Abstract In addition to the gravitational-wave (GW) tensor modes of general relativity, more general metric theories of gravity allow up to four additional polarization states. Sensitivity curves for these additional GW polarizations are key quantities for assessing how well a detector can constrain such theories. In this work, we derive analytical expressions and high-accuracy approximate formulas for the sensitivity curves for the vector- x , vector- y , breathing, and longitudinal modes of the second-generation time-delay interferometry (TDI). Our analysis covers the TDI Michelson, $$(\alpha ,\beta ,\gamma ),$$ ( α , β , γ ) , Monitor, Beacon, Relay, and Sagnac combinations, together with the orthogonal A , E , T channels constructed from them. The validity of analytical expressions is confirmed by Monte Carlo integration. We find that, in the high-frequency limit, the sensitivity curves for the tensor and breathing modes scale as $$f^{2},$$ f 2 , whereas those for the vector and longitudinal modes approach the explicit asymptotic forms $$\frac{c^{\textrm{op}}f^{2}}{\ln f}$$ c op f 2 ln f and $$\frac{4c^{\textrm{op}}}{\pi ^{2} L^{2}} f,$$ 4 c op π 2 L 2 f , respectively. In the low-frequency limit, for all GW modes, the sensitivity curves of the $$\zeta $$ ζ combination and of the T channel scale as $$f^{-6},$$ f - 6 , whereas those of the other TDI combinations and of the A , E channels scale as $$f^{-4}.$$ f - 4 . In this limit, the sensitivity curves for the tensor and vector modes coincide, and likewise for the breathing and longitudinal modes. For the breathing mode, the sensitivity curves of the $$(\alpha ,\beta ,\gamma )$$ ( α , β , γ ) and $$\zeta $$ ζ combinations and of the T channel exhibit singularities at frequencies $$f = k/L$$ f = k / L and do not exhibit a frequency range with nearly flat sensitivity. LISA and Taiji exhibit better sensitivity for $$f \lesssim 0.01$$ f ≲ 0.01 Hz due to their longer arm lengths, whereas TianQin performs better at $$f \gtrsim 0.1$$ f ≳ 0.1 Hz. We also highlight the advantage of utilizing the uncorrelated A , E , T channels to maximize the signal-to-noise ratio. These analytical formulas are useful for estimating the capability of future space-based GW detectors to constrain additional GW polarizations.

Publisher Correction: The Newtonian limit of orthonormal frames in metric theories of gravity

Abstract We extend well-known results on the Newtonian limit of Lorentzian metrics to orthonormal frames. Concretely, we prove that, given a one-parameter family of Lorentzian metrics that in the Newtonian limit converges to a Galilei structure, any family of orthonormal frames for these metrics converges pointwise to a Galilei frame, assuming that the two obvious necessary conditions are satisfied: the spatial frame must not rotate indefinitely as the limit is approached, and the frame’s boost velocity with respect to some fixed reference observer needs to converge.

That light propagating in a gravitational field gets frequency-shifted is one of the basic consequences of any metric theory of gravity rooted in the equivalence principle. At the same time, also a time dependent material’s refractive index can frequency-shift light propagating in it. The mathematical analogy between the two effects is such that the latter has been used to study the optical analogue of a black-hole spacetime. Here, we combine these two effects by showing that light propagation in non-linear media in the presence of a moving refractive index perturbation can lead to a gravity-dependent blueshift. We find that the predicted blueshift surpasses the gravitational redshift even if the medium is considered to be perfectly stiff. In realistic scenarios, by far the strongest frequency shift arises due to the deformation of the dielectric medium and the corresponding photoelastic change of refractive index. This has the potential to facilitate optical sensing of small gravity gradients.

