finite numerical resolution (cf. Baumgarte and Shapiro 142). To quantify
numerical resolution errors, simulations are performed for a series of
increasing numerical resolutions, checking for convergence of the resulting
gravitational waveforms. The error for a given waveform is reported as the
mismatch between the waveform and the waveform from the next-lowest resolution
simulation (see, e.g., the methods in Boyle et al. 244 for technical details).
For gravity theories which admit a well-posed initial value problem no
approximation to the theory is necessary, and hence numerical accuracy is
still the biggest source of systematic errors. For example, this is the case
in the recent evolutions in EdM [649], which admits a well-posed initial value
problem, and in EsGB [484], where a well-posed modified Harmonic formulation
of the theory is adopted.

BBH simulations beyond-GR, such as in dCS gravity [927] and Einstein dilaton
Gauss–Bonnet gravity [923] have numerical resolution errors comparable to
those of GR (see the convergence analyses in Witek et al. 1202, Okounkova et
al. 925, 927), and thus the codes to generate these waveforms must improve in
the same way as GR codes in order to be ready for LISA. However, in addition
to numerical resolution errors (comparable to those of GR), present beyond-GR
NR simulations have additional sources of systematic errors. In order to
ensure a well-posed initial value problem (which may not exist in EdGB [959,
958, 1025, 1024] and dCS [445]), BBH mergers in quadratic gravity are
performed in an order-reduction scheme, where the spacetime metric is expanded
about a GR BBH merger spacetime, and one computes the leading-order beyond-GR
correction to the metric and gravitational waveform perturbatively. One then
obtains the ‘full’ gravitational waveform by combining the background GR
metric with the leading-order correction, for a suitable choice of EdGB or dCS
coupling constant. The order-reduction scheme operates under the assumption
that corrections to GR are small, and that higher-order corrections (such as
the second-order correction to the spacetime metric) are negligible. This
assumption governs an instantaneous regime of validity, where there is a
maximum allowed value of the coupling constant that is allowed in order for
the corrections to the spacetime metric to be smaller than the background
metric (see Okounkova et al. 926 for technical details). Thus, the NR
simulations performed in this scheme are not valid for all possible physical
values of coupling constant.

Additionally, the order-reduction scheme introduces systematic errors through
secular dephasing. Since the location of the BHs in the perturbative scheme is
governed by GR (with no back-reaction from the beyond-GR fields), the phase of
the BBH inspiral in the order-reduction scheme is different from that of the
full theory, hence leading to secular dephasing. This is a common feature of
perturbative approximations to dynamical systems [166], and in particular, PN
approximations [1198] and self-force BH perturbation theory [986] encounter
these challenges when applying perturbative approaches to long-duration
inspirals. Hence, the present beyond-GR waveforms focus primarily on the
merger (the phase that’s crucial to numerically simulate for tests of GR
[1235]) and ringdown phases, ensuring that the beyond-GR simulation starts
late enough so that the secular systematic errors are negligible [927, 923].
Finally, the extraction of GWs themselves is another potential source of
systematic errors. For example, it is unclear whether the Newman–Penrose
scalars that are commonly adopted in GR, have the same physical interpretation
in modified gravity.

Another modified theory where some work has been done is Moffat’s tensor-
vector-scalar theory [867]. The theory admits a well-posed initial value
problem, but for sufficiently small scales, it becomes mathematically
equivalent to Einstein–Maxwell theory in vacuum, where BHs carry a
“gravitational" (not electric) charge, that is determined by the theory’s
coupling constant. First simulations in this context have been performed by
Bozzola and Paschalidis [245]. However, a systematic error in this case is
whether the approximation of the theory as Einstein–Maxwell in vacuum holds
true. Given that the theory admits a well-posed initial value problem, it is
straightforward to move beyond this approximation.

