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This summary of the second Terrestrial Very-Long-Baseline Atom Interferometry (TVLBAI) Workshop provides a comprehensive overview of our meeting held in London in April 2024, building on the initial discussions during the inaugural workshop held at CERN in March 2023. Like the summary of the first workshop, this document records a critical milestone for the international atom interferometry community. It documents our concerted efforts to evaluate progress, address emerging challenges, and refine strategic directions for future large-scale atom interferometry projects. Our commitment to collaboration is manifested by the integration of diverse expertise and the coordination of international resources, all aimed at advancing the frontiers of atom interferometry physics and technology, as set out in a Memorandum of Understanding signed by over 50 institutions., Comment: Summary of the second Terrestrial Very-Long-Baseline Atom Interferometry Workshop held at Imperial College London: https://indico.cern.ch/event/1369392/

It has recently been noted that deformations of extremal AdS black holes in four and higher dimensions are generically non-smooth or singular on the horizon. Further, it was found that certain deformations of asymptotically flat extremal black holes are marginal, causing the nature of the horizon to be strongly sensitive to UV corrections -- in particular, Wilson coefficients associated with consistent UV completions generically lead to worse behaviour on the horizon. In this work, we extend the discussion of deformed horizons in the presence of UV corrections to extremal charged black holes in AdS, where we find a tower of marginal perturbations for different black hole masses. We argue that the apparent UV sensitivity of marginal modes is, in fact, a feature of the UV theory which is correctly reproduced by the EFT, and illustrate this with explicit UV completions confirming the validity of the EFT. We demonstrate that the same holds for a scalar-Maxwell EFT with known UV completion. In the gravitational case, the sign of EFT corrections to marginal perturbations is generally connected with the signs implied by positivity bounds, with UV completions generically leading to worse behaviour on the horizon. We conjecture that this result is more generic, and use this to derive more general positivity bounds motivated by the weak gravity conjecture, which we illustrate with further evidence., Comment: 35 pages + appendices, 2 figures; v2: made clarifications, updated references, corrected typos

The class of Galileon scalar fields theories encapsulate the Vainshtein screening mechanism which is characteristic of a large range of infrared modified theories of gravity. Such theories can lead to testable departures from General Relativity through fifth forces and new scalar modes of gravitational radiation. However, the inherent non-linearity of the Vainshtein mechanism has limited analytic attempts to describe Galileon theories with both cubic and quartic interactions. To improve on this, we perform direct numerical simulations of the quartic Galileon model for a rotating binary source and infer the power spectrum of given multipoles. To tame numerical instabilities we utilize a low-pass filter, extending previous work on the cubic Galileon. Our findings show that the multipole expansion is well-defined and under control. Moreover, our results confirm that despite being a non-linear scalar, the dominant Galileon radiation is quadrupole, and we find a new scaling behaviour deep inside the Vainshtein region., Comment: 12 pages, 6 figures

We investigate quantum decoherence in a class of models which interpolates between expanding (inflation) and contracting (ekpyrosis) scenarios. For the cases which result in a scale-invariant power spectrum, we find that ekpyrotic universes lead to complete decoherence of the curvature perturbation before the bounce. This is in stark contrast to the inflationary case, where recoherence has been previously observed in some situations. Although the purity can be computed for couplings of all sizes, we also study the purity perturbatively and observe that late-time (secular growth) breakdown of perturbation theory often occurs in these cases. Instead, we establish a simple yet powerful late-time purity resummation which captures the exact evolution to a remarkable level, while maintaining analytical control. We conclude that the cosmological background plays a crucial role in the decoupling of the heavy fields during inflation and alternatives., Comment: 27 pages + appendices, 10 figures

This document presents a summary of the 2023 Terrestrial Very-Long-Baseline Atom Interferometry Workshop hosted by CERN. The workshop brought together experts from around the world to discuss the exciting developments in large-scale atom interferometer (AI) prototypes and their potential for detecting ultralight dark matter and gravitational waves. The primary objective of the workshop was to lay the groundwork for an international TVLBAI proto-collaboration. This collaboration aims to unite researchers from different institutions to strategize and secure funding for terrestrial large-scale AI projects. The ultimate goal is to create a roadmap detailing the design and technology choices for one or more km-scale detectors, which will be operational in the mid-2030s. The key sections of this report present the physics case and technical challenges, together with a comprehensive overview of the discussions at the workshop together with the main conclusions., Comment: Summary of the Terrestrial Very-Long-Baseline Atom Interferometry Workshop held at CERN: https://indico.cern.ch/event/1208783/