Recently we found compelling evidence for a gravitational-wave background with Hellings and Downs (HD) correlations in our 15 yr data set. These correlations describe gravitational waves as predicted by general relativity, which has two transverse polarization modes. However, more general metric theories of gravity can have additional polarization modes, which produce different interpulsar correlations. In this work, we search the NANOGrav 15 yr data set for evidence of a gravitational-wave background with quadrupolar HD and scalar-transverse (ST) correlations. We find that HD correlations are the best fit to the data and no significant evidence in favor of ST correlations. While Bayes factors show strong evidence for a correlated signal, the data does not strongly prefer either correlation signature, with Bayes factors ∼2 when comparing HD to ST correlations, and ∼1 for HD plus ST correlations to HD correlations alone. However, when modeled alongside HD correlations, the amplitude and spectral index posteriors for ST correlations are uninformative, with the HD process accounting for the vast majority of the total signal. Using the optimal statistic, a frequentist technique that focuses on the pulsar-pair cross-correlations, we find median signal-to-noise ratios of 5.0 for HD and 4.6 for ST correlations when fit for separately, and median signal-to-noise ratios of 3.5 for HD and 3.0 for ST correlations when fit for simultaneously. While the signal-to-noise ratios for each of the correlations are comparable, the estimated amplitude and spectral index for HD are a significantly better fit to the total signal, in agreement with our Bayesian analysis.

We find the conditions under which scale-invariant Einstein-Cartan gravity with scalar matter fields leads to an approximate conformal invariance of the flat space particle theory up to energies of the order of the Planck mass. In the minimal setup, these models, in addition to the fields of the Standard Model and the graviton, contain only one extra particle — a massless dilaton. Theories of this type can pave the way for a self-completion all the way up the Planck scale and lead to rather universal inflationary predictions, close to those of the simplest Higgs-inflation scenario in the metric theory of gravity.

Accretion of magnetized gas on compact astrophysical objects such as black holes (BHs) has been successfully modeled using general relativistic magnetohydrodynamic (GRMHD) simulations. These simulations have largely been performed in the Kerr metric, which describes the spacetime of a vacuum and stationary spinning BH in general relativity (GR). The simulations have revealed important clues to the physics of accretion flows and jets near the BH event horizon and have been used to interpret recent Event Horizon Telescope images of the supermassive BHs M87* and Sgr A*. The GRMHD simulations require the spacetime metric to be given in horizon-penetrating coordinates such that all metric coefficients are regular at the event horizon. Only a few metrics, notably the Kerr metric and its electrically charged spinning analog, the Kerr–Newman metric, are currently available in such coordinates. We report here horizon-penetrating forms of a large class of stationary, axisymmetric, spinning metrics. These can be used to carry out GRMHD simulations of accretion on spinning, nonvacuum BHs and non-BHs within GR, as well as accretion on spinning objects described by non-GR metric theories of gravity.

Metric-affine gravity: Nonmetricity of space as dark matter/energy ?

Using the general parametrization of spherically symmetric and asymptotically flat black holes in arbitrary metric theories of gravity and implying that: a) the post-Newtonian constraints are taken into account and b) basic astrophysically relevant characteristics (such as, dominant quasinormal modes, frequency at the innermost stable circular orbit, binding energy, radius of the shadow etc.) are indistinguishable from their Schwarzschild values, we propose a simple metric which depends on three independent parameters (coefficients of the parametrization). Variation of these three parameters can, nevertheless, lead to the two distinctive features. The first is the black-hole temperature, and consequently the Hawking radiation, which can differ a lot from its Schwarzschild limit. The second is the outburst of overtones which become extremely sensitive to small changes of the parameters.

We investigate the possibility of reducing the number of degrees of freedom (d.o.f.) starting from generic metric theories of gravity by introducing multiple auxiliary constraints (ACs), under the restriction of retaining spatial covariance as a gauge symmetry. Arbitrary numbers of scalar-, vector- and tensor-type ACs are considered a priori, yet we find that no vector- and tensor-type constraints should be introduced, and that scalar-type ACs should be no more than four for the purpose of constructing minimally modified gravity (MMG) theories which propagate only two tensorial d.o.f., like general relativity (GR). Through a detailed Hamiltonian analysis, we exhaust all the possible classifications of ACs and find out the corresponding minimalizing and symmetrizing conditions for obtaining the MMG theories. In particular, no condition is required in the case of four ACs, hence in this case the theory can couple with matter consistently and naturally. To illustrate our formalism, we build a concrete model for this specific case by using the Cayley-Hamilton theorem and derive the dispersion relation of the gravitational waves, which is subject to constraints from the observations.

