Standard General Relativity Research Articles

Cosmic microwave background with Brans-Dicke gravity. I. Covariant formulation

In the covariant cosmological perturbation theory, a 1+3 decomposition ensures that all variables in the frame-independent equations are covariant, gauge-invariant and have clear physical interpretations. We develop this formalism in the case of Brans-Dicke gravity, and apply this method to the calculation of cosmic microwave background (CMB) anisotropy and large scale structures (LSS). We modify the publicly available Boltzmann code CAMB to calculate numerically the evolution of the background and adiabatic perturbations, and obtain the temperature and polarization spectra of the Brans-Dicke theory for both scalar and tensor mode, the tensor mode result for the Brans-Dicke gravity are obtained numerically for the first time. We first present our theoretical formalism in detail, then explicitly describe the techniques used in modifying the CAMB code. These techniques are also very useful to other gravity models. Next we compare the CMB and LSS spectra in Brans-Dicke theory with those in the standard general relativity theory. At last, we investigate the ISW effect and the CMB lensing effect in the Brans-Dicke theory. Constraints on Brans-Dicke model with current observational data is presented in a companion paper (paper II).

Non-Linear Equation of State and Effective Phantom Divide in Brane Models

Here, DGP model of brane-gravity is analyzed and compared with the standard general relativity and Randall-Sundrum cases using non-linear equation of state. Phantom fluid is known to violate the weak energy condition. In this paper, it is found that this characteristic of phantom energy is affected drastically by the negative brane-tension $\lambda$ of the RS-II model. It is found that in DGP model strong energy condition(SEC) is always violated and the universe accelerates only where as in RS-II model even SEC is not violated for $1 < \rho/\lambda < 2$ and the universe decelerates.

Semiclassical unimodular gravity

Classically, unimodular gravity is known to be equivalent to General Relativity (GR),except for the fact that the effective cosmological constant Λ has the status of an integration constant. Here, we explore various formulations of unimodular gravity beyond the classical limit.We first consider the non-generally covariant action formulation in which the determinantof the metric is held fixed to unity. We argue that the corresponding quantum theory is also equivalent to General Relativity for localized perturbative processes which take place in generic backgrounds of infinite volume (such as asymptotically flat spacetimes). Next, using the same action, we calculate semiclassical non-perturbative quantities, which we expect will be dominated by Euclidean instanton solutions. We derive the entropy/area ratio for cosmological and black hole horizons, finding agreement with GR for solutions in backgrounds of infinite volume, but disagreement for backgrounds with finite volume.In deriving the above results, the path integral is taken over histories with fixed 4-volume. We point out that the results are different if we allow the 4-volume of the different histories to vary over a continuum range. In this "generalized" version of unimodular gravity, one recovers the full set of Einstein's equations in the classical limit, including the trace,so Λ is no longer an integration constant. Finally, we consider the generally covariant theory due to Henneaux and Teitelboim, which is classically equivalent to unimodular gravity. In this case, the standard semiclassical GR results are recovered provided that the boundary term in the Euclidean action is chosen appropriately.

ΛCDMuniverse inf(R)gravity

Several different explicit reconstructions of f(R) gravity are obtained from the background FRW expansion history. It is shown that the only theory whose Lagrangian is a simple function of the Ricci scalar R, that admits an exact LCDM expansion history is standard General Relativity with a positive cosmological constant and the only way to obtain this behaviour of the scale factor for more general functions of R is to add additional degrees of freedom to the matter sector.

MacDowell–Mansouri gravity and Cartan geometry

The geometric content of the MacDowell–Mansouri formulation of general relativity is best understood in terms of Cartan geometry. In particular, Cartan geometry gives clear geometric meaning to the MacDowell–Mansouri trick of combining the Levi-Civita connection and coframe field, or soldering form, into a single physical field. The Cartan perspective allows us to view physical spacetime as tangentially approximated by an arbitrary homogeneous ‘model spacetime’, including not only the flat Minkowski model, as is implicitly used in standard general relativity, but also de Sitter, anti-de Sitter or other models. A ‘Cartan connection’ gives a prescription for parallel transport from one ‘tangent model spacetime’ to another, along any path, giving a natural interpretation of the MacDowell–Mansouri connection as ‘rolling’ the model spacetime along physical spacetime. I explain Cartan geometry, and ‘Cartan gauge theory’, in which the gauge field is replaced by a Cartan connection. In particular, I discuss MacDowell–Mansouri gravity, as well as its more recent reformulation in terms of BF theory, in the context of Cartan connections.

