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The wolensing Python package offers a solution for gravitational wave lensing computations within the full wave-optics regime. This tool is primarily designed to calculate the gravitational lensing amplification factor including diffractive effects, an essential component for generating accurate lensed gravitational wave waveforms. These waveforms are integral to astrophysical and cosmological studies related to gravitational-wave lensing. Integrating with lensingGW (Pagano, Hannuksela, and Li 2020), wolensing provides solutions for image positions in the high-frequency regime where wave and geometrical optics converge. This functionality allows the amplification factor to be applicable across a wider frequency range. Another key feature of wolensing is its ability to plot time delay contours on the lens plane, offering researchers a visual tool to better understand the relationship between the lens system and the amplification factor. wolensing is compatible with various lens models in lenstronomy (Birrer et al. 2021). There are also built-in lens models including point mass, singular isothermal sphere (SIS),and nonsingular isothermal ellipsoid (NIE) with jax (Bradbury et al. 2018) supporting GPU computation. Users can accommodate different lens models in the code with jax. wolensing is available as an open-source package on PyPI and can be installed via pip.

Wave-optics phenomena in gravitational lensing occur when the signal's wavelength is commensurate to the gravitational radius of the lens. Although potentially detectable in lensed gravitational waves, fast radio bursts and pulsars, accurate numerical predictions are challenging to compute. Here we present novel methods for wave-optics lensing that allow the treatment of general lenses. In addition to a general algorithm, specialized methods optimize symmetric lenses (arbitrary number of images) and generic lenses in the single-image regime. We also develop approximations for simple lenses (point-like and singular isothermal sphere) that drastically outperform known solutions without compromising accuracy. These algorithms are implemented in Gravitational Lensing of Waves (GLoW): an accurate, flexible, and fast code. GLoW efficiently computes the frequency-dependent amplification factor for generic lens models and arbitrary impact parameters in O(1 ms) to O(10 ms) depending on the lens configuration and complexity. GLoW is readily applicable to model lensing diffraction on gravitational-wave signals, offering new means to investigate the distribution of dark-matter and large-scale structure with signals from ground and space detectors., Comment: 21 pages, 10 figures. GLoW is publicly available at https://github.com/miguelzuma/GLoW_public . All the scripts needed to reproduce the figures in this paper can also be found in the repository

Gravitational lensing offers a powerful tool for exploring the matter distribution in the Universe. Thanks to their low frequencies and phase coherence, gravitational waves (GWs) allow for the observation of novel wave-optics features (WOFs) in lensing, inaccessible to electromagnetic signals. Combined with the existing accurate source models, lensed GWs can be used to infer the properties of gravitational lenses. The prospect is particularly compelling for space-borne detectors, where the high signal-to-noise ratio expected from massive black hole binary mergers allows WOFs to be distinguished deep into the weak lensing regime, drastically increasing the detection probability. Here, we investigate in detail the capacity of the LISA mission to detect WOFs caused by dark matter halos, galaxies and the supermassive black holes (SMBHs) within them. We estimate the total optical depth to be $\lambda_{\rm tot} \sim 6 \times 10 ^{-3}$ for the loudest binaries of total mass $M_{\rm BBH} \sim 10^6 M_{\odot}$, with the dominant contribution coming from SMBHs. We also find that WOFs in low-mass binaries $M_{\rm BBH} \sim 10^4 M_{\odot}$ are more likely due to the central galaxies. Within our model of gravitational lenses, we predict $\mathcal{O}(0.1)-\mathcal{O}(1)$ weakly-lensed events to be detectable during the 5 years of LISA mission, depending on the source population models. We show that WOFs signatures are very sensitive to the properties of dark-matter halos with $M_{\rm vir}\in (10^6-10^8)M_\odot$: increasing the compactness parameter by $\sim 3$ in that range raises the detection rate by $\sim 26$. Additionally, we show that collective effects from the complex inner halo structure can further enhance detectability. This suggests that lensed GWs in LISA will be an excellent probe of dark-matter theories, baryonic and halo sub-structures., Comment: 26 pages, 19 figures, 3 tables

Every signal propagating through the universe is diffracted by the gravitational fields of intervening objects, aka gravitational lenses. Diffraction is most efficient when caused by compact lenses, which invariably produce additional images of a source. The signals associated with additional images are generically faint, but their collective effect may be detectable with coherent sources, such as gravitational waves (GWs), where both amplitude and phase are measured. Here, I describe lens stochastic diffraction (LSD): Poisson-distributed fluctuations after GW events caused by compact lenses. The amplitude and temporal distribution of these signals encode crucial information about the mass and abundance of compact lenses. Through the collective stochastic signal, LSD offers an order-of-magnitude improvement over single lens analysis for objects with mass $\gtrsim 10^3 M_\odot$. This gain can improve limits on compact dark-matter halos and allows next-generation instruments to detect supermassive black holes, given the abundance inferred from quasar luminosity studies., Comment: 5 pages, 4 figures. Comments welcome!