Future space gravitational-wave detectors will detect gravitational waves with high sensitivity in the mHz frequency band. One possible source is the stochastic gravitational-wave background (SGWB), possibly from astronomy and cosmology. Detecting SGWB could provide an opportunity to directly examine the polarization of gravitational waves. While general relativity predicts only two tensor modes for gravitational-wave polarization, general metric theories of gravity allow up to four additional modes, including two vector and two scalar modes. Observing other polarization modes of gravitational waves would directly indicate that general relativity needs to be modified. However, the application of polarization identification methods developed for ground-borne detectors to space-borne detectors will require improvement. In this paper, we design a new statistic for the characteristics of space-borne detectors and perform Bayesian analysis. We analyze the performance of the new statistics, including the signal-to-noise ratio, ability to identify SGWB, and parameter estimation. The results show that the Bayesian method with new statistics is good enough to meet the needs of space detectors to identify polarized SGWB. In particular, space-borne gravitational-wave detectors will have the ability to distinguish the scalar-breathing mode and the scalar-longitudinal mode with this Bayesian method, which ground-based detectors cannot.

We introduce a way of measuring ${H}_{0}$ from a combination of independent geometrical data sets, without the need for calibration or the choice of a cosmological model. We build on the distance duality relation, which sets the ratio of the luminosity and angular diameter distance to a fixed scaling in redshift for any metric theory of gravity with standard photon propagation and constitutes a founding block of any theory describing our Universe. Our method allows us to determine ${H}_{0}$ from first principles, unleashing the measurement of this fundamental constant from calibration and the assumption of a cosmological model. We find ${H}_{0}=69.5\ifmmode\pm\else\textpm\fi{}1.7\text{ }\text{ }\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$ at 68% C.L., showing that the Hubble constant can be constrained at the percent level with minimal assumptions.

ABSTRACT The Etherington distance duality relation is well-established for metric theories of gravity, and confirms the duality between the luminosity distance and the angular diameter distance through the conservation of surface brightness. A violation of the Etherington distance duality due to lensing in a non-metric space–time would lead to fluctuations in surface brightness of galaxies. Likewise, fluctuations of the surface brightness can arise in classical astrophysics as a consequence of intrinsic tidal interaction of galaxies with their environment. Therefore, we study these in two cases in detail: First, for intrinsic size fluctuations and the resulting changes in surface brightness, and secondly, for an area-metric space–time as an example of a non-metric space–time, where the distance duality relation itself acquires modifications. The aim of this work is to quantify whether a surface brightness fluctuation effect due to area-metric gravity would be resolvable compared to the similar effect caused by intrinsic alignment. We thus compare the auto- and cross-correlations of the angular spectra in these two cases and show that the fluctuations in intrinsic brightness can potentially be measured with a cumulative signal-to-noise ratio Σ(ℓ) ≥ 3 in a Euclid-like survey. The measurement in area-metric space–times, however, depends on the specific parameter choices, which also determine the shape and amplitude of the spectra. While lensing surveys do have sensitivity to lensing-induced surface brightness fluctuations in area-metric space–times, the measurement does not seem to be possible for natural values of the Etherington-breaking parameters.

We consider the gravity assist maneuver, that is, a correction of spacecraft motion at its passing near a planet, as a tool for evaluating the Eddington post-Newtonian parameters $\beta$ and $\gamma$, characterizing vacuum spherically symmetric gravitation fields in metric theories of gravity. We estimate the effect of variation in $\beta$ and $\gamma$ on a particular trajectory of a probe launched from the Earth's orbit and passing closely near Venus, where relativistic corrections slightly change the impact parameter of probe scattering in Venus's gravitational field. It is shown, in particular, that a change of $10^{-4}$ in $\beta$ or $\gamma$ leads to a shift of about 50 km in the probe's aphelion position.

Hod's proposal claims that the least damped quasinormal mode of a black hole must have the imaginary part smaller than half of the surface gravity at the event horizon.The Strong Cosmic Censorship in General Relativity implies that this bound must be even weaker: half of the surface gravity at the Cauchy horizon. The appealing question is whether these bounds are limited by the Einstein theory only?Here we will present numerical evidence that once the black hole size is much smaller than then the radius of the cosmological horizon, both the Hod's proposal and the strong cosmic censorship bound for quasinormal modes are satisfied for general spherically symmetric black holes in an arbitrary metric theory of gravity. The low-lying quasinormal frequencies have the universal behavior in this regime and do not depend on the near-horizon geometry, but only on the asymptotic parameters: the value of the cosmological constant and black hole mass.

