A clarification on prevailing misconceptions in unimodular gravity

Unimodular Gravity (UG) is a restricted version of General Relativity (GR) in which the determinant of the metric is a fixed function and the field equations are given by the trace-free part of the full Einstein equations. The background equations in UG and GR are identical. It was recently claimed that, the first order contribution in the temperature fluctuation of the Cosmic Microwave Background (CMB) in UG is different from GR. In this work, we calculate the first order perturbation equations in UG and show that the Sachs-Wolfe effect in UG, in terms of gauge invariant variables, is identical to GR. We also show that the second order perturbation equation of Mukhnanov-Sasaki variable in UG, is identical to GR. The only difference comes from the gauge choices due the constraint on the metric determinant. Hence, UG and GR are identical and indistinguishable in CMB data on large scales.

We present constraints on testing general relativity (GR) at cosmological scales using recent data sets and assess the impact of galaxy intrinsic alignment (IA) in the CFHTLenS lensing data on those constraints. We consider CMB temperature data from Planck, the galaxy power spectrum from WiggleZ, weak lensing tomography from the CFHTLenS, ISW-galaxy cross correlations, and BAO data from 6dF, SDSS DR7, and BOSS DR9. We use a parameterization of the modified gravity (MG) that is binned in redshift and scale, a parameterization that evolves monotonically in scale but is binned in redshift, and a functional parameterization that evolves only in redshift. We present the results in terms of the MG parameters $Q$ and $\Sigma$. We employ an IA model with an amplitude $A_{CFHTLenS}$ that is included in the parameter analysis. We find an improvement in the constraints on the MG parameters corresponding to $40-53\%$ increase on the figure of merit compared to previous studies, and GR is found consistent with the data at the $95\%$ CL. The bounds found on $A_{CFHTLenS}$ are sensitive to the MG parameterization used, and the correlations between $A_{CFHTLenS}$ and MG parameters are found to be weak to moderate. For all 3 MG parameterizations $A_{\rm CFHTLenS}$ is found consistent with zero when the whole lensing sample is used, however, when using the optimized early-type galaxy sample a significantly non-zero $A_{\rm CFHTLenS}$ is found for GR and the third MG parameterization. We find that the tensions observed in previous studies persist, and there is an indication that CMB data and lensing data prefer different values for MG parameters, particularly for the parameter $\Sigma$. The analysis of the confidence contours and probability distributions suggest that the bimodality found follows that of the known tension in the $\sigma_8$ parameter. (Abridged)

We constrain deviations from general relativity (GR) including both redshift and scale dependencies in the modified gravity (MG) parameters. In particular, we employ the under-used binning approach and compare the results to functional forms. We use available datasets such as Cosmic Microwave Background (CMB) from Planck 2018, Baryonic Acoustic Oscillations (BAO) and Redshift Space Distortions (BAO/RSD) from the BOSS Data Release 12, the 6DF Galaxy Survey, the SDSS Data Release 7 Main Galaxy Sample, the correlation of Lyman-α forest absorption and quasars from SDSS-DR14, Supernova Type Ia (SNe) from the Pantheon compilation, and DES Y1 data. Moreover, in order to maximize the constraining power from available datasets, we analyze MG models where we alternatively set some of the MG parameters to their GR values and vary the others. Using functional forms, we find an up to 3.5-σ tension with GR in Σ (while μ is fixed) when using Planck+SNe+BAO+BAO/RSD; this goes away when lensing data is included, i.e. CMB lensing and DES Y1 (CMBL+DES) . Using different binning methods, we find that a tension with respect to GR above 2-σ in the (high-z, high-k) bin is persistent even when including CMBL+DES to Planck+SNe+BAO+BAO/RSD . Also, we find another tension above 2-σ in the (low-z, high-k) bin, but that can be reduced with the addition of lensing data. Furthermore, we perform a model comparison using the Deviance Information Criterion statistical tool and find that the MG model (μ=1, Σ) is weakly favored by the data compared to ΛCDM, except when DES data is included. Another noteworthy result is that we find that the binning methods do not agree with the widely-used functional parameterization where the MG parameters are proportional to ΩDE(a), and this is clearly apparent in the high-z and high-k regime where this parameterization underestimates the deviations from GR.

