Degenerate Higher-order Scalar-tensor Theories Research Articles

Strong gravitational lensing by DHOST black holes

The gravitational deflection of light in the strong field limit is an important test for alternative theories of gravity. However, solutions for the metric that allow for analytic computations are not always available. We implement a hybrid analytic-numerical approximation to determine the deflection angle in static, spherically symmetric spacetimes. We apply this to a set of numerical black hole solutions within the class of modified gravity theories known as degenerate higher order scalar–tensor theories (DHOST). Comparing our results to a more time consuming full numerical integration, we find that we can accurately describe the deflection angle for light rays passing at arbitrary distances from the photon sphere with a combination of two analytic-numerical approximations. Furthermore, we find a range of parameters where our DHOST black holes predict strong lensing effects whose size is comparable with the uncertainty in the properties of the supermassive black hole in M87 reported by the event horizon telescope, showing that strong lensing is a viable alternative to put constraints on these models of modified gravity.

Stealth dark energy in scordatura DHOST theory

A stealth de Sitter solution in scalar-tensor theories has an exact de Sitter background metric and a nontrivial scalar field profile. Recently, in the context of Degenerate Higher-Order Scalar-Tensor (DHOST) theories it was shown that stealth de Sitter solutions suffer from either infinite strong coupling or gradient instability for scalar field perturbations. The sound speed squared is either vanishing or negative. In the first case, the strong coupling scale is zero and thus lower than the energy scale of any physical phenomena. From the viewpoint of effective field theory, this issue is naturally resolved by introducing a controlled detuning of the degeneracy condition dubbed scordatura, recovering a version of ghost condensation. In this paper we construct a viable dark energy model in the scordatura DHOST theory based on a stealth cosmological solution, in which the metric is the same as in the standard ΛCDM model and the scalar field profile is linearly time-dependent. We show that the scordatura mechanism resolves the strong coupling and gradient instability. Further, we find that the scordatura is also necessary to make the quasi-static limit well-defined, which implies that the subhorizon observables are inevitably affected by the scordatura. We derive the effective gravitational coupling and the correction to the friction term for the subhorizon evolution of the linear dark matter energy density contrast as well as the Weyl potential and the gravitational slip parameter. In the absence of the scordatura, the quasi-static approximation would break down at all scales around stealth cosmological solutions even if the issue of the infinite strong coupling is unjustly disregarded. Therefore previous estimations of the subhorizon evolution of matter density contrast in modified gravity in the literature need to be revisited by taking into account the scordatura effect.

Horndeski theories and beyond from higher dimensions

The Einstein–Hilbert action with a cosmological constant is the most general local four-dimensional action leading to second-order derivative equations of motion that are symmetric and divergence free. In higher dimensions, additional terms can appear. We investigate a generalised metric decomposition involving a scalar degree of freedom to express the higher-dimensional action as an effective four-dimensional scalar–tensor theory. From the higher-dimensional Ricci scalar alone and a subclass of our metric ansatz, we recover the subset of Horndeski theories with luminal speed of gravitational waves. More generally, beyond-Horndeski terms appear. When including a Gauss–Bonnet scalar in the higher-dimensional action, we generate contributions to all cubic-order second-derivative terms present in the degenerate higher-order scalar–tensor theory (DHOST) as well as higher-derivative terms beyond that. We discuss this technique as a way to generate healthy four-dimensional gravity theories with an extra scalar degree of freedom and outline further generalisations of our method.

UV sensitive one-loop matter power spectrum in degenerate higher-order scalar-tensor theories

We study matter density perturbations up to third order and the one-loop matter power spectrum in degenerate higher-order scalar-tensor (DHOST) theories beyond Horndeski. We systematically solve gravitational field equations and fluid equations order by order and find three novel shape functions characterizing the third-order solution in DHOST theories. A complete form of the one-loop matter power spectrum is then obtained using the resultant second- and third-order solutions. We confirm the previous result that the convergence condition of the loop integrals in the infrared limit becomes more stringent than that of the standard one in general relativity. We show that also in the ultraviolet limit the convergence condition becomes more stringent and the one-loop matter power spectrum is thus sensitive to the short-wavelength behavior of the linear power spectrum.