Einstein-aether theory provides a model to test the validity of local Lorentz invariance in gravitational interactions. The speed of gravitational waves as measured from the binary neutron star event GW170817 sets stringent limits on Einstein-aether theory, but only on a combination of the theory's free parameters. For this reason, a significant part of the theory's parameter space remains unconstrained by observations. Motivated by this, we explore the propagation of gravitational waves in Einstein-aether theory over an inhomogeneous background (i.e., gravitational wave lensing) as a potential mechanism to break the degeneracies between the theory's free parameters, and hence enable new constraints on the theory to be obtained. By bringing the field equations into the form of the so-called kinetic matrix and applying a formalism known as the propagation eigenstate framework, we find that the speed of gravitational waves is modified by inhomogeneities in the aether field. However, the modification is common to both gravitational polarizations and vanishes in the limit in which gravitational waves propagate with luminal speed. This lens-dependent gravitational wave speed contrasts with the lens-induced birefringence observed in other theories beyond general relativity, like Horndeski's theory. While the potential to improve tests based on gravitational-wave speed is limited, our formalism sets the basis to fully describe signal propagation over inhomogeneous spacetimes in Einstein-aether theory and other extensions of general relativity., Comment: 19 pages plus appendices, 4 figures

When gravitational waves pass near a gravitating object, they are deflected, or lensed. If the object is massive, such that the wavelength of the waves is small compared to its gravitational size, lensed gravitational wave events can be identified when multiple signals are detected at different times. However, when the wavelength is long, wave-optics diffraction effects will be important, and a lensed event can be identified by looking for frequency-dependent modulations to the gravitational waveform, without having to associate multiple signals. For current ground-based gravitational wave detectors observing stellar-mass binary compact object mergers, wave-optics effects are important for lenses with masses $\lesssim 1000 M_{\odot}$. Therefore, minihalos below this mass range could potentially be identified by lensing diffraction. The challenge with analyzing these events is that the frequency-dependent lensing modulation, or the amplification factor, is prohibitively expensive to compute for Bayesian parameter inference. In this work, we use a novel time-domain method to construct interpolators of the amplification factor for the Navarro-Frenk-White (NFW), generalized singular isothermal sphere (gSIS) and cored isothermal sphere (CIS) lens models. Using these interpolators, we perform Bayesian inference on gravitational-wave signals lensed by minihalos injected in mock detector noise, assuming current sensitivity of ground-based detectors. We find that we could potentially identify an event when it is lensed by minihalos and extract the values of all lens parameters in addition to the parameters of the GW source. All of the methods are implemented in Glworia, the accompanying open-source Python package, and can be generalized to study lensed signals detected by current and next-generation detectors., Comment: 19 pages, 9 figures

Wave packets propagating in inhomogeneous media experience a coupling between internal and external degrees of freedom and, as a consequence, follow spin-dependent trajectories. These phenomena, well known in optics and condensed matter physics, are referred to as spin Hall effects. Similarly, the gravitational spin Hall effect is expected to affect the propagation of gravitational waves on curved spacetimes. In this general-relativistic setup, the curvature of spacetime acts as impurities in a semiconductor or inhomogeneities in an optical medium, leading to a frequency- and polarization-dependent propagation of wave packets. In this letter, we study this effect for strong-field lensed gravitational waves generated in hierarchical triple black hole systems in which a stellar-mass binary merges near a more massive black hole. We calculate how the gravitational spin Hall effect modifies the gravitational waveforms and show its potential for experimental observation. If detected, these effects will have profound implications for astrophysics and tests of general relativity.

Every signal propagating through the universe is at least weakly lensed by the intervening gravitational field. In some situations, wave-optics phenomena (diffraction, interference) can be observed as frequency-dependent modulations of the waveform of gravitational waves (GWs). We will denote these signatures as Wave-Optics Features (WOFs) and analyze them in detail. Our framework can efficiently and accurately compute WOF in the single-image regime, of which weak lensing is a limit. The phenomenology of WOF is rich and offers valuable information: the dense cusps of individual halos appear as peaks in Green's function for lensing. If resolved, these features probe the number, effective masses, spatial distribution and inner profiles of substructures. High signal-to-noise GW signals reveal WOFs well beyond the Einstein radius, leading to a fair probability of observation by upcoming detectors such as LISA. Potential applications of WOF include reconstruction of the lens' projected density, delensing standard sirens and inferring large-scale structure morphology and the halo mass function. Because WOF are sourced by light halos with negligible baryonic content, their detection (or lack thereof) holds promise to test dark matter scenarios., Comment: 26 pages, 12 figures

Theories beyond general relativity (GR) modify the propagation of gravitational waves (GWs). In some, inhomogeneities (aka. gravitational lenses) allow interactions between the metric and additional fields to cause lens-induced birefringence (LIB): a different speed of the two linear GW polarisations ($+$ and $\times$). Inhomogeneities then act as non-isotropic crystals, splitting the GW signal into two components whose relative time delay depends on the theory and lens parameters. Here we study the observational prospects for GW scrambling, i.e when the time delay between both GW polarisations is smaller than the signal's duration and the waveform recorded by a detector is distorted. We analyze the latest LIGO-Virgo-KAGRA catalog, GWTC-3, and find no conclusive evidence for LIB. The highest log Bayes factor that we find in favour of LIB is $3.21$ for GW$190521$, a particularly loud but short event. However, when accounting for false alarms due to (Gaussian) noise fluctuations, this evidence is below 1-$\sigma$. The tightest constraint on the time delay is $<0.51$ ms (90% C.L.) from GW$200311\_115853$. From the non-observation of GW scrambling, we constrain the optical depth for LIB, accounting for the chance of randomly distributed lenses (eg. galaxies) along the line of sight. Our LIB constraints on a (quartic) scalar-tensor Horndeski theory are more stringent than solar system tests for a wide parameter range and comparable to GW170817 in some limits. Interpreting GW190521 as an AGN binary (i.e. taking an AGN flare as a counterpart) allows even more stringent constraints. Our results demonstrate the potential and high sensitivity achievable by tests of GR, based on GW lensing., Comment: 18 pages, 10 figures

