simulations using this formalism in scalar-Gauss-
Bonnet theories that allow for BH scalarization [126, 125] demonstrated that
evolution ceases to be well-posed for larger values of the coupling. How
different gauge choices and different models perform in this respect is not
well explored yet.

Franchini  _et al._ [133], motivated by going beyond the small-coupling
approximation, explored _fixing-the-equations_ approach to perform numerical
simulations, an approach inspired by works in dissipative relativistic
hydrodynamics [70]. The main idea is to modify the evolution equations of the
theory, such that the short wavelength modes (with respect to some chosen
scale) in the solution are somewhat controlled, while longer wavelength modes
(associated to strong-coupling and the breakdown of well-posedness) become
tractable. Franchini  _et al._ [133] were able to use the formalism to study
scalar field collapse in spherical symmetry in the scalarization model of
Silva  _et al._ [311]. This allowed them to evolve the system past situations
were the original equations fail, due to appearance of regions in spacetime
where the evolution changes character from hyperbolic to elliptic, which
prevents further evolution of the system in time [283]. Moreover, the “fixed-
equations” system evolves towards static BHs predicted by the original system
of equations, thus indicating that it may provide a suitable “completion” of
the original theory. Whether this approach remains valid in a situation with
less symmetry, for example the case of BH binaries, remains to be explored.

Progress has also been made in modeling compact binary dynamics in the wider
separation and lower velocities, i.e., in the domain of validity of PN theory.
In particular, Shiralilou  _et al._ [306, 307] extended to a general
$f(\varphi)$ coupling the calculations of Yagi  _et al._ [364], that applied
for the shift-symmetric scalar-Gauss-Bonnet theory, i.e.,
$f(\varphi)\sim\varphi$. These works calculated the leading-order corrections
due to curvature nonlinearities in the GW and scalar waveforms, finding that
corrections due to the Gauss-Bonnet term appear at 1PN order in GWs. They also
obtained the GW polarization and phasing. In addition, Julié and Berti [179]
argued that during the adiabatic inspiral of two BHs, the Wald entropy of each
BH is constant. This allows a precise definition of the “sensitivities” of BHs
in scalar-Gauss-Bonnet gravity in terms of the variation of the ADM mass as a
function of the ambient scalar field the BH is embedded in, at fixed Wald
entropy. This parallels the notion that the baryonic mass of NSs is constant
during the inspiral [88]. They also derived the two-body Lagrangian at 1PN
order. Julié  _et al._ [180] used this result in conjunction with the
numerical calculation of sensitivities of spontaneously scalarized BHs to
study binaries. They found that, in principle, BHs can evolve towards a
situation in which the inner singularity approaches the event horizon of the
BH before they merge. This suggest the possibility that, dependent on the
model of scalar-Gauss-Bonnet gravity and the binary parameters, the field
equations might simply not be able to predict a full binary evolution from
inspiral to merger.

###  IV.2 Black hole scalarization in the presence of matter

Stefanov  _et al._ [335] suggested the earliest model of BH scalarization
which consisted of the DEF model coupled to nonlinear electromagnetism [334,
333]. They introduced a coupling between the scalar field and the Born-Infeld
Lagrangian, and found that BH solutions with scalar hair branch-off from the
general relativistic sequence of solutions. These BHs violate the no-scalar-
hair theorems by Mayo and Bekenstein [216], Sen and Banerjee [301] which are
valid for charged, nonrotating BHs in scalar-tensor theory. The bifurcation
point occurs where the GR sequence of solutions becomes unstable to scalar
perturbations [119].

A variety of scalarization models have been studied in Einstein-Maxwell theory
following Herdeiro  _et al._ [163], which introduced a coupling between the
scalar field and the Lorentz invariant ${\cal F}^{2}=F_{\mu\nu}F^{\mu\nu}$ of
the form $\exp(-\alpha\varphi^{2}){\cal F}^{2}$, where $\alpha$ is a
dimensionless constant. In this model, BHs can scalarize, with ${\cal F}^{2}$
playing the role of the trace of the energy-momentum tensor in the NS
scalarization in DEF-like models, or the Gauss-Bonnet invariant in BH
scalarization models.

The simpler nature of the model allowed Herdeiro  _et al._ [163] to perform
numerical relativity simulations to study the scalarization process in the
time-domain by expanding upon already available code bases [296, 295]. This
allowed Herdeiro  _et al._ [163] to show that the endpoint of the instability
of the Reissner-Nordström BH is indeed a BH with scalar hair. Additional
studies of this model were done by Fernandes  _et al._ [132], Astefanesei  _et
al._ [29], Herdeiro  _et al._ [159]. Within this model, Myung and Zou [240]
studied the instability of the Reissner-Nordström solution, while Myung and
Zou [238, 241], Blázquez-Salcedo  _et al._ [51] analyzed the stability of the
scalarized solutions. Fernandes  _et al._ [131], Herdeiro and Oliveira [160]
explored other variations of this theme, including scalar-axion couplings,
i.e., $\propto h(\varphi)F^{\mu\nu}\tilde{F}_{\mu\nu}$, where
$\tilde{F}_{\mu\nu}=(1/2)\epsilon\indices{{}_{\mu\nu}^{\alpha\beta}}F_{\alpha\beta}$
In addition, Ikeda  _et al._ [175] considered the coupling between the double-
dual of the Riemann tensor and $F_{\mu\nu}$.

Cardoso  _et al._ [67, 68] explored a relevant scenario from an astrophysical
perspective: matter in the vicinity of a BH could trigger scalarization and,
in turn, endow the BH with scalar hair. They showed that this “matter-induced”
scalarization (see Sec. II.3.3) is, in principle, possible for the DEF model.
The viablility of this proposal in realistic matter configurations, such as an
accretion disk or a dark matter halo, has not been explored yet. Matter-
induced scalarization seems unlike to happen in light of the severe
constraints on the DEF model. However, this might not be the case for other
scalarization models, and the topic deserves further work.

After it was understood that in scalar-Gauss-Bonnet theories the Schwarzschild
solution can scalarize, it was natural to ask if the same can happen to its
charged counterpart, i.e., the Reissner-Nordström solution. Doneva  _et al._
[106] analyzed this and found the existence of two bifurcation points, one at
larger masses where the scalarized solutions bifurcated from the Reissner-
Nordström one, and one at smaller masses where the scalar charge of the
solution decreases again to zero and the branch merges again with the GR one.
Scalarized charged BHs (in the fundamental mode) are also thermodynamically
favored over the Reissner-Nordström solution. Brihaye and Hartmann [60] showed
that for near-extremal Reissner-Nordström case, scalarization can happen for
either sign of the scalar-to-Gauss-Bonnet couplings constant (35b). Other
works varying the choice of the coupling function $f$ between the scalar field
(or the axion field) and the Maxwell invariant, and studying the scalarization
of dyonic BHs include Fernandes  _et al._ [132], Fernandes  _et al._ [131],
Blázquez-Salcedo  _et al._ [52] and Blázquez-Salcedo  _et al._ [51].