On rotating black holes in DHOST theories

Using the disformal solution-generating method, we construct new axisymmetric solutions in Degenerate Higher Order Scalar Tensor (DHOST) theories. The method consists in first considering a “seed” known solution in DHOST theories and then performing a disformal transformation of the metric to obtain a new solution. In vacuum, the two solutions are equivalent but they become physically inequivalent when one considers coupling to matter. In that way, we “disform” the stealth Kerr black hole solution and we obtain a first analytic rotating non-stealth solution in DHOST theories, while the associated scalar field is time-dependent with a constant kinetic density. The new solution is characterized by three parameters: the mass, the spin and the disformal parameter which encodes the deviation with respect to the Kerr geometry. We explore some geometrical properties of the novel disformed Kerr geometry which is no more Ricci flat but has the same singularity as the Kerr metric, admits an ergoregion, and is asymptotically flat. Moreover, the hidden symmetry of the Kerr solution is broken, providing an example of a non-circular geometry in a higher order theory of gravity. We also discuss geodesic motions and compute its (disformed) null directions which are interesting tools to understand the causal structure of the geometry. In addition, to illustrate again the potentiality of the disformal solution-generating method, we present another axisymmetric solution for DHOST theories obtained from a disformal transformation of the generalized Kerr solution of Einstein-Scalar gravity.

Testing gravity theories with cosmic microwave background in the degenerate higher-order scalar-tensor theory

We study the cosmic microwave background (CMB) in the framework of the degenerate higher-order scalar-tensor (DHOST) theory to test gravity theories. This theoretical framework includes the wide class of dark energy models such as the Horndeski theory and its extensions as certain limits, and the general relativity can be also recovered. In this study, to test gravity theories with CMB, we formulate the linear perturbations of gravity and matter in the theory and their effective description parameterised by time-dependent effective field theory (EFT) parameters, $\alpha_i$ $(i=B,K,T,M,H,L)$ and $\beta_i$ $(i=1,2,3)$. Based on the resultant DHOST framework, we develop a numerical code to solve Boltzmann equations consistently. We then show that the angular power spectra of the CMB temperature anisotropies, E-mode and lensing potential as a demonstration and find that the parameter characterising the DHOST theory, $\beta_1$, provides the larger modifications of the spectra, compared with other EFT parameters. We also show the results in the case of a specific model in which the cosmic expansion as well as the EFT parameters are consistently determined.

Higher order spectra of weak lensing convergence maps in parametrized theories of modified gravity

ABSTRACT We compute the low-ℓ limit of the family of higher order spectra for projected (2D) weak lensing convergence maps. In this limit these spectra are computed to an arbitrary order using tree-level perturbative calculations. We use the flat-sky approximation and Eulerian perturbative results based on a generating function approach. We test these results for the lower order members of this family, i.e. the skew- and kurt-spectra against state-of-the-art simulated all-sky weak lensing convergence maps and find our results to be in very good agreement. We also show how these spectra can be computed in the presence of a realistic sky-mask and Gaussian noise. We generalize these results to 3D and compute the equal-time higher order spectra. These results will be valuable in analysing higher order statistics from future all-sky weak lensing surveys such as the Euclid survey at low-ℓ modes. As illustrative examples, we compute these statistics in the context of the Horndeski and beyond Horndeski theories of modified gravity. They will be especially useful in constraining theories such as the Gleyzes–Langlois–Piazza–Vernizzi (GLPV) theories and degenerate higher order scalar-tensor theories as well as the commonly used normal-branch of Dvali–Gabadadze–Porrati model, clustering quintessence models and scenarios with massive neutrinos.