Lensed gravitational waves will offer new means to probe the distribution of matter in the universe, complementary to electromagnetic signals. Lensed continuous gravitational waves provide new challenges and opportunities beyond those of transient compact binary coalescence. Here we consider continuous gravitational waves emitted by isolated neutron stars and lensed by Sgr A$^*$, the supermassive black hole at the center of our galaxy, a system observable by the next generation of gravitational wave detectors. We analyze the signatures of this system in detail, addressing parameter estimation and model selection. Future detectors can distinguish lensed continuous waves and measure their parameters with precision $\sim 1 - 10\%$ for sources within $2-4$ Einstein radii of Sgr A$^*$, depending on the source distance, thanks to the relative motion of the observer-lens-source system. The chances of observing strongly-lensed neutron stars increase by one order of magnitude relative to previous estimates, thanks to the possibility of detecting lensed systems at several Einstein radii. Multiple images can be resolved with an angular accuracy $\sim 10$mas, comparable to the best optical telescopes. Image localization probes deviations from axial symmetry and the existence of companions to Sgr A$^*$ in regions complementary to stellar orbits and black hole imaging. Our methods and many of our results extend to other lenses (e.g. galactic substructure) and sources (e.g. long-lived inspiralling binaries), rendering lensed continuous gravitational waves into versatile probes of astrophysics and fundamental physics., Comment: 15 pages, 10 figures

Just like light, gravitational waves (GWs) are deflected and magnified by gravitational fields as they propagate through the Universe. However, their low frequency, phase coherence and feeble coupling to matter allow for distinct lensing phenomena, such as diffraction and central images, that are challenging to observe through electromagnetic sources. Here we explore how these phenomena can be used to probe features of gravitational lenses. We focus on two variants of the singular isothermal sphere, with 1) a variable slope of the matter density and 2) a central core. We describe the imprints of these features in the wave- and geometric-optics regimes, including the prospect of detecting central images. We forecast the capacity of LISA and advanced LIGO to study strongly lensed signals and measure the projected lens mass, impact parameter and slope or core size. A broad range of lens masses allows all parameters to be measured with precision up to $\sim 1/{\rm SNR}$, despite large degeneracies. Thanks to wave-optics corrections, all parameters can be measured, even when no central image forms. Although GWs are sensitive to projected quantities, we compute the probability distribution of lens redshift, virial mass and projection scale given a cosmology. As an application, we consider the prospect of constraining self-interacting and ultra-light dark matter, showing the regions of parameter space accessible to strongly-lensed GWs. The distinct GW signatures will enable novel probes of fundamental physics and astrophysics, including the properties of dark matter and the central regions of galactic halos., Comment: 43 pages, 27 figures. Matches PRD version

Gravitational wave (GW) astronomy offers the potential to probe the wave-optics regime of gravitational lensing. Wave optics (WO) effects are relevant at low frequencies, when the wavelength is comparable to the characteristic lensing time delay multiplied by the speed of light, and are thus often negligible for electromagnetic signals. Accurate predictions require computing the conditionally convergent diffraction integral, which involves highly oscillatory integrands and is numerically difficult. We develop and implement several methods to compute lensing predictions in the WO regime valid for general gravitational lenses. First, we derive approximations for high and low frequencies, obtaining explicit expressions for several analytic lens models. Next, we discuss two numerical methods suitable in the intermediate frequency range: 1) Regularized contour flow yields accurate answers in a fraction of a second for a broad range of frequencies. 2) Complex deformation is slower, but requires no knowledge of solutions to the geometric lens equation. Both methods are independent and complement each other. We verify sub-percent accuracy for several lens models, which should be sufficient for applications to GW astronomy in the near future. Apart from modelling lensed GWs, our method will also be applicable to the study of plasma lensing of radio waves and tests of gravity., Comment: 21 pages, 9 figures. Matches PRD version

The propagation of gravitational waves can be described in terms of null geodesics by using the geometrical optics approximation. However, at large but finite frequencies the propagation is affected by the spin-orbit coupling corrections to geometrical optics, known as the gravitational spin Hall effect. Consequently, gravitational waves follow slightly different frequency- and polarization-dependent trajectories, leading to dispersive and birefringent phenomena. We study the potential for detecting the gravitational spin Hall effect in hierarchical triple black hole systems, consisting of an emitting binary orbiting a more massive body acting as a gravitational lens. We calculate the difference in time of arrival with respect to the geodesic propagation and find that it follows a simple power-law dependence on frequency with a fixed exponent. We calculate the gravitational spin Hall-corrected waveform and its mismatch with respect to the original waveform. The waveform carries a measurable imprint of the strong gravitational field if the source, lens, and observer are sufficiently aligned, or for generic observers if the source is close enough to the lens. We present constraints on dispersive time delays from GWTC-3, translated from limits on Lorentz invariance violation. Finally, we address the sensitivity of current and future ground detectors to dispersive lensing. Our results demonstrate that the gravitational spin Hall effect can be detected, providing a novel probe of general relativity and the environments of compact binary systems.

The Laser Interferometer Space Antenna (LISA) has the potential to reveal wonders about the fundamental theory of nature at play in the extreme gravity regime, where the gravitational interaction is both strong and dynamical. In this white paper, the Fundamental Physics Working Group of the LISA Consortium summarizes the current topics in fundamental physics where LISA observations of GWs can be expected to provide key input. We provide the briefest of reviews to then delineate avenues for future research directions and to discuss connections between this working group, other working groups and the consortium work package teams. These connections must be developed for LISA to live up to its science potential in these areas., Comment: Accepted in: Living Reviews in Relativity

The Laser Interferometer Space Antenna (LISA) has two scientific objectives of cosmological focus: to probe the expansion rate of the universe, and to understand stochastic gravitational-wave backgrounds and their implications for early universe and particle physics, from the MeV to the Planck scale. However, the range of potential cosmological applications of gravitational wave observations extends well beyond these two objectives. This publication presents a summary of the state of the art in LISA cosmology, theory and methods, and identifies new opportunities to use gravitational wave observations by LISA to probe the universe.