Herdeiro and Radu [162] put forward a motivation to study these models. They
noted that quantum effects can break the scale invariance and vanishing
energy-momentum trace properties of electro-vacuum GR BHs. As a consequence,
even the simplest nonminimally scalar field coupling $\xi\varphi^{2}R$ can
result in BH scalarization when these quantum effects are taken into account.
Within these models, they discussed the scalarization of the Reissner-
Nördstrom solution and a noncommutative generalization of the Schwarzschild
solution. They found that resultant scalarized BHs are, in general,
entropically favored over the GR solutions.

###  IV.3 Variation of the curvature-induced scalarization model

In light of the nontrivial effect of rotation on scalarization, it is quite
natural to consider what happens when Gauss-Bonnet invariant is replaced with
the Pontryagin invariant ${}^{\ast}RR$ as the “curvature source”. The
Pontryagin density is known to vanish in spherically symmetric spacetimes
(such as that of Schwarzschild BH) and becomes nonzero for stationary and
axisymmetric spacetimes (such as that of a Kerr BH). In this sense, all
possible scalarized BHs in this theory are necessarily “spin induced”; a fact
naturally shared with BH solutions in dynamical Chern-Simons gravity (see
e.g., Yunes and Pretorius [373], Konno  _et al._ [190], and Alexander and
Yunes [9] for a review) which features a linear coupling between the scalar
field and the Pontryagin invariant. We remark that the coupling between scalar
field and Pontryagin density results in field equations with higher order
derivatives (contrary to the case of Gauss-Bonnet) and, as such, these models
are usually treated as EFTs, see e.g., Motohashi and Suyama [235], Delsate
_et al._ [98]. Myung and Zou [239] studied the combined effect of Gauss-Bonnet
and Pontryagin densities. In the test field limit, the scalar field dynamics
with an effective mass proportional to $\phi^{2}\,{}^{\ast}RR$ was also
studied in a Kerr background [139, 117] and in the Schwarzschild-NUT
background [62].

Another extension of the original “Gauss-Bonnet” spontaneous scalarization
model involves considering $n>1$ scalar fields $\varphi^{a}$, whose
interaction is determined by their “target-space” $\gamma_{ab}(\varphi^{c})$,
an $n$-dimensional Riemannian manifold [86, 169]. These are similar to the
tensor-multi-scalar DEF models for NSs scalarization presented in Sec. III.4.
In these models, the scalar field dynamics is determined by the quantity
$\gamma_{ab}(\varphi)g^{\rho\sigma}\partial_{\rho}\varphi^{a}\partial_{\sigma}\varphi^{b}$.
Doneva  _et al._ [109] numerically obtained scalarized BH for the case $n=3$
and maximally symmetric target-space geometries. An important feature of these
solution is that the scalar fields do not decay asymptotically as $1/r$ (i.e.,
the BHs do not have a monopole scalar charge). This shows that BHs in these
theories will not emit dipole scalar radiation when placed in binaries.

##  V Generalizations of scalarization to other fields and instabilities

We identified the underlying reason for spontaneous scalarization in its
various different forms to be a tachyonic instability. However, the scalar
nature of the field did not have any special role in the mechanism. This
suggests that other fields, e.g. vectors, might also spontaneously develop
nontrivial configurations around compact objects when they exhibit suitable
couplings to curvature. In this section, we will investigate this idea, which
leads to the so-called _spontaneous tensorization_ theories.

A second type of generalization of the spontaneous scalarization considers new
types of instabilities as opposed to new types of fields. For instance,
instead of replacing the scalar field with, say, a vector, replacing the
tachyonic instability with, say, a ghost instability. We will see that a
surprising key result of spontaneous tensorization is that these two paths are
intimately connected. Namely, even if we only intend to have a theory of
tachyons living on nonscalar fields, ghost and gradient instabilities
necessarily arise in almost all models.

###  V.1 Spontaneous vectorization

What happens if we replace the scalar field of spontaneous scalarization with
a vector? In scalarization, we started with the most general context in Sec.
II.2.1, considering all the allowed couplings to the metric. Here we will
follow the reverse path, starting with individual models of vector-tensor
theories, and considering the more encompassing theory later. This exposition
is preferable because the study of specific models, that are straightforward
generalizations of known scalarization models, reveal some pathologies, and
provides guidance for further model building.

The idea and the first specific model of vectorization was introduced in
analogy with a massive version of the DEF model in Eq. (18). Consider the
vector-tensor theory action [43, 266]

$\displaystyle S$ $\displaystyle=\frac{1}{16\pi G_{*}}\int{\rm
d}^{4}x\,\sqrt{-g}\,\left[R-F^{\mu\nu}F_{\mu\nu}-2\mu_{X}^{2}x\right]$
$\displaystyle\quad+S_{{\rm m}}\left[\Psi_{{\rm m}};\,{\cal
A}^{2}(x)g_{\mu\nu}\right],$ (36)

where $F_{\mu\nu}=\nabla_{\mu}X_{\nu}-\nabla_{\nu}X_{\mu}$,
$x=g^{\mu\nu}X_{\mu}X_{\nu}$. The vector field $X_{\mu}$ has the canonical
kinetic term in this frame, and the matter degrees of freedom couple to the
metric $\tilde{g}_{\mu\nu}={\cal A}^{2}(x)g_{\mu\nu}$ which is conformally
scaled with respect to $g_{\mu\nu}$. The vector field equation is

$\displaystyle\nabla_{\rho}F^{\rho\mu}$ $\displaystyle=[-8\pi G_{*}{\cal
A}^{4}\Lambda\tilde{T}+\mu_{X}^{2}]\,X^{\mu},$ (37)

where $\Lambda\equiv{\rm d}\ln{\cal A}/{\rm d}x$, $T_{\mu\nu}$ is the energy-
momentum tensor in the Einstein frame, and $\tilde{T}$ is the trace of the
stress-energy tensor in the Jordan frame, i.e., with respect to the metric
$\tilde{g}_{\mu\nu}$.

Equation (37) is that of a massive vector (Proca) field where the expression
inside the square brackets acts as an effective mass $\mu_{\rm eff}^{2}$. In
parallel to the DEF model, when $\cal A$ has an appropriate form, e.g. ${\cal
A}=\exp(\beta_{0}x/2)$ with sufficiently negative $\beta_{0}$, $\mu_{\rm
eff}^{2}$ becomes negative. Furthermore, for dense enough objects such as NSs,
this occurs for order-of-unity values of $\beta_{0}$. The expectation is that
the vector field grows from arbitrarily small perturbations around $X_{\mu}=0$
due to this tachyonic behavior, which can be called spontaneous vectorization,
in exact analogy to spontaneous scalarization. However, we will later see that
there are many subtle points in this narrative.