Relativistic causality constrains the $S$-matrix both through its analyticity, and by imposing lower bounds on the scattering time delay. These bounds are easiest to determine for spacetimes which admit either a timelike or null Killing vector. We revisit a class of pp-wave spacetimes and carefully determine the scattering time delay for arbitrary incoming states in the eikonal, semi-classical, and Born approximations. We apply this to the EFT of gravity in arbitrary dimensions. It is well-known that higher-dimension operators such as the Gauss-Bonnet term, when treated perturbatively at low energies, can appear to make both positive and negative contributions to the time delays of the background geometry. We show that even when multiple shockwaves are stacked, the corrections to the scattering time delay relative to the background are generically unresolvable within the regime of validity of the effective field theory so long as the Wilson coefficients are of order unity. This is in agreement with previously derived positivity/bootstrap bounds and the requirement that infrared causality be maintained in consistent low-energy effective theories, irrespective of the UV completion., Comment: 68 pages, 3 figures; v2: references updated, typos corrected

Effective field theories (EFT) are strongly constrained by fundamental principles such as unitarity, locality, causality, and Lorentz invariance. In this paper, we consider the EFT of photons (or other $U(1)$ gauge field) and compare different approaches to obtain bounds on its Wilson coefficients. We present an analytic derivation of the implications of unitarity (linear and non-linear positivity bounds) and compare these constraints with the requirement of causal propagation of the photon modes around non-trivial backgrounds generated by external sources. We find that the low energy causality condition can give complementary constraints to the positivity bounds. Applying both constraints together can significantly reduce the allowed region of the photon EFT parameters., Comment: 37 pages + appendices and references, 23 figures

It has recently been argued that half degrees of freedom could emerge in Lorentz and parity invariant field theories, using a non-linear Proca field theory dubbed Proca-Nuevo as a specific example. We provide two proofs, using the Lagrangian and Hamiltonian pictures, that the theory possesses a pair of second class constraints, leaving $D-1$ degrees of freedom in $D$ spacetime dimensions, as befits a consistent Proca model. Our proofs are explicit and straightforward in two dimensions and we discuss how they generalize to an arbitrary number of dimensions. We also clarify why local Lorentz and parity invariant field theories cannot hold half degrees of freedom.

We present a formulation of ghost-free massive gravity with flat reference metric that exhibits the full non-linear constraint algebraically, in a way that can be directly implemented for numerical simulations. Motivated by the presence of higher order operators in the low-energy effective description of massive gravity, we show how the inclusion of higher-order gradient (dissipative) terms leads to a well-posed formulation of its dynamics. While the formulation is presented for a generic combination of the minimal and quadratic mass terms on any background, for concreteness, we then focus on the numerical evolution of the minimal model for spherically symmetric gravitational collapse of scalar field matter. This minimal model does not carry the relevant interactions to switch on an active Vainshtein mechanism, at least in spherical symmetry, thus we do not expect to recover usual GR behaviour even for small graviton mass. Nonetheless we may ask what the outcome of matter collapse is for this gravitational theory. Starting with small initial data far away from the centre, we follow the matter through a non-linear regime as it falls towards the origin. For sufficiently weak data the matter disperses. However for larger data we generally find that the classical evolution breaks down due to the theory becoming infinitely strongly coupled without the presence of an apparent horizon shielding this behaviour from an asymptotic observer., Comment: 40 pages, 15 figures

We consider positivity constraints applicable to the Effective Field Theory (EFT) of gravity in arbitrary dimensions. By considering scattering of indefinite initial and final states, we highlight the existence of a gravitational scattering amplitude for which full crossing symmetry is manifest and utilize the recently developed crossing symmetric dispersion relations to derive compact non-linear bounds. We show that the null constraints built into these dispersion relations lead to a finite energy sum rule for gravity which may be extended to a one-parameter family of continuous moment sum rules. These sum rules enforce a UV-IR relation which imposes constraints on both the Regge trajectory and residue. We also highlight a situation where the Regge trajectory is uniquely determined in terms of the sub-Regge scale amplitude. Generically the Regge behaviour may be split into an IR sensitive part calculable within a given EFT which mainly depends on the lightest fields in Nature, and an IR independent part which are subject to universal positivity constraints following from unitarity and analyticity., Comment: 38 pages + appendices

We summarize recent progress in the development, application, and understanding of effective field theories and highlight promising directions for future research. This Report is prepared as the TF02 "Effective Field Theory" topical group summary for the Theory Frontier as part of the Snowmass 2021 process., Comment: 12 pages

Effective field theories (EFT) of dark energy (DE) -- built to parameterise the properties of DE in an agnostic manner -- are severely constrained by measurements of the propagation speed of gravitational waves (GW). However, GW frequencies probed by ground-based interferometers lie around the typical strong coupling scale of the EFT, and it is likely that the effective description breaks down before even reaching that scale. We discuss how this leaves the possibility that an appropriate ultraviolet completion of DE scenarios, valid at scales beyond an EFT description, can avoid present constraints on the GW speed. Instead, additional constraints in the lower frequency LISA band would be harder to escape, since the energies involved are orders of magnitude lower. By implementing a method based on GW multiband detections, we show indeed that a single joint observation of a GW150914-like event by LISA and a terrestrial interferometer would allow one to constrain the speed of light and gravitons to match to within $10^{-15}$. Multiband GW observations can therefore firmly constrain scenarios based on the EFT of DE, in a robust and unambiguous way., Comment: 11 pages, 4 figures. Updated to match the version published in JCAP