In this paper we will list a few important goals that need to be addressed in the next decade, also taking into account the current discordances between the different cosmological probes, such as the disagreement in the value of the Hubble constant $H_0$, the $\sigma_8$--$S_8$ tension, and other less statistically significant anomalies. While these discordances can still be in part the result of systematic errors, their persistence after several years of accurate analysis strongly hints at cracks in the standard cosmological scenario and the necessity for new physics or generalisations beyond the standard model. In this paper, we focus on the $5.0\,\sigma$ tension between the {\it Planck} CMB estimate of the Hubble constant $H_0$ and the SH0ES collaboration measurements. After showing the $H_0$ evaluations made from different teams using different methods and geometric calibrations, we list a few interesting new physics models that could alleviate this tension and discuss how the next decade's experiments will be crucial. Moreover, we focus on the tension of the {\it Planck} CMB data with weak lensing measurements and redshift surveys, about the value of the matter energy density $\Omega_m$, and the amplitude or rate of the growth of structure ($\sigma_8,f\sigma_8$). We list a few interesting models proposed for alleviating this tension, and we discuss the importance of trying to fit a full array of data with a single model and not just one parameter at a time. Additionally, we present a wide range of other less discussed anomalies at a statistical significance level lower than the $H_0$--$S_8$ tensions which may also constitute hints towards new physics, and we discuss possible generic theoretical approaches that can collectively explain the non-standard nature of these signals.[Abridged], Comment: Contribution to Snowmass 2021. 224 pages, 27 figures. Accepted for publication in JHEAp

The next generation of ground-based gravitational-wave detectors will observe coalescences of black holes and neutron stars throughout the cosmos, thousands of them with exceptional fidelity. The Science Book is the result of a 3-year effort to study the science capabilities of networks of next generation detectors. Such networks would make it possible to address unsolved problems in numerous areas of physics and astronomy, from Cosmology to Beyond the Standard Model of particle physics, and how they could provide insights into workings of strongly gravitating systems, astrophysics of compact objects and the nature of dense matter. It is inevitable that observatories of such depth and finesse will make new discoveries inaccessible to other windows of observation. In addition to laying out the rich science potential of the next generation of detectors, this report provides specific science targets in five different areas in physics and astronomy and the sensitivity requirements to accomplish those science goals. This report is the second in a six part series of reports by the GWIC 3G Subcommittee: i) Expanding the Reach of Gravitational Wave Observatories to the Edge of the Universe, ii) The Next Generation Global Gravitational Wave Observatory: The Science Book (this report), iii) 3G R&D: R&D for the Next Generation of Ground-based Gravitational Wave Detectors, iv) Gravitational Wave Data Analysis: Computing Challenges in the 3G Era, v) Future Ground-based Gravitational-wave Observatories: Synergies with Other Scientific Communities, and vi) An Exploration of Possible Governance Models for the Future Global Gravitational-Wave Observatory Network., Comment: 69 pages, 18 figures

The N-body gauge allows the introduction of relativistic effects in Newtonian cosmological simulations. Here we extend this framework to general Horndeski gravity theories, and investigate the relativistic effects that the scalar field introduces in the matter power spectrum at intermediate and large scales. In particular, we show that the kineticity function at these scales enhances the amplitude of the signal of contributions coming from the extra degree of freedom. Using the Quasi-Static Approximation (QSA), we separate modified gravity effects into two parts: one that only affects small-scale physics, and one that is due to relativistic effects. This allows our formalism to be readily implemented in modified gravity N-body codes in a straightforward manner, e.g., relativistic effects can be included as an additional linear density field in simulations. We identify the emergence of gravity acoustic oscillations (GAOs) in the matter power spectrum at large scales, $k \sim 10^{-3}-10^{-2}$ Mpc$^{-1}$. GAO features have a purely relativistic origin, coming from the dynamical nature of the scalar field. GAOs may be enhanced to detectable levels by the rapid evolution of the dark energy sound horizon in certain modified gravity models and can be seen as a new test of gravity at scales probed by future galaxy and intensity-mapping surveys., Comment: 26 pages, 11 figures and 1 table. To appear in JCAP

Attempts at constraining theories of late time accelerated expansion often assume broad priors for the parameters in their phenomenological description. Focusing on shift-symmetric scalar-tensor theories with standard gravitational wave speed, we show how a more careful analysis of their dynamical evolution leads to much narrower priors. In doing so, we propose a simple and accurate parametrisation of these theories, capturing the redshift dependence of the equation of state, $w(z)$, and the kinetic braiding parameter, $\alpha_{\rm B}(z)$, with only two parameters each, and derive their statistical distribution (a.k.a. theoretical priors) that fit the cosmology of the underlying model. We have considered two versions of the shift-symmetric model, one where the energy density of dark energy is given solely by the scalar field, and another where it also has a contribution from the cosmological constant. By including current data, we show how theoretical priors can be used to improve constraints by up to an order of magnitude. Moreover, we show that shift-symmetric theories without a cosmological constant are observationally viable. We work up to quartic order in first derivatives of the scalar in the action and our results suggest this truncation is a good approximation to more general shift-symmetric theories. This work establishes an actionable link between phenomenological parameterisations and Lagrangian-based theories, the two main approaches to test cosmological gravity and cosmic acceleration., Comment: 18 pages, 13 figures; Version as accepted in PRD - minor changes

