Laplacian Instabilities Research Articles

Dynamical system analysis at background and perturbation levels: Quintessence in severe disadvantage comparing to ΛCDM

We perform for the first time a dynamical system analysis of both the background and perturbation equations, of $\Lambda$CDM cosmology and quintessence scenario with an exponential potential. In the former case the perturbations do not change the stability of the late-time attractor of the background equations, and the system still results in the dark-energy dominated, de Sitter solution, having passed from the correct dark-matter era with $\gamma\approx6/11$. However, in the case of quintessence the incorporation of perturbations changes the stability and properties of the background evolution, and the only conditionally stable points present either an exponentially increasing matter clustering not favored by observations, or Laplacian instabilities, and thus not physically interesting. This result is a severe disadvantage of quintessence cosmology comparing to $\Lambda$CDM paradigm.

Most general cubic-order Horndeski Lagrangian allowing for scaling solutions and the application to dark energy

In cubic-order Horndeski theories where a scalar field $\phi$ is coupled to nonrelativistic matter with a field-dependent coupling $Q(\phi)$, we derive the most general Lagrangian having scaling solutions on the isotropic and homogenous cosmological background. For constant $Q$ including the case of vanishing coupling, the corresponding Lagrangian reduces to the form $L=Xg_2(Y)-g_3(Y)\square \phi$, where $X=-\partial_{\mu}\phi\partial^{\mu}\phi/2$ and $g_2, g_3$ are arbitrary functions of $Y=Xe^{\lambda \phi}$ with constant $\lambda$. We obtain the fixed points of the scaling Lagrangian for constant $Q$ and show that the $\phi$-matter-dominated-epoch ($\phi$MDE) is present for the cubic coupling $g_3(Y)$ containing inverse power-law functions of $Y$. The stability analysis around the fixed points indicates that the $\phi$MDE can be followed by a stable critical point responsible for the cosmic acceleration. We propose a concrete dark energy model allowing for such a cosmological sequence and show that the ghost and Laplacian instabilities can be avoided even in the presence of the cubic coupling.

Gauge-ready formulation of cosmological perturbations in scalar-vector-tensor theories

In scalar-vector-tensor (SVT) theories with parity invariance, we perform a gauge-ready formulation of cosmological perturbations on the flat Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW) background by taking into account a matter perfect fluid. We derive the second-order action of scalar perturbations and resulting linear perturbation equations of motion without fixing any gauge conditions. Depending on physical problems at hand, most convenient gauges can be chosen to study the development of inhomogeneities in the presence of scalar and vector fields coupled to gravity. This versatile framework, which encompasses Horndeski and generalized Proca theories as special cases, is applicable to a wide variety of cosmological phenomena including nonsingular cosmology, inflation, and dark energy. By deriving conditions for the absence of ghost and Laplacian instabilities in several different gauges, we show that, unlike Horndeski theories, it is possible to evade no-go arguments for the absence of stable nonsingular bouncing/genesis solutions in both generalized Proca and SVT theories. We also apply our framework to the case in which scalar and vector fields are responsible for dark energy and find that the separation of observables relevant to the evolution of matter perturbations into tensor, vector, and scalar sectors is transparent in the unitary gauge. Unlike the flat gauge chosen in the literature, this result is convenient to confront SVT theories with observations associated with the cosmic growth history.

Dark energy in scalar-vector-tensor theories

The scalar-vector-tensor theories with second-order equations of motion can accommodate both Horndeski and generalized Proca theories as specific cases. In the presence of a perfect fluid, we study the cosmology in such a most general scheme of scalar-vector-tensor theories with parity invariance by paying particular attention to the application to dark energy. We obtain a closed-form expression of the background equations of motion by using coefficients appearing in the second-order action of scalar perturbations. We also derive conditions for the absence of ghost and Laplacian instabilities of tensor and vector perturbations and show that the existence of matter does not substantially modify the stabilities of dynamical degrees of freedom in the small-scale limit. On the other hand, the sound speed of scalar perturbations is affected by the presence of matter. Employing the quasi-static approximation for scalar perturbations deep inside the sound horizon, we derive analytic expressions of Newtonian and weak lensing gravitational potentials as well as two scalar perturbations arising from the scalar and vector fields. We apply our general framework to dark energy theories with the tensor propagation speed equivalent to the speed of light and show that the observables associated with the growth of matter perturbations and weak lensing potentials are generally affected by intrinsic vector modes and by interactions between scalar and vector fields.