Assuming unitarity, locality, causality, and Lorentz invariance of the, otherwise unknown, UV completion, we derive a new set of constraints on the effective field theory coefficients for the most general, ghost-free Generalized Proca and Proca Nuevo massive vector models. For the Generalized Proca model, we include new interactions that had not been previously considered in the context of positivity bounds and find these additional terms lead to a widened parameter space for the previously considered interactions. Although, the Generalized Proca and Proca Nuevo models are inequivalent, we find interesting analogues between the coefficients parameterizing the two models and the roles they play in the positivity bounds., Comment: 40 pages, 9 figures, 1 table, references and funding agencies updated

Physical principles such as unitarity, causality, and locality can constrain the space of consistent effective field theories (EFTs) by imposing two-sided bounds on the allowed values of Wilson coefficients. In this paper, we consider the bounds that arise from the requirement of low-energy causality alone, without appealing to any assumptions about UV physics. We focus on shift-symmetric theories, and consider bounds that arise from the propagation around both a homogeneous and a spherically-symmetric background. We find that low-energy causality, namely the requirement that there are no resolvable time advances within the regime of validity of the EFT, produces two-sided bounds in agreement with compact positivity constraints previously obtained from $2 \rightarrow 2$ scattering amplitude dispersion relations using full crossing symmetry., Comment: 38 pages, 5 figures (updated in v2)

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

Fundamental principles of local quantum field theory or of quantum gravity can enforce consistency requirements on the space of consistent low-energy effective field theories. We survey the various techniques that have been used to put UV constraints on IR physics, including those from causality considerations in the form of S-matrix positivity and bootstrap bounds, scattering time delays, conformal field theory and holographic methods, together with those that arise from landscape/swampland criteria such as the weak gravity conjecture. We review recent applications of these constraints to corrections to Standard Model physics, corrections to Einstein gravity, and cosmological theories and highlight promising future directions., Comment: Contribution to Snowmass 2021; 19 pages + references

We consider the effective field theory of gravity around black holes, and show that the coefficients of the dimension-8 operators are tightly constrained by causality considerations. Those constraints are consistent with -- but tighter than -- previously derived causality and positivity bounds and imply that the effects of one of the dimension-8 operators by itself cannot be observable while remaining consistent with causality. We then establish in which regime one can expect the generic dimension-8 and lower order operators to be potentially observable while preserving causality, providing a theoretical prior for future observations. We highlight the importance of "infrared causality" and show that the requirement of "asymptotic causality" or net (sub)luminality would fail to properly diagnose violations of causality., Comment: 12 pages, 4 figures, typos corrected, constraints on the dim-6 operators updated

We distinguish between the notions of asymptotic causality and infrared causality for gravitational effective field theories, and show that the latter gives constraints consistent with gravitational positivity bounds. We re-explore the scattering of gravitational waves in a spherically symmetric background in the EFT of gravity in $D\ge 5$, for which the leading-order correction to Einstein gravity is determined by the Gauss-Bonnet operator. We reproduce the known result that the truncated effective theory exhibits apparent time advances relative to the background geometry for specific polarisations, which naively signal a violation of causality. We show that by properly identifying the regime of validity of the effective theory, the apparent time advance can be shown to be unresolvable. To illustrate this, we identify specific higher-dimension operators in the EFT expansion which become large for potentially resolvable time advances, rendering the EFT expansion invalid. Our results demonstrate how staying within the confines of the EFT, neither infrared nor asymptotic causality are ever violated for Einstein-Gauss-Bonnet gravity, no matter how low the scale, and furthermore its causality can be understood without appealing to a precise UV completion such as string theory., Comment: 28 pages + appendices; v2: typos corrected

S-matrix bootstrap and positivity bounds are usually viewed as constraints on low-energy theories imposed by the requirement of a standard UV completion. By considering graviton--photon scattering in the Standard Model, we argue that the low-energy theory can be used to put constraints on the UV behaviour of the gravitational scattering amplitudes., Comment: Published in PRL. 5 pages + supplementary material and 1 figure