Violation of the consistency relations for large-scale structure with dark energy

We study infrared effects in perturbation theory for large-scale structure coupled to the effective field theory of dark energy, focusing on, in particular, Degenerate Higher-Order Scalar-Tensor (DHOST) theories. In the subhorizon, Newtonian limit, DHOST theories introduce an extra large-scale velocity viπ which is in general different from the matter velocity vi. Contrary to the case in Horndeski theories, the presence of this extra large-scale velocity means that one cannot eliminate the long-wavelength effects of both vi and viπ with a single coordinate transformation, and thus the standard ΛCDM consistency relations for large-scale structure are violated by terms proportional to the relative velocity vi − viπ. We show, however, that in non-linear quantities this violation is determined by the linear equations and the symmetries of the fluid system. We find that the size of the baryon acoustic oscillations in the squeezed limit of the bispectrum is modified, that the bias expansion contains extra terms which contribute to the squeezed limit of the galaxy bispectrum, that infrared modes in the one-loop power spectrum no longer cancel, and that the equal-time double soft limit of the tree-level trispectrum is non-vanishing. {We also show how one can resum the effects of one of the large-scale velocities on the power spectrum, and how to include the relative velocity perturbatively in the resummation. This result is also applicable to other theories with a large-scale relative velocity, such as dark matter and baryons.} Many of our computations are also relevant for perturbation theory in ΛCDM with exact time dependence.

Late-time cosmological evolution in degenerate higher-order scalar-tensor models

We study the late cosmological evolution, from the nonrelativistic matter dominated era to the dark energy era, in modified gravity models described by Degenerate Higher-Order Scalar-Tensor (DHOST) theories. They represent the most general scalar-tensor theories propagating a single scalar degree of freedom and include Horndeski and Beyond Horndeski theories. We provide the homogeneous evolution equations for any quadratic DHOST theory, without restricting ourselves to theories where the speed of gravitational waves coincides with that of light since the present constraints apply to wavelengths much smaller than cosmological scales. To illustrate the potential richness of the cosmological background evolution in these theories, we consider a simple family of shift-symmetric models, characterized by three parameters and compute the evolution of dark energy and of its equation of state. We also identify the regions in parameter space where the models are perturbatively stable.

Cosmological evolution of viable models in the generalized scalar-tensor theory

We investigate the parameter distributions of the viable generalized scalar-tensor theory with conventional dust matter after GW170817 in a model-independent way. We numerically construct the models by computing the time evolution of a scalar field, which leads to a positive definite second-order Hamiltonian and are consistent with the observed Hubble parameter. We show the model parameter distributions in the degenerate higher-order scalar-tensor (DHOST) theory, and its popular subclasses, e.g., Horndeski and GLPV theories, etc.. We find that 1) the Planck mass run rate, $\alpha_M$, is insensitive to distinguish the theories. 2) the kinetic-braiding parameter, $\alpha_B$, clearly discriminates the models from those of the Horndeski theory, 3) the parameters for the higher-order theories, $\alpha_H$ and $\beta_1$, are relatively smaller in magnitude (by several factors) than $\alpha_M$ and $\alpha_B$, but can still be used for discriminating the theories except for the GLPV theory. Based on the above three facts, we propose a minimal set of parameters that sensibly distinguishes the subclasses of DHOST theories, ($\alpha_M$, $\alpha_B-\alpha_M/2$, $\beta_1$).

Regular black holes via the Kerr-Schild construction in DHOST theories

We extend the standard Kerr-Schild solution generating method to higher order scalar tensor theories that are shift-invariant for the scalar field. Certain degeneracy conditions, crucial for the absence of Ostrogradski ghosts, are found to be required for the validity of the Kerr-Schild ansatz while on the other hand, theories with no parity symmetry are excluded from the solution generating method. The extended Kerr-Schild symmetry turns out to be a very useful tool to easily construct black hole solutions from simple seed configurations. In particular, the generating method developed is adapted to construct generic black holes but also regular black hole solutions within Degenerate Higher Order Scalar Tensor (DHOST) theories. As a particular example we show how to construct explicitly the Hayward metric as a solution to a specific DHOST theory.