New scaling solutions in cubic Horndeski theories

We propose a viable dark energy scenario in the presence of cubic Horndeski interactions and a standard scalar-field kinetic term with two exponential potentials. We show the existence of new scaling solutions along which the cubic coupling $G_3$ provides an important contribution to the field density that scales in the same way as the background fluid density. The solutions finally exit to the epoch of cosmic acceleration driven by a scalar-field dominated fixed point arising from the second exponential potential. We clarify the viable parameter space in which all the theoretically consistent conditions including those for the absence of ghost and Laplacian instabilities are satisfied on scaling and scalar-field dominated critical points. In comparison to Quintessence with the same scalar potential, we find that the cubic coupling gives rise to some novel features: (i) the allowed model parameter space is wider in that a steeper potential can drive the cosmic acceleration; (ii) the dark energy equation of state $w_{\phi}$ today can be closer to $-1$ relative to Quintessence; (iii) even if the density associated with the cubic coupling dominates over the standard field density in the scaling era, the former contribution tends to be suppressed at low redshifts. We also compute quantities associated with the growth of matter perturbations and weak lensing potentials under the quasi-static approximation in the sub-horizon limit and show that the cubic coupling leads to the modified evolution of perturbations which can be distinguished from Quintessence.

Cosmology in scalar-vector-tensor theories

We study the cosmology on the Friedmann-Lema\^{\i}tre-Robertson-Walker background in scalar-vector-tensor theories with a broken $U(1)$ gauge symmetry. For parity-invariant interactions arising in scalar-vector-tensor theories with second-order equations of motion, we derive conditions for the absence of ghosts and Laplacian instabilities associated with tensor, vector, and scalar perturbations at linear order. This general result is applied to the computation of the primordial tensor power spectrum generated during inflation as well as to the speed of gravity relevant to the late-time cosmic acceleration. We also construct a concrete inflationary model in which a temporal vector component ${A}_{0}$ contributes to the dynamics of cosmic acceleration besides a scalar field $\ensuremath{\phi}$ through their kinetic mixings. In this model, we show that all the stability conditions of perturbations can be consistently satisfied during inflation and subsequent reheating.

Odd-parity stability of hairy black holes in U(1) gauge-invariant scalar-vector-tensor theories

In scalar-vector-tensor theories with $U(1)$ gauge invariance, it was recently shown that there exists a new type of hairy black hole (BH) solutions induced by a cubic-order scalar-vector interaction. In this paper, we derive conditions for the absence of ghosts and Laplacian instabilities against odd-parity perturbations on a static and spherically symmetric background for most general $U(1)$ gauge-invariant scalar-vector-tensor theories with second-order equations of motion. We apply those conditions to hairy BH solutions arising from the cubic-order coupling and show that the odd-parity stability in the gravity sector is always ensured outside the event horizon with the speed of gravity equivalent to that of light. We also study the case in which quartic-order interactions are present in addition to the cubic coupling and obtain conditions under which black holes are stable against odd-parity perturbations.

Black holes in quartic-order beyond-generalized Proca theories

The generalized Proca theories with second-order equations of motion can be healthily extended to a more general framework in which the number of propagating degrees of freedom remains unchanged. In the presence of a quartic-order nonminimal coupling to gravity arising in beyond-generalized Proca theories, the speed of gravitational waves ct on the Friedmann–Lemaître–Robertson–Walker (FLRW) cosmological background can be equal to that of light c under a certain condition. By using this condition alone, we show that the speed of gravitational waves in the vicinity of static and spherically symmetric black holes is also equivalent to c for the propagation of odd-parity perturbations along both radial and angular directions. As a by-product, the black holes arising in our beyond-generalized Proca theories are plagued by neither ghost nor Laplacian instabilities against odd-parity perturbations. We show the existence of both exact and numerical black hole solutions endowed with vector hairs induced by the quartic-order coupling.