Even though we presented spontaneous vectorization in the Einstein frame with
a nonminimal coupling to matter, it can be converted to a theory of minimal
matter coupling and vector-curvature couplings as in spontaneous scalarization
[270]. Even more directly, there are theories of spontaneous vectorization
that are purely conceived through curvature coupling, the first example being
the Hellings-Nordtvedt theory [158] studied by Annulli  _et al._ [18]

$\displaystyle S$ $\displaystyle=\frac{1}{16\pi G}\int{\rm
d}^{4}x\,\sqrt{-g}\,\left[R-F^{\mu\nu}F_{\mu\nu}-\Omega xR\right.$
$\displaystyle\quad\left.-\,\eta X^{\mu}X^{\nu}R_{\mu\nu}\right]+S_{{\rm
m}}[\Psi_{{\rm m}};\,g_{\mu\nu}],$ (38)

where $\Omega$ and $\eta$ are coupling constants. Scalarization through
coupling to the Gauss-Bonnet term also has a vector analog as in the action
[271]

$\displaystyle S$ $\displaystyle=\frac{1}{16\pi G}\int
d^{4}x\sqrt{-g}\left[R-F^{\mu\nu}F_{\mu\nu}+f(x)\mathscr{G}\right]$
$\displaystyle\quad+S_{{\rm m}}[\Psi_{{\rm m}};\,g_{\mu\nu}],$ (39)

which has a similar structure to Eq. (II.2.3). Action (V.1) leads to BH
vectorization as well, unlike the other models we have seen so far.

The models of vectorization we have examined are only specific examples, and
any theory of scalarization can be generalized to vector fields in principle.
In this regard, Brihaye and Verbin [63] studied the coupling of a vector field
to a scalar, and Oliveira and Pombo [249] and Brihaye  _et al._ [61] studied
the coupling between two vector fields to obtain vectorized compact objects.
Kase  _et al._ [183] considered generalized Proca theories [157, 343] with
various different couplings of a massive vector field, and showed that
spontaneous vectorization occurs in these models as well. Ramazanoğlu [268]
investigated models where the effective mass of the vector field is generated
by the Higgs-mechanism, preserving the gauge symmetry. On a separate front,
conformal scaling of the metric in the matter action (V.1) can be generalized
to disformal transformations, and spontaneous vectorization still occurs [274,
227]. In terms of approximate solutions, Garcia-Saenz  _et al._ [141] recently
calculated the Schwarzschild QNMs of nonminimally coupled vector fields. There
have also been efforts to study all these phenomena in a unified manner in
compact binaries, using more generic EFT tools [185].

To summarize, coupling of the vector field to any non-vanishing term in the
action can be considered for spontaneous vectorization [271], and most options
have been considered at least in a preliminary sense. The case of all possible
couplings, an analog of the minimal action of scalarization in Sec. II.2.1,
has also been studied by Garcia-Saenz  _et al._ [140]. This work also revealed
some of the fundamental problems of vectorization, as we will discuss more
broadly later.

Vectorizations models strongly resemble scalarization models in terms of their
actions, but this can be misleading. One major difference is in the number of
additional degrees of freedom. Scalar fields contribute a single new degree of
freedom irrespective of whether they are part of spontaneous scalarization or
not, provided that the field equations are second order in derivatives. For
vectorization, this is not the case. The action (37) and all the vector-tensor
theory actions we have discussed break the gauge freedom found in massless
vector fields.121212Even though we have an intrinsic mass $\mu_{X}$ for the
vector field in our discussion, vectorization can occur without this term, as
is the case for scalarization. As a result, the vectors of vectorization
models carry three degrees of freedom instead of the two found in
electromagnetism. This is not an immediate reason of concern, since minimally
coupled massive vectors, known as Proca fields [265], also break the gauge
symmetry, and they still provide a classical field theory with well-behaved
dynamics. However, as we will see, the extra degree of freedom is far more
problematic in vectorization.

Although the vectorization models are designed to have a linear tachyonic
instability in analogy with scalarization, the presence of the third degree of
freedom, whose dynamics is not obvious, casts doubt on the success of this
design. Namely, whether vectorization can indeed be described as a tachyonic
instability of the vector modes, that is then quenched nonlinearly is not
apparent. This concern is amplified by the fact that vectorized compact
objects in the models studied so far seem to have some striking differences
with respect to their scalarized counterparts.

As an example, consider vectorized BHs in the theory described by action
(V.1). Entropy can be a measure of stability for BHs, where higher entropy
solutions are favored in terms of stability. Entropy of spherically symmetric
vectorized BHs were numerically computed by Barton  _et al._ [35], and GR BHs
were shown to be entropically favored over vectorized ones in all cases. The
verdict of stability is not definite without time evolution, but this is a
strong indication for the instability of vectorized BHs. In contrast,
scalarized BHs in scalar-Gauss-Bonnet theories, e.g., Eq. (II.2.3), can be
entropically favored to the ones in GR for appropriate coupling functions
[113], and their stability is also indicated by other methods (see Sec.
IV.1.2), which makes the stability of vectorized objects even more suspect.

Figure 24:  The strength of vectorization measured by ${\cal A}(r=0)-1$ as a
function the NS mass $M$ for various values of $\beta_{0}$ and $\mu_{X}$ in
Eq. (37). Rows: upper $\mu_{X}=1.6\times 10^{-11}$ eV, middle
$\mu_{X}=8.0\times 10^{-12}$ eV, lower $\mu_{X}=4.8\times 10^{-12}$ eV.
Columns: left $\beta_{0}=-4$, middle $\beta_{0}=-5$, right $\beta_{0}=-6$.
Dashed and dot-dashed lines indicate solutions where the energy density
$\varepsilon$ or $\Phi=-n^{\mu}X_{\mu}$ ($n^{\mu}$ being the normal vector
field to spatial hypersurfaces) do not monotonically decrease with radius,
respectively. Credit: Ramazanoğlu [266].

Scalarized and vectorized NSs also show major differences. For example, the
dependence of the strength of vectorization on the NS mass has no discernible
pattern as can be seen in Fig. 24, whereas scalarization occurs in a finite NS
mass interval, and vanishes at the boundary mass values (see Fig. 1). It is
also known that while scalarized NSs are continuously connected to those of GR
as the theory parameters are smoothly changed, e.g. in the limit $\beta\to 0$
in Fig. 1, this is not the case in vectorization [266, 225]. The latter is
known to be an indication of instability in certain scalarization theories,
see Mendes and Ortiz [219]. Thus, qualitative differences between scalarized
and vectorized objects are also apparent for NSs, and moreover, point to the
instability of the vectorized ones.

These observations suggest that the vectorized compact object solutions
obtained so far may not be stable, hence they are not astrophysically
relevant, at least for the case of spherically symmetric spacetimes. A close
look at the instability mechanism of vectorization is going to reveal the
underlying reasons for this in the following section.

###  V.2 Vectorization and ghosts

Despite the fact that the vectorization phenomena we have presented are
directly inspired by scalarization, we have seen noteworthy differences
between the two mechanisms. Even though it was puzzling for a time, this is
now strongly suspected to be related to the fact that vectorization models
suffer from ghost and/or gradient instabilities in addition to tachyonic ones,
as demonstrated by Garcia-Saenz  _et al._ [140] and Silva  _et al._ [308].
Here, we will first follow the latter, and later outline the methodology of
the former.131313Esposito-Farèse  _et al._ [130] is an earlier similar study
that specifically investigates cosmological scenarios.

