The tachyonic instability is associated with the unboundedness of the Hamiltonian from below and results in an unstable low-$k$ regime. In the cosmological exploration of modified gravity, it is seldom taken into account, with more focus given to the popular no-ghost and no-gradient conditions. The latter though are intrinsically high-$k$ statements. Here we combine all three conditions into a full set of requirements that we show to guarantee stability on the whole range of cosmological scales. We then explore the impact of the different conditions on the parameter space of scalar-tensor gravity, with particular emphasis on the no-tachyon one. We focus on Horndeski gravity and also consider separately the two subclasses of $f(R)$ and Generalized Brans Dicke theories. We identify several interesting features, for instance in the parameter space of designer $f(R)$ on a $w$CDM background, shedding light on previous findings. When looking at the phenomenological functions $Σ$ and $μ$, associated to the weak lensing and clustering potential respectively, we find that in the case of Generalized Brans Dicke the no-tachyon condition clearly cuts models with $μ\,,\,Σ>1$. This effect is less prevalent in the Horndeski case due to the larger amount of free functions in the theory.
Shift-symmetric Horndeski theories admit an interesting class of Schwarzschild black hole solutions exhibiting time-dependent scalar hair. By making use of Lemaître coordinates, we analyze perturbations around these types of black holes, and demonstrate that scalar perturbations around black hole backgrounds inevitably have gradient instabilities. Taken together with previously established results, this newly-discovered instability rules out black holes with time-dependent scalar hair in Horndeski theories.
Theories with higher order time derivatives generically suffer from ghost-like instabilities, known as Ostrogradski instabilities. This fate can be avoided by considering "degenerate" Lagrangians, whose kinetic matrix cannot be inverted, thus leading to constraints between canonical variables and a reduced number of physical degrees of freedom. In this work, we derive in a systematic way the degeneracy conditions for scalar-tensor theories that depend quadratically on second order derivatives of a scalar field. We thus obtain a classification of all degenerate theories within this class of sca…
A number of stability criteria exist for dark energy theories, associated with requiring the absence of ghost, gradient and tachyonic instabilities. Tachyonic instabilities are the least well explored of these in the dark energy context and we here discuss and derive criteria for their presence and size in detail. Our findings suggest that, while the absence of ghost and gradient instabilities is indeed essential for physically viable models and so priors associated with the absence of such instabilities significantly increase the efficiency of parameter estimations without introducing unphysi…
It was confirmed that the negative modes of the Euclidean section for asymptotically flat black holes reveal the thermodynamic instability of these black holes in the grand canonical ensemble (GCE). These include Schwarzschild, Reissner-Nordström, Kerr, and Kerr-Newman black holes. In this work, we develop the relation between thermodynamic instability in the GCE and tachyonic instability for asymptotically flat black holes, where the latter is the onset for obtaining black holes with scalar hair. This implies that the tachyonic instability of black holes when introducing scalar coupling to th…
The running of the Higgs self coupling may lead to numerous phenomena in early universe cosmology. In this paper we introduce a scenario where the Higgs running induces turns in the trajectory passing a region with tachyonic mass, leading to a temporal tachyonic growth in the curvature power spectrum. This effect induced by the Higgs leaves phenomena in the form of primordial black holes and stochastic gravitational waves, where proposed GW observatories will be able to probe in the near future.
In the framework of Einstein-Maxwell-scalar theory, we studied scalar field perturbations of Reissner-Nordström-Melvin (RNM) black holes, which describes the RN black holes immersed in a uniform magnetic field. Due to the coupling to the Maxwell term, the scalar field acquires an effective mass whose square, in the presence of the magnetic field, will become negative somewhere outside the horizon for either sign of the coupling constant $α$, thus triggering the tachyonic instability and leading to spontaneous scalarization when $α$ is large enough. The magnetic field has significant influences…
Recently it was pointed out that in shift-symmetric scalar-tensor theories a black hole can have nontrivial scalar hair which depends linearly on time. We develop black hole perturbation theory for such solutions and compute the quadratic action of odd-parity perturbations. We show that around all the solutions known so far with such time-dependent scalar hair the perturbations trigger instabilities or are presumably strongly coupled.
