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In the context of Chameleon gravity, we present a semi-analytical solution of the chameleon field profile in accurately modelled galaxy cluster's mass components, namely: the stellar mass of the Brightest Cluster Galaxy (BCG), the baryonic mass in galaxies (other than the BCG), the mass of the Intra-Cluster Medium (ICM) and the diffuse cold dark matter (CDM). The obtained semi-analytic profile is validated against the numerical solution of the chameleon field equation and implemented in the \textsc{MG-MAMPOSSt} code for kinematic analyses of galaxy clusters in modified gravity scenarios. By means of mock halos, simulated both in GR and in modified gravity, we show that the combination of velocities and positions of cluster member galaxies, along with data of the stellar velocity dispersion profile of the BCG, can impose constraints on the parameter space of the Chameleon model; for a cluster generated in GR, these constraints are at the same level as a joint lensing+kinematics analysis of a cluster modelled with a single mass profile, without the BCG data., Comment: 22 pages, 10 figures

The chameleon screening mechanism has been constrained many a time using dynamic and kinematic galaxy cluster observables. Current constraints are, however, insensitive to different mass components within galaxy clusters and have been mainly focused on a single mass density profile, the Navarro-Frenk-While mass density model. In this work, we extend the study of the Chameleon screening mechanism in galaxy clusters by considering a series of mass density models, namely: generalized-Navarro-Frenk-While, b-Navarro-Frenk-While, Burket, Isothermal and Einasto. The coupling strength ($\beta$) and asymptotic value of the chameleon field ($\phi_\infty$) are constrained by using kinematics analyses of simulated galaxy clusters, generated both assuming General Relativity and a strong chameleon scenario. By implementing a Bayesian analysis we comprehensively show that the biases introduced due to an incorrect assumption of the mass model are minimal. Similarly, we also demonstrate that a spurious detection of evidence for modifications to gravity is highly unlikely when utilizing the kinematics of galaxy clusters., Comment: 25 pages, 7 Figures, to be submitted to JCAP

Virial-like identities obtained through Derrick's scaling argument are powerful, multi-purpose tools to study general relativistic models. Applications comprise establishing no-go/hair theorems and numerical accuracy tests. In the presence of a horizon (\textit{aka} boundary), the spacetime can be divided into regions, each with its own identity. So far, such identities have only been computed in the region outside the event horizon; however, adding a positive cosmological constant endows an additional boundary (the cosmological horizon), with the region between the latter and the former of particular interest. In this letter, by performing a radial coordinate transformation, we generalise Derrick's scaling argument to compute virial identities \textit{across the whole} non-asymptotically flat spacetimes. The developed method is applied to the entire Reissner-Nordstrom-de Sitter spacetime. A convenient gauge that trivialises the gravitational contribution to the identity between horizons is also found., Comment: 9 pages, 1 Figure

Scalarized black holes (BH) have been shown to form dynamically in extended-scalar-tensor theories, either through spontaneous scalarization -- when the BH is unstable against linear perturbations -- or through a non-linear scalarization. In the latter, linearly stable BHs can ignite scalarization when sufficiently perturbed. These phenomena are, however, not incompatible and mixed scalarization is also possible. We explore two aspects of the Einstein-Maxwell-Scalar model: solutions containing, simultaneously, linear (\textit{aka} standard) and non-linear scalarization; and the effects of having one of the coupling constants with an 'opposite sign' to the one leading to scalarization. Both points are addressed by constructing and examining the mixed scalarization's domain of existence. An overall dominance of the spontaneous scalarization over the non-linear scalarization is observed. Thermodynamically, an entropical preference for mixed over the standard scalarization (spontaneous or non-linear) exists. In the presence of counter scalarization, a quench of the scalarization occurs, mimicking the effect of a scalar particle's mass/positive self-interaction term., Comment: 15 pages, 5 figures