Let us first review the basic aspects of different types of instabilities
[99]. Recalling our discussion of the tachyon in Secs. II.1.1 and II.1.2,
consider the linearized scalar field equation (5) in $1+1$ dimensions

$\displaystyle
g^{tt}\partial_{t}^{2}\delta\varphi+g^{xx}\partial_{x}^{2}\delta\varphi=\mu^{2}\varphi,$
(40)

where we assume the metric is diagonal with constant components for
simplicity. The common case without instabilities is when $g^{tt}<0$,
$g^{xx}>0$, and $\mu^{2}>0$. What happens when each of these three terms
changes its sign, while the other two stay the same?

For a plain wave mode $\delta\varphi(t,x)\propto\exp[{\rm i}\left(\omega
t-kx\right)]$, the dispersion relation is given by

$\displaystyle\omega(k)=\sqrt{(\mu^{2}+g^{xx}k^{2})/(-g^{tt})},$ (41)

We saw that for $\mu^{2}<0$ the mode behaves as a tachyon, where $\omega(k)$
becomes imaginary for sufficiently small $|k|$. This leads to exponential
growth as discussed, and the fastest growing mode behaves as
$\sim\exp[\sqrt{\mu^{2}/g^{tt}}\,t]$, i.e., the growth rate has an upper
limit.

A _ghost_ occurs when $g^{tt}>0$. There is also exponential growth in this
case, however the growth rate diverges with increasing wave number as
$\sim\exp[\sqrt{g^{xx}/g^{tt}}kt]$, hence has no upper bound. The case
$g^{xx}<0$ also gives the same asymptotic growth rate as the ghost, and is
called a _gradient instability_.141414For the simple equation (40), a
simultaneous change of sign of two of the coefficients is equivalent to
changing the sign of the third one, i.e. $g^{tt}>0$, $g^{xx}<0$ and
$\mu^{2}>0$ is also a tachyon. Similarly, all three coefficients changing
signs gives a stable theory. This is due to the fact that the equation as a
whole or the action that leads to it can be multiplied by an overall factor of
$-1$ without changing physics. However, the scalar, or any field we will
investigate which carries instabilities, is always coupled to the metric and
other terms in the action, which means the overall sign is meaningful. Hence,
we will use the aforementioned classification, and say that e.g. a theory with
$g^{tt}>0$, $g^{xx}<0$ and $\mu^{2}>0$ carries both ghost and gradient
instabilities following the literature [99]. In summary, ghost and gradient
instabilities as we presented them here have qualitative differences from
tachyons, a main one being the arbitrarily fast growth rates. This means a
well-defined time evolution can be problematic in such theories, and it is
harder to devise nonlinear quenching mechanisms.

The spontaneous vectorization model of Eq. (V.1) modifies the “mass term” in
the field equation for the vector field. Hence, the first impression might be
that it leads to a tachyonic instability as in scalarization. However, this
approach is naive, and a thorough analysis reveals that theories of
vectorization necessarily carry ghost instabilities in addition to tachyonic
ones. Following Silva  _et al._ [308], one way to understand this is using the
Stueckelberg trick [287] where we introduce a scalar field $\psi$ in the
action (V.1) by the substitution

$\displaystyle X_{\alpha}\to X_{\alpha}+\mu_{X}^{-1}\nabla_{\alpha}\psi.$ (42)

This leads to the scalar equation of motion

$\displaystyle\overline{\Box}\psi=-\bar{g}^{\mu\nu}\,[\,\mu_{X}\overline{\nabla}_{\mu}X_{\nu}+\tfrac{1}{2}(\overline{\nabla}_{\mu}\log\hat{z})(\overline{\nabla}_{\nu}\psi+\mu_{X}X_{\nu})\,].$

where the covariant derivative $\overline{\nabla}$ is compatible with the new
metric

$\bar{g}^{\mu\nu}=\hat{z}\,g^{\mu\nu},\qquad\hat{z}\equiv\frac{\mu^{2}_{\rm
eff}}{\mu_{X}^{2}}=1-\frac{8\pi G_{*}{\cal
A}^{4}\Lambda\tilde{T}}{\mu_{X}^{2}}.$ (44)

The signature of $\bar{g}_{\alpha\beta}$ changes with the sign of $\hat{z}$,
which controls the change of sign of the effective mass of $X_{\mu}$ in Eq.
(37). In other words, when we try to instigate a tachyon on $X_{\mu}$ by
having negative $\hat{z}$ in parts of the spacetime, $\psi$ carries ghost and
gradient instabilities [cf. Eq. (40)]. This is a degree of freedom whose
dynamics is governed by the _effective metric_ $\bar{g}_{\mu\nu}$, not the
spacetime metric $g_{\mu\nu}$.

It is curious to see that the Stueckelberg trick reveals the ghost, even
though it is not apparent in the vector field equation (37). However, a more
careful analysis can clarify this issue [308]. First note that the vector
field equation (37) is not manifestly hyperbolic, i.e. its principal part, the
term with the highest derivatives, is not the wave operator, since

$\nabla_{\mu}F^{\mu\nu}=\nabla_{\mu}\nabla^{\mu}X^{\nu}-\nabla_{\mu}\nabla^{\nu}X^{\mu}.$
(45)

The second term on the right hand side of Eq. (37) vanishes for the Proca
theory, ${\cal A}=1$, making the principal part the wave operator. This is not
the case in general due to the so-called (generalized) _Lorenz condition_

$\nabla_{\mu}\left(\hat{z}X^{\mu}\right)=0,$ (46)

which is obtained by acting on both sides of Eq. (37) by $\nabla_{\mu}$, and
recalling the antisymmetry of $F^{\rho\mu}$. We can use Eq. (46) to obtain a
manifestly hyperbolic form of the linearized field equations on a fixed
background of vanishing vector fields as

$\displaystyle\nabla_{\mu}\nabla^{\mu}X_{\nu}+(\nabla_{\mu}\ln\hat{z})\nabla_{\nu}X^{\mu}=\mathcal{M}_{\nu\mu}X^{\mu},$
(47)

where we defined the mass-squared tensor

$\displaystyle\mathcal{M}_{\nu\mu}=\hat{z}\mu_{X}^{2}g_{\nu\mu}+R_{\nu\mu}-\nabla_{\nu}\nabla_{\mu}\ln|\hat{z}|.$
(48)

Here, the metric and $\hat{z}$ have their fixed background GR values. This is
a generalized massive wave equation.

Since $\hat{z}=1$ in the absence of matter, and it becomes negative in a
continuous manner inside matter for an instability to exist at all, $\hat{z}$
necessarily vanishes at some points of a NS spacetime. The
$\nabla_{\nu}\nabla_{\mu}\ln\hat{z}$ term introduced by the vectorization
contains powers of $\hat{z}^{-1}$, which means the effective mass-squared
tensor diverges at such points. Alternatively, if we move the $\hat{z}^{-1}$
terms to the other side of the equation, the principle part becomes
$\hat{z}\Box X_{\nu}$, i.e. it changes its sign with $\hat{z}$ the same way as
the $\psi$ field in equation (V.2). Hence, if we properly rewrite the vector
field equation into a manifestly hyperbolic form, the ghost is
apparent.151515There is no change of sign in the case of intrinsically
massless vector fields, $\mu_{X}=0$, but it can be shown that the coefficients
in the field equations still diverge [308].