Gravitational signature of Schwarzschild black holes in dynamical Chern-Simons gravity

Dynamical Chern-Simons gravity is an extension of General Relativity in which the gravitational field is coupled to a scalar field through a parity-violating Chern-Simons term. In this framework, we study perturbations of spherically symmetric black hole spacetimes, assuming that the background scalar field vanishes. Our results suggest that these spacetimes are stable, and small perturbations die away as a ringdown. However, in contrast to standard General Relativity, the gravitational waveforms are also driven by the scalar field. Thus, the gravitational oscillation modes of black holes carry imprints of the coupling to the scalar field. This is a smoking gun for Chern-Simons theory and could be tested with gravitational-wave detectors, such as LIGO or LISA. For negative values of the coupling constant, ghosts are known to arise, and we explicitly verify their appearance numerically. Our results are validated using both time evolution and frequency domain methods.

Probing modifications of general relativity using current cosmological observations

We test General Relativity (GR) using current cosmological data: the cosmic microwave background (CMB) from WMAP5 (Komatsu et al. 2009), the integrated Sachs-Wolfe (ISW) effect from the cross-correlation of the CMB with six galaxy catalogs (Giannantonio et al. 2008), a compilation of supernovae Type Ia (SNe) including the latest SDSS SNe (Kessler et al. 2009), and part of the weak lensing (WL) data from CFHTLS (Fu et al. 2008, Kilbinger et al. 2009) that probe linear and mildly non-linear scales. We first test a model where the effective Newton's constant, mu, and the ratio of the two gravitational potentials, eta, transit from the GR value to another constant at late times; in this case, we find that standard GR is fully consistent with the combined data. The strongest constraint comes from the ISW effect which would arise from this gravitational transition; the observed ISW signal imposes a tight constraint on a combination of mu and eta that characterizes the lensing potential. Next, we consider four pixels in time and space for each function mu and eta, and perform a Principal Component Analysis (PCA) finding that seven of the resulting eight eigenmodes are consistent with GR within the errors. Only one eigenmode shows a 2-sigma deviation from the GR prediction, which is likely to be due to a systematic effect. However, the detection of such a deviation demonstrates the power of our time- and scale-dependent PCA methodology when combining observations of structure formation and expansion history to test GR.

Black holes inf(R)theories

In the context of $f(R)$ theories of gravity, we address the problem of finding static and spherically symmetric black hole solutions. Several aspects of constant curvature solutions with and without electric charge are discussed. We also study the general case (without imposing constant curvature). Following a perturbative approach around the Einstein-Hilbert action, it is found that only solutions of the Schwarzschild-(anti) de Sitter type are present up to second order in perturbations. Explicit expressions for the effective cosmological constant are obtained in terms of the $f(R)$ function. Finally, we have considered the thermodynamics of black holes in anti-de Sitter space-time and found that this kind of solutions can only exist provided the theory satisfies $R_0+f(R_0)<0$. Interestingly, this expression is related to the condition which guarantees the positivity of the effective Newton's constant in this type of theories. In addition, it also ensures that the thermodynamical properties in $f(R)$ gravities are qualitatively similar to those of standard General Relativity.

The breaking of the Equivalence Principle in theories with varying α

AbstractThe Standard Model and General Relativity provide a good description of phenomena at low energy. These theories, which agree very well with the experiment, contain a set of parameters called “fundamental constants”, that are assumed invariant under changes in location and reference system. However, their possible variation has been studied since Dirac made the large numbers hypothesis (LNH). Moreover, unified field theory and extra dimensions theories such as Kaluza-Klein or Superstring theories, state not only the variation of these constants, but also the simultaneity of the variations.The Eötvös effect is one of the most sensitive indicators of changes in fundamental constants. Bekenstein (2002) showed that in his theory, using a classical static particle model of matter, there is no Eötvös effect and therefore met the Universality of Free Fall and the Principle of Equivalence.We present different results than those obtained by Bekenstein, Kraiselburd, Vucetich (2009). Modifying his theory, taking more realistic models of matter and using the model THεμ techniques (Ligtman-Lee (1975) and Haugan (1979), not used before to analyze this model), very small but measurable effects have been found.

Power-law cosmic expansion inf(R)gravity models

We show that within the class of $f(R)$ gravity theories, Friedmann-Lema\^{\i}tre-Robertson-Walker power-law perfect fluid solutions only exist for ${R}^{n}$ gravity. This significantly restricts the set of exact cosmological solutions which have similar properties to what is found in standard general relativity.