Gravitational waves (GW), as light, are gravitationally lensed by intervening matter, deflecting their trajectories, delaying their arrival and occasionally producing multiple images. In theories beyond general relativity (GR), new gravitational degrees of freedom add an extra layer of complexity and richness to GW lensing. We develop a formalism to compute GW propagation beyond GR over general space-times, including kinetic interactions with new fields. Our framework relies on identifying the dynamical propagation eigenstates (linear combinations of the metric and additional fields) at leading order in a short-wave expansion. We determine these eigenstates and the conditions under which they acquire a different propagation speed around a lens. Differences in speed between eigenstates cause birefringence phenomena, including time delays between the metric polarizations (orthogonal superpositions of $h_+,h_\times$) observable without an electromagnetic counterpart. In particular, GW echoes are produced when the accumulated delay is larger than the signal's duration, while shorter time delays produce a scrambling of the wave-form. We also describe the formation of GW shadows as non-propagating metric components are sourced by the background of the additional fields around the lens. As an example, we apply our methodology to quartic Horndeski theories with Vainshtein screening and show that birefringence effects probe a region of the parameter space complementary to the constraints from the multi-messenger event GW170817. In the future, identified strongly lensed GWs and binary black holes merging near dense environments, such as active galactic nuclei, will fulfill the potential of these novel tests of gravity., Comment: Matches PRD version (only minor changes w.r.t. v1)

Discrepant measurements of the Universe's expansion rate ($H_0$) may signal physics beyond the standard cosmological model. Here I describe two early modified gravity mechanisms that reconcile the value of $H_0$ by increasing the expansion rate in the era of matter-radiation equality. These mechanisms, based on viable Horndeski theories, require significantly less fine-tuned initial conditions than early dark energy with oscillating scalar fields. In Imperfect Dark Energy at Equality (IDEE), the initial energy density dilutes slower than radiation but faster than matter, naturally peaking around the era of equality. The minimal IDEE model, a cubic Galileon, is too constrained by the cosmic microwave background (Planck) and baryon acoustic oscillations (BAO) to relieve the $H_0$ tension. In Enhanced Early Gravity (EEG), the scalar field value modulates the cosmological strength of gravity. The minimal EEG model, an exponentially coupled cubic Galileon, gives a Planck+BAO value $H_0=68.7 \pm 1.5$ (68\% c.l.), reducing the tension with SH0ES from $4.4\sigma$ to $2.6\sigma$. Additionally, Galileon contributions to cosmic acceleration may reconcile $H_0$ via Late-Universe Phantom Expansion (LUPE). Combining LUPE, EEG and $\Lambda$ reduces the tension between Planck, BAO and SH0ES to $2.5\sigma$. I will also describe additional tests of coupled Galileons based on local gravity tests, primordial element abundances and gravitational waves. While further model building is required to fully resolve the $H_0$ problem and satisfy all available observations, these examples show the wealth of possibilities to solve cosmological tensions beyond Einstein's General Relativity., Comment: Updated discussion of S8 tension. Summary for busy readers. 26 pages + appendices, 17 figures. Accepted in PRD

The late time acceleration of the Universe can be characterized in terms of an extra, time dependent, component of the universe -- dark energy. The simplest proposal for dark energy is a scalar-tensor theory -- quintessence -- which consists of a scalar field, $\phi$, whose dynamics is solely dictated by its potential, $V(\phi)$. Such a theory can be uniquely characterized by the equation of state of the scalar field energy momentum-tensor. We find the time dependence of the equation of state for a broad family of potentials and, using this information, we propose an analytic prior distribution for the most commonly used parametrization. We show that this analytic prior can be used to accurately predict the distribution of observables for the next generation of cosmological surveys. Including the theoretical priors in the comparison with observations considerably improves the constraints on the equation of state., Comment: V1: 13 pages, 11 figures, comments welcome; V2: 14 pages, 11 figures, minor clarifications added, accepted for publication in PRD

Cosmological datasets have great potential to elucidate the nature of dark energy and test gravity on the largest scales available to observation. Theoretical predictions can be computed with hi_class (www.hiclass-code.net), an accurate, fast and flexible code for linear cosmology, incorporating a wide range of dark energy theories and modifications to general relativity. We introduce three new functionalities into hi_class: (1) Support for models based on covariant Lagrangians, including a constraint-preserving integration scheme for the background evolution and a series of worked-out examples: Galileon, nKGB, quintessence (monomial, tracker) and Brans-Dicke. (2) Consistent initial conditions for the scalar-field perturbations in the deep radiation era, identifying the conditions under which modified-gravity isocurvature perturbations may grow faster than adiabatic modes leading to a loss of predictivity. (3) An automated quasistatic approximation scheme allowing order-of-magnitude improvement in computing performance without sacrificing accuracy for wide classes of models. These enhancements bring the treatment of dark energy and modified gravity models to the level of detail comparable to software tools restricted to standard $\Lambda$CDM cosmologies. The hi_class code is publicly available (https://github.com/miguelzuma/hi_class_public), ready to explore current data and prepare for next-generation experiments., Comment: 35 pages + appendices, 11 figures. Updated discussions. Code available on https://github.com/miguelzuma/hi_class_public

Since the very beginning of astronomy the location of objects on the sky has been a fundamental observational quantity that has been taken for granted. While precise two dimensional positional information is easy to obtain for observations in the electromagnetic spectrum, the positional accuracy of current and near future gravitational wave detectors is limited to between tens and hundreds of square degrees, which makes it extremely challenging to identify the host galaxies of gravitational wave events or to confidently detect any electromagnetic counterparts. Gravitational wave observations provide information on source properties and distances that is complementary to the information in any associated electromagnetic emission and that is very hard to obtain in any other way. Observing systems with multiple messengers thus has scientific potential much greater than the sum of its parts. A gravitational wave detector with higher angular resolution would significantly increase the prospects for finding the hosts of gravitational wave sources and triggering a multi-messenger follow-up campaign. An observatory with arcminute precision or better could be realised within the Voyage 2050 programme by creating a large baseline interferometer array in space and would have transformative scientific potential. Precise positional information of standard sirens would enable precision measurements of cosmological parameters and offer new insights on structure formation; a high angular resolution gravitational wave observatory would allow the detection of a stochastic background and resolution of the anisotropies within it; it would also allow the study of accretion processes around black holes; and it would have tremendous potential for tests of modified gravity and the discovery of physics beyond the Standard Model., Comment: 26 pages, 2 figures. White paper submitted to ESA's Voyage 2050 call on behalf of the LISA Consortium 2050 Task Force

