Gravitational waves and geometrical optics in scalar-tensor theories
Alice Garoffolo, Gianmassimo Tasinato, Carmelita Carbone, Daniele Bertacca, Sabino Matarrese
备注：
42 pages. V2: Improved discussion, in particular on gauge conditions, and references added. Results unchanged. V3: Updated version to be published in JCAP
The detection of gravitational waves (GWs) propagating through cosmic structures can provide invaluable information on the geometry and content of our Universe, as well as on the fundamental theory of gravity. In order to test possible departures from General Relativity, it is essential to analyse, in a modified gravity setting, how GWs propagate through a perturbed cosmological space-time. Working within the framework of geometrical optics, we develop tools to address this topic for a broad class of scalar-tensor theories, including scenarios with non-minimal, derivative couplings between scalar and tensor modes. We determine the corresponding evolution equations for the GW amplitude and polarization tensor. The former satisfies a generalised evolution equation that includes possible effects due to a variation of the effective Planck scale; the latter can fail to be parallely transported along GW geodesics unless certain conditions are satisfied. We apply our general formulas to specific scalar-tensor theories with unit tensor speed, and then focus on GW propagation on a perturbed space-time. We determine corrections to standard formulas for the GW luminosity distance and for the evolution of the polarization tensor, which depend both on modified gravity and on the effects of cosmological perturbations. Our results can constitute a starting point to disentangle among degeneracies from different sectors that can influence GW propagation through cosmological space-times.
We derive a Higgs inflationary model in the context of a complex geometrical scalar-tensor theory of gravity. In this model the Higgs inflaton scalar field has geometrical origin playing the role of the Weyl scalar field in the original non-riemannian background geometry. The energy scale enough to generate inflation from the Higgs energy scale is achieved due to the compatibility of the theory with its background complex Weyl-integrable geometry. We found that for a number of e-foldings $N=63$, a nearly scale invariant spectrum for the inflaton is obtained with an spectral index $n_s\simeq 0.…
In this paper we obtain some cosmological solutions that describe the present period of accelerating expansion of the universe in the framework of a geometrical gauge scalar-tensor theory of gravity. The background geometry in the model is the Weyl integrable and we found a class of power law solutions for the Weyl scalar field when an invariant metric is employed in a power law expanding universe. We obtain a deceleration and an equation of state parameters in agreement with PLANCK 2018 observational data for some specific parameters of the model. The deceleration parameter tends asymptotical…
We investigate, in the framework of a recently introduced new class of invariant geometrical scalar-tensor theory of gravity, the possibility that a viscous dark fluid can be described in a unified manner by a single scalar field. Thus we developed a model in which both the metric tensor and the scalar field have geometrical origin. The scalar field is characterized by a non-canonical kinetic term and the scalar viscosity of the dark fluid appears as soon the kinetic energy of the scalar field is no longer canonical. The scalar viscosity is considered as a function of the Hubble and the decele…
In this letter we investigate gauge invariant scalar fluctuations of the metric in a non-perturbative formalism for a Higgs inflationary model recently introduced in the framework of a geometrical scalar-tensor theory of gravity. In this scenario the Higgs inflaton field has its origin in the Weyl scalar field of the background geometry. We found a nearly scale invariance of the power spectrum for linear scalar fluctuations of the metric. For certain parameters of the model we obtain values for the scalar spectral index $n_s$ and the scalar to tensor ratio $r$ that fit well with the Planck 201…
We investigate cosmological inflationary scenarios from a gravitoelectromagnetic theory. Our work is formulated in the light of a recently introduced geometrical Weyl-Invariant scalar-tensor theory of gravity, where the nature of both the electromagnetic potential and the inflaton field is attributed to the space-time geometry. We obtain a Harrison-Zeldovich power spectrum for quantum fluctuations of the inflaton field. In our model the electromagnetic fields have also a nearly scale invariant power spectrum for a power-law inflation. We found that the the seed magnetic fields have a nearly sc…
We consider an approach to Brans-Dicke theory of gravity in which the scalar field has a geometrical nature. By postulating the Palatini variation, we find out that the role played by the scalar field consists in turning the space-time geometry into a Weyl integrable manifold. This procedure leads to a scalar-tensor theory that differs from the original Brans-Dicke theory in many aspects and presents some new features.
We consider scalar-tensor theories of gravity defined in Weyl integrable space-time and show that in the ADM formalism Weyl transformations corresponding to change of frames induce canonical transformations between different representations of the phase space. In this context, we discuss the physical equivalence of two distinct Weyl frames at the classical level.