Proca-Nuevo is a non-linear theory of a massive spin-1 field which enjoys a non-linearly realized constraint that distinguishes it among other generalized vector models. We show that the theory may be extended by the addition of operators of the Generalized Proca class without spoiling the primary constraint that is necessary for consistency, allowing to interpolate between Generalized Proca operators and Proca-Nuevo ones. The constraint is maintained on flat spacetime and on any fixed curved background. Upon mixing extended Proca-Nuevo dynamically with gravity, we show that the constraint gets broken in a Planck scale suppressed way. We further prove that the theory may be covariantized in models that allow for consistent and ghost-free cosmological solutions. We study the models in the presence of perfect fluid matter, and show that they describe the correct number of dynamical variables and derive their dispersion relations and stability criteria. We also exhibit, in a specific set-up, explicit hot Big Bang solutions featuring a late-time self-accelerating epoch, and which are such that all the stability and subluminality conditions are satisfied and where gravitational waves behave precisely as in General Relativity., Comment: 34 pages + appendices, 8 figures

The construction of general derivative self-interactions for a massive Proca field relies on the well-known condition for constrained systems of having a degenerate Hessian. The nature of the existing constraints algebra will distinguish among different classes of interactions. Proca-Nuevo interactions enjoy a non-trivial constraint by mixing terms of various order whereas Generalized Proca interactions satisfy the degeneracy condition order by order for each individual Lagrangians. In both cases the vector field propagates at most three degrees of freedom. It has been shown that the scattering amplitudes of Proca-Nuevo arising at the tree level always differ from those of the Generalized Proca, implying their genuinely different nature and a lack of relation by local field redefinitions. In this work, we show the quantum stability of the Proca-Nuevo theory below a specific UV cut-off. Although Proca-Nuevo and Generalized Proca are different inherently in their classical structure, both have the same high energy behaviour when quantum corrections are taken into account. The arising counter terms have the exact same structure and scaling. This might indicate that whatever UV completion they may come from, we expect it to be of similar nature., Comment: 15 pages, 2 figures

Amplitude methods have proven to be a promising technique to perform Post-Minkowskian calculations used as inputs to construct gravitational waveforms. In this paper, we show how these methods can be extended beyond the standard calculations in General Relativity with a minimal coupling to matter. As proof of principle, we consider spinless particles conformally coupled to a gravitational helicity-0 mode. We clarify the subtleties in the matching procedure that lead to the potential for conformally coupled matter. We show that in the probe particle limit, we can reproduce well known results for the field profile. With the scattering amplitudes at hand, we compute the conservative potential and scattering angle for the binary system. We find that the result is a non trivial expansion that involves not only the coupling strengths, but also a non trivial dependence on the energy/momentum of the scattered particles., Comment: 31 pages plus appendices, 4 figures; v2: typos fixed and references added; v3: 1 appendix added, matches published version

Positivity bounds - constraints on any low-energy effective field theory imposed by the fundamental axioms of unitarity, causality and locality in the UV - have recently been used to constrain scalar-tensor theories of dark energy. However, the coupling to matter fields has so far played a limited role. We show that demanding positivity when including interactions with standard matter fields leads to further constraints on the dark energy parameter space. We demonstrate how implementing these bounds as theoretical priors affects cosmological parameter constraints and explicitly illustrate the impact on a specific Effective Field Theory for dark energy. We also show in this model that the existence of a standard UV completion requires that gravitational waves must travel superluminally on cosmological backgrounds., Comment: 14+6 pages, 2 figures; v2: minor clarifications added, matches published version

The metric of a spacetime can be greatly simplified if the spacetime is circular. We prove that in generic effective theories of gravity, the spacetime of a stationary, axisymmetric and asymptotically flat solution must be circular if the solution can be obtained perturbatively from a solution in the general relativity limit. This result applies to a broad class of gravitational theories that include arbitrary scalars and vectors in their light sector, so long as their nonstandard kinetic terms and nonmininal couplings to gravity are treated perturbatively., Comment: 7+5 pages, 1 figure, text and references updated, v2 matches version published in PRL

We apply positivity bounds directly to a $U(1)$ gauge theory with charged scalars and charged fermions, i.e. QED, minimally coupled to gravity. Assuming that the massless $t$-channel pole may be discarded, we show that the improved positivity bounds are violated unless new physics is introduced at the parametrically low scale $\Lambda_{\rm new} \sim (e m M_{\rm pl})^{1/2}$, consistent with similar results for scalar field theories, far lower than the scale implied by the weak gravity conjecture. This is sharply contrasted with previous treatments which focus on the application of positivity bounds to the low energy gravitational Euler-Heisenberg effective theory only. We emphasise that the low-cutoff is a consequence of applying the positivity bounds under the assumption that the pole may be discarded. We conjecture an alternative resolution that a small amount of negativity, consistent with decoupling limits, is allowed and not in conflict with standard UV completions, including weakly coupled ones., Comment: 17 pages, 7 figures, 13 pages appendices