Moving forward one would like to obtain NR beyond-GR mergers that are free of
these systematic effects by being valid for all values of coupling constants
and avoiding secular dephasing. In order to mitigate dephasing growth, one can
‘stitch’ together results for simulations with different starting points of
beyond-GR effects (hence different points of zero dephasing), as discussed for
the case of EMRIs in Pound and Poisson [985], Warburton et al. [1185].
Similarly, one can apply the multiscale and dynamical renormalization group
methods of Kunihiro [764], Galley and Rothstein [550], Hinderer and Flanagan
[644], Pound [984] to get rid of secular dephasing effects. This is an active
area of work.

In order to produce beyond-GR BBH waveforms valid for all values of coupling
parameter, one must move beyond perturbative methods and consider simulating
BBHs in beyond-GR theories with well-posed initial value problems. Such
approaches have been discussed using methods from fluid dynamics in Allwright
and Lehner [51], Cayuso et al. [330]. In particular, Witek et al. [1203]
recently derived a 3+1 split for the (non-perturbative) equations of motion of
EdGB in a step towards NR simulation.

####  VII.4.2 PN waveforms

Incomplete inspiral waveforms in theories beyond GR may give rise to
systematic errors in tests of GR with GWs. The inspiral waveform is computed
via the PN method. In most non-GR theories, only the leading PN correction has
been computed, which can be mapped to the PPE framework (see Sect. VI.2.1).
However, there are certain theories in which corrections to the waveform have
been computed also to higher PN orders [1234, 1063, 1239, 1080, 1081]. One
such example is the Brans–Dicke theory, which is one of the simplest forms of
scalar-tensor theories. The effect of higher PN corrections in this theory to
constrain the theoretical coupling constant (Brans–Dicke parameter) has been
studied in Yunes et al. [1235]. The authors showed that the bound on the
Brans–Dicke parameter with GW150914 only changes by $\sim 10\%$ when one
includes higher PN corrections or not. This shows that the higher-order
corrections are not important when constraining Brans–Dicke theory, though
there may be other non-GR theories where sub-leading PN corrections become
important, and thus deriving such corrections is an important direction for
future research (see Sect. VI.4).

####  VII.4.3 Effects of bosonic dark matter

The effects of bosonic dark matter in the context of GR are discussed in Sect.
IV. Here we discuss effects related to modified gravity.

Ultralight bosonic dark matter around spinning BHs can trigger superradiant
instabilities, forming long-lived bosonic condensates outside the horizon
which can alter the inspiral GW signal. A number of modified gravity theories
with additional (scalar and/or vector) degrees of freedom exhibit phenomena
such as dynamical scalarization (see e.g., Benkel et al. 167, Witek et al.
1202, Ripley and Pretorius 1024, East and Ripley 484, Berti et al. 187 for
Einstein-Scalar-Gauss–Bonnet Theories and Hirschmann et al. 649, Herdeiro et
al. 627, Fernandes et al. 512 for Einstein-Scalar-Maxwell theories), where BHs
in those theories spontaneously or dynamically acquire scalar hair. How BHs in
modified gravity behave around ultra-light bosonic dark matter environments,
and what new effects could arise as well as our ability to place constraints
on deviations from GR and/or on dark matter has not yet been explored.

####  VII.4.4 EMRI waveforms

With EMRI signals, we expect to be able to detect higher order-independent
multipole moments of the central body [1032, 1033] and measure (if any)
deviations from GR [122]. Most of the work in this direction, has so far
considered BH spacetimes with a multipolar structure different from Kerr, such
as bumpy BHs [361], and constructed approximate “kludge" waveforms [581, 351]
generated considering geodesics in a perturbed Kerr spacetime with orbital
parameters evolved using PN equations that assume radiation-reaction effects
as in GR [578, 107, 547, 74, 874].

This approach reproduces the orbit’s main features but not to the precision
required to determine that the inspiral is indeed an inspiral into a Kerr BH
or not [1051, 547]. Thus, further work on EMRI waveform modelling will require
the consistent inclusion of radiation reaction not described as in GR,
additional parameters related to the modification from GR, and environmental
effects. Note that the presence of environmental effects may limit our ability
to perform proper parameter estimation for some events [119, 118] if these are
not properly taken into account.