In this work we consider a scale-tensor theory in which the space-time is endowed with a Weyl integrable geometrical structure due to the Palatini variational method. Since the scalar field has a geometrical nature (related to non-metricity), the theory is known as \textit{Geometrical Scalar-Tensor}. On the framework of Weyl transformations, a non-minimally coupled scalar-tensor theory on the Jordan frame corresponds to a minimally coupled Einstein-Hilbert action on the Einstein frame. The scalar potential is selected by the Noether symmetry approach in order to obtain conserved quantities for…
We obtain vacuum solutions in the presence of a cosmological constant in the context of the Weyl geometrical scalar-tensor theory. We investigate the limit when $ω$ goes to infinity and show by working out the solutions that in this limit there are some cases in which the scalar field tends to a constant (with the implicit consequence of the geometry becoming Riemannian), although the solutions do not reduce to the corresponding Einstein solutions. We have also extended a previous result, known in the literature, by showing that in the case of vacuum with cosmological constant the field equati…
We derive an interacting quintessence model on the framework of a recently introduced new class of geometrical scalar-tensor theories of gravity formulated on a Weyl-Integrable geometry, where the gravitational sector is described by both a scalar and a tensor metric field. By using a Palatini variational principle we construct a scalar-tensor action invariant under the Weyl symmetry group of the background geometry, which in the Einstein-Riemann frame leads to a gravitoelectromagnetic theory. We use the gauge freedom of the theory and the fact that the Weyl scalar field couples with matter fi…
We investigate cosmological models in a recently proposed geometrical theory of gravity, in which the scalar field appears as part of the space-time geometry. We extend the previous theory to include a scalar potential in the action. We solve the vacuum field equations for different choices of the scalar potential and give a detailed analysis of the solutions. We show that in some cases a cosmological scenario is found that seems to suggest the appearance of a geometric phase transition. We build a toy model, in which the accelerated expansion of the early universe is driven by pure geometry.
A pressing problem in comparing inflationary models with observation is the accurate calculation of correlation functions. One approach is to evolve them using ordinary differential equations ("transport equations"), analogous to the Schwinger-Dyson hierarchy of in-out quantum field theory. We extend this approach to the complete set of momentum space correlation functions. A formal solution can be obtained using raytracing techniques adapted from geometrical optics. We reformulate inflationary perturbation theory in this language, and show that raytracing reproduces the familiar "delta N" Tay…
A general affine connection has enough degrees of freedom to describe the classical gravitational and electromagnetic fields in the metric-affine formulation of gravity. The gravitational field is represented in the Lagrangian by the symmetric part of the Ricci tensor, while the classical electromagnetic field is represented geometrically by the tensor of homothetic curvature. We introduce matter as the four-velocity field subject to the kinematical constraint in which the Lagrange multiplier represents the energy density. A coupling between the four-velocity and the trace of the nonmetricity …
The scalar-tensor theories of gravity in spacetime dimensions $D+1>2$ are studied. By doing Hamiltonian analysis, we obtain the geometrical dynamics of the theories from their Lagrangian. The Hamiltonian formalism indicates that the theories are naturally divided into two sectors by the coupling parameter $ω$. The Hamiltonian structure in both sectors are similar to the corresponding structure of 4-dimensional cases. It turns out that there is a symplectic reduction from the canonical structure of $so(D+1)$ Yang-Mills theories coupled to the scalar field to the canonical structure of the geome…
We study the geometrical-optics expansion for circularly-polarized electromagnetic waves propagating on a curved spacetime in general relativity. We show that higher-order corrections to the Faraday and stress-energy tensors may be found via a system of transport equations, in principle. At sub-leading order, the stress-energy tensor possesses terms proportional to the wavelength whose sign depends on the handedness of the circular polarization. Due to such terms, the direction of energy flow is not aligned with the gradient of the eikonal phase, in general, and the wave may carry a transverse…
A spherical planetary nebula is described as a geometric model. The nebula itself is considered as a thin-shell which visualized as a boundary of two spacetimes. The inner and outer curvature tensors of the thin-shell are found in order to get an expression of the energy-momentum tensor on the thin-shell. The energy density and pressure expressions are derived using the energy-momentum tensor. The time evolution of the radius of the thin-shell is obtained in terms of the energy density. The model is tested by using a simple power function for decreasing energy density and the evolution pattern…
The Tolman~VII solution, an exact analytic solution to the spherically symmetric, static Einstein equations with a perfect fluid source, has many characteristics that make it interesting for modelling high density physical astronomical objects. Here we supplement those characteristics with the geometrical tensors that this solution possess, and find that the Weyl, Riemann, and Ricci tensor components show unexpected mathematical behaviour that change depending on physically motivated parameters, even though the mass of the modelled objects is fixed. We show these features firstly through tenso…