The recently reported gravitational wave events GW$150914$ and GW$151226$ caused by the mergers of binary black holes [arXiv:1602.03841],[arXiv:1606.04855],[arXiv:1606.04856] provide a formidable way to set constraints on alternative metric theories of gravity in the strong field regime. In this paper, we develop an approach where an arbitrary theory of gravity can be parametrised by an effective coupling $G_{eff}$ and an effective gravitational potential $\Phi(r)$. The standard Newtonian limit of General Relativity is recovered as soon as $G_{eff}\rightarrow G_N$ and $\Phi(r)\rightarrow \Phi_{N}$. The upper bound on the graviton mass and the gravitational interaction length, reported by the LIGO-VIRGO collaboration, can be directly recast in terms of the parameters of the theory which allows an analysis where the gravitational wave frequency modulation sets constraints on the range of possible alternative models of gravity. Numerical results based on published parameters for the binary black hole mergers are also reported. Comparison of the observed phase of the GW$150914$ and GW$151226$ with the modulated phase in alternative theories of gravity does not give reasonable constraints due the large uncertainties in the estimated parameters for the coalescing black holes. In addition to these general considerations, we obtain limits for the frequency dependence of the $\alpha$ parameter in scalar tensor theories of gravity.

Gravitational wave (GW) observations of coalescing compact binaries will be unique probes of strong-field, dynamical aspects of relativistic gravity. We present a short review of various schemes proposed in the literature to test general relativity (GR) and alternative theories of gravity using inspiral waveforms. Broadly these schemes may be classified into two types: model dependent and model independent. In the model dependent category, GW observations are compared against a specific waveform model representative of a particular theory or a class of theories such as scalar-tensor theories, dynamical Chern–Simons theory and massive graviton theories. Model independent tests are attempts to write down a parametrized gravitational waveform where the free parameters take different values for different theories and (at least some of) which can be constrained by GW observations. We revisit some of the proposed bounds in the case of downscaled LISA configuration (eLISA) and compare them with the original LISA configuration. We also compare the expected bounds on alternative theories of gravity from ground-based and space-based detectors and find that space-based GW detectors can test GR and other theories of gravity with unprecedented accuracies. We then focus on a recent proposal to use singular value decomposition of the Fisher information matrix to improve the accuracies with which post-Newtonian theory can be tested. We extend those results to the case of space-based detector eLISA and discuss its implications.

We review recent developments and results in testing general relativity (GR) at cosmological scales. The subject has witnessed rapid growth during the last two decades with the aim of addressing the question of cosmic acceleration and the dark energy associated with it. However, with the advent of precision cosmology, it has also become a well-motivated endeavor by itself to test gravitational physics at cosmic scales. We overview cosmological probes of gravity, formalisms and parameterizations for testing deviations from GR at cosmological scales, selected modified gravity (MG) theories, gravitational screening mechanisms, and computer codes developed for these tests. We then provide summaries of recent cosmological constraints on MG parameters and selected MG models. We supplement these cosmological constraints with a summary of implications from the recent binary neutron star merger event. Next, we summarize some results on MG parameter forecasts with and without astrophysical systematics that will dominate the uncertainties. The review aims at providing an overall picture of the subject and an entry point to students and researchers interested in joining the field. It can also serve as a quick reference to recent results and constraints on testing gravity at cosmological scales.