These enhanced waveforms will increase model degeneracies that will impact our
ability to learn about fundamental physics in the strong-field regime of
gravity. That is why it is paramount to fully understand possible discernible
features, such as characteristic variations of the amplitude and the energy
emission rate [1114] or the appearance of prolonged resonances [74] in EMRI
signals, as well as the development of novel data analysis techniques to
extract information.

###  VII.5 Burning Questions and Needed Developments

In this chapter we have discussed how different systematics in the modelling
of waveforms and induced by astrophysical environment can affect tests of GR.
Below, we summarize the most important salient points.

  * •

How does (residual) eccentricity induced by binary-disk or hierarchical
triplet interactions limit our ability to perform fundamental physics tests
with BH binaries?

  * •

Can accretion, plasma effects or other stellar compact objects in the vicinity
of an EMRI induce observable changes in the GW frequency evolution during the
inspiral and/or ringdown that can spoil fundamental physics tests?

  * •

Self-force calculations for generic BBHs in vacuum or embedded in a background
(e.g. dark matter boson cloud) at second-order are necessary for proper
modelling of EMRIs, so that reliable waveforms are available to test for
fundamental physics.

  * •

What PN accuracy is required for supermassive comparable-mass BH binaries so
that we can perform precision tests of fundamental physics with LISA?

  * •

Development of PN waveforms in beyond-GR theories may need to move beyond
leading order corrections. The order to which one should go to constrain
theories should be investigated.

  * •

What is the accuracy level that numerical relativity simulations must reach in
order to produce high-fidelity waveforms for LISA?

  * •

Numerical relativity must push through existing non-convergence problems to
achieve high-fidelity waveforms near merger. Moreover, code efficiency and
scaling must increase considerably to achieve resolutions necessary to supress
systematic errors and to achieve template indistinguishability.

  * •

Phenom and EOS models must be improved and calibrated against numerical
relativity simulations through the entire parameter space of BBHs, and include
eccentricity, arbitrary spins, and substantially different masses. Efforts
must be made to include environmental effects such as dynamical friction or
interactions with clouds of ultralight fields. Additionally, Phenom and EOB
models must be extended to modified gravity.

  * •

Numerical relativity efforts must expand to include more modified theories of
gravity (especially those that admit a well-posed initial value problem).
Efforts must be made to understand systematic errors of existing approaches
that tackle ill-posed modified gravity theories.

  * •

A combination of the effects of bosonic dark matter and modified gravity must
be considered in order to be able to understand how more complex deviations
from standard general relativity can take place.

  * •

Development of waveforms for EMRIs in modified gravity must take place. In
particular, further work on EMRI waveform modelling requires the consistent
inclusion of radiation reaction in beyond-GR theories, additional parameters
related to the modification from GR, and any environmental effects.

  * •

An important potential source of systematics that has not been included in
this section (because no work has been done on this yet), but that is surely
important to ensure the robustness of tests of GR are instrumental
systematics. These are uncertainties due to calibration errors, data gaps, or
other issues with the instrument itself.

##  VIII Future Directions

This white paper has been focused on the fundamental physics we can extract
from LISA data. The white paper first discussed tests of the theoretical
pillars of GR, such as the speed of propagation of GWs and the GW memory. The
white paper then moved on to tests of the nature of BHs, discussing
constraints on deviations from the Kerr hypothesis and other tests. Next, the
white paper summarized tests of GR related to fundamental questions in
cosmology, such as tests probing models that attempt to explain dark matter
and others that attempt to explain dark energy (as an alternative to a
cosmological constant). The white paper then concluded with a discussion of
generic tests of GR, and the effects of systematics in all of these tests due
to mismodelling and the influence of astrophysical environments.