An interesting mechanism for the formation of hairy black holes occurs when a vector field, non-minimally coupled to a source term, grows from a perturbation of the vacuum black hole, \textit{aka} vectorization. Its study has, however, been lacking, in part due to the constant threat of ghost instabilities that have plagued vector fields. In this work, we show evidence that, in a generic family of extended-vector-tensor theories where the vector field is non-minimally coupled to the model's invariant (source term), a spherically symmetric, vectorized black hole always suffers from ghost instabilities. These ultimately turn the process of vectorization astrophysically unviable., Comment: 12 pages

It was recently found that in certain flavours of scalar-Gauss-Bonnet gravity linearly stable bald black holes can co-exist with stable scalarized solutions. The transition between both can be ignited by a large nonlinear perturbation, thus the process was dubbed non-linear scalarization, and it happens with a jump that leads to interesting astrophysical implications. Generalizing these results to the case of nonzero scalar field potential is important because a massive self-interacting scalar field can have interesting theoretical and observational consequences, e.g. reconcile scalar-Gauss-Bonnet gravity with binary pulsar observation, stabilize black hole solutions, etc. That is why in the present paper, we address this open problem. We pay special attention to the influence of a scalar field mass and self-interaction on the existence of scalarized phases and the presence of a jump between stable bald and hairy back holes. Our results show that both the addition of a mass and positive self-interaction of the scalar field result in suppression or quenching of the overall scalarization phenomena. A negative scalar field self-interaction results in an increase of the scalarization. The presence and the size of the jump, though, are not so sensitive to the scalar field potential., Comment: 18 pages, 9 figures

We construct and explore the physical properties of \textit{scalaroca stars}: spherically symmetric solitonic solutions made of a complex scalar field $\Phi$ and a complex Proca field $A^\mu$. We restrict our attention to configurations in which both fields are in the fundamental state and possess an equal mass, focusing on the cases when ($i$) the scalar and Proca fields are (non--linearly) super--imposed and do not interact with each other; and ($ii$) the scalar and Proca fields interact through the term $\alpha |\Phi| ^2 A^\mu A_\mu$. The solutions are found numerically for the non--interacting case ($\alpha=0$) as well as for both signs of the interaction coupling constant $\alpha$. While pure ($i.e.$ single--field) Proca/scalar boson stars are the most/least massive for weakly--interacting fields, one can obtain more massive solutions for a sufficiently strong interaction. Besides, in the latter case, solutions can be either in a synchronized state -- in which both fields have the same frequency -- or in a non--synchronized state. In addition, we observe that the coupling between the two fields allows solitonic solutions with a real scalar field. We further comment on the possibility of spontaneous scalarization and vectorization of the interacting solitonic solution., Comment: 21 pages, 13 figures, this project was started before the recently published work ArXiv:2304.08019

The high-precision astrometric mission GAIA recently reported the remarkable discovery of a Sun-like star closely orbiting a dark object, with a semi-major axis and period of $1.4\, \rm{AU}$ and $187.8$ days respectively. While the plausible expectation for the central dark object is a black hole, the evolutionary mechanism leading to the formation of such a two-body system is highly challenging. Here, we challenge the scenario of a central black hole and show that the observed orbital dynamics can be explained under fairly general assumptions if the central dark object is a stable clump of bosonic particles of spin-0, or spin-1, known as a boson star. We further explain how future astrometric measurements of similar systems will provide an exciting opportunity to probe the fundamental nature of compact objects and test compact alternatives to black holes., Comment: 11 pages, 4 figures. Comments are very welcome

Virial identities are a useful mathematical tool in General Relativity. Not only have they been used as a numerical accuracy tool, but they have also played a significant role in establishing no-go and no-hair theorems while giving some physical insight into the considered system from an energy balance perspective. While the calculation of these identities tends to be a straightforward application of Derrick's scaling argument~\cite{derrick1964comments}, the complexity of the resulting identity is system dependent. In particular, the contribution of the Einstein-Hilbert action, due to the presence of second-order derivatives of the metric functions, becomes increasingly complex for generic metrics. Additionally, the Gibbons-Hawking-York term needs to be taken into account \cite{Herdeiro:2021teo}. Thankfully, since the gravitational action only depends on the metric, it is expected that a ``convenient'' gauge that trivializes the gravitational action contribution exists. While in spherical symmetry such a gauge is known (the $m-\sigma$ parametrization), such has not been found for axial symmetry. In this letter, we propose a ``convenient'' gauge for axial symmetry and use it to compute an identity for Kerr black holes with scalar hair., Comment: 10 pages