A new cosmological theory is proposed in the theoretical framework of modified gravity theories which is based on a tachyonic field non-minimally coupled with a specific topological invariant constructed with third order contractions of the Riemann tensor. After proposing the action of the cosmological model, the modified Friedmann relations and the Klein-Gordon equations are obtained, describing the corresponding geometrical corrections to the Einstein-Hilbert action. The physical features of the cosmological system are investigated by adopting the dynamical system analysis in the case of an …
The Geometrical Localization mechanism in Randall-sundrum (RS) scenarios is extended by considering the coupling between a quadratic mass term and geometrical tensors. Since the quadratic term is symmetric, tensors with two symmetric indices have to be taken into account. These are the Ricci and the Einstein tensors. For the Ricci tensor it is shown that a localized zero mode exists while that is not possible for the Einstein tensor. It is already known that the Ricci scalar generates a localized solution but the metrics do not. Therefore, it can be conclude that divergenceless tensors do not …
A gravitational theory of a scalar field non-minimally coupled with torsion and boundary term is considered with the aim to construct Lorentzian wormholes. Geometrical parameters including shape and redshift functions are obtained for these solutions. We adopt the formalism of Noether Gauge Symmetry Approach in order to find symmetries, Lie brackets and invariants (conserved quantities). Furthermore by imposing specific forms of potential function, we are able to calculate metric coefficients and discuss their geometrical behavior.
In this paper we adopt a global and non-entropic approach to the problem of the arrow of time, according to which the arrow of time is an intrinsic geometrical property of spacetime. Our main aim is to show the double role played by the energy-momentum tensor in the context of our approach. On the one hand, the energy-momentum tensor is the intermediate step that permits to turn the geometrical time-asymmetry of the universe into a local arrow of time manifested as a time-asymmetric energy flow. On the other hand, the energy-momentum tensor supplies the basis for deducing the time-asymmetry of…
We consider some aspects of spontaneous breaking of Lorentz Invariance in field theories, discussing the possibility that the certain tensor operators may condensate in the ground state in which case the tensor Goldstone particles would appear. We analyze their dynamics and discuss to which extent such a theory could imitate the gravity. We are also interested if the universality of coupling of such `gravitons' with other particles can be achieved in the infrared limit. Then we address the more complicated models when such tensor Goldstones coexist with the usual geometrical gravitons. At the …
A non-minimal photon-torsion axial coupling in the quantum electrodynamics (QED) framework is considered. The geometrical optics in Riemannian-Cartan spacetime is considering and a plane wave expansion of the electromagnetic vector potential is considered leading to a set of the equations for the ray congruence. Since we are interested mainly on the torsion effects in this first report we just consider the Riemann-flat case composed of the Minkowskian spacetime with torsion. It is also shown that in torsionic de Sitter background the vacuum polarisation does alter the propagation of individual…
The material tensor of linear response in electrodynamics is constructed out of products of two symmetric second rank tensor fields which in the approximation of geometrical optics and for uniaxial symmetry reduce to "optical" metrics, describing the phenomenon of birefringence. This representation is interpreted in the context of an underlying internal geometrical structure according to which the symmetric tensor fields are vectorial elements of an associated two-dimensional space.
Fulling, King, Wybourne and Cummings (FKWC) have proposed to expand systematically the Riemann polynomials encountered in the context of field theories in curved spacetime on standard bases constructed from group theoretical considerations. They have also displayed such bases for scalar Riemann polynomials of order eight or less in the derivatives of the metric tensor and for tensorial Riemann polynomials of order six or less. Here we provide a slightly modified version of the FKWC-bases as well as an important list of geometrical relations we have used in recent works. These relations are ind…
We provide irreducible expressions for the metric variations of the gravitational action terms constructed from the 17 curvature invariants of order six in derivatives of the metric tensor i.e. from the geometrical terms appearing in the diagonal heat-kernel or Gilkey-DeWitt coefficient $a_3$. We then express, for a four dimensional spacetime, the approximated stress-energy tensor constructed from the renormalized DeWitt-Schwinger effective action associated with a massive scalar field. We also construct, for higher dimensional spacetimes, the infinite counterterms of order six in derivatives …
In a space-time with torsion, the action for the gravitational field can be extended with a parity-violating piece. We show how to obtain such a piece from geometry itself, by suitably modifying the affine connection so as to include a pseudo-tensorial part. A consistent method is thus suggested for incorporating parity-violation in the Lagrangians of all matter fields with spin in a space-time background with torsion.