As we hope this white paper demonstrates, there is a tremendous amount of work
yet to be done before LISA flies in order to fully realize and exploit LISA’s
scientific potential in this area. The most pressing questions have been
discussed at the end of each of the sections summarized above, but let us here
touch on a few of the key points. Regarding tests of GR, the most burning
questions all involve the development of more detailed models for the GWs
emitted in specific modified theories, including not just for the inspiral
phases, but also the merger and the ringdown for comparable-mass binaries and
EMRIs.

Much more work is also still needed in the context of LISA tests on the nature
of BHs. Of particular note is the development of a systematic method to carry
out Kerr hypothesis tests through the extraction of multipole moments of SMBHs
in binaries. Moreover, the development of accurate waveform models to study
the effect of ultralight bosonic fields around SMBHs, as well as the inspiral,
merger and ringdown of exotica is needed. Regarding the latter, further
studies to determine whether all exotica are a priori possible or whether some
theoretical restrictions should be imposed is also important to limit the
number of possibilities to study.

The extraction of fundamental physics information related to cosmology with
LISA data will also require that we work hard to address several burning
questions. Perhaps one of the most important ones is the construction of
accurate waveform models that incorporate the effect of ultralight and heavier
dark matter fields in the inspiral, merger and ringdown of compact objects of
various mass ratios. Another important aspect that needs further studying is
the effect of screening in the generation of propagating GWs, which would be
particularly important when attempting to constrain modified gravity dark
energy models.

Let us now discuss the burning questions related to model-independent tests of
GR. As discussed in Sect. VI, perhaps the most important action item here is
the development of model-independent modifications to waveform models in all
three phases of coalescence. The merger phase, in particular, is not well-
understood in enough modified theories to allow for a model-independent
parameterization. Another important action item is related to the extension of
LIGO/Virgo tests [14] to the LISA realm, such as the residual test and
polarization tests. These tests will be different with LISA because LISA will
be sensitive to a large number of sources that will be on during the entire
observation, and because of the motion of LISA around the Sun.

Perhaps one of the most important points to consider is to what extent tests
of GR will be affected by waveform systematics or other systematics that could
be induced by astrophysical environments. Waveform systematics will be
particularly problematic for SMBH signals, because of their expected high
SNRs. But this could be a problem even for intermediate mass-ratio systems,
for which we may not have sufficiently accurate generic waveforms yet.
Waveform systematics may also be a problem for EMRIs, although one expects
second-order waveform calculations to be complete before LISA flies. However,
astrophysical systematics may still be a particular problem for EMRIs,
creating a floor to our ability to test GR.

Given all of this, it is hopefully clear that a great amount of work is still
needed to extract the most fundamental physics from LISA data and to ensure
such inferences are robust. A close collaboration between the LISA fundamental
physics WG, and other WGs, such as the waveform modelling and the cosmology,
is called for and will be paramount to successfully overcome all these
difficulties. We remain nonetheless optimistic that through the infrastructure
of the LISA Consortium these collaborations can be organized and structured,
so that we can get the most science out of the data, when LISA flies.