The successful miniaturisation of extremely accurate atomic clocks and atom interferometers invites prospects for satellite missions to perform precision experiments. We discuss the e ects predicted by general relativity and alternative theories of gravity that can be detected by a clock, which orbits the Earth. Our experiment relies on the pre- cise tracking of the spacecraft using its observed tick-rate. The spacecraft's reconstructed four-dimensional trajectory will reveal the nature of gravitational perturbations in Earth's gravitational field, potentially di erentiating between di erent theories of gravity. This mission can measure multiple relativistic e ects all during the course of a single exper- iment, and constrain the Parametrized Post-Newtonian Parameters around the Earth. A satellite carrying a clock of fractional timing inaccuracy of f=f 10 16 in an elliptic orbit around the Earth would constrain the PPN parametersj 1j;j 1j . 10 6 . We also briefly review potential constraints by atom interferometers on scalar tensor theories and in particular on Chameleon and dilaton models.

We carry out the canonical analysis of a covariant version of unimodular gravity in terms of the connection representation. We then proceed to quantize this theory by implementing the Dirac procedure. We confirm whether and how the Kodama state, which is a solution of quantum general relativity, can be extended into covariant unimodular gravity. Finally, we discuss the difference of quantum states between covariant unimodular gravity, the original unimodular gravity, and general relativity.

The direct detection of gravitational waves opens new perspectives for measuring properties of gravitationally bound compact objects. It is then important to investigate black holes and neutron stars in alternative theories of gravity, since they can have features that make them observationally distinguishable from their general relativity (GR) counterparts. In this work, we examine a special case of vector Galileons, a vector-tensor theory of gravity with interesting cosmological properties, which consists of a one parameter modification of the Einstein–Maxwell action. Within this theory, we study configurations describing asymptotically flat, spherically symmetric black holes and neutron stars. The set of black hole solutions in this theory is surprisingly rich, generalising results found in GR or in related scalar-tensor theories. We investigate the properties and conserved charges of black holes, using both analytical and numerical techniques, highlighting configurations that are more compact than in GR. We then study properties of neutron stars, showing how the vector profile can influence the star internal structure. Depending on properties of matter and fields inside the star, neutron stars can be more massive than in GR, and they can be even more compact than Schwarzschild black holes, making these objects observationally interesting. We also comment on possible extensions of our configurations to magnetically charged or rotating configurations.

Regge-Teitelboim embedding gravity is the modified gravity based on a simple string-inspired geometrical principle—our spacetime is considered here as a 4-dimensional surface in a flat bulk. This theory is similar to the recently popular theory of mimetic gravity—the modification of gravity appears in both theories as a result of the change of variables in the action of General Relativity. Embedding gravity, as well as mimetic gravity, can be used in explaining the dark matter mystery since, in both cases, the modified theory can be presented as General Relativity with additional fictitious matter (embedding matter or mimetic matter). For the general case, we obtain the equations of motion of embedding matter in terms of embedding function as a set of first-order dynamical equations and constraints consistent with them. Then, we construct a non-relativistic limit of these equations, in which the motion of embedding matter turns out to be slow enough so that it can play the role of cold dark matter. The non-relativistic embedding matter turns out to have a certain self-interaction, which could be useful in the context of solving the core-cusp problem that appears in the Λ-Cold Dark Matter (ΛCDM) model.

The tidal response of compact objects in an inspiraling binary system is measured by a set of tidal Love and dissipation numbers imprinted in the gravitational waveforms. While a four-dimensional black hole in vacuum within general relativity has vanishing Love numbers, a black hole in alternative theories of gravity can acquire nonvanishing Love numbers. The dissipation numbers may quantify Planckian corrections at the horizon scale. These properties will allow a test of classical theories of gravity in the strong-field regime with gravitational-wave observation. Since black holes are not in the exact vacuum environment in astrophysical situations, the following question arises: Can the environment affect the tidal response? In this paper, we investigate the stability of the tidal response of a Schwarzschild black hole for frequency-dependent tidal-field perturbations against a small modification of the background. Our analysis relies on the scattering theory, which overcomes difficulties in computing the relativistic tidal Love numbers. The tidal Love and dissipation numbers can be understood from the property of sufficiently low-frequency scattering waves. We show that the tidal Love numbers are sensitive to the property of the modification. Therefore, we need careful consideration of the environment around the black hole in assessing the deviation of the underlying theory of gravity from general relativity with the Love numbers. The modification has less impact on the dissipation numbers, indicating that quantifying the existence of the event horizon with them is not spoiled. We also demonstrate that in a composite system, i.e., a compact object with environmental effects, the Love and dissipation numbers are approximately determined by the sum of the numbers of each component.