Spontaneous symmetry breaking usually occurs due to the tachyonic (spinodal) instability of a scalar field near the top of its effective potential at $ϕ= 0$. Naively, one might expect the field $ϕ$ to fall from the top of the effective potential and then experience a long stage of oscillations with amplitude O(v) near the minimum of the effective potential at $ϕ= v$ until it gives its energy to particles produced during these oscillations. However, it was recently found that the tachyonic instability rapidly converts most of the potential energy V(0) into the energy of colliding classical wave…
Dense enough compact objects were recently shown to lead to an exponentially fast increase of the vacuum energy density for some free scalar fields properly coupled to the spacetime curvature as a consequence of a tachyonic-like instability. Once the effect is triggered, the star energy density would be overwhelmed by the vacuum energy density in a few milliseconds. This demands that eventually geometry and field evolve to a new configuration to bring the vacuum back to a stationary regime. Here, we show that the vacuum fluctuations built up during the unstable epoch lead to particle creation …
We study the stability under linear odd-parity perturbations of static spherically symmetric black holes in Horndeski gravity. We derive the master equation for these perturbations and obtain the conditions of no-ghost and Laplacian instability. In order for the black hole solutions to be stable, we study their generalized "Regge-Wheeler potential". It turns out that the problem is reduced to an algebraic problem where three functions characterizing the black hole should be positive outside the horizon to prove the stability. We found that these conditions are similar to the no-ghost and Lapla…
Spontaneous scalarization is a mechanism that endows relativistic stars and black holes with a nontrivial configuration only when their spacetime curvature exceeds some threshold. The standard way to trigger spontaneous scalarization is via a tachyonic instability at the linear level, which is eventually quenched due to the effect of non-linear terms. In this paper, we identify all of the terms in the Horndeski action that contribute to the (effective) mass term in the linearized equations and, hence, can cause or contribute to the tachyonic instability that triggers scalarization.
We constrain the viable models of Horndeski gravity, written in its equivalent Generalised Galileon version, by resorting to the Witten positive energy theorem. We find that the free function $G_3(ϕ, X)$ in the Lagrangian is constrained to be a function solely of the scalar field, $G_3(ϕ)$, and relations among the free functions are found. Other criterion for stability are also analysed, such as the attractiveness of gravity, the Dolgov-Kawasacki instability and the energy conditions. Some applications for Cosmology are discussed
The vector-tensor Horndeski theory is supplemented by a real, massless scalar field non-minimally coupled to the Horndeski interaction term. The generic dyonic Reissner-Nordstrom solutions characterized by electric and magnetic charges, acquire an instability and bifurcate into electromagnetically charged solutions presenting an horizon and a cloud of scalar hairs at specific values of the non-minimal coupling constant. These hairy black holes are studied in terms of the physical parameters of the model and the critical phenomena limiting their domain of existence are discussed.
A Hamiltonian density bounded from below implies that the lowest-energy state is stable. We point out, contrary to common lore, that an unbounded Hamiltonian density does not necessarily imply an instability: Stability is indeed a coordinate-independent property, whereas the Hamiltonian density does depend on the choice of coordinates. We discuss in detail the relation between the two, starting from k-essence and extending our discussion to general field theories. We give the correct stability criterion, using the relative orientation of the causal cones for all propagating degrees of freedom.…
We study a metric cubic gravity theory considering odd-parity modes of linear inhomogeneous perturbations on a spatially homogeneous Bianchi type I manifold close to the isotropic de Sitter spacetime. We show that in the regime of small anisotropy, the theory possesses new degrees of freedom compared to General Relativity, whose kinetic energy vanishes in the limit of exact isotropy. From the mass dispersion relation we show that such theory always possesses at least one ghost mode as well as a very short-time-scale (compared to the Hubble time) classical tachyonic (or ghost-tachyonic) instabi…
All Quiet on the Vector Front: Scalar Growth and Perturbative Control in Horndeski--Weyl-squared Gravity
Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）；Institute for Theoretical Physics, Leibniz University Hannover（汉诺威莱布尼茨大学理论物理研究所）；Max Planck Institute for Gravitational Physics, Albert Einstein Institute（马克斯·普朗克引力物理研究所爱因斯坦研究所）；Dipartimento di Fisica e Astronomia “G. Galilei”, Università degli Studi di Padova（帕多瓦大学伽利略物理与天文系）；INFN, Sezione di Padova（意大利国家核物理研究所帕多瓦分部）