Dark energy scenario consistent with GW170817 in theories beyond Horndeski gravity

The Gleyzes-Langlois-Piazza-Vernizzi (GLPV) theories up to quartic order are the general scheme of scalar-tensor theories allowing the possibility for realizing the tensor propagation speed $c_t$ equivalent to 1 on the isotropic cosmological background. We propose a dark energy model in which the late-time cosmic acceleration occurs by a simple k-essence Lagrangian analogous to the ghost condensate with cubic and quartic Galileons in the framework of GLPV theories. We show that a wide variety of the variation of the dark energy equation of state $w_{\rm DE}$ including the entry to the region $w_{\rm DE}<-1$ can be realized without violating conditions for the absence of ghosts and Laplacian instabilities. The approach to the tracker equation of state $w_{\rm DE}=-2$ during the matter era, which is disfavored by observational data, can be avoided by the existence of a quadratic k-essence Lagrangian $X^2$. We study the evolution of nonrelativistic matter perturbations for the model $c_t^2=1$ and show that the two quantities $\mu$ and $\Sigma$, which are related to the Newtonian and weak lensing gravitational potentials respectively, are practically equivalent to each other, such that $\mu \simeq \Sigma>1$. For the case in which the deviation of $w_{\rm DE}$ from $-1$ is significant at a later cosmological epoch, the values of $\mu$ and $\Sigma$ tend to be larger at low redshifts. We also find that our dark energy model can be consistent with the bounds on the deviation parameter $\alpha_{\rm H}$ from Horndeski theories arising from the modification of gravitational law inside massive objects.

Black hole perturbations in vector-tensor theories: the odd-mode analysis

In generalized Proca theories with vector-field derivative couplings, a bunch of hairy black hole solutions have been derived on a static and spherically symmetric background. In this paper, we formulate the odd-parity black hole perturbations in generalized Proca theories by expanding the corresponding action up to second order and investigate whether or not black holes with vector hair suffer ghost or Laplacian instabilities. We show that the models with cubic couplings G3(X), where X=−AμAμ/2 with a vector field Aμ, do not provide any additional stability condition as in General Relativity. On the other hand, the exact charged stealth Schwarzschild solution with a nonvanishing longitudinal vector component A1, which originates from the coupling to the Einstein tensor GμνAμ Aν equivalent to the quartic coupling G4(X) containing a linear function of X, is unstable in the vicinity of the event horizon. The same instability problem also persists for hairy black holes arising from general quartic power-law couplings G4(X) ⊃ β4 Xn with the nonvanishing A1, while the other branch with A1=0 can be consistent with conditions for the absence of ghost and Laplacian instabilities. We also discuss the case of other exact and numerical black hole solutions associated with intrinsic vector-field derivative couplings and show that there exists a wide range of parameter spaces in which the solutions suffer neither ghost nor Laplacian instabilities against odd-parity perturbations.

Unification of dark matter-dark energy in generalized Galileon theories

We present a unified description of the dark matter and the dark energy sectors, in the framework of shift-symmetric generalized Galileon theories. Considering a particular combination of terms in the Horndeski Lagrangian in which we have not introduced a cosmological constant or a matter sector, we obtain an effective unified cosmic fluid whose equation of state wU is zero during the whole matter era, namely from redshifts z ∼ 3000 up to z ∼ 2–3. Then at smaller redshifts it starts decreasing, passing the bound wU = −1/3, which marks the onset of acceleration, at around z ∼ 0.5. At present times it acquires the value wU = −0.7. Finally, it tends toward a de-Sitter phase in the far future. This behaviour is in excellent agreement with observations. Additionally, confrontation with Supernovae type Ia data leads to a very efficient fit. Examining the model at the perturbative level, we show that it is free from pathologies such as ghosts and Laplacian instabilities, at both scalar and tensor sectors, at all times.