Dynamics of a self-interacting scalar field trapped in the braneworld for a wide variety of self-interaction potentials

We apply the dynamical systems tools to study the linear dynamics of a self-interacting scalar field trapped in the braneworld, for a wide variety of self-interaction potentials. We focus on Randall-Sundrum and on Dvali-Gabadadze-Porrati braneworld models exclusively. These models are complementary to each other: while the Randall-Sundrum brane produces UV) corrections to general relativity, the Dvali-Gabadadze-Porrati braneworld modifies Einstein's theory at large scales, i e., produces IR modifications of general relativity. This study of the asymptotic properties of both braneworld models, shows---in the phase space---the way the dynamics of a scalar field trapped in the brane departs from standard general relativity behavior.

Exact analysis of scaling and dominant attractors beyond the exponential potential

By considering the potential parameter Γ as a function of another potential parameter λ (Zhou et al 2008 Phys. Lett. B 660 7–12), we successfully extend the analysis of a two-dimensional autonomous dynamical system of a quintessence scalar field model to the analysis of a three-dimensional system, which enables us to study the critical points of a large number of potentials beyond the exponential potential exactly. We find that there are ten critical points in all, three points P3,5,6 are general points which are possessed by all quintessence models regardless of the form of potentials and the rest of the points are closely connected to the concrete potentials. It is quite surprising that, apart from the exponential potential, there are a large number of potentials which can give a scaling solution when the function f(λ)(=Γ(λ) − 1) equals zero for one or some values of λ∗ and if the parameter λ∗ also satisfies condition (16) or (17) at the same time. We give the differential equations to derive these potentials V(ϕ) from f(λ). We also find that, if some conditions are satisfied, the de-Sitter-like dominant point P4 and the scaling solution point P9 (or P10) can be stable simultaneously unlike P9 and P10. Although we survey scaling solutions beyond the exponential potential for ordinary quintessence models in standard general relativity, this method can be applied to other extensively scaling solution models studied in the literature (Copeland et al 2006 Int. J. Mod. Phys. D 15 1753) including coupled quintessence, (coupled-)phantom scalar field, k-essence and even beyond the general relativity case H2 ∝ ρnT. We also discuss the disadvantage of our approach.

Standard general relativity from Chern–Simons gravity

Chern–Simons models for gravity are interesting because they provide a truly gauge-invariant action principle in the fiber-bundle sense. So far, their main drawback has largely been its perceived remoteness from standard General Relativity, based on the presence of higher powers of the curvature in the Lagrangian (except, remarkably, for three-dimensional spacetime). Here we report on a simple model that suggests a mechanism by which standard General Relativity in five-dimensional spacetime may indeed emerge at a special critical point in the space of couplings, where additional degrees of freedom and corresponding “anomalous” Gauss–Bonnet constraints drop out from the Chern–Simons action. To achieve this goal, both the Lie algebra g and the symmetric g-invariant tensor that define the Chern–Simons Lagrangian are constructed by means of the Lie algebra S-expansion method with a suitable finite Abelian semigroup S. The results are generalized to arbitrary odd dimensions, and the possible extension to the case of eleven-dimensional supergravity is briefly discussed.

GRAVITATIONAL WAVES IN THE HYPERSPACE?

In the framework of the debate on high-frequency gravitational waves (GWs), after a review of GWs in standard General Relativity, which is due for completness, the possibility of merging such a traditional analysis with the hyperspace formalism that has been recently introduced in some papers in the literature, with the goal of a better understanding of manifolds dimensionality also in a cosmological framework, is discussed. Using the concept of refractive index in the hyperspace, spherical solutions are given and the propagation of GWs in a region of the hyperspace with a unitary refractive index is also discussed. Propagation phenomena associated to the higher dimensionality are proposed, possibly including nonlinear effects. Further and accurate studies in this direction are needed.

Cascading gravity and degravitationThis paper was presented at the Theory CANADA 4 conference, held at the Centre de Recherches Mathématiques at the Université de Montréal, Québec, Canada on 4–7 June 2008.

Cascading gravity is an explicit realization of the idea of degravitation, where gravity behaves as a high-pass filter. This could explain why a large cosmological constant does not backreact as much as anticipated from standard General Relativity. The model relies on the presence of at least two infinite extra dimensions while our world is confined on a four-dimensional brane. Gravity is then four-dimensional at short distances and becomes weaker and weaker at larger distances.

F(R) Cosmology with torsion

f(R)-gravity with geometric torsion (not related to any spin fluid) is considered in a cosmological context. We derive the field equations in vacuum and in the presence of perfect-fluid matter and discuss the related cosmological models. Torsion vanishes in vacuum for almost all arbitrary functions f(R) leading to standard general relativity. Only for f(R)=R2, torsion gives a contribution in the vacuum leading to accelerated behavior. When material sources are considered, we find that the torsion tensor is different from zero even with spinless material sources. This tensor is related to the logarithmic derivative of f′ (R), which can be expressed also as a nonlinear function of the trace of the matter energy–momentum tensor Σμν. We show that the resulting equations for the metric can always be arranged to yield effective Einstein equations. When the homogeneous and isotropic cosmological models are considered, terms originated by torsion can lead to accelerated expansion. This means that, in f(R)-gravity, torsion can be a geometric source for acceleration.