Modified Gravity (MG) scenarios have been advocated to account for the dark energy phenomenon in the universe. These models predict departures from General Relativity on large cosmic scales that can be tested through a variety of probes such as observations of galaxy clusters among others. Here, we investigate the imprint of MG models on the internal mass distribution of cluster-like halos as probed by the dark matter halo sparsity. To this purpose we perform a comparative analysis of the properties of the halo sparsity using N-body simulation halo catalogs of a standard flat $\Lambda$CDM model and MG scenarios from the DUSTGRAIN-pathfinder simulation suite. We find that the onset of the screening mechanism leaves a distinct signature in the redshift evolution of the ensemble average halos sparsity. Measurements of the sparsity of galaxy clusters from currently available mass estimates are unable to test MG models due to the large uncertainties on the cluster masses. We show that this should be possible in the future provided large cluster samples with cluster masses determined to better than $30\%$ accuracy level.

To evaluate a potential usually one analyzes trajectories of test particles. For the Galactic Center case astronomers use bright stars or photons, so there are two basic observational techniques to investigate a gravitational potential, namely, (a) monitoring the orbits of bright stars near the Galactic Center as it is going on with 10m Keck twin and four 8m VLT telescopes equipped with adaptive optics facilities (in addition, recently the IR interferometer GRAVITY started to operate with VLT); (b) measuring the size and shape of shadows around black hole with VLBI-technique using telescopes operating in mm-band. At the moment, one can use a small relativistic correction approach for stellar orbit analysis, however, in the future the approximation will not be precise enough due to enormous progress of observational facilities and recently the GRAVITY team found that the first post-Newtonian correction has to be taken into account for the gravitational redshift in the S2 star orbit case. Meanwhile for smallest structure analysis in VLBI observations one really needs a strong gravitational field approximation. We discuss results of observations and their interpretations. In spite of great efforts there is a very slow progress to resolve dark matter (DM) and dark energy (DE) puzzles and in these circumstances in last years a number of alternative theories of gravity have been proposed. Parameters of these theories could be effectively constrained with of observations of the Galactic Center. We show some cases of alternative theories of gravity where their parameters are constrained with observations, in particular, we consider massive theory of gravity. We choose the alternative theory of gravity since there is a significant activity in this field and in the last years theorists demonstrated an opportunity to create such theories without ghosts, on the other hand, recently, the joint LIGO & Virgo team presented an upper limit on graviton mass such as mg< 1:2 × 10-22eV [1] analyzing gravitational wave signal in their first paper where they reported about the discovery of gravitational waves from binary black holes as it was suggested by C. Will [2]. So, the authors concluded that their observational data do not indicate a significant deviation from classical general relativity. We show that an analysis of bright star trajectories could estimate a graviton mass with a commensurable accuracy in comparison with an approach used in gravitational wave observations and the estimates obtained with these two approaches are consistent. Therefore, such an analysis gives an opportunity to treat observations of bright stars near the Galactic Center as a useful tool to obtain constraints on the fundamental gravity law. We showed that in the future graviton mass estimates obtained with analysis of trajectories of bright stars would be better than current LIGO bounds on the value, therefore, based on a potential reconstruction at the Galactic Center we obtain bounds on a graviton mass and these bounds are comparable with LIGO constraints. Analyzing size of shadows around the supermassive black hole at the Galactic Center (or/and in the center of M87) one could constrain parameters of different alternative theories of gravity as well.