Dark energy survivals in massive gravity after GW170817: SO(3) invariant

The recent detection of the gravitational wave signal GW170817 together with an electromagnetic counterpart GRB 170817A from the merger of two neutron stars puts a stringent bound on the tensor propagation speed. This constraint can be automatically satisfied in the framework of massive gravity. In this work, we consider a general SO(3)-invariant massive gravity with five propagating degrees of freedom and derive the conditions for the absence of ghosts and Laplacian instabilities in the presence of a matter perfect fluid on the flat Friedmann-Lemaître-Robertson-Walker (FLRW) cosmological background. The graviton potential containing the dependence of three-dimensional metrics and a fiducial metric coupled to a temporal scalar field gives rise to a scenario of the late-time cosmic acceleration in which the dark energy equation of state wDE is equivalent to −1 or varies in time. We find that the deviation from the value wDE=−1 provides important contributions to the quantities associated with the stability conditions of tensor, vector, and scalar perturbations. In concrete models, we study the dynamics of dark energy arising from the graviton potential and show that there exist viable parameter spaces in which neither ghosts nor Laplacian instabilities are present for both wDE>−1 and wDE<−1. We also generally obtain the effective gravitational coupling Geff with non-relativistic matter as well as the gravitational slip parameter ηs associated with the observations of large-scale structures and weak lensing. We show that, apart from a specific case, the two quantities Geff and ηs are similar to those in general relativity for scalar perturbations deep inside the sound horizon.

Instabilities in Horndeski-Yang-Mills inflation

A non-abelian $SU(2)$ gauge field with a non-minimal Horndeski coupling to gravity gives rise to a de Sitter solution followed by a graceful exit to a radiation-dominated epoch. In this Horndeski Yang-Mills (HYM) theory we derive the second-order action for tensor perturbations on the homogeneous and isotropic quasi de Sitter background. We find that the presence of the Horndeski non-minimal coupling to the gauge field inevitably introduces ghost instabilities in the tensor sector during inflation. Moreover, we also find Laplacian instabilities for the tensor perturbations deep inside the Hubble radius during inflation. Thus, we conclude that the HYM theory does not provide a consistent inflationary framework due to the presence of ghosts and Laplacian instabilities.

Cosmology in generalized Proca theories

We consider a massive vector field with derivative interactions that propagates only the 3 desired polarizations (besides two tensor polarizations from gravity) with second-order equations of motion in curved space-time. The cosmological implications of such generalized Proca theories are investigated for both the background and the linear perturbation by taking into account the Lagrangian up to quintic order. In the presence of a matter fluid with a temporal component of the vector field, we derive the background equations of motion and show the existence of de Sitter solutions relevant to the late-time cosmic acceleration. We also obtain conditions for the absence of ghosts and Laplacian instabilities of tensor, vector, and scalar perturbations in the small-scale limit. Our results are applied to concrete examples of the general functions in the theory, which encompass vector Galileons as a specific case. In such examples, we show that the de Sitter fixed point is always a stable attractor and study viable parameter spaces in which the no-ghost and stability conditions are satisfied during the cosmic expansion history.

Beyond generalized Proca theories

We consider higher-order derivative interactions beyond second-order generalized Proca theories that propagate only the three desired polarizations of a massive vector field besides the two tensor polarizations from gravity. These new interactions follow the similar construction criteria to those arising in the extension of scalar-tensor Horndeski theories to Gleyzes-Langlois-Piazza-Vernizzi (GLPV) theories. On the isotropic cosmological background, we show the existence of a constraint with a vanishing Hamiltonian that removes the would-be Ostrogradski ghost. We study the behavior of linear perturbations on top of the isotropic cosmological background in the presence of a matter perfect fluid and find the same number of propagating degrees of freedom as in generalized Proca theories (two tensor polarizations, two transverse vector modes, and two scalar modes). Moreover, we obtain the conditions for the avoidance of ghosts and Laplacian instabilities of tensor, vector, and scalar perturbations. We observe key differences in the scalar sound speed, which is mixed with the matter sound speed outside the domain of generalized Proca theories.

