Merger, numerics, and complete waveforms beyond General Relativity

The merger of binary compact objects will test the highly dynamical and strongly non-linear regime of gravity, that can only be modelled by using numerical simulations. As

and of the NSB merger [21] with an associated short GRB [23] and a plethora of

concurrent electromagnetic signals from the same source [22] has already provided

constraints on deviations from GR in various forms and contexts (e.g. [17, 674–

677, 1489, 1490, 1504] respectively. However, most of these constraints use either

partial, potentially parametrised, waveforms or rely on propagation effects. In fact, the true potential of testing GR is currently limited by the lack of knowledge of GW emission

during the merger phase in alternatives to GR [1504]. This problem is particularly acute

for heavier BH mergers, where only a short part of the inspiral is detected.

This suggests that constraints could be strengthened significantly in most cases if one had complete, theory-specific, waveforms. However, performing stable and accurate numerical simulations that would produce such waveforms requires an understanding of several complex issues. Probably the first among them is the well-posedness of the system of equations that describe evolution of a given alternative scenario to GR. In the next section we briefly describe the issue of well-posedness and discuss some techniques that have been used so far to obtain a well-posed evolution system in alternative theories.

In Sec.3.4.3 we overview some recent numerical studies of nonlinear evolution beyond

GR.

3.4.1. Initial value formulation and predictivity beyond GR Modelling the evolution

of a binary system for given initial data is a type of initial value problem (IVP). An IVP is well-posed if the solution exists, is unique and does not change abruptly for small changes in the data. A theory with an ill-posed IVP cannot make predictions. The IVP

is well-posed in GR [1577] but it is not clear if this is true for most of its contenders. This

is a vastly overlooked issue and a systematic exploration of the IVP in many interesting

alternative theories, such as those discussed in Sec. 2.1, is pending.

A class of alternative theories of gravity that are known to be well-posed is scalar-

tensor theories described by action (21). As discussed in Sec. 2.2, after suitable field

redefinitions, these theories take the form of GR plus a scalar field with a canonical kinetic term and potential non-minimal couplings between the scalar and standard model

fields, see action (22). Since these couplings do not contain more than two derivatives,

one can argue for well-posedness using the known results for GR and the fact that lower-order derivative terms are not relevant for this discussion. Interestingly, most alternative theories actually include modifications to the leading order derivative terms in the modified Einstein’s equations, so a theory-specific study is necessary. This has been attempted in very few cases and results are mostly very preliminary. In particular, there

is evidence that dynamical Chern-Simons gravity is ill-posed [1110]. Certain generalised

scalar-tensor theories appear to be ill-posed in a generalised harmonic gauge when

linearised over a generic, weak field background [1578]; however, note that this result is

gauge-dependent, and hence not conclusive. Finally, in certain Lorentz-violating theories that exhibit instantaneous propagation, describing evolution might require solving a mixture of hyperbolic and elliptic equations (where the latter are not constraints as in

3.4.2. Well-posedness and effective field theories One is tempted to use well-posedness as a selection criterion for alternative theories of gravity, as a physical theory certainly needs to be predictive (in an appropriate sense). However, alternatives to GR can be thought of as effective field theories — truncations of a larger theory and hence inherently limited in their range of validity. This complicates the question of what one should do when a given theory turns out to be ill-posed. In fact, it even affects how one should view its field equations and the dynamics they describe in the first place. Effective field theories (EFTs) can often contain spurious degrees of freedom (e.g. ghosts) that lead to pathological dynamics. In linearised theories it is easy to remove such degrees of freedom and the corresponding pathologies, but there is no unique prescription to doing so in general. Hence, instead of setting aside theories that appear to be ill-posed when taken at face value, perhaps one should be looking for a way to ‘cure’ them and render them predictive at nonlinear level.