We propose a space-based interferometer surveying the gravitational wave (GW) sky in the milli-Hz to $\mu$-Hz frequency range. By the 2040s', the $\mu$-Hz frequency band, bracketed in between the Laser Interferometer Space Antenna (LISA) and pulsar timing arrays, will constitute the largest gap in the coverage of the astrophysically relevant GW spectrum. Yet many outstanding questions related to astrophysics and cosmology are best answered by GW observations in this band. We show that a $\mu$-Hz GW detector will be a truly overarching observatory for the scientific community at large, greatly extending the potential of LISA. Conceived to detect massive black hole binaries from their early inspiral with high signal-to-noise ratio, and low-frequency stellar binaries in the Galaxy, this instrument will be a cornerstone for multimessenger astronomy from the solar neighbourhood to the high-redshift Universe., Comment: 28 pages, 8 figures; White Paper submitted to ESA's Voyage 2050 call for papers on behalf of the LISA Consortium 2050 Task Force

Future gravitational-wave observations will enable unprecedented and unique science in extreme gravity and fundamental physics answering questions about the nature of dynamical spacetimes, the nature of dark matter and the nature of compact objects., Comment: 14 pages, 2 figures, White Paper submitted to the Astro-2020 (2020 Astronomy and Astrophysics Decadal Survey) by GWIC-3G Science Case Team (GWIC: Gravitational-Wave International Committee); replaced Figure 1 with a revised version that incorporates Gravity IR Flare data point

No Slip Gravity is a simple modified gravity theory with only one free function and the interesting characteristic that -- unlike many modified gravity theories -- it suppresses growth. This allows it to fit current redshift space distortion and $\sigma_8$ mass fluctuation amplitude data better than $\Lambda$CDM in general relativity, while retaining a $\Lambda$CDM background expansion and hence distances. Since it has no gravitational slip it alters equally CMB lensing and matter density growth, and in addition affects the CMB gravitational wave B-mode polarization power spectrum. We investigate and compute the effects of No Slip Gravity for CMB lensing and B-modes, and present a simple analytic approximation. Using a Monte Carlo analysis, we place constraints on the theory from current CMB data., Comment: 9 pages, 9 figures

Gravitational waves (GWs) provide a new tool to probe the nature of dark energy (DE) and the fundamental properties of gravity. We review the different ways in which GWs can be used to test gravity and models for late-time cosmic acceleration. Lagrangian-based gravitational theories beyond general relativity (GR) are classified into those breaking fundamental assumptions, containing additional fields and massive graviton(s). In addition to Lagrangian based theories we present the effective theory of DE and the $\mu$-$\Sigma$ parametrization as general descriptions of cosmological gravity. Multi-messenger GW detections can be used to measure the cosmological expansion (standard sirens), providing an independent test of the DE equation of state and measuring the Hubble parameter. Several key tests of gravity involve the cosmological propagation of GWs, including anomalous GW speed, massive graviton excitations, Lorentz violating dispersion relation, modified GW luminosity distance and additional polarizations, which may also induce GW oscillations. We summarize present constraints and their impact on DE models, including those arising from the binary neutron star merger GW170817. Upgrades of LIGO-Virgo detectors to design sensitivity and the next generation facilities such as LISA or Einstein Telescope will significantly improve these constraints in the next two decades., Comment: 28 pages, 13 figures, 1 table. Matches published version

The possibility of linking inflation and late cosmic accelerated expansion using the $\alpha$-attractor models has received increasing attention due to their physical motivation. In the early universe, $\alpha$-attractors provide an inflationary mechanism compatible with Planck satellite CMB observations and predictive for future gravitational wave CMB modes. Additionally $\alpha$-attractors can be written as quintessence models with a potential that connects a power law regime with a plateau or uplifted exponential, allowing a late cosmic accelerated expansion that can mimic behavior near a cosmological constant. In this paper we study a generalized dark energy $\alpha$-attractor model. We thoroughly investigate its phenomenology, including the role of all model parameters and the possibility of large-scale tachyonic instability clustering. We verify the relation that $1+w\sim 1/\alpha$ (while the gravitational wave power $r\sim\alpha$) so these models predict that a signature should appear in either the primordial B-modes or in late time deviation from a cosmological constant. We constrain the model parameters with current datasets, including the cosmic microwave background (Planck 2015 angular power spectrum, polarization and lensing), baryon acoustic oscillations (BOSS DR12) and supernovae (Pantheon compressed). Our results show that expansion histories close to a cosmological constant exist in large regions of the parameter space, not requiring a fine-tuning of the parameters or initial conditions., Comment: V1: 23 pages, 14 figures, comments welcome. V2: 23 pages, 14 figures, added some minor corrections and a major modification: used the Full Planck Likelihood instead of the compressed one. This version is accepted for publication in JCAP

The nature of dark matter (DM) remains unknown despite very precise knowledge of its abundance in the universe. An alternative to new elementary particles postulates DM as made of macroscopic compact halo objects (MACHO) such as black holes formed in the very early universe. Stellar-mass primordial black holes (PBHs) are subject to less robust constraints than other mass ranges and might be connected to gravitational-wave signals detected by the Laser Interferometer Gravitational-Wave Observatory (LIGO). New methods are therefore necessary to constrain the viability of compact objects as a DM candidate. Here we report bounds on the abundance of compact objects from gravitational lensing of type Ia supernovae (SNe). Current SNe datasets constrain compact objects to represent less than 35.2% (Joint Lightcurve Analisis) and 37.2% (Union 2.1) of the total matter content in the universe, at 95% confidence-level. The results are valid for masses larger than $\sim 0.01M_\odot$ (solar-masses), limited by the size SNe relative to the lens Einstein radius. We demonstrate the mass range of the constraints by computing magnification probabilities for realistic SNe sizes and different values of the PBH mass. Our bounds are sensitive to the total abundance of compact objects with $M \lesssim 0.01M_\odot$ and complementary to other observational tests. These results are robust against cosmological parameters, outlier rejection, correlated noise and selection bias. PBHs and other MACHOs are therefore ruled out as the dominant form of DM for objects associated to LIGO gravitational wave detections. These bounds constrain early-universe models that predict stellar-mass PBH production and strengthen the case for lighter forms of DM, including new elementary particles., Comment: 16 pages, 9 figures. Discussion on finite SNe size and outliers expanded. Version accepted for publication in PRL