Effective field theory of modified gravity on the spherically symmetric background: Leading order dynamics and the odd-type perturbations

We consider perturbations of a static and spherically symmetric background endowed with a metric tensor and a scalar field in the framework of the effective field theory of modified gravity. We employ the previously developed 2+1+1 canonical formalism of a double Arnowitt-Deser-Misner (ADM) decomposition of space-time, which singles out both time and radial directions. Our building block is a general gravitational action that depends on scalar quantities constructed from the 2+1+1 canonical variables and the lapse. Variation of the action up to first-order in perturbations gives rise to three independent background equations of motion, as expected from spherical symmetry. The dynamical equations of linear perturbations follow from the second-order Lagrangian after a suitable gauge fixing. We derive conditions for the avoidance of ghosts and Laplacian instabilities for the odd-type perturbations. We show that our results not only incorporates those derived in the most general scalar-tensor theories with second-order equations of motion (the Horndeski theories) but they can be applied to more generic theories beyond Horndeski.

Effective field theory of modified gravity with two scalar fields: Dark energy and dark matter

We present a framework for discussing the cosmology of dark energy and dark matter based on two scalar degrees of freedom. An effective field theory of cosmological perturbations is employed. A unitary gauge choice renders the dark energy field into the gravitational sector, for which we adopt a generic Lagrangian depending on three-dimensional geometrical scalar quantities arising in the ADM decomposition. We add to this dark-energy associated gravitational sector a scalar field $\phi$ and its kinetic energy $X$ as dark matter variables. Compared to the single-field case, we find that there are additional conditions to obey in order to keep the equations of motion for linear cosmological perturbations at second order. For such a second-order multi-field theory we derive conditions under which ghosts and Laplacian instabilities of the scalar and tensor perturbations are absent. We apply our general results to models with dark energy emerging in the framework of the Horndeski theory and dark matter described by a k-essence Lagrangian $P(\phi,X)$. We derive the effective coupling between such an imperfect-fluid dark matter and the gravitational sector under the quasi-static approximation on sub-horizon scales. By considering the purely kinetic Lagrangian $P(X)$ as a particular case, the formalism is verified to reproduce the gravitational coupling of a perfect-fluid dark matter.

Stability of Horndeski vector-tensor interactions

We study the Horndeski vector-tensor theory that leads to second order equations of motion and contains a non-minimally coupled abelian gauge vector field. This theory is remarkably simple and consists of only 2 terms for the vector field, namely: the standard Maxwell kinetic term and a coupling to the dual Riemann tensor. Furthermore, the vector sector respects the U(1) gauge symmetry and the theory contains only one free parameter, M2, that controls the strength of the non-minimal coupling.We explore the theory in a de Sitter spacetime and study the presence of instabilities and show that it corresponds to an attractor solution in the presence of the vector field. We also investigate the cosmological evolution and stability of perturbations in a general FLRW spacetime. We find that a sufficient condition for the absence of ghosts is M2 > 0. Moreover, we study further constraints coming from imposing the absence of Laplacian instabilities. Finally, we study the stability of the theory in static and spherically symmetric backgrounds (in particular, Schwarzschild and Reissner-Nordström-de Sitter). We find that the theory, quite generally, do have ghosts or Laplacian instabilities in regions of spacetime where the non-minimal interaction dominates over the Maxwell term.We also calculate the propagation speed in these spacetimes and show that superluminality is a quite generic phenomenon in this theory.

Reheating in three-form inflation

We consider the three-form field, which has been considered as a candidate for realizing inflation, coupled to a scalar field which models the relativistic matter particles produced during the reheating epoch. We have investigated the stability conditions for this theory and found that introducing such a coupling does not lead to any ghosts or Laplacian instabilities. We have also investigated the reheating temperature and the production of particles due to parametric resonances. We have found that this process is more efficient in this theory compared to the result of the standard-scalar-field inflationary scenario.

Stability of the 3-form field during inflation

We consider the minimally coupled 3-form field, which has been considered as a candidate for realizing inflation. We have studied the conditions to avoid ghosts and Laplacian instabilities and found that some classes of potentials, e.g., the Mexican-hat, will in general be unstable. We then propose other classes of potentials which are instead free from any instability, drive a long enough slow-roll regime followed by an oscillatory epoch, and as a consequence, can provide successful inflation. Finally, we also provide stable potentials that lead to a small enough propagation speed for the scalar perturbations, making it possible for these models to produce non-Gaussianities.