The presence of a massless spin-2 field in an effective field theory results in a $t$-channel pole in the scattering amplitudes that precludes the application of standard positivity bounds. Despite this, recent arguments based on compactification to three dimensions have suggested that positivity bounds may be applied to the $t$-channel pole subtracted amplitude. If correct this would have deep implications for UV physics and the Weak Gravity Conjecture. Within the context of a simple renormalizable field theory coupled to gravity we find that applying these arguments would constrain the low-energy coupling constants in a way which is incompatible with their actual values. This contradiction persists on deforming the theory. Further enforcing the $t$-channel pole subtracted positivity bounds on such generic renormalizable effective theories coupled to gravity would imply new physics at a scale parametrically smaller than expected, with far reaching implications. This suggests that generically the standard positivity bounds are inapplicable with gravity and we highlight a number of issues that impinge on the formulation of a three-dimensional amplitude which simultaneously satisfies the required properties of analyticity, positivity and crossing symmetry. We conjecture instead a modified bound that ought to be satisfied independently of the precise details of the high energy completion., Comment: 28 pages + appendices

Within the low-energy effective field theories of QED and gravity, the low-energy speed of light or that of gravitational waves can typically be mildly superluminal in curved spacetimes. Related to this, small scattering time advances relative to the curved background can emerge from known effective field theory coefficients for photons or gravitons. We clarify why these results are not in contradiction with causality, analyticity or Lorentz invariance, and highlight various subtleties that arise when dealing with superluminalities and time advances in the gravitational context. Consistent low-energy effective theories are shown to self-protect by ensuring that any time advance and superluminality calculated within the regime of validity of the effective theory is necessarily unresolvable, and cannot be argued to lead to a macroscopically larger lightcone. Such considerations are particularly relevant for putting constraints on cosmological and gravitational effective field theories and we provide explicit criteria to be satisfied so as to ensure causality., Comment: references added, typos corrected

We investigate the propagation of gravitational waves on a black hole background within the low energy effective field theory of gravity, where effects from heavy fields are captured by higher dimensional curvature operators. Depending on the spin of the particles integrated out, the speed of gravitational waves at low energy can be either superluminal or subluminal as compared to the causal structure observed by other species. Interestingly however, gravitational waves are always exactly luminal at the black hole horizon, implying that the horizon is identically defined for all species. We further compute the corrections on quasinormal frequencies caused by the higher dimensional curvature operators and highlight the corrections arising from the low energy effective field., Comment: 21 pages, 2 figures

We propose a new class of Proca interactions that enjoy a non-trivial constraint and hence propagates the correct number of degrees of freedom for a healthy massive spin-1 field. We show that the scattering amplitudes always differ from those of the Generalized Proca. This implies that the new class of interactions proposed here are genuinely different from the Generalized Proca and there can be no local field redefinitions between the two. In curved spacetime, massive gravity is the natural covariantization but we show how other classes of covariantizations can be considered., Comment: 28 pages, 1 figure Version published in Physical Review D. The title has been changed to match the one of prd

We explore the effective field theory for single and multiple interacting pseudo-linear spin-2 fields. By applying forward limit positivity bounds, we show that among the parameters contributing to elastic tree level scattering amplitude, there is no region of compatibility of the leading interactions with a standard local UV completion. Our result generalizes to any number of interacting pseudo-linear spin-2 fields. These results have significant implications for the organization of the effective field theory expansion for pseudo-linear fields., Comment: 18 pages

The consistency of the EFT of two interacting spin-2 fields is checked by applying forward limit positivity bounds on the scattering amplitudes to exclude the region of parameter space devoid of a standard UV completion. We focus on two classes of theories that have the highest possible EFT cutoff, namely those theories modelled on ghost-free interacting theories of a single massive spin-2 field. We find that the very existence of interactions between the spin-2 fields implies more stringent bounds on all the parameters of the EFT, even on the spin-2 self-interactions. This arises for two reasons. First, with every new field included in the low-energy EFT, comes the `knowledge' of an extra pole to be subtracted, hence strengthening the positivity bounds. Second, while adding new fields increases the number of free parameters from the new interactions, this is rapidly overcome by the increased number of positivity bounds for different possible scattering processes. We also discuss how positivity bounds appear to favour relations between operators that effectively raise the cutoff of the EFT., Comment: 32 pages plus appendices, 16 figures

We consider the effective field theory of multiple interacting massive spin-2 fields. We focus on the case where the interactions are chosen so that the cutoff is the highest possible, and highlight two distinct classes of theories. In the first class, the mass eigenstates only interact through potential operators that carry no derivatives in unitary gauge at leading order. In the second class, a specific kinetic mixing between the mass eigenstates is included non-linearly. Performing a decoupling and ADM analysis, we point out the existence of a ghost present at a low scale for the first class of interactions. For the second class of interactions where kinetic mixing is included, we derive the full $\Lambda_3$ decoupling limit and confirm the absence of any ghosts. Nevertheless both formulations can be used to consistently describe an EFT of interacting massive spin-2 fields which, for a suitable technically natural tuning of the EFT, have the same strong coupling scale $\Lambda_3$. We identify the generic form of EFT corrections in each case. By using Galileon Duality transformations for the specific case of two massive spin-2 fields with suitable couplings, the decoupling limit theory is shown to be a bi-Galileon., Comment: 39 pages + appendices, 1 figure