It has recently been suggested that at the post-inflationary stage of the mixed Higgs-$R^2$ model of inflation efficient particle production can arise from the tachyonic instability of the Higgs field. It might complete the preheating of the Universe if appropriate conditions are satisfied, especially in the Higgs-like regime. In this paper, we study this behavior in more depth, including the conditions for occurrence, analytical estimates for the maximal efficiency, and the necessary degree of fine-tuning among the model parameters to complete preheating by this effect. We find that the param…
Nonlinearities in a realistic axion field potential may play an important role in the cosmological dynamics. In this paper we use the Boltzmann code CLASS to solve the background and linear perturbations evolution of an axion field and contrast our results with those of CDM and the free axion case. We conclude that there is a slight delay in the onset of the axion field oscillations when nonlinearities in the axion potential are taken into account. Besides, we identify a tachyonic instability of linear modes resulting in the presence of a bump in the power spectrum at small scales. Some commen…
It was recently shown that a scalar field suitably coupled to the Gauss-Bonnet invariant $\mathcal{G}$ can undergo a spin-induced linear tachyonic instability near a Kerr black hole. This instability appears only once the dimensionless spin $j$ is sufficiently large, that is, $j \gtrsim 0.5$. A tachyonic instability is the hallmark of spontaneous scalarization. Focusing, for illustrative purposes, on a class of theories that do exhibit this instability, we show that stationary, rotating black hole solutions do indeed have scalar hair once the spin-induced instability threshold is exceeded, whi…
We present a thorough stability analysis of modified gravity theories in the presence of matter fields. We use the Effective Field Theory framework for Dark Energy and Modified Gravity to retain a general approach for the gravity sector and a Sorkin-Schutz action for the matter one. Then, we work out the proper viability conditions to guarantee in the scalar sector the absence of ghosts, gradient and tachyonic instabilities. The absence of ghosts can be achieved by demanding a positive kinetic matrix, while the lack of a gradient instability is ensured by imposing a positive speed of propagati…
This paper is a pedagogical introduction to models of gravity and how to constrain them through cosmological observations. We focus on the Horndeski scalar-tensor theory and on the quantities that can be measured with a minimum of assumptions. Alternatives or extensions of General Relativity have been proposed ever since its early years. Because of Lovelock theorem, modifying gravity in four dimensions typically means adding new degrees of freedom. The simplest way is to include a scalar field coupled to the curvature tensor terms. The most general way of doing so without incurring in the Ostr…
It was found recently that the anisotropies in the homogeneous Bianchi I cosmology considered within the context of a specific Horndeski theory are damped near the initial singularity instead of being amplified. In this work we extend the analysis of this phenomenon to cover the whole of the Horndeski family. We find that the phenomenon is absent in the K-essence and/or Kinetic Gravity Braiding theories, where the anisotropies grow as one approaches the singularity. The anisotropies are damped at early times only in more general Horndeski models whose Lagrangian includes terms quadratic and cu…
We show that a massless canonical scalar field minimally coupled to general relativity can become a tachyonic ghost at low energies around a background in which the scalar's gradient is spacelike. By performing a canonical transformation we demonstrate that this low energy ghost can be recast, at the level of the action, in a form of a fluid that undergoes a Jeans-like instability affecting only modes with large wavelength. This illustrates that low-energy tachyonic ghosts do not lead to a catastrophic quantum vacuum instability, unlike the usual high-energy ghost degrees of freedom.
We study massive scalar field perturbation on Kerr black holes in dynamical Chern-Simons gravity by performing a $(2+1)$-dimensional simulation. Object pictures of the wave dynamics in time domain are obtained. The tachyonic instability is found to always occur for any nonzero black hole spin and any scalar field mass as long as the coupling constant exceeds a critical value. The presence of the mass term suppresses or even quenches the instability. The quantitative dependence of the onset of the tachyonic instability on the coupling constant, the scalar field mass and the black hole spin is g…
We present a detailed study of second-order matter perturbations for the general Horn- deski class of models. Being the most general scalar-tensor theory having second-order equations of motion, it includes many known gravity and dark energy theories and General Relativity with a cosmological constant as a specific case. This enables us to estimate the leading order dark matter bispectrum generated at late-times by gravitational instability. We parametrize the evolution of the first and second-order equations of motion as proposed by Bellini and Sawicki (2014), where the free functions of the …
In this paper, we propose a massive gravity theory with 5 degrees of freedom. The mass term is constructed by 3 Stuckelberg scalar fields, which respects SO(3) symmetry in the fields' configuration. By the analysis on the linear cosmological perturbations, we found that such 5 d.o.f are free from ghost instability, gradiant instability, and tachyonic instability.