Equations (47) and (48) show that calling $\mu^{2}_{\rm
eff}=\hat{z}\mu_{X}^{2}$ the effective mass squared of the theory is
misleading, unlike the case of scalarization. Even though $\mu^{2}_{\rm eff}$
replaces the mass term in the vector field equation of a minimally coupled
Proca field, it is not actually the effective mass of physically propagating
degrees of freedom. The _physical mass_ is determined by Eq. (48), which
diverges at points where $\hat{z}=0$.

One might at first think that the ghost and gradient instabilities can be
regularized by nonlinearities. After all, the tachyon of scalarization is also
an instability, but it is eventually quenched by nonlinear effects as
explained in Sec. II.1. In such a scenario, one would expect the vectorized
objects to be free of instabilities, even though the GR solutions are
unstable. However, this is not the case. Demirboğa  _et al._ [99] showed that
spherically symmetric vectorized NSs in the theory (V.1) are ghost and
gradient unstable at the perturbative level.

Even if stable solutions can be found in other vectorization models, there is
a more fundamental problem due to the modification to the principal part of
the vector equations in Eq. (V.2) or (47). The coefficient of the wave
operator vanishes at certain points in spacetime, or equivalently, the
physical effective mass diverges if we move these coefficients to the other
side of the equation. In either case, this leads to divergent time derivatives
for arbitrarily small perturbative vector field values, which renders the
initial assumptions of a perturbative approach invalid. Overall, it is not
clear whether one can even define a well-posed time evolution, let alone one
that can somehow suppress the exponential growth of the vector field [308].
Further elucidation of these issues requires a more rigorous mathematical
investigation, which has not been attempted yet.

We have so far investigated the ghost and gradient instabilities of the
specific vectorization theory in action (V.1), but similar results are known
to exist for all theories of vectorization [140, 308] due to similar
mechanisms. To summarize, replacing the scalar of scalarization with a vector
field results in theories where the central issue is not the nature of the
field the instability lives on, but the type of instabilities that arise. We
cannot construct vectorization models that only exhibit the tachyonic
instability commonly associated with the scalarization mechanism, which can be
benign and tamed by nonlinear effects. Rather, vectorization also exhibits far
more threatening instabilities.

Above, we closely followed Silva  _et al._ [308], but Garcia-Saenz  _et al._
[140] reach similar conclusions via an alternative approach. The authors
consider the most general action that contains a vector field and the metric
[157, 343], and truncate its action at the quadratic order around GR to obtain
the linearized field equations. The main qualitative difference from the
scalar minimal action of Sec. II.2.1 is that one can demonstrate that if any
of the physical degrees of freedom has a tachyonic instability, then there are
also degrees of freedom with ghost or gradient instabilities, which are
manifested by change of sign of coefficients in the dispersion relationship.
The conclusion is the same: vectorization based on tachyonic instabilities
necessarily has ghost or gradient instabilities as well.

Recent works showed that the issues with vectorization also happen in simpler
theories, an example being minimally coupled self-interacting vector field
theories. For instance, for $\mathcal{A}(x)=1$ the action (V.1) becomes the
minimally coupled Proca theory, which is well-posed. However, generalizations
of the potential beyond a mass term, that is $2\mu_{X}^{2}x\to 4V(x)$ for some
generic function $V$, results in ill-posedness [236, 79, 81]. The underlying
reason is the fact that there is an analog of the generalized Lorenz condition
(46) in this case as well, which ultimately leads to the same result: there is
a degree of freedom in the theory whose dynamics is governed by an effective
metric, an analog of $\bar{g}_{\mu\nu}$ in Eq. (44). This new metric depends
on the vector field itself, and can lose its Lorentzian nature depending on
how $X_{\mu}$ evolves in time [81].

Esposito-Farèse  _et al._ [130] provided some of the earliest examples of the
ill-posedness of self-interacting vector field theories we mentioned above in
a cosmological context, and it was recently demonstrated that initially
healthy configurations of self-interacting vectors naturally evolve to a point
where hyperbolic evolution becomes impossible [236, 79, 81]. This means,
nonlinear extensions of the Proca theory are more delicate than their scalar
counterparts in terms of providing physical theories. It seems that the
underlying reason for the “failure” of vectorization is a much more general
phenomenon in theories with conditions such as Eq. (46), at least for the
models conceived thus far. In this sense, exploration of vectorization has
been leading to a deeper understanding of vector fields in general, which has
implications beyond theories of gravitation.

We close this section with a pertinent comment on ghost instabilities. We
encountered them as an unappealing artefact in vectorization theories, but one
could consider making them the driver of a spontaneous growth mechanism. This
option has indeed been considered prior to their discovery in vectorization
models. The simplest example is another analog of the DEF model where the
conformal scaling function ${\cal A}$ depends on derivative terms [267]

$\displaystyle S$ $\displaystyle=\frac{1}{16\pi G_{*}}\int
d^{4}x\sqrt{-g}\left[R-2g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi-2\mu^{2}\varphi^{2}\right]$
$\displaystyle\quad+S_{{\rm m}}[\Psi_{{\rm m}};{\cal A}^{2}(K)g_{\mu\nu}],$
(49)

with $K=-g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi/2$. The scalar
equation of motion is

$\displaystyle\nabla_{\mu}\left(\hat{z}\nabla^{\mu}\varphi\right)=\mu^{2}\varphi,$
(50)

where $\hat{z}\equiv 1+4\pi G_{*}\tilde{T}{\cal A}^{4}\left({\rm d}\ln{\cal
A}/{\rm d}K\right)$. Unlike the tachyonic instability where the mass term is
modified, the derivative terms are modified in this case.

The nature of Eq. (50) can be better understood when we consider the
linearized equation for perturbative values of the scalar around a GR
background, and highlight the principal part as

$\displaystyle\hat{z}\,\Box\delta\varphi+\dots=0.$ (51)

For an appropriate choice such as ${\cal A}=\exp(\beta_{0}K)$ with large
enough $\beta_{0}>0$, the sign of $\hat{z}$ can become negative in the
presence of matter. In all spontaneous scalarization theories considered so
far, the source of the instability was the change of sign of the effective
mass term, which resulted in a tachyon. Here, the overall sign of the wave
operator changes, hence we have both ghost and gradient instabilities,
recalling Eq. (40) and the subsequent discussion. Spontaneous scalarization
that might arise from this new mechanism is called _ghost-based spontaneous
scalarization_ [267]. It has similar problems as vectorization, in that it is
hard to tame the ghost and other instabilities, hence we do not end up with a
theory with sensible dynamics.