Within the standard effective field theory of General Relativity, we show that the speed of gravitational waves deviates, ever so slightly, from luminality on cosmological and other spontaneously Lorentz-breaking backgrounds. This effect results from loop contributions from massive fields of any spin, including Standard Model fields, or from tree level effects from massive higher spins $s \ge 2$. We show that for the choice of interaction signs implied by S-matrix and spectral density positivity bounds suggested by analyticity and causality, the speed of gravitational waves is in general superluminal at low-energies on NEC preserving backgrounds, meaning gravitational waves travel faster than allowed by the metric to which photons and Standard Model fields are minimally coupled. We show that departure of the speed from unity increases in the IR and argue that the speed inevitably returns to luminal at high energies as required by Lorentz invariance. Performing a special tuning of the EFT so that renormalization sensitive curvature-squared terms are set to zero, we find that finite loop corrections from Standard Model fields still lead to an epoch dependent modification of the speed of gravitational waves which is determined by the precise field content of the lightest particles with masses larger than the Hubble parameter today. Depending on interpretation, such considerations could potentially have far-reaching implications on light scalar models, such as axionic or fuzzy cold dark matter., Comment: references added, minor typos corrected

Among the Scalar-Tensor modified theories of gravity, DHOST models could play a special role for dark energy while being consistent with current observations, notably those constraining the speed of gravitational waves. Schwarzschild-de Sitter black holes were shown to be exact solutions of a particular subclass of quadratic DHOST theories, while carrying a nontrivial scalar profile that linearly evolves in time and hence potentially providing exciting new phenomenological windows to explore this model. We investigate the physical perturbations about such black holes and find that the odd-parity tensor perturbations behave in a way indistinguishable to GR. On the other hand, the effective metric for the (even-parity) scalar perturbations is singular, indicating that those exact black hole solutions are infinitely strongly coupled and cannot be trusted within the regime of validity of the DHOST effective field theory. We show how this strong coupling result is generalizable to a whole class of solutions with arbitrary manifolds both for DHOST and Horndeski., Comment: 18 pages, discussion and references added

We reconsider the construction of general derivative self-interactions for a massive Proca field. The constructed Lagrangian is such that the vector field propagates at most three degrees of freedom, thus avoiding the ghostly nature of a fourth polarisation. The construction makes use of the well-known condition for constrained systems of having a degenerate Hessian. We briefly discuss the casuistry according to the nature of the existing constraints algebra. We also explore various classes of interesting new interactions that have been recently raised in the literature. For the sixth order Lagrangian that satisfies the constraints by itself we prove its topological character, making such a term irrelevant. There is however a window of opportunity for exploring other classes of fully-nonlinear interactions that satisfy the constraint algebra by mixing terms of various order., Comment: 7 pages

Recently, it has been argued that application of the Weak Gravity Conjecture (WGC) to spin-2 fields implies a universal upper bound on the cutoff of the effective theory for a single spin-2 field. We point out here that these arguments are largely spurious, because of the absence of states carrying spin-2 Stuckelberg $U(1)$ charge, and because of incorrect scaling assumptions. Known examples such as Kaluza-Klein theory that respect the usual WGC do so because of the existence of a genuine $U(1)$ field under which states are charged, as in the case of the Stuckelberg formulation of spin-1 theories, for which there is an unambiguously defined $U(1)$ charge. Theories of bigravity naturally satisfy a naive formulation of the WGC, $M_W< M_{\rm Pl}$, since the force of the massless graviton is always weaker than the massive spin-2 modes. It also follows that theories of massive gravity trivially satisfies this form of the WGC. We also point out that the identification of a massive spin-2 state in a truncated higher derivative theory, such as Einstein-Weyl-squared or its supergravity extension, bears no relationship with massive spin-2 states in the UV completion, contrary to previous statements in the literature. We also discuss the conjecture from a swampland perspective and show how the emergence of a universal upper bound on the cutoff relies on strong assumptions on the scale of the couplings between the spin-2 and other fields, an assumption which is known to be violated in explicit examples., Comment: Improved discussions

We develop a full four-dimensional numerical code to study scalar gravitational radiation emitted from binary systems and probe the Vainshtein mechanism in situations that break the static and spherical symmetry, relevant for binary pulsars as well as black holes and neutron stars binaries. The present study focuses on the cubic Galileon which arises as the decoupling limit of massive theories of gravity. Limitations associated with the numerical methods prevent us from reaching a physically realistic hierarchy of scales; nevertheless, within this context we observe the same power law scaling of the radiated power as previous analytic estimates, and confirm a strong suppression of the power emitted in the monopole and dipole as compared with quadrupole radiation. Following the trend to more physically realistic parameters, we confirm the suppression of the power emitted in scalar gravitational radiation and the recovery of General Relativity with good accuracy. This paves the way for future numerical work, probing more generic, physically relevant situations and sets of interactions that may exhibit the Vainshtein mechanism., Comment: 27 pages, 9 figures, 1 table. Minor typos corrected and refs added