We search for isotropic stochastic gravitational-wave background (SGWB) in the International Pulsar Timing Array second data release. By modeling the SGWB as a power-law, we find very strong Bayesian evidence for a common-spectrum process, and further this process has scalar transverse (ST) correlations allowed in general metric theory of gravity as the Bayes factor in favor of the ST-correlated process versus the spatially uncorrelated common-spectrum process is 30 ± 2. The median and the 90% equal-tail amplitudes of ST mode are ST=1.29−0.44+0.51×10−15 , or equivalently the energy density parameter per logarithm frequency is ΩGWST=2.31−1.30+2.19×10−9 , at frequency of 1 year−1. However, we do not find any statistically significant evidence for the tensor transverse (TT) mode and then place the 95% upper limits as TT<3.95×10−15 , or equivalently ΩGWTT<2.16×10−9 , at frequency of 1 year−1.

Abstract In general metric theory of gravity, a gravitational wave is allowed to have up to six polarizations: two scalar and two vector modes in addition to tensor modes. In case the number of laser‐interferometric gravitational wave telescopes is larger than the number of polarizations of a gravitational wave, all the polarizations can be individually reconstructed. Since it depends on theories of gravity which polarizations the gravitational waves have, the investigation of polarizations is important for the test of theories of gravity. In order to test the scalar–tensor gravity theory, one of important alternative theories of gravity, the scalar mode of GW170817 observed by LIGO Livingstone, Hanford and Virgo is reconstructed without prior information about any tensor–scalar gravity theories. The upper limit of the scalar mode in term of the band‐limited root‐sum‐square of the amplitude is with the time window of 2 [s] and frequency window of ≈60–120 [Hz]. It is also studied how much the tensor modes are leaked into the reconstructed scalar mode, and it is found that the reconstructed scalar mode contains roughly 30% of energy leaked from the tensor modes.

General Relativity predicts only two tensor polarization modes for gravitational waves while at most six possible polarization modes are allowed in the general metric theory of gravity. The number of polarization modes is determined by the specific modified theory of gravity. Therefore, the determination of polarization modes can be used to test gravitational theory. We introduce a concrete data analysis pipeline for a space-based detector such as LISA to detect the polarization modes of gravitational waves. This method can be used for monochromatic gravitational waves emitted from any compact binary system with a known sky position and frequency to detect mixtures of tensor and extra polarization modes. We use the source $\mathrm{J}0806.3+1527$ with one year of simulation data as an example to show that this approach is capable of probing pure and mixed polarizations without knowing the exact polarization modes. We also find that the ability of detection of extra polarization depends on the gravitational-wave source location and the amplitude of nontensorial components.

The general parametrization of spherically symmetric and asymptotically flat black-hole spacetimes in arbitrary metric theories of gravity was suggested in [3]. The parametrization is based on the continued fraction expansion in terms of the compact radial coordinate and has superior convergence and strict hierarchy of parameters. It is known that some observable quantities, related to particle motion around the black hole, such as the eikonal quasinormal modes, radius of the shadow, frequency at the innermost stable circular orbit, and others, depend mostly on only a few of the lowest coefficients of the parametrization. Here we continue this approach by studying the dominant (low-lying) quasinormal modes for such generally parametrized black holes. We show that, due to the hierarchy of parameters, the dominant quasinormal frequencies are also well determined by only the first few coefficients of the expansion for the so-called moderate black-hole geometries. The latter are characterized by a relatively slow change of the metric functions in the radiation zone near the black hole. The nonmoderate metrics, which change strongly between the event horizon and the innermost stable circular orbit are usually characterized by echoes or by the distinctive (from the Einstein case) quasinormal ringing which does not match the current observational data. Therefore, the compact description of a black-hole spacetime in terms of the truncated general parametrization is an effective formalism for testing strong gravity and imposing constraints on allowed black-hole geometries.

We study a Born-Infeld inspired model of gravity and electromagnetism in\nwhich both types of fields are treated on an equal footing via a determinantal\napproach in a metric-affine formulation. Though this formulation is a priori in\nconflict with the postulates of metric theories of gravity, we find that the\nresulting equations can also be obtained from an action combining the\nEinstein-Hilbert action with a minimally coupled nonlinear electrodynamics. As\nan example, the dynamics is solved for the charged static black hole.\n