Enlarging minimal-supergravity parameter space by decreasing pre-nucleosynthesis Hubble rate in scalar-tensor cosmologies

We determine under what conditions Scalar Tensor cosmologies predict an expansion rate which is reduced as compared to the standard General Relativity case. We show that ST theories with a single matter sector typically predict an enchanced Hubble rate in the past, as a consequence of the requirement of an attractive xed point towards General Relativity at late times. Instead, when additional matter sectors with dieren t conformal factors are added, the late time convergence to General Relativity is mantained and at the same time a reduced expansion rate in the past can be driven. For suitable choices of the parameters which govern the scalar eld evolution, a sizeable reduction (up to about 2 orders of magnitude) of the Hubble rate prior to Big Bag Nucleosynthesis can be obtained. We then discuss the impact of these cosmological models on the relic abundance of dark matter is minimal Supergravity models: we show that the cosmologically allowed regions in parameter space are signican tly enlarged, implying a change in the potential reach of LHC on the neutralino phenomenology.

Model of dark matter and dark energy based on gravitational polarization

A model of dark matter and dark energy based on the concept of gravitational polarization is investigated. We propose an action in standard general relativity for describing, at some effective or phenomenological level, the dynamics of a dipolar medium, i.e. one endowed with a dipole moment vector, and polarizable in a gravitational field. Using first-order cosmological perturbations, we show that the dipolar fluid is undistinguishable from standard dark energy (a cosmological constant �) plus standard dark matter (a pressureless perfect fluid), and therefore benefits from the successes of the �-CDM (�-cold dark matter) scenario at cosmological scales. Invoking an argument of “weak clusterisation” of the mass distribution of dipole moments, we find that the dipolar dark matter reproduces the phenomenology of the modified Newtonian dynamics (MOND) at galactic scales. The dipolar medium action naturally contains a cosmological constant, and we show that if the

Summary of session C1: experimental gravitation

The fact that gravity is a metric theory follows from the Einstein equivalence principle. This principle consists of (i) the universality of free fall, (ii) the universality of the gravitational redshift and (iii) the local validity of Lorentz invariance. Many experiments searching for deviations from standard general relativity test the various aspects of the Einstein equivalence principle. Here we report on experiments covering the whole Einstein equivalence principle. Until now all experiments have been in agreement with the Einstein equivalence principle. As a consequence, gravity has to be described by a metric theory. Any metric theory of gravity leads to effects such as perihelion shift, deflection of light, gravitational redshift, gravitational time delay, Lense–Thirring effect, Schiff effect, etc. A particular theory of that sort is Einstein's general relativity. For weak gravitational fields which are asymptotically flat any deviation from Einstein's general relativity can be parametrized by a few constants, the PPN parameters. Many astrophysical observations and space experiments are devoted to a better measurement of the effects and, thus, of the PPN parameters. It is clear that gravity is best tested for intermediate ranges, that is, for distances between 1 m and several astronomical units. It is highly interesting to push forward our domain of experience and to strengthen the experimental foundation of gravity also beyond these scales. This point is underlined by the fact that many quantum gravity and unification-inspired theories suggest deviation from the standard laws of gravity at very small or very large scales. In this session summary we briefly outline the status and report on the talks presented in session C1 about experimental gravitation.

Detecting a Lorentz-violating field in cosmology

We consider cosmology in the Einstein-\AE{}ther theory (the generally covariant theory of gravitation coupled to a dynamical timelike Lorentz-violating vector field) with a linear \AE{}-Lagrangian. The $3+1$ spacetime splitting approach is used to derive covariant and gauge invariant perturbation equations which are valid for a general class of Lagrangians. Restricting attention to the parameter space of these theories which is consistent with local gravity experiments, we show that there are tracking behaviors for the \AE{} field, both in the background cosmology and at the linear perturbation level. The primordial power spectrum of scalar perturbations in this model is shown to be the same as that predicted by standard general relativity. However, the power spectrum of tensor perturbation is different from that in general relativity, but has a smaller amplitude and so cannot be detected at present. We also study the implications for late-time cosmology and find that the evolution of photon and neutrino anisotropic stresses can source the \AE{} field perturbation during the radiation and matter dominated epochs, and as a result the CMB and matter power spectra are modified. However, these effects are degenerate with respect to other cosmological parameters, such as neutrino masses and the bias parameter in the observed galaxy spectrum.