Derrick-type virial identities, obtained via dilatation (scaling) arguments, have a variety of applications in field theories. We deconstruct such virial identities in relativistic gravity showing how they can be recast as self-evident integrals of appropriate combinations of the equations of motion. In spherical symmetry, the appropriate combination and gauge choice guarantee the geometric part can be integrated out to yield a master form of the virial identity as a non-trivial energy-momentum balance condition, valid for both asymptotically flat black holes and self-gravitating solitons, for any matter model. Specifying the matter model we recover previous results obtained via the scaling procedure. We then discuss the more general case of stationary, axi-symmetric, asymptotically flat black hole or solitonic solutions in General Relativity, for which a master form for their virial identity is proposed, in a specific gauge but regardless of the matter content. In the flat spacetime limit, the master virial identity for both the spherical and axial cases reduces to a balance condition for the principal pressures, discussed by Deser., Comment: 14 pages

The general relativity vacuum black holes (BHs) can be scalarised in models where a scalar field non-minimally couples to the Gauss-Bonnet (GB) invariant. Such GB scalarisation comes in two flavours, depending on the GB sign that triggers the phenomenon. Hereafter these two cases are termed GB$^\pm$ scalarisation. For vacuum BHs, only GB$^+$ scalarisation is possible in the static case, while GB$^-$ scalarisation is $spin$ induced. But for electrovacuum BHs, GB$^-$ is also $charged$ induced. We discuss the GB$^-$ scalarisation of Reissner-Nordstr\"om and Kerr-Newman BHs, discussing zero modes and constructing fully non-linear solutions. Some comparisons with GB$^+$ scalarisation are given. To assess the generality of the observed features, we also briefly consider the GB$^\pm$ scalarisation of stringy dilatonic BHs and coloured BHs which provide qualitative differences with respect to the electrovacuum case., Comment: 16 pages, 6 figures

Virial (aka scaling) identities are integral identities that are useful for a variety of purposes in non-linear field theories, including establishing no-go theorems for solitonic and black hole solutions, as well as for checking the accuracy of numerical solutions. In this paper, we provide a pedagogical rationale for the derivation of such integral identities, starting from the standard variational treatment of particle mechanics. In the framework of one-dimensional (1D) effective actions, the treatment presented here yields a set of useful formulas for computing virial identities in any field theory. Then, we propose that a complete treatment of virial identities in relativistic gravity must take into account the appropriate boundary term. For General Relativity this is the Gibbons-Hawking-York boundary term. We test and confirm this proposal with concrete examples. Our analysis here is restricted to spherically symmetric configurations, which yield 1D effective actions (leaving higher-D effective actions and in particular the axially symmetric case to a companion paper). In this case, we show that there is a particular "gauge" choice, $i.e.$ a choice of coordinates and parameterizing metric functions, that simplifies the computation of virial identities in General Relativity, making both the Einstein-Hilbert action and the Gibbons-Hawking-York boundary term non-contributing. Under this choice, the virial identity results exclusively from the matter action. For generic "gauge" choices, however, this is not the case., Comment: 27 pages, 1 figure