Multi-messenger gravitational wave (GW) astronomy has commenced with the detection of the binary neutron star merger GW170817 and its associated electromagnetic counterparts. The almost coincident observation of both signals places an exquisite bound on the GW speed $|c_g/c-1|\leq5\cdot10^{-16}$. We use this result to probe the nature of dark energy (DE), showing that a large class of scalar-tensor theories and DE models are highly disfavored. As an example we consider the covariant Galileon, a cosmologically viable, well motivated gravity theory which predicts a variable GW speed at low redshift. Our results eliminate any late-universe application of these models, as well as their Horndeski and most of their beyond Horndeski generalizations. Three alternatives (and their combinations) emerge as the only possible scalar-tensor DE models: 1) restricting Horndeski's action to its simplest terms, 2) applying a conformal transformation which preserves the causal structure and 3) compensating the different terms that modify the GW speed (to be robust, the compensation has to be independent on the background on which GWs propagate). Our conclusions extend to any other gravity theory predicting varying $c_g$ such as Einstein-Aether, Ho\v{r}ava gravity, Generalized Proca, TeVeS and other MOND-like gravities., Comment: 6 pages, 2 figures. Matches PRL version

Cosmological models with Galileon gravity are an alternative to the standard $\Lambda {\rm CDM}$ paradigm with testable predictions at the level of its self-accelerating solutions for the expansion history, as well as large-scale structure formation. Here, we place constraints on the full parameter space of these models using data from the cosmic microwave background (CMB) (including lensing), baryonic acoustic oscillations (BAO) and the Integrated Sachs-Wolfe (ISW) effect. We pay special attention to the ISW effect for which we use the cross-spectra, $C_\ell^{\rm T g}$, of CMB temperature maps and foreground galaxies from the WISE survey. The sign of $C_\ell^{\rm T g}$ is set by the time evolution of the lensing potential in the redshift range of the galaxy sample: it is positive if the potential decays (like in $\Lambda {\rm CDM}$), negative if it deepens. We constrain three subsets of Galileon gravity separately known as the Cubic, Quartic and Quintic Galileons. The cubic Galileon model predicts a negative $C_\ell^{\rm T g}$ and exhibits a $7.8\sigma$ tension with the data, which effectively rules it out. For the quartic and quintic models the ISW data also rule out a significant region of the parameter space but permit regions where the goodness-of-fit is comparable to $\Lambda {\rm CDM}$. The data prefers a non zero sum of the neutrino masses ($\sum m_\nu\approx 0.5$eV) with $ \sim \! 5\sigma$ significance in these models. The best-fitting models have values of $H_0$ consistent with local determinations, thereby avoiding the tension that exists in $\Lambda {\rm CDM}$. We also identify and discuss a $\sim \! 2\sigma$ tension that Galileon gravity exhibits with recent BAO measurements. Our analysis shows overall that Galileon cosmologies cannot be ruled out by current data but future lensing, BAO and ISW data hold strong potential to do so., Comment: 26 pages, 7 Figures; published version

We study the role of field redefinitions in general scalar-tensor theories. In particular, we first focus on the class of field redefinitions linear in the spin-2 field and involving derivatives of the spin-0 mode, generically known as disformal transformations. We start by defining the action of a disformal transformation in the tangent space. Then, we take advantage of the great economy of means of the language of differential forms to compute the full transformation of Horndeski's theory under general disformal transformations. We obtain that Horndeski's action maps onto itself modulo a reduced set of non-Horndeski Lagrangians. These new Lagrangians are found to be invariant under disformal transformation that depend only in the first derivatives of the scalar. Moreover, these combinations of Lagrangians precisely appear when expressing in our basis the constraints of the recently proposed Extended Scalar-Tensor (EST) theories. These results allow us to classify the different orbits of scalar-tensor theories invariant under particular disformal transformations, namely the special disformal, kinetic disformal and disformal Horndeski orbits. In addition, we consider generalizations of this framework. We find that there are possible well-defined extended disformal transformations that have not been considered in the literature. However, they generically cannot link Horndeski theory with EST theories. Finally, we study further generalizations in which extra fields with different spin are included. These field redefinitions can be used to connect different gravity theories such as multi-scalar-tensor theories, generalized Proca theories and bi-gravity. We discuss how the formalism of differential forms could be useful for future developments in these lines., Comment: 23+10 pages, 1 figure and 2 tables

The next generation of surveys will greatly improve our knowledge of cosmological gravity. In this paper we focus on how Stage IV photometric redshift surveys, including weak lensing and multiple tracers of the matter distribution and radio experiments combined with measurements of the cosmic microwave background will lead to precision constraints on deviations from General Relativity. We use a broad subclass of Horndeski scalar-tensor theories to forecast the accuracy with which we will be able to determine these deviations and their degeneracies with other cosmological parameters. Our analysis includes relativistic effects, does not rely on the quasi-static evolution and makes conservative assumptions about the effect of screening on small scales. We define a figure of merit for cosmological tests of gravity and show how the combination of different types of surveys, probing different length scales and redshifts, can be used to pin down constraints on the gravitational physics to better than a few percent, roughly an order of magnitude better than present probes. Future cosmological experiments will be able to constrain the Brans-Dicke parameter at a level comparable to Solar System and astrophysical tests, Comment: 16 pages, 8 figures, accepted for publication in PRD (corrected minor typos)