A very well-known EFT, viscous relativistic hydrodynamics, requires such ‘curing’ to control undesirable effects of higher order derivatives and a prescription for doing

so has been given long ago [1579–1581]. A similar prescription applicable to gravity

theories has been given recently [1582]. Roughly speaking, this approach treats the

theory as a gradient expansion and, hence, it considers as the cause of the pathologies runaway energy transfer to the ultraviolet, that in turn renders the gradient expansion inapplicable. As a results, it attempts to modify the equations so as to prevent such transfer and to ensure that the system remains within the regime of validity of the effective descriptions throughout the evolution.

Another approach is to consider the theory as arising from a perturbative expansion is a certain parameter, say λ. If that were the case, then higher order corrections in λ have been neglected and, consequently, the solutions can only be trusted only up to a certain order in λ. Moreover, they have to be perturbatively close (in λ) to solutions with λ = 0, which are solution of GR. Hence, one can iteratively generate fully dynamical solutions order-by-order in λ. This process is effectively an order-reduction algorithm and yields a well-posed system of equations. It has a long history in gravity theories in

different contexts (e.g. [1583–1585]), but it has only recently been used for nonlinear,

dynamical evolution in alternative theories [1108, 1586, 1587].

These two ‘cures’ do not necessarily give the same results. Moreover, there is no reason to believe that there cannot be other ways to address this problem, so this remains a crucial open question. In principle, the way that an EFT is obtained from a more fundamental theory should strongly suggest which is the way forward when considering nonlinear evolution. However, in practice one often starts with an EFT and hopes to eventually relate it to some fundamental theory. Hence, it seems wise to consider all possible approaches. In principle different theories might require different approaches. 3.4.3. Numerical simulations in alternative theories As mentioned above, one can straightforwardly argue that the IVP is well-posed in standard scalar-tensor theories

this class of theories are generically identical to GR and carry no scalar configuration. Hence, even though gravitational radiation in these theories can in principle contain a

longitudinal component, it is highly unlikely it will get excited in a BBH. In Sec. 4.2

we outline the conditions under which BHs can differ from their GR counterparts in

standard scalar-tensor theories.∗ When these conditions are satisfied there should be

an imprint in GWs from BH binaries. It is worth highlighting here a couple of cases where numerical simulations have to be used to address this question. The first has to do with the role of asymptotics. It has been shown that time-dependent asymptotic

for the scalar could lead to scalar radiation during the coalescence [1107], though this

emission would probably be undetectable for realistic asymptotic values of the scalar field gradient. The second case has to do with whether matter in the vicinity of a BH can force it to develop a non-trivial configuration. This has been shown in idealised setups

only in Refs. [1555, 1556]. However, numerical simulations for the same phenomenon in

NSs have been performed in [1226, 1227, 1397].

f(R) theories of gravity (see Ref. [1588] for a review), which are dynamically

equivalent to a specific subclass of scalar-tensor theories, are also well-posed [1588,1589].

A comparative study of NSB mergers in GR with those of a one-parameter model of

f (R) = R + aR2 _{gravity is performed in [1590].}

A well-posed extension of scalar-tensor theories is Einstein-Maxwell-Dilaton gravity. It has its origin in low energy approximations of string theory and it includes, apart from the metric and a scalar, a U(1) gauge field. The scalar field couples exponentially to the gauge field, allowing for deviations from GR even for BHs with asymptotically constant scalar field. BBH simulations have shown that these deviations are rather small

for reasonable values of the hidden charge [1513], leading to weak constraints on the free

parameters of the theory.

Finally, some first numerical results have recently appeared in scalar-Gauss–Bonnet

gravity, and specifically the theory described by action (30), and in Chern-Simons gravity

(see Sec. 2.2for more details on the theories). Refs. [1586, 1587] studied scalar evolution

in scalar-Gauss–Bonnet gravity and Ref. [1108] performed the first binary evolution in

Chern-Simons gravity. These results are notable for using a perturbative expansion in the free parameter of the theory, described in the previous section, in order to circumvent potential issues with well-posedness (as discussed above, if the field equation are taken

at face value, well-posedness is known to be an issue for Chern-Simons gravity [1110]

and it might be an issue for scalar-Gauss–Bonnet gravity [1578]).

4. The nature of Compact Objects