## Acknowledgments

E. Berti is supported by NSF Grants No. PHY-1912550 and AST-2006538, NASA ATP
Grants No. 17-ATP17-0225 and 19-ATP19-0051, NSF-XSEDE Grant No. PHY-090003,
and NSF Grant PHY-20043. D. Blas is supported by a ‘Ayuda Beatriz Galindo
Senior’ from the Spanish ‘Ministerio de Universidades’, grant BG20/00228. IFAE
is partially funded by the CERCA program of the Generalitat de Catalunya. The
research leading of to these results has received funding from the Spanish
Ministry of Science and Innovation
(PID2020-115845GB-I00/AEI/10.13039/501100011033). K. Clough is supported by
funding from the European Research Council (ERC) under the European Unions
Horizon 2020 research and innovation programme (grant agreement No 693024). A.
Cárdenas-Avendaño acknowledges funding from the Fundación Universitaria Konrad
Lorenz (Project 5INV1) and from Will and Kacie Snellings. M. Crisostomi and E.
Barausse are supported by the European Union’s H2020 ERC Consolidator Grant
“GRavity from Astrophysical to Microscopic Scales” (Grant No. GRAMS-815673).
P. Fleury received the support of a fellowship from “la Caixa” Foundation (ID
100010434). The fellowship code is LCF/BQ/PI19/11690018. C. Herdeiro thanks
the support of the Center for Research and Development in Mathematics and
Applications (CIDMA) through the Portuguese Foundation for Science and
Technology (FCT - Fundação para a Ciência e a Tecnologia), references
UIDB/04106/2020 and UIDP/04106/2020, the projects PTDC/FIS-OUT/28407/2017,
CERN/FIS-PAR/0027/2019, PTDC/FIS-AST/3041/2020 and the European Union’s
Horizon 2020 research and innovation (RISE) programme H2020-MSCA-RISE-2017
Grant No. FunFiCO-777740. P. Pani and E. Maggio acknowledge financial support
provided under the European Union’s H2020 ERC, Starting Grant agreement no.
DarkGRA–757480, and under the MIUR PRIN and FARE programmes (GW-NEXT, CUP:
B84I20000100001), and support from the Amaldi Research Center funded by the
MIUR program "Dipartimento di Eccellenza" (CUP: B81I18001170001). N.Frusciante
was supported by Fundação para a Ciência e a Tecnologia (FCT) through the
research grants UIDB/04434/2020, UIDP/04434/2020, PTDC/FIS-OUT/29048/2017,
CERN/FIS-PAR/0037/2019, the FCT project “CosmoTests – Cosmological tests of
gravity theories beyond General Relativity" with ref. number
CEECIND/00017/2018 and the FCT project “BEYLA –BEYond LAmbda" with ref. number
PTDC/FIS-AST/0054/2021. L.Lombriser was supported by a Swiss National Science
Foundation Professorship grant (No. 170547). S.N. acknowledges support from
the Alexander von Humboldt Foundation. D.N. acknowledges support from the NSF
Grant No. PHY-2011784. R.B. acknowledges financial support from FCT – Fundação
para a Ciência e a Tecnologia, I.P., under the Scientific Employment Stimulus
- Individual Call - 2020.00470.CEECIND. V. Paschalidis acknowledges support
from NSF Grant PHY-1912619 and NASA Grant 80NSSC20K1542 to the University of
Arizona. B.S.S. is supported by NSF grants No. AST-2006384 and PHY-2012083.
C.F.S. is supported by contracts ESP2017-90084-P and
PID2019-106515GB-I00/AEI/10.13039/501100011033 (Spanish Ministry of Science
and Innovation) and 2017-SGR-1469 (AGAUR, Generalitat de Catalunya). T. P. S.
acknowledges partial support from the STFC Consolidated Grant No.
ST/P000703/1. M. Ruiz acknowledges support from NASA Grant 80NSSC17K0070 to
the University of Illinois at Urbana-Champaign. I.D. Saltas is supported by
the Czech Science Foundation GAČR, Grant No. 21-16583M. N. Stergioulas is
supported by the ESA Prodex grant PEA:4000132310 "LISA Stochastic Signals
Analysis Pipeline". F.V. acknowledges partial support from CNES. K.Y.
acknowledges support from NSF Grant PHY-1806776, NASA Grant 80NSSC20K0523, a
Sloan Foundation Research Fellowship and the Owens Family Foundation. K.Y.
would like to also acknowledge support by the COST Action GWverse CA16104 and
JSPS KAKENHI Grants No. JP17H06358. N. Yunes acknowledges support from NASA
Grants No. NNX16AB98G, 80NSSC17M0041 and 80NSSC18K1352, NSF Award No. 1759615,
and the Simons Foundation through MPS Award Number 896696. D.D. acknowledge
financial support via an Emmy Noether Research Group funded by the German
Research Foundation (DFG) under grant no. DO 1771/1-1.

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