###  V.3 Spontaneous tensorization

The basic mechanism of spontaneous scalarization can be generalized beyond
vector fields, resulting in _spontaneous tensorization_. However, all known
such theories also feature ghosts and suffer from similar problems with
vectorization, aside from a single possible example of spinor-tensor theory.

The simplest mathematical generalization of scalarization beyond vectors
occurs for $p$-form fields, i.e., totally antisymmetric tensors
$X_{\mu_{1}\dots\mu_{p}}$ of rank $(0,p)$. This is due to the fact that a
vector field $X_{\mu}$ is a $1$-form field, and the actions we encountered in
spontaneous vectorization can be readily generalized to this class of fields
as [272]

$\displaystyle S$ $\displaystyle=\frac{1}{16\pi}\int{\rm
d}^{4}x\,\sqrt{-g}\,\left[R-F_{\mu_{1}\dots\mu_{p+1}}F^{\mu_{1}\dots\mu_{p+1}}\right]$
$\displaystyle\quad+S_{{\rm m}}\left[\Psi_{{\rm m}};\,{\cal
A}^{2}(x)g_{\mu\nu}\right],$ (52)

where
$F_{\mu_{1}\dots\mu_{p+1}}=(p+1)\nabla_{[\mu_{1}}X_{\mu_{2}\dots\mu_{p+1}]}$,
with square brackets denoting antisymmetrization, and
$x=X_{\mu_{1}\dots\mu_{p}}X^{\mu_{1}\dots\mu_{p}}$. It is then straightforward
to show that the analog of $\mu^{2}_{\rm eff}$ becomes negative inside NSs for
an appropriate choice of ${\cal A}(x)$. However, these theories carry ghost
instabilities similar to those of vectors when we try to establish spontaneous
growth from tachyonic instabilities, since they possess an analog of the
Lorenz condition (46) [308].

Spontaneous tensorization of a symmetric rank $(0,2)$ tensor field
$f_{\mu\nu}$, which is spin-$2$, is a natural avenue to investigate after
spin-$0$ (scalars) and spin-$1$ (vectors). Ghosts appear in this case once
more since the metric in our gravity theories, $g_{\mu\nu}$, is also a
spin-$2$ field. Note that spontaneous tensorization theories include terms
where the field that tensorizes, in this case $f_{\mu\nu}$, is coupled to the
metric $g_{\mu\nu}$. However, interacting spin-$2$ fields are known to
generically lead to ghost degrees of freedom, rendering most such theories
unphysical [58, 277], and ghost-free theories being discovered more recently
in the form of massive gravity and bigravity [278, 155, 277]. Matter can
couple to one of the spin-$2$ fields in the novel ghost-free theories, but
coupling to both metrics generically invoke a ghost again [369]. For example,
trying to mimic the DEF model by having a conformal scaling function that
depends on $f_{\mu\nu}$, i.e., changing the matter action as
$g_{\mu\nu}\to{\cal A}(f_{\mu\nu},g_{\mu\nu})\ g_{\mu\nu}$, leads to ghosts.
Overall, there is no known form of bigravity theory that features an analog of
spontaneous scalarization.

Spinor fields present the most interesting case of generalizing scalarization,
perhaps aside from vectors.161616Although a spinor is not a tensor in the
technical sense, we will classify spinorization as an example of
tensorization. The classical Lagrangian for a Dirac spinor is

$\displaystyle\mathcal{L}_{\psi}=\tfrac{1}{2}\left[\bar{\psi}\gamma^{\mu}(\nabla_{\mu}\psi)-\nabla_{\mu}\hat{\gamma}^{5}\gamma^{\mu}\psi\right]-\mu\bar{\psi}\psi,$
(53)

where the conventions for gamma matrices and the effect of covariant
derivatives on spinors can be found in Ramazanoğlu [269]. This action provides
the usual dispersion relation $\omega^{2}=k_{i}k^{i}+\mu^{2}$ for a plane wave
of the form $\psi\propto\exp[{\rm i}(\omega t-k_{i}x^{i})]$. Since the mass
term appears linearly, not quadratically, changing its sign results in the
same dispersion relation. However, a tachyonic spinor action is still possible
in the form of

$\displaystyle\mathcal{L}_{\psi}^{5}=\tfrac{1}{2}\left[\bar{\psi}\hat{\gamma}^{5}\gamma^{\mu}(\nabla_{\mu}\psi)-(\nabla_{\mu}\bar{\psi})\hat{\gamma}^{5}\gamma^{\mu}\psi\right]-\mu\bar{\psi}\psi.$
(54)

with the field equation

$\displaystyle\gamma^{\mu}\nabla_{\mu}\psi-\hat{\gamma}^{5}\mu\psi=0.$ (55)

Here, $\omega^{2}=k_{i}k^{i}-\mu^{2}$, that is, the dispersion relation is
tachyonic [76, 178].

Ramazanoğlu [269] used Eq. (54) to obtain the first _spontaneous
spinorization_ theory where spinor fields are unstable to growth around GR
backgrounds of NSs. Even though the meaning of a ghost as opposed to a tachyon
is a subtle issue for spinors, the equation of motion in this theory has
divergent coefficients as in the vector field case (47), hence this form of
spinorization suffers from similar problems to vectorization.

Minamitsuji [226] proposed an alternative model of spinorization given by the
action

$\displaystyle S$ $\displaystyle=\frac{1}{16\pi G_{*}}\int{\rm
d}^{4}x\,\sqrt{-g}\,\left\\{R+\tfrac{1}{2}\left[\bar{\psi}\hat{\gamma}^{5}\gamma^{\mu}(\nabla_{\mu}\psi)\right.\right.$
$\displaystyle\quad\left.\left.-\,(\nabla_{\mu}\bar{\psi})\hat{\gamma}^{5}\gamma^{\mu}\psi\right]\right\\}+S_{\rm
m}\left[\Psi_{{\rm m}},\,{\cal A}^{2}(\bar{\psi}\psi)g_{\mu\nu}\right].$ (56)

This model has the unusual feature that the derivative part of the spinor
action is not the canonical one in Eq. (53), but rather the “tachyonic” one in
Eq. (54). The resulting tachyonic equation of motion is

$\displaystyle\gamma^{\mu}\nabla_{\mu}\psi-\hat{\gamma}^{5}(4\pi{\cal
A}^{4}\Lambda\tilde{T})\psi=0,$ (57)

where $\Lambda={\rm d}\ln{\cal A}/{\rm d}(\bar{\psi}\psi)$. The interpretation
is straightforward, we have a tachyonic field with an effective mass $\mu_{\rm
eff}=4\pi{\cal A}^{4}\Lambda\tilde{T}$ [cf. Eq. (55)]. The most important
aspect of this theory is the fact that the effective mass term and all
coefficients of the field equation are regular everywhere, hence, the dynamics
does not have any signs of ill-posedness that we have found in all
generalizations of scalarization so far. In other words, action (57) is the
only known analog of spontaneous scalarization for non-scalar fields that does
not suffer from ghost or gradient instabilities.