The geometrical-optics expansion reduces the problem of solving wave equations to one of solving transport equations along rays. Here we consider scalar, electromagnetic and gravitational waves propagating on a curved spacetime in general relativity. We show that each is governed by a wave equation with the same principal part. It follows that: each wave propagates at the speed of light along rays (null generators of hypersurfaces of constant phase); the square of the wave amplitude varies in inverse proportion to the cross section of the beam; and the polarization is parallel-propagated along…
We study a locally anisotropic model of General Relativity in the framework of a more general geometrical structure than the Riemannian one. In this model the observable anisotropy of the CMBR (WMAP) is represented by a tensor of anisotropy and it is included in the metric structure of space-time. As well, some interesting special cases of spaces are considered.
Geometrical optics describes, with good accuracy, the propagation of high-frequency plane waves through an electromagnetic medium. Under such approximation, the behaviour of the electromagnetic fields is characterised by just three quantities: the temporal frequency $ω$, the spatial wave (co)vector $k$, and the polarisation (co)vector $a$. Numerous key properties of a given optical medium are determined by the Fresnel surface, which is the visual counterpart of the equation relating $ω$ and $k$. For instance, the propagation of electromagnetic waves in a uniaxial crystal, such as calcite, is r…
The main purpose of this contribution is to determine physical and geometrical characterizations of whole classes of stationary cyclic symmetric gravitational fields coupled to Maxwell electromagnetic fields within the $(2+1)$-dimensional gravity. The physical characterization is based on the determination of the local and global energy-momentum-mass quantities using the Brown-York approach. As far as to the algebraic-geometrical characterization is concerned, the eigenvalue problem for the electromagnetic field, energy-momentum and Cotton tensors is solved and their types are established. The…
The geometrical approximation of the extended Maxwell equation in curved spacetime incorporating interactions induced by the vacuum polarization effects is considered. Taking into account these QED interactions and employing the analogy between eikonal equation in geometrical optics and Hamilton-Jacobi equation for the particle motion, we study the phase structure of the modified theory. There is a complicated, local induced phase which is believed to be responsible for the modification of the classical picture of light ray. The main features of QGO could be obtained through the study of this …
A geometrical-optics analogy for gravitational lensing using axicon-type lenses
Institut für Astronomie und Astrophysik, Eberhard Karls Universität Tübingen（蒂宾根大学天文与天体物理研究所）；Department of Optics, Faculty of Physical Sciences, Universidad Complutense de Madrid（马德里康普顿斯大学物理科学学院光学系）
We present some new derivations of the effect of a plane gravitational wave on a light ray. A simple interpretation of the results is that a gravitational wave causes a phase modulation of electromagnetic waves. We arrive at this picture from two contrasting directions, namely null geodesics and Maxwell's equations, or, geometric and wave optics. Under geometric optics, we express the geodesic equations in Hamiltonian form and solve perturbatively for the effect of gravitational waves. We find that the well-known time-delay formula for light generalizes trivially to massive particles. We also …
This document contains a description of physics entirely based on a geometric presentation: all of the theory is described giving only a pseudo-riemannian manifold (M, g) of dimension n > 5 for which the g tensor is, in studied domains, almost everywhere of signature (-, -, +, ..., +). No object is added to this space-time, no general principle is supposed. The properties we impose to some domains of (M, g) are only simple geometric constraints, essentially based on the concept of "curvature". These geometric properties allow to define, depending on considered cases, some objects (frequently d…
Determining cosmological field equations represents a still very debated matter and implies a wide discussion around different theoretical proposals. A suitable conceptual scheme could be represented by gravity models that naturally generalize Einstein Theory like higher order gravity theories and higher dimensional ones. Both of these two different approaches allow to define, at the effective level, Einstein field equations equipped with source-like energy momentum tensors of geometrical origin. In this paper, it is discussed the possibility to develop a five dimensional fourth order gravity …
Symmetries of geometrical and physical quantities in general relativity provide important information about the curvature structure of the spacetimes. Symmetries of the curvature and the Weyl tensors, known as curvature and Weyl collineations respectively, are two of such important symmetries. Some results on these symmetries for Bianchi type V spacetimes are discussed.