We consider various decoupling limits of ghost-free massive gravity on (A)dS. The first is a decoupling limit on AdS space where the mass goes to zero while the AdS radius is held fixed. This results in an interacting massive Proca vector theory with a $\Lambda_2\sim (M_{\rm Pl} m)^{1/2}$ strong coupling scale which is ghost-free by construction and yet can not be put in the form of the generalized Proca theories considered so far. We comment on the existence of a potential duality between this Proca theory and a CFT on the boundary. The second decoupling limit we consider is a new limit on dS, obtained by sending the mass towards the finite partially massless value. We do this by introducing the scalar St\"uckelberg field which restores the partially massless symmetry. For generic values of the parameters, only a finite number of operators enter the partially massless decoupling limit and take the form of dS Galileons. If the interactions are chosen to be precisely those of the `candidate' non-linear partially massless theory, the resulting strong coupling scale has a higher value and the resulting decoupling limit includes an infinite number of interactions which we give in closed form. These interactions preserve both the linear partially massless symmetry and the dS version of the Galileon shift symmetry., Comment: 51 pages, 2 figures. v2 refs added

The recent direct detection of a neutron star merger with optical counterpart has been used to severely constrain models of dark energy that typically predict a modification of the speed of gravitational waves. We point out that the energy scales observed at LIGO, and the particular frequency of the neutron star event, lie very close to the strong coupling scale or cutoff associated with many dark energy models. While it is true that at very low energies one expects gravitational waves to travel at a speed different than light in these models, the same is no longer necessarily true as one reaches energy scales close to the cutoff. We show explicitly how this occurs in a simple model with a known partial UV completion. Within the context of Horndeski, we show how the operators that naturally lie at the cutoff scale can affect the speed of propagation of gravitational waves and bring it back to unity at those scales. We discuss how further missions including LISA and PTAs could play an essential role in testing such models., Comment: 6 pages, 1 figure, small edits and references added

We apply the recently developed positivity bounds for particles with spin, applied away from the forward limit, to the low energy effective theories of massive spin-1 and spin-2 theories. For spin-1 theories, we consider the generic Proca EFT which arises at low energies from a heavy Higgs mechanism, and the special case of a charged Galileon for which the EFT is reorganized by the Galileon symmetry. For spin-2, we consider generic $\Lambda_5$ massive gravity theories and the special `ghost-free' $\Lambda_3$ theories. Remarkably we find that at the level of 2-2 scattering, the positivity bounds applied to $\Lambda_5$ massive gravity theories impose the special tunings which generate the $\Lambda_3$ structure. For $\Lambda_3$ massive gravity theories, the island of positivity derived in the forward limit appears relatively stable against further bounds., Comment: 38 pages, 2 figures

Theories such as massive Galileons and massive gravity can satisfy the presently known improved positivity bounds provided they are weakly coupled. We discuss the form of the EFT Lagrangian for a weakly coupled UV completion of massive gravity which closely parallels the massive Galileon, and perform the power counting of corrections to the scattering amplitude and the positivity bounds. The Vainshtein mechanism which is central to the phenomenological viability of massive gravity is entirely consistent with weak coupling since it is classical in nature. We highlight that the only implication of the improved positivity constraints is that EFT cutoff is lower than previous assumed, and discuss the observable implications, emphasizing that these bounds are not capable of ruling out the model contrary to previous statements in the literature., Comment: 30 pages,2 figures. Typos corrected, minor additions

For a low energy effective theory to admit a standard local, unitary, analytic and Lorentz-invariant UV completion, its scattering amplitudes must satisfy certain inequalities. While these bounds are known in the forward limit for real polarizations, any extension beyond this for particles with nonzero spin is subtle due to their non-trivial crossing relations. Using the transversity formalism (i.e. spin projections orthogonal to the scattering plane), in which the crossing relations become diagonal, these inequalities can be derived for 2-to-2 scattering between any pair of massive particles, for a complete set of polarizations at and away from the forward scattering limit. This provides a set of powerful criteria which can be used to restrict the parameter space of any effective field theory, often considerably more so than its forward limit subset alone., Comment: 29 pages + 6 appendices, 3 figures