Can a dynamically robust bosonic star (BS) produce an (effective) shadow that mimics that of a black hole (BH)? The BH shadow is linked to the existence of light rings (LRs). For free bosonic fields, yielding mini-BSs, it is known that these stars can become ultra-compact - i.e., possess LRs - but only for perturbatively unstable solutions. We show this remains the case even when different self-interactions are considered. However, an effective shadow can arise in a different way: if BSs reproduce the existence of an innermost stable circular orbit (ISCO) for timelike geodesics (located at $r_{\rm ISCO}=6M$ for a Schwarzschild BH of mass M), the accretion flow morphology around BHs is mimicked and an effective shadow arises in an astrophysical environment. Even though spherical BSs may accommodate stable timelike circular orbits all the way down to their centre, we show the angular velocity along such orbits may have a maximum away from the origin, at $R_{\Omega}$; this scale was recently observed to mimic the BH's ISCO in some scenarios of accretion flow. Then: (i) for free scalar fields or with quartic self-interactions, $R_{\Omega}\neq 0$ only for perturbatively unstable BSs; (ii) for higher scalar self-interactions, e.g. axionic, $R_{\Omega}\neq 0$ is possible for perturbatively stable BSs, but no solution with $R_{\Omega}=6M$ was found in the parameter space explored; (iii) but for free vector fields, yielding Proca stars (PSs), perturbatively stable solutions with $R_{\Omega}\neq 0$ exist, and indeed $R_{\Omega}=6M$ for a particular solution. Thus, dynamically robust spherical PSs can mimic the shadow of a (near-)equilibrium Schwarzschild BH with the same M, in an astrophysical environment, despite the absence of a LR, at least under some observation conditions, as we confirm by comparing the lensing of such PSs and Schwarzschild BHs., Comment: Abstract abridged due to arXiv length limit; 22 pages, 9 figures

In this work, we generalise the spontaneous scalarization phenomena in Einstein-Maxwell-Scalar models to a higher spin field. The result is an Einstein-Maxwell-Vector model wherein a vector field is non-minimally coupled to the Maxwell invariant by an exponential coupling function. We show that the latter guarantees the circumvention of an associated no-hair theorem when the vector field has the form of an electric field. Different than its scalar counterpart, the new spontaneously vectorized Reissner-Nordstr\"om (RN) black holes are, always, undercharged while being entropically preferable. The solution profile and domain of existence are presented and analysed., Comment: 12 pages, 5 figures

Einstein-Maxwell-scalar models allow for different classes of black hole solutions, depending on the non-minimal coupling function $f(\phi)$ employed, between the scalar field and the Maxwell invariant. Here, we address the linear mode stability of the black hole solutions obtained recently for a quartic coupling function, $f(\phi)=1+\alpha\phi^4$ [1]. Besides the bald Reissner-Nordstr\"om solutions, this coupling allows for two branches of scalarized black holes, termed cold and hot, respectively. For these three branches of black holes we calculate the spectrum of quasinormal modes. It consists of polar scalar-led modes, polar and axial electromagnetic-led modes, and polar and axial gravitational-led modes. We demonstrate that the only unstable mode present is the radial scalar-led mode of the cold branch. Consequently, the bald Reissner-Nordstr\"om branch and the hot scalarized branch are both mode-stable. The non-trivial scalar field in the scalarized background solutions leads to the breaking of the degeneracy between axial and polar modes present for Reissner-Nordstr\"om solutions. This isospectrality is only slightly broken on the cold branch, but it is strongly broken on the hot branch., Comment: 21 pages, 8 figures

The phenomenon of spontaneous scalarisation of charged black holes (BHs) has recently motivated studies of various Einstein-Maxwell-scalar models. Within these models, different classes of BH solutions are possible, depending on the non-minimal coupling function $f(\phi)$, between the scalar field and the Maxwell invariant. Here we consider the class wherein both the (bald) electrovacuum Reissner-Nordstr\"om (RN) BH and new scalarised BHs co-exist, and the former are never unstable against scalar perturbations. In particular we examine the model, within this subclass, with a quartic coupling function: $f(\Phi) = 1+\alpha \Phi ^4$. The domain of existence of the scalarised BHs, for fixed $\alpha$, is composed of two branches. The first branch (cold scalarised BHs) is continuously connected to the extremal RN BH. The second branch (hot scalarised BHs) connects to the first one at the minimum value of the charge to mass ratio and it includes overcharged BHs. We then assess the perturbative stability of the scalarised solutions, focusing on spherical perturbations. On the one hand, cold scalarised BHs are shown to be unstable by explicitly computing growing modes. The instability is quenched at both endpoints of the first branch. On the other hand, hot scalarised BHs are shown to be stable by using the S-deformation method. Thus, in the spherical sector this model possesses two stable BH local ground states (RN and hot scalarised). We point out that the branch structure of BHs in this model parallels the one of BHs in five dimensional vacuum gravity, with [Myer-Perry BHs, fat rings, thin rings] playing the role of [RN, cold scalarised, hot scalarised] BHs., Comment: 12 pages, 6 figures