Null hypersurfaces are a mathematical consequence of the Lorentzian signature of general relativity; singularities in mathematical models usually indicate where the interesting physics takes place. This book discusses what happens when you combine these ideas.Right from the preface, this is a no-nonsense book. There are two principalapproaches to singular shells, one distributional and the other 'cut andpaste'; both are treated in detail. A working knowledge of GR is assumed,including familiarity with null tetrads, differential forms, and 3 + 1decompositions. Despite my own reasonably extensive, closely relatedknowledge, there was material unfamiliar to me already in chapter 3, althoughI was reunited with some old friends in later chapters.The exposition is crisp, with a minimum of transition from chapter to chapter.In fact, my main criticism is that there is no clear statement of theorganization of the book, nor is there an index. Everything is here, and thestory is compelling if you know what to look for, although it is less easy tofollow the story if you are not already familiar with it.But this is really a book for experts, and the authors certainly qualify,having played a significant role in developing and extending the results theydescribe. It is also entirely appropriate that the book is dedicated toWerner Israel, who pioneered the thin-shell approach to (non-null) singularsurfaces and later championed the use of similar methods for analysing nullshells.After an introductory chapter on impulsive signals, the authors show how theBianchi identities can be used to classify spacetimes with singular nullhypersurfaces. This approach, due to the authors, generalizes the frameworkoriginally proposed by Penrose [1]. While astrophysical applicationsare discussed only briefly, the authors point out that detailed physicalcharacteristics of signals from isolated sources can be determined in thismanner. In particular, they describe the behaviour of test particles in such aspacetime, in an initial attempt to outline a framework for the detection ofimpulsive gravitational waves.Subsequent chapters describe the singular null hypersurfaces obtained byboosting isolated gravitational sources, building on the work of Aichelburgand Sexl [2], and by colliding impulsive waves, building onthe work of Khan and Penrose [3]. In between, the special caseof spherical symmetry is considered, both with and without collisions. Thereis also a short chapter discussing the effect of replacing GR by alternativetheories of gravity, and an appendix which briefly summarizes the non-nullcase.The references are reasonably complete, from Synge and Penrose to the recentwork of the authors. However, there are a few relatively minor errors andomissions. For instance, the results in chapter 3 about shells of matter inboth Schwarzschild and Reissner–Nordström geometries are presented without reference or derivation. And I was disappointed to see that my own work with 't Hooft [4] on the horizon shift due to the impulsive wave of amassless particle at the horizon of a Schwarzschild black hole—a directgeneralization of the work by Aichelburg and Sexl—is not mentioned.But none of these minor complaints detracts from my appreciation of having acomplete discussion of singular null hypersurfaces all in one place. Thethree fundamental papers [1–3] whichstarted this area of research all appeared at essentially the same time, 35years ago; it is high time there was a unified presentation of the entirefield. This book fills that need admirably, and could serve as the core of agraduate seminar for students having already taken a course in generalrelativity, or as a reference. My copy will have a treasured place in mylibrary. ReferencesPenrose R 1972 The geometry of impulsive gravitational waves General Relativity: Papers in Honour of J L Syngeed L Ó Raifeartaigh (Oxford: Clarendon) pp 101–30Aichelburg P C and Sexl R U 1971 On the gravitational field of a massless particle Gen. Rel. Grav. 2 303–12 Khan K A and Penrose R 1971 Scattering of two impulsive gravitational plane waves Nature 229 185–6 Dray T and 't Hooft G 1985 The gravitational shock wave of a massless particle Nucl. Phys. B 253 173–88

The General theory of relativity is one of the most successful theories ofgravity. Despite its successful applications, it has some difficulties in examiningthe behaviour of particles precisely in strong gravitational fields. Bi-metric typetheories of gravity are classified as alternative theories of gravity that describingsuch strong gravitational fields, such as the gravitational field formed at the coreof our galaxy. In order to obtain the equations of motion for spinning fluids, weuse the Weyssenhoff tensor to express the spin fluid. The equations of motion forspinning fluids are derived using Euler-Lagrange equation. We present theequations of motion for spinning fluids and their corresponding spin deviationequations in some classes of Bi-metric type theories. Also, we obtain equationsof motion for spinning fluids and their corresponding spin deviation for avariable mass. Moreover, we extend our study to examine the status of motionfor spinning charged fluids and their corresponding spin deviation equations.