Black holes with a nonconstant kinetic term in degenerate higher-order scalar tensor theories

We investigate static and spherically symmetric black hole (BH) solutions in shift-symmetric quadratic-order degenerate higher-order scalar-tensor (DHOST) theories. We allow a nonconstant kinetic term $X=g^{\mu\nu} \partial_\mu\phi\partial_\nu \phi$ for the scalar field $\phi$ and assume that $\phi$ is, like the spacetime, a pure function of the radial coordinate $r$, namely $\phi=\phi(r)$. First, we find analytic static and spherically symmetric vacuum solutions in the so-called {\it Class Ia} DHOST theories, which include the quartic Horndeski theories as a subclass. We consider several explicit models in this class and apply our scheme to find the exact vacuum BH solutions. BH solutions obtained in our analysis are neither Schwarzschild or Schwarzschild (anti-) de Sitter. We show that a part of the BH solutions obtained in our analysis are free of ghost and Laplacian instabilities and are also mode stable against the odd-parity perturbations. Finally, we argue the case that the scalar field has a linear time dependence $\phi=qt+\psi (r)$ and show several simple examples of nontrivial BH solutions with a nonconstant kinetic term obtained analytically and numerically.

Perturbations of stealth black holes in degenerate higher-order scalar-tensor theories

Among the Scalar-Tensor modified theories of gravity, DHOST models could play a special role for dark energy while being consistent with current observations, notably those constraining the speed of gravitational waves. Schwarzschild-de Sitter black holes were shown to be exact solutions of a particular subclass of quadratic DHOST theories, while carrying a nontrivial scalar profile that linearly evolves in time and hence potentially providing exciting new phenomenological windows to explore this model. We investigate the physical perturbations about such black holes and find that the odd-parity tensor perturbations behave in a way indistinguishable to GR. On the other hand, the effective metric for the (even-parity) scalar perturbations is singular, indicating that those exact black hole solutions are infinitely strongly coupled and cannot be trusted within the regime of validity of the DHOST effective field theory. We show how this strong coupling result is generalizable to a whole class of solutions with arbitrary manifolds both for DHOST and Horndeski.

Obtaining Precision Constraints on Modified Gravity with Helioseismology.

We propose helioseismology as a new, precision probe of fifth forces at astrophysical scales, and apply it on the most general scalar-tensor theories for dark energy, known as degenerate higher-order scalar-tensor theories. We explain how the effect of the fifth force on the solar interior leaves an observable imprint on the acoustic oscillations, and under certain assumptions we numerically compute the nonradial pulsation eigenfrequencies within modified gravity. We illustrate its constraining power by showing that helioseismic observations have the potential to improve constraints on the strength of the fifth force by more than 2 orders of magnitude, as -1.8×10^{-3}≤Y≤1.2×10^{-3} (at 2σ). This in turn would suggest constraints of similar order for the theory's free functions around a cosmological background (α_{H}, β_{1}).

Black hole solutions in shift-symmetric degenerate higher-order scalar-tensor theories

We show that most classes of shift-symmetric degenerate higher-order scalar-tensor (DHOST) theories which satisfy certain degeneracy conditions are not compatible with the conditions for the existence of exact black hole solutions with a linearly time-dependent scalar field whose canonical kinetic term take a constant value. Combined with constraints from the propagation speed of gravitational waves, our results imply that cubic DHOST theories are strongly disfavored and that pure quadratic theories are likely to be the most viable class of DHOST theories. We find exact static and spherically symmetric (Schwarzschild and Schwarzschild-(anti-)de Sitter) black hole solutions in all shift-symmetric higher-derivative scalar-tensor theories which contain up to cubic order terms of the second-order derivatives of the scalar field, especially the full class of Horndeski and Gleyzes-Langlois-Piazza-Vernizzi theories. After deriving the conditions for the coupling functions in the DHOST Lagrangian that allow the exact solutions, we clarify their compatibility with the degeneracy conditions.

Linear stability analysis of hairy black holes in quadratic degenerate higher-order scalar-tensor theories: Odd-parity perturbations

We study static spherically symmetric black hole solutions with a linearly time-dependent scalar field and discuss their linear stability in the shift- and reflection-symmetric subclass of quadratic degenerate higher-order scalar-tensor (DHOST) theories. We present the explicit forms of the reduced system of background field equations for a generic theory within this subclass. Using the reduced equations of motion, we show that in several cases the solution is forced to be of the Schwarzschild or Schwarzschild-(anti-)de Sitter form. We consider odd-parity perturbations around general static spherically symmetric black hole solutions, and derive the concise criteria for the black holes to be stable. Our analysis also covers the case with a static or constant profile of the scalar field.