A non-singular cosmological bounce in the Einstein frame can only take place if the Null Energy Condition (NEC) is violated. We explore situations where a single scalar field drives the NEC violation and derive the constraints imposed by demanding tree level unitarity on a cosmological background. We then focus on the explicit constraints that arise in P(X) theories and show that constraints from perturbative unitarity make it impossible for the NEC violation to occur within the region of validity of the effective field theory without also involving irrelevant operators that arise at a higher scale that would enter from integrating out more massive degrees of freedom. Within the context of P(X) theories we show that including such operators allows for a bounce that does not manifestly violate tree level unitarity, but at the price of either imposing a shift symmetry or involving technically unnatural small operator coefficients within the low-energy effective field theory., Comment: 35 pages, 1 figure

The EFT coefficients in any gapped, scalar, Lorentz invariant field theory must satisfy positivity requirements if there is to exist a local, analytic Wilsonian UV completion. We apply these bounds to the tree level scattering amplitudes for a massive Galileon. The addition of a mass term, which does not spoil the non-renormalization theorem of the Galileon and preserves the Galileon symmetry at loop level, is necessary to satisfy the lowest order positivity bound. We further show that a careful choice of successively higher derivative corrections are necessary to satisfy the higher order positivity bounds. There is then no obstruction to a local UV completion from considerations of tree level 2-to-2 scattering alone. To demonstrate this we give an explicit example of such a UV completion., Comment: 31 pages

Assuming the existence of a local, analytic, unitary UV completion in a Poincar\'{e} invariant scalar field theory with a mass gap, we derive an infinite number of positivity requirements using the known properties of the amplitude at and away from the forward scattering limit. These take the form of bounds on combinations of the pole subtracted scattering amplitude and its derivatives. In turn, these positivity requirements act as constraints on the operator coefficients in the low energy effective theory. For certain theories these constraints can be used to place an upper bound on the mass of the next lightest state that must lie beyond the low energy effective theory if such a UV completion is to ever exist., Comment: 5 pages

We study the development of caustics in shift-symmetric scalar field theories by focusing on simple waves with an $SO(p)$-symmetry in an arbitrary number of space dimensions. We show that the pure Galileon, the DBI-Galileon, and the extreme-relativistic Galileon naturally emerge as the unique set of caustic-free theories, highlighting a link between the caustic-free condition for simple $SO(p)$-waves and the existence of either a global Galilean symmetry or a global (extreme-)relativistic Galilean symmetry., Comment: 25 pages. v2: References added and included discussion on extreme-relativistic case. v3: Matches published version in PRD, appendix expanded

Recently, aLIGO has announced the first direct detections of gravitational waves, a direct manifestation of the propagating degrees of freedom of gravity. The detected signals GW150914 and GW151226 have been used to examine the basic properties of these gravitational degrees of freedom, particularly setting an upper bound on their mass. It is timely to review what the mass of these gravitational degrees of freedom means from the theoretical point of view, particularly taking into account the recent developments in constructing consistent massive gravity theories. Apart from the GW150914 mass bound, a few other observational bounds have been established from the effects of the Yukawa potential, modified dispersion relation and fifth force that are all induced when the fundamental gravitational degrees of freedom are massive. We review these different mass bounds and examine how they stand in the wake of recent theoretical developments and how they compare to the bound from GW150914., Comment: A review commissioned by Reviews of Modern Physics; Published in Reviews of Modern Physics

Euclid is a European Space Agency medium class mission selected for launch in 2020 within the Cosmic Vision 2015 2025 program. The main goal of Euclid is to understand the origin of the accelerated expansion of the universe. Euclid will explore the expansion history of the universe and the evolution of cosmic structures by measuring shapes and redshifts of galaxies as well as the distribution of clusters of galaxies over a large fraction of the sky. Although the main driver for Euclid is the nature of dark energy, Euclid science covers a vast range of topics, from cosmology to galaxy evolution to planetary research. In this review we focus on cosmology and fundamental physics, with a strong emphasis on science beyond the current standard models. We discuss five broad topics: dark energy and modified gravity, dark matter, initial conditions, basic assumptions and questions of methodology in the data analysis. This review has been planned and carried out within Euclid's Theory Working Group and is meant to provide a guide to the scientific themes that will underlie the activity of the group during the preparation of the Euclid mission., Comment: This article provides an update of arXiv:1206.1225, with different authors. Forecasts are not updated in this version

We review how the (absence of) Ostrogradsky instability manifests itself in theories with multiple fields. It has recently been appreciated that when multiple fields are present, the existence of higher derivatives may not automatically imply the existence of ghosts. We discuss the connection with gravitational theories like massive gravity and beyond Horndeski which manifest higher derivatives in some formulations and yet are free of Ostrogradsky ghost. We also examine an interesting new class of Extended Scalar-Tensor Theories of gravity which has been recently proposed. We show that for a subclass of these theories, the tensor modes are either not dynamical or are infinitely strongly coupled. Among the remaining theories for which the tensor modes are well-defined one counts one new model that is not field-redefinable to Horndeski via a conformal and disformal transformation but that does require the vacuum to break Lorentz invariance. We discuss the implications for the effective field theory of dark energy and the stability of the theory., Comment: references updated