The direct detection of gravitational waves (GWs) is an invaluable new tool to probe gravity and the nature of cosmic acceleration. A large class of scalar-tensor theories predict that GWs propagate with velocity different than the speed of light, a difference that can be $\mathcal{O}(1)$ for many models of dark energy. We determine the conditions behind the anomalous GW speed, namely that the scalar field spontaneously breaks Lorentz invariance and couples to the metric perturbations via the Weyl tensor. If these conditions are realized in nature, the delay between GW and electromagnetic (EM) signals from distant events will run beyond human timescales, making it impossible to measure the speed of GWs using neutron star mergers or other violent events. We present a robust strategy to exclude or confirm an anomalous speed of GWs using eclipsing binary systems, whose EM phase can be exquisitely determined. he white dwarf binary J0651+2844 is a known example of such system that can be used to probe deviations in the GW speed as small as $c_g/c-1\gtrsim 2\cdot 10^{-12}$ when LISA comes online. This test will either eliminate many contender models for cosmic acceleration or wreck a fundamental pillar of general relativity., Comment: 7 pages, comments welcome

Many theories of modified gravity with higher order derivatives are usually ignored because of serious problems that appear due to an additional ghost degree of freedom. Most dangerously, it causes an immediate decay of the vacuum. However, breaking Lorentz invariance can cure such abominable behavior. By analyzing a model that describes a massive graviton together with a remaining Boulware-Deser ghost mode we show that even ghostly theories of modified gravity can yield models that are viable at both classical and quantum levels and, therefore, they should not generally be ruled out. Furthermore, we identify the most dangerous quantum scattering process that has the main impact on the decay time and find differences to simple theories that only describe an ordinary scalar field and a ghost. Additionally, constraints on the parameters of the theory including some upper bounds on the Lorentz-breaking cutoff scale are presented. In particular, for a simple theory of massive gravity we find that a breaking of Lorentz invariance is allowed to happen even at scales above the Planck mass. Finally, we discuss the relevance to other theories of modified gravity., Comment: 18 pages, 3 figures, version published in JHEP

We present the public version of hi_class (www.hiclass-code.net), an extension of the Boltzmann code CLASS to a broad ensemble of modifications to general relativity. In particular, hi_class can calculate predictions for models based on Horndeski's theory, which is the most general scalar-tensor theory described by second-order equations of motion and encompasses any perfect-fluid dark energy, quintessence, Brans-Dicke, $f(R)$ and covariant Galileon models. hi_class has been thoroughly tested and can be readily used to understand the impact of alternative theories of gravity on linear structure formation as well as for cosmological parameter extraction., Comment: 17 pages + appendices, 4 figures, code available on https://github.com/miguelzuma/hi_class_public

We address the impact of consistent modifications of gravity on the largest observable scales, focusing on relativistic effects in galaxy number counts and the cross-correlation between the matter large scale structure (LSS) distribution and the cosmic microwave background (CMB). Our analysis applies to a very broad class of general scalar-tensor theories encoded in the Horndeski Lagrangian and is fully consistent on linear scales, retaining the full dynamics of the scalar field and not assuming quasi-static evolution. As particular examples we consider self-accelerating Covariant Galileons, Brans-Dicke theory and parameterizations based on the effective field theory of dark energy, using the \hiclass\, code to address the impact of these models on relativistic corrections to LSS observables. We find that especially effects which involve integrals along the line of sight (lensing convergence, time delay and the integrated Sachs-Wolfe effect -- ISW) can be considerably modified, and even lead to $\mathcal{O}(1000\%)$ deviations from General Relativity in the case of the ISW effect for Galileon models, for which standard probes such as the growth function only vary by $\mathcal{O}(10\%)$. These effects become dominant when correlating galaxy number counts at different redshifts and can lead to $\sim 50\%$ deviations in the total signal that might be observable by future LSS surveys. Because of their integrated nature, these deep-redshift cross-correlations are sensitive to modifications of gravity even when probing eras much before dark energy domination. We further isolate the ISW effect using the cross-correlation between LSS and CMB temperature anisotropies and use current data to further constrain Horndeski models (abridged)., Comment: 30 pages plus appendices, 9 figures. References added. Accepted for publication in JCAP

We use a description based on differential forms to systematically explore the space of scalar-tensor theories of gravity. Within this formalism, we propose a basis for the scalar sector at the lowest order in derivatives of the field and in any number of dimensions. This minimal basis is used to construct a finite and closed set of Lagrangians describing general scalar-tensor theories invariant under Local Lorentz Transformations in a pseudo-Riemannian manifold, which contains ten physically distinct elements in four spacetime dimensions. Subsequently, we compute their corresponding equations of motion and find which combinations are at most second order in derivatives in four as well as arbitrary number of dimensions. By studying the possible exact forms (total derivatives) and algebraic relations between the basis components, we discover that there are only four Lagrangian combinations producing second order equations, which can be associated with Horndeski's theory. In this process, we identify a new second order Lagrangian, named kinetic Gauss-Bonnet, that was not previously considered in the literature. However, we show that its dynamics is already contained in Horndeski's theory. Finally, we provide a full classification of the relations between different second order theories. This allows us to clarify, for instance, the connection between different covariantizations of Galileons theory. In conclusion, our formulation affords great computational simplicity with a systematic structure. As a first step we focus on theories with second order equations of motion. However, this new formalism aims to facilitate advances towards unveiling the most general scalar-tensor theories., Comment: 28 pages, 1 figure, version published in PRD (minor changes)