Vainshtein regime in scalar-tensor gravity: Constraints on degenerate higher-order scalar-tensor theories

We study the screening mechanism in the most general scalar-tensor theories that leave gravitational waves unaffected and are thus compatible with recent LIGO/Virgo observations. Using the effective field theory of dark energy approach, we consider the general action for perturbations beyond linear order, focussing on the quasi-static limit. When restricting to the subclass of theories that satisfy the gravitational wave constraints, the fully nonlinear effective Lagrangian contains only three independent parameters. One of these, $\beta_1$, is uniquely present in degenerate higher-order theories. We compute the two gravitational potentials for a spherically symmetric matter source and we find that for $\beta_1 \ge 0$ they decrease as the inverse of the distance, as in standard gravity, while the case $\beta_1 < 0$ is ruled out. For $\beta_1 > 0$, the two potentials differ and their gravitational constants are not the same on the inside and outside of the body. Generically, the bound on anomalous light bending in the Solar System constrains $\beta_1 \lesssim 10^{-5}$. Standard gravity can be recovered outside the body by tuning the parameters of the model, in which case $\beta_1 \lesssim 10^{-2}$ from the Hulse-Taylor pulsar.

Generalized conformal transformation and inflationary attractors

We investigate the inflationary attractors in models of inflation inspired from general conformal transformation of general scalar-tensor theories to the Einstein frame. The coefficient of the conformal transformation in our study depends on both the scalar field and its kinetic term. Therefore the relevant scalar-tensor theories display the subset of the class I of the degenerate higher-order scalar-tensor theories in which both the scalar field and its kinetic term can non-minimally couple to gravity. We find that if the conformal coefficient $\Omega$ takes a multiplicative form such that $\Omega \equiv w(\phi)W(X)$ where $X$ is the kinetic term of the field $\phi$, the theoretical predictions of the proposed model can have usual universal attractor independent of any functions of $W(X)$. For the case where $\Omega$ takes an additive form, such that $\Omega \equiv w(\phi) + k(\phi) \Xi(X)$, we find that there are new $\xi$ attractors in addition to the universal ones. We analyze the inflationary observables of these models and compare them to the latest constraints from the Planck collaboration. We find that the observable quantities associated to these new $\xi$ attractors do not satisfy the constraints from Planck data at a strong coupling limit.

Screening mechanism in degenerate higher-order scalar-tensor theories evading gravitational wave constraints

We consider a subclass of degenerate higher-order scalar-tensor (DHOST) theories in which gravitational waves propagate at the speed of light and do not decay into scalar fluctuations. The screening mechanism in DHOST theories evading these two gravitational wave constraints operates very differently from that in generic DHOST theories. We derive a spherically symmetric solution in the presence of nonrelativistic matter. General relativity is recovered in the vacuum exterior region provided that functions in the Lagrangian satisfy a certain condition, implying that fine-tuning is required. Gravity in the matter interior exhibits novel features: although the gravitational potentials still obey the standard inverse power law, the effective gravitational constant is different from its exterior value, and the two metric potentials do not coincide. We discuss possible observational constraints on this subclass of DHOST theories, and argue that the tightest bound comes from the Hulse-Taylor pulsar.

Constraining degenerate higher-order scalar-tensor theories with linear growth of matter density fluctuations

We investigate the potential of cosmological observations, such as galaxy surveys, for constraining degenerate higher-order scalar-tensor (DHOST) theories, focusing in particular on the linear growth of the matter density fluctuations. We develop a formalism to describe the evolution of the matter density fluctuations during the matter dominated era and in the early stage of the dark energy dominated era in DHOST theories, and give an approximate expression for the gravitational growth index in terms of several parameters characterizing the theory and the background solution under consideration. By employing the current observational constraints on the growth index, we obtain a new constraint on a parameter space of DHOST theories. Combining our result with other constraints obtained from the Newtonian stellar structure, we show that the degeneracy between the effective parameters of DHOST theories can be broken without using the Hulse-Taylor pulsar constraint.