Quintessence with tachyonic resonance and late-time cosmic-microwave-background and gravitational-wave signals
Nagoya University（名古屋大学）；Kobayashi-Maskawa Institute for the Origin of Particles and the Universe, Nagoya University（小林-益川粒子宇宙起源研究所）；Institute for Advanced Research, Nagoya University（名古屋大学高等研究院）；Rikkyo University（立教大学）
A scalar field non-minimally coupled to certain geometric [or matter] invariants which are sourced by [electro]vacuum black holes (BHs) may spontaneously grow around the latter, due to a tachyonic instability. This process is expected to lead to a new, dynamically preferred, equilibrium state: a scalarised BH. The most studied geometric [matter] source term for such spontaneous BH scalarisation is the Gauss-Bonnet quadratic curvature [Maxwell invariant]. This phenomenon has been mostly analysed for asymptotically flat spacetimes. Here we consider the impact of a positive cosmological constant,…
We consider cosmological models with a dynamical dark energy field, and study the presence of three types of commonly found instabilities, namely ghost (when fields have negative kinetic energy), gradient (negative momentum squared) and tachyon (negative mass squared). In particular, we study the linear scalar perturbations of theories with two interacting scalar fields as a proxy for a dark energy and matter fields, and explicitly show how canonical transformations relate these three types of instabilities with each other. We generically show that low-energy ghosts are equivalent to tachyonic…
We consider a variant of the Nojiri-Odintsov covariant Horava-like gravitational model, where diffeomorphism invariance is broken dynamically via a non-standard coupling to a perfect fluid. The theory allows to address some of the potential instability problems present in Horava-Lifshitz gravity due to explicit diffeomorphism invariance breaking. The fluid is instead constructed from a scalar field constrained by a Lagrange multiplier. In fact, the Lagrange multiplier construction allows for an extension of the Horava-like model to include the scalar field of mimetic gravity, an extension whic…
It is well known that the Klein Gordon (KG) equation $\Box Φ+ m^2Φ=0$ has tachyonic unstable modes on large scales ($k^2<\vert m \vert^2$) for $m^2<m_{cr}^2=0$ in a flat Minkowski spacetime with maximum growth rate $Ω_{F}(m)= \vert m \vert$ achieved at $k=0$. We investigate these instabilities in a Reissner-Nordström-deSitter (RN-dS) background spacetime with mass $M$, charge $Q$, cosmological constant $Λ>0$ and multiple horizons. By solving the KG equation in the range between the event and cosmological horizons, using tortoise coordinates $r_*$, we identify the bound states of the emerging S…
In a subclass of scalar-tensor theories, it has been shown that standard general relativity solutions of neutron stars and black holes with trivial scalar field profiles are unstable. Such an instability leads to solutions which are different from those of general relativity and have non-trivial scalar field profiles, in a process called scalarization. In the present work we focus on scalarization due to a non-minimal coupling of the scalar field to the Gauss-Bonnet curvature invariant. The coupling acts as a tachyonic mass for the scalar mode, thus leading to the instability of general relati…
Under the same spirit of the Galileon-Horndeski theories and their more modern extensions, the generalized SU(2) Proca theory was built by demanding that its action may be free of the Ostrogradski's instability. Nevertheless, the theory must also be free of other instability problems in order to ensure its viability. As a first approach to address this issue, we concentrate on a quite general variant of the theory and investigate the general conditions for the absence of ghost and gradient instabilities in the tensor sector without the need for resolving the dynamical background. The phenomeno…
Universal Non-Gaussian Signatures from Transient Instabilities
RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences（理化学研究所交叉理论科学中心）；Van Swinderen Institute for Particle Physics and Gravity, University of Groningen（格罗宁根大学范斯温德恩粒子物理与引力研究所）；Institut d’Astrophysique de Paris, Sorbonne Université, CNRS（巴黎天体物理学研究所，索邦大学，法国国家科学研究中心）；Max Planck Institute for Physics, Werner-Heisenberg-Institut（马克斯·普朗克物理研究所，维尔纳·海森堡研究所）；Max Planck-IAS-NTU Center for Particle Physics, Cosmology and Geometry（马克斯·普朗克-国际先进研究-南洋理工大学粒子物理、宇宙学与几何中心）