Our results on spontaneous tensorization suggest a no-go theorem for
generalizing scalarization beyond scalar fields, at least if we want to avoid
ghosts. It is curious to find the only exception to this trend in the
relatively exotic case of spinorization, which invites studies of the deeper
reasons that make spontaneous scalarization hard to replicate for other types
of fields.

##  VI Open problems and future perspectives

We chose to review the literature by starting from the theoretical
underpinning of the scalarization mechanism, then moving on to discussing NSs
and BHs separately, and then returning to model building in order to discuss
generalization of the mechanism to other fields. Most, but not all, open
problems have come up already within one or more of the previous sections and
discussed to some extent. Nevertheless, we will use this section to revisit
them and cover any additional areas that require further development, opting
to a concise summary of future perspectives.

###  VI.1 Scalarization and cosmology

As already discussed in Sec. II.2.4, one of the key challenges for
scalarization is understanding whether it is compatible with cosmology. Recall
that the main premise of scalarization is that there exists some constant
value of the scalar field, $\varphi_{0}$, for which spacetimes of GR become
admissible solutions to the field equations and it is these solutions that
describe stationary objects that we expect (from an observational perspective)
to carry no scalar charge. For this to be true, cosmic evolution needs to
comply and drive $\varphi$ to $\varphi_{0}$ in the late universe, or else the
whole universe will in effect be “scalarized.” It has been pointed out early
on by Damour and Nordtvedt [91] and more recently by Anderson  _et al._ [15],
Franchini and Sotiriou [135] that reaching this preferred value for the scalar
fields in the late universe is not generic for simple models of scalarization
and instead requires severe fine-tuning of initial conditions.

This could well be an artefact of not having the complete theory and, ideally,
one would expect the need to fine-tuned initial conditions to disappear by
addressing further corrections to the model. A first step in this direction
has been made by Antoniou  _et al._ [24], inspired by Damour and Nordtvedt
[91], by showing that the “mixed” model of scalarization discussed in Sec.
II.2.4 has GR with a constant scalar as a late-time cosmic attractor for the
right sign of the coupling between the scalar and the Ricci scalar. This
demonstrated that, in principle, scalarization models can be made compatible
with late-time cosmology and provided a recipe for doing so: include in the
action terms that will dominate in late cosmology (e.g., Ricci coupling) over
the terms that control the onset of scalarization (e.g., Gauss-Bonnet
invariant) and hence impose the desired cosmological behavior. However, the
specific model discussed as an example in Antoniou  _et al._ [24] is by no
means unique, and there are many other attempts to address similar concerns
[72, 231, 128]. Perhaps more importantly, understanding the efficiency of the
attractor behavior in reproducing the behavior of GR (with a cosmological
constant) quantitatively, down to evolution of perturbations, structure
formations, etc. certainly deserves further investigations and could lead to
constraints on realistic models of scalarization.

A second point of friction between scalarization and cosmology relates to the
early universe. Broadly speaking, scalarization is controlled by the coupling
between the scalar field and curvature invariants, so it is inevitable that
these couplings will become increasingly important as one runs cosmic
evolution backwards and move to higher and higher curvatures. In particular,
they will tend to dominate when the size of the universe will become
comparable to the size of the compact objects we want scalarization to affect
today (see e.g. Antoniou  _et al._ [24]). It has been pointed out by Anson
_et al._ [20, 21] that quantum fluctuations could then seed a scalarization-
like instability during inflation and that it would be hard to prevent this
(linear) instability by adding corrections to the action. It should however be
stressed that one does not need the scalar to necessarily remain constant or
have its current value through the evolution of the universe. Hence one might
not need to prevent such an instability, but instead just quench it
nonlinearly, exactly as it happens in scalarization itself. The scalar could
then smoothly evolve away from the vacuum that leads to this instability
before it becomes relevant, as one moves backwards in cosmic time. The key
question here is whether there exists an EFT applicable to the early universe,
which contains the scalarization models as late universe limits, but can also
host an inflationary scenario compatible with observations.

As a final remark in relation to cosmology, we should emphasize that we have
discussed here only the cosmological implications of known scalarization
models and only under the assumption that the scalar field is cosmologically
subdominant at late times. There is clearly a plethora of generalized scalar-
tensor theories that have been used in the context of inflation or dark
energy, which exhibit couplings between the scalar and curvature invariants
similar to those employed for scalarization models, but in most cases the
nature of the couplings differ quite significantly from those of scalarization
models. Reviewing such attempts goes well beyond the scope of this manuscript.
Exploring whether a scalar field that exhibits scalarization around compact
objects could also play a role on inflation or account for dark energy would
certainly be interesting, but putting together such a model (with a single
scalar) seems particularly challenging. It is worth mentioning in this context
that evading constraints on the coupling between a scalar and the Gauss-Bonnet
invariant coming from the speed of GWs does require the scalar to be
cosmologically subdominant [135].

###  VI.2 Dynamical evolution

The study of the dynamical nonlinear regime of scalarization requires
nonperturbative time-evolution schemes, whose availability varies between
models. Finding formulations of gravity theories that are amenable to
numerical time evolution, i.e., numerical relativity, is nontrivial even in GR
[264, 66, 31], and the issue can become even more complicated for alternative
theories.

Time-domain, nonlinear evolution of DEF-like models have been done since the
early days by Novak [246, 245], Novak and Ibanez [247], and binary inspirals
have been performed for nearly a decade [33]. This was possible thanks to the
relative simplicity of the field equations that allows one to establish their
well-posedness [290, 291]. However, detailed and fully nonlinear numerical
results in relation to GWs, a primary tool to test scalarization, are still
few, such as the study by Shibata  _et al._ [305].

The picture is quite different for BH scalarization where the coupling to the
Gauss-Bonnet term can result in a complicated causal structure of the
spacetime in dynamical situations. It was demonstrated both at linear [49, 47,
48] and nonlinear [125, 126, 283] levels that, part, but not all, of the
parameter space of spontaneous scalarization where scalarized BHs can be found
loses hyperbolicity when the theory is evolved in time in a certain set of
gauges, which means that no predictions can be obtained. For the moment, there
is no consensus on whether this loss of hyperbolicity can be cured by a better
gauge choice or if it is intrinsic to the evolution equations, which calls for
further work on the issue. Furthermore, there are no detailed studies of
binary evolution and merger even for the part of the parameter space for which
hyperbolic time evolution has been shown to exist. The dynamics of more
general models where various coupling terms are present as in Eq. (II.2.1) are
not available either. The lack of results on all these fronts presents
important future research directions.

It can be the case that an EFT, obtained as a certain limit or truncation of a
more fundamental theory, is ill-posed while the latter is or is expected to be
well-posed. A typical example is relativistic hydrodynamics once viscocity has
been taken into account. Indeed, it has been suggested to introduce techniques
used in hydrodynamics to gravity in order to render the problematic time
evolution of some theories hyperbolic by suitably modifying the equations,
while keeping the end point of evolution the same [70]. This approach has seen
increasing use in alternatives to GR, including the specific case of scalar-
Gauss-Bonnet theories under certain symmetries [133], and it is another avenue
to explore the dynamics of scalarization in cases where current methods are
inadequate. The applicability and power of such methods to general cases,
e.g., fully nonlinear 3+1 evolution of a highly dynamical system, are yet to
be confirmed.