We consider an augmented Einstein-Maxwell-scalar model including an axionic-type coupling between the scalar and electromagnetic field. We study dyonic black hole solutions in this model. For the canonical axionic coupling emerging from high energy physics, all charged black holes have axion hair. We present their domain of existence and investigate some physical properties. For other axionic-type couplings, two classes of black hole solutions may co-exist in the model: scalar-free Reissner-Nordstr\"om black holes and scalarised black holes. We show that in some region of the parameter space, the scalar-free solutions are unstable. Then, there is non-uniqueness since new scalarised black hole solutions with the same global charges also exist, which are entropically preferred over the scalar-free solutions and, moreover, emerge dynamically from the instability of the former., Comment: 23 pages, 13 figures

Spontaneous scalarisation of electrically charged, asymptotically flat Reissner-Nordstr\"om black holes (BHs) has been recently demonstrated to occur in Einstein-Maxwell-Scalar (EMS) models. EMS BH scalarisation presents a technical simplification over the BH scalarisation that has been conjectured to occur in extended Scalar-Tensor Gauss-Bonnet (eSTGB) models. It is then natural to ask: 1) how universal are the conclusions extracted from the EMS model? And 2) how much do these conclusions depend on the choice of the non-minimal coupling function? Here we perform a comparative analysis of different forms for the coupling function including: exponential, hyperbolic, power-law and a rational function (fraction) couplings. In all of them we obtain and study the domain of existence of fundamental, spherically symmetric, scalarised BHs and compute, in particular, their entropy. The latter shows that scalarised EMS BHs are always entropically preferred over the RN BHs with the same total charge to mass ratio $q$. This contrasts with the case of eSTGB, where for the same power-law coupling the spherical, fundamental scalarised BHs are not entropically preferred over Schwarzschild. Also, while the scalarised solutions in the EMS model for the exponential, hyperbolic and power-law coupling are very similar, the rational function coupling leads to a transition in the domain of existence. Furthermore, fully non-linear dynamical evolutions of unstable RN BHs with different values of $q$ are presented. These show: 1) for sufficiently small $q$, scalarised solutions with (approximately) the same $q$ form dynamically; 2) for large $q$, spontaneous scalarisation visibly decreases $q$; thus evolutions are non-conservative; 3) despite the existence of non-spherical, static scalarised solutions, the evolution of unstable RN BHs under non-spherical perturbations leads to a spherical scalarised BH., Comment: 18 pages, 8 figures; Some visualizations of the numerical evolutions can be seen here: http://gravitation.web.ua.pt/node/1657; v2. minor changes, matches the version published in CQG

The existence of localized, approximately stationary, lumps of the classical gravitational and electromagnetic field -- $geons$ -- was conjectured more than half a century ago. If one insists on exact stationarity, topologically trivial configurations in electro-vacuum are ruled out by no-go theorems for solitons. But stationary, asymptotically flat geons found a realization in scalar-vacuum, where everywhere non-singular, localized field lumps exist, known as (scalar) boson stars. Similar geons have subsequently been found in Einstein-Dirac theory and, more recently, in Einstein-Proca theory. We identify the common conditions that allow these solutions, which may also exist for other spin fields. Moreover, we present a comparison of spherically symmetric geons for the spin $0,1/2$ and $1$, emphasising the mathematical similarities and clarifying the physical differences, particularly between the bosonic and fermonic cases. We clarify that for the fermionic case, Pauli's exclusion principle prevents a continuous family of solutions for a fixed field mass; rather only a discrete set exists, in contrast with the bosonic case., Comment: 14 pages, 4 figures