We study the stability of static as well as of rotating and charged black holes in (4+1)-dimensional Anti-de Sitter space-time which possess spherical horizon topology. We observe a non-linear instability related to the condensation of a charged, tachyonic scalar field and construct "hairy" black hole solutions of the full system of coupled Einstein, Maxwell and scalar field equations. We observe that the limiting solution for small horizon radius is either a hairy soliton solution or a singular solution that is not a regular extremal solution. Within the context of the gauge/gravity duality t…
Locally AdS$_{d-1}\times\mathbb{R}$ uniform black strings (UBS) in the presence of a massless scalar field are believed to avoid the onset of the Gregory-Laflamme (GL) instability in $d\ge4$ as no tachyonic modes exist in the spectrum of the Laplace-Beltrami operator. We present analytic and numerical evidence of GL modes in the Lichnerowicz spectrum indicating that AdS$_{d-1}$ UBSs are classically and thermodynamically unstable at the linear level in $d>4$. In $d=4$, we confirm that uniform BTZ$_3$ strings are indeed stable as previously suggested. We propose that linear instabilities of blac…
We show that the $R^{(3)}δK$ operator in effective field theory is significant for avoiding the instability of nonsingular bounce, where $R^{(3)}$ and $K_{μν}$ are the three-dimensional Ricci scalar and the extrinsic curvature on the spacelike hypersurface, respectively. We point out that the covariant Lagrangian of $R^{(3)}δK$, i.e., $L_{R^{(3)}δK}$, has the second order derivative couplings of scalar field to gravity which do not appear in Horndeski theory or its extensions, but does not bring the Ostrogradski ghost. We also discuss the possible effect of $L_{R^{(3)}δK}$ on the primordial sc…
We show that the five-dimensional Maxwell theory with the Chern-Simons term is tachyonic in the presence of a constant electric field. When coupled to gravity, a sufficiently large Chern-Simons coupling causes instability of the Reissner-Nordstrom black holes in anti-de Sitter space. The instability happens only at non-vanishing momenta, suggesting a spatially modulated phase in the holographically dual quantum field theory in 3+1 dimensions, with spontaneous current generation in a helical configuration. The three-charge extremal black hole in the type IIB superstring theory on AdS_5 x S^5 ba…
We consider cosmological inflation generated by a scalar field slowly rolling off from a de Sitter maximum of its potential. The models belong to the class of hilltop models and represent the most general model of this kind in which the scalar potential can be written as the sum of two exponentials. The minimally coupled Einstein-scalar gravity theory obtained in this way is the cosmological version of a two-scale generalization of known holographic models, allowing for solitonic solutions interpolating between an AdS spacetime in the infrared and scaling solutions in the ultraviolet. We then …
We investigate the stability of asymptotically anti-de Sitter gravity coupled to tachyonic scalar fields with mass at or slightly above the Breitenlohner-Freedman bound. The boundary conditions in these ``designer gravity'' theories are defined in terms of an arbitrary function W. Previous work had suggested that the energy in designer gravity is bounded below if i) W has a global minimum and ii) the scalar potential admits a superpotential P. More recently, however, certain solutions were found (numerically) to violate the proposed energy bound. We resolve the discrepancy by observing that a …
We study two types of linear instabilities in Einstein-Born-Infeld-scalar field theory and show that small Born-Infeld (BI) black holes suffer from superradiant instability within a particular range of frequencies; the larger the BI coupling parameter, the slower the growth rate of the instability. It is predicted that a small BI black hole decays into a small hairy black hole that has a charged scalar condensate floating near the horizon. We numerically show that there is a phase transition between hairy and BI black holes. The metric solution shows the existence of a hairy black hole for $T<…