###  VI.3 Model building in and beyond scalarization

We have covered several models of spontaneous scalarization and their
phenomenology for NSs and BHs. What they all have in common is that
scalarization is triggered by a tachyonic instability, whose threshold is
controlled by less than a handful of terms in the action [17]. Yet, the
properties of the final scalarized configuration depend quite delicately on
the nonlinearities and coupling terms that contribute beyond the linear level.
The specific choices of these nonlinear interactions have been shown to
crucially affect the stability [49, 311, 26] and the scalar charge [311, 212,
25, 106] of scalarized solutions, as well as the cosmological behavior of the
corresponding theory [24, 20, 128]. Thus it is clear that such choices affect
viability and observability of the models. As a result, although certain
models of scalarization, such as the original DEF model, have been studied
fairly exhaustively, the exploration of the broader class of theories that
exhibit scalarization, and their phenomenology, has only started.

Furthermore, one could consider introducing new classes of scalarization based
on instabilities that are not tachyonic in nature. As discussed in Sec. V, so
far success in this direction is limited but it is nevertheless an existing
prospect. Alternatively, one can abandon the idea of a linear instability
quenched by nonlinearities altogether as we discussed in Sec. II.3.5. It was
recently demonstrated in Doneva and Yazadjiev [118] that a theory that
features a $\varphi^{4}$ coupling to the Gauss-Bonnet invariant can still have
scalarized BHs below a certain mass threshold, although the Kerr solution is
linearly stable with respect to (massless) scalar perturbations [50]. Such
observation remains true in more general theories, such as Gauss-Bonnet
gravity with multiple scalar fields [329]. This strongly suggests the
existence of a class of theories in which scalarization is a purely nonlinear
phenomenon.

Scalarization models that have been considered so far typically assume that
the scalar field does not couple to matter directly in a suitable choice of
variables, usually called the Jordan frame, which suffices for the theory to
satisfy the WEP. However, as pointed out in Coates  _et al._ [80], this
assumption might not be necessary because scalarization itself forces the
scalar field into a trivial configuration away from a specific compact object.
Therefore, scalarization models can be indistinguishable from GR for what
regards the WEP for all current observations. It was further argued in Coates
_et al._ [80] and Franchini  _et al._ [134] that particular couplings to
matter that do not disrupt scalarization could introduce a Higgs-like
mechanism in gravity: the scalar field changes value only near compact
objects, and its coupling to matter changes the properties of the Standard
Model in these regions. This is a largely unexplored possibility.

As was discussed in detail in Sec. V, another important open question is
whether one can generalize the scalarization mechanism to other fields. Doing
so in a context of a theory that is free of pathologies has so far proven to
be quite challenging, except perhaps for the case of spinors [226]. This is in
part because controlling the dynamics of the extra degree of freedom is
notoriously difficult in general in gravity theories with additional vector or
tensor fields [277], and vectorization or tensorization models are no
exception.

We should stress that the work on scalarization, or its extensions, has
focused on isolated compact objects. In fact, theoretical considerations for
putting together scalarization models have also been heavily influence by
studies of isolated BHs and NSs. However, as we discussed already above,
scalarization could happen dynamically in a binary [33, 251, 305, 342, 127,
314, 110]. Apart from finding ways to model this effect [184] and potentially
constraining it with GW observations (see also below), it would be
particularly interesting to explore further what properties of a binary
control dynamical scalarization, and how this differs between different
scalarization models.

###  VI.4 Observational prospects

Perhaps the most attractive feature of scalarization is that the tachyonic
instability typically leads to large scalar field amplitudes before it is
quenched, which leads to nonperturbative deviations from GR in high curvature
regions. At the same time, the theory mimics GR closely for weak gravity,
easily satisfying existing tests. This suggests that new fundamental scalar
that exhibit this behavior could be discovered with strong gravity
observations.

Current observations put stringent bounds on scalar radiation from pulsars,
which have already almost completely ruled out the original (massless) DEF
model [378]. Electromagnetic observations of compact objects in binaries or
surrounded by some form of matter also have the potential to extend such
results to more general models of scalarization. For example, x-ray data from
NS surfaces have been recently considered as a possible way to detect
scalarized objects [316, 315]. The masses, radii and moment of inertia of NSs
can be inferred through such observations [55, 56, 57]. Future observation
could be used to distinguish scalarized NSs from nonscalarized ones, for
instance, using EOS-independent relations. The astrophysical signatures of
most extensions of the DEF model, especially those that allow BH scalarization
have not been studied in details yet, because they are recent.

The most exciting current development in strong gravity is the advent of GW
astronomy, which is an arena for exploring spontaneous scalarization. Effects
of scalarization on GWs of coalescing NSs and/or BHs have been investigated to
some extend, mostly for the DEF model [302, 185, 184, 244] and scalar-Gauss-
Bonnet gravity [361]. The most pressing issue in this front is the development
of accurate waveform models, that ideally would cover the inspiral, merger and
ringdown stages of coalescing binaries. Developing such waveform models in GR
has been (and remains) a formidable task that combines tools from post-
Newtonian theory, numerical relativity, black-hole perturbation theory, and
gravitational self-force. Progress in all these fronts are necessary in order
for one to have accurate “scalarized” waveform models. Stars that undergo core
collapse are another interesting source of GWs. Albeit expected to be rare,
such events may produce large bursts of scalar radiation [328, 284, 200] and
may contribute to the stochastic GW background which could, in principle, be
probed by pulsar timing arrays. In addition, some model of scalarization (but
not exclusively) predict the existence of extra polarizations in GWs, which
provides another way to test them.

Last but not the least, future space-borne GW detectors such as LISA will
bring another channel for testing modified gravity. Especially interesting
with respect to scalarization is the case of extreme mass-ratio inspirals when
the secondary object is scalarized [214, 215]. Data from future GW detectors,
such as LISA, or third generation ground-based detectors, such as the Einstein
Telescope and the Cosmic Explorer, combined with more precise electromagnetic
observations, such as those from improved x-ray observatories, will be crucial
for detecting or ruling out scalar fields that exhibit scalarization.

###### Acknowledgements.

D.D acknowledges financial support via an Emmy Noether Research Group funded
by the German Research Foundation (DFG) under grant no. DO 1771/1-1. S.Y would
like to thank the University of Tübingen for the financial support and
acknowledges the partial support by the Bulgarian NSF Grant KP-06-H28/7. T.S.
acknowledges partial support from the STFC Consolidated Grants No.
ST/T000732/1 and No. ST/V005596/1. F.M.R acknowledges support from 2020 Bilim
Akademisi GEBİP award and TÜBİTAK Project No. 122F097. H.O.S. acknowledges
funding from the Deutsche Forschungsgemeinschaft (DFG) - project number:
386119226. The authors would also like to acknowledge networking support by
the COST Actions CA16104 and CA15117.

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