This document summarizes a lecture on binary black holes and tests of general relativity. The lecture covers topics from current gravitational wave detections, including parameters inferred from detections like GW150914, GW151226, and others. It discusses using observations of binary black hole mergers to test predictions of general relativity like the existence of an innermost stable circular orbit and properties of the final black hole like its mass and spin.

1. Binary Black Holes & Tests of GR
Luis Lehner
Perimeter Institute for Theoretical
Physics

2. Summary of lectures’ philosophy
• BBHs, will discuss some topics from current detections, but starting
from a few years before them and going through them
• Covering just a partial (limited, biased?) topics. Philosophy will be to go
in-and-out of GR, drawing lessons from it and contrasting what may be
out there
• Some specific examples will be mentioned, these only reflect what I
know (to various degrees of depth)
• Will also reflect my personal fears, struggles and ideas of what could be
done as we move forward
• Unless asked, I will not be describing ideas/steps of how to do
‘numerical relativity’ [feel free to ask though]
• Warning! I’ll be jumping back and forth, do ask questions! I do not know
what you know/don’t know

3. General Relativity
• Theory of gravity based on fundamental principles:
diffeomorphism invariance + massless spin-2 field GR
(at the linear level. e.g. Weinberg)
• Fundamental ingredients:
– relativity: no special frame
– equivalence: inertial effects indistinguishable from gravitational
ones (e.g. equivalence between inertial and gravitational mass)
– covariance: equations invariant under spacetime
diffeomorphism
– causality: each point admits a notion of past, present and future

4. Where tested ?
[figure from Yunes,Yagi,Pretorius’16]

5. Why bother?
• UV regime: singularities, nonrenormalizable
• IR regime: need for dark matter, dark energy
• Hierarchy: why is gravity so much ‘weaker’ than
EM?
• How to try look for alternatives (better called
‘extension’ perhaps)
– An option: Effective field theory:
• consider a hierarchy of corrections and different scales
[what scale?], with GR being the lowest order of the
hierarchy

6. A possible path…
• Can one consider something else beyond ܵ = ‫׬‬ ܴ ܸ݀ ?
• Lovelock theorem: ‘in d=4, the only divergence free
symmetric rank-2 tensor constructed solely from the
metric (and derivs up to 2nd order) and preserving
diffeomorphism invariant is the Einstein tensor plus a
cosmological term’ (see also Cartan ‘20s)
• Thus… one should: give up 2nd order (Lec 4!), add
further fields, give up a symmetry, give up d=4, give up
gauge invariance via Stuckelberg trick restored it with
further fields…[all options pursued, we’ll discuss some]

7. ‘Stuckelberg trick’
[Renaux-Petel ‘15]

8. Higher dimensions?
• Can analyze this option already at Newtonian levels,
• ߮ ~ ‫ݎ‬ଷିௗ
• But angular momentum barrier ~ r -2
no circular orbits allowed in d>4. No Kerr bound in d>5
(and delicate in d=4); cosmic censorship generically
violated [LL-Pretorius ‘10 – Figueras+ ‘17] )
must make other dimensions ‘special’ in some way
(branes, small size, compactified with flux stabilitization,
etc)

9. Giving up Lorentz invariance
[preferred time direction]
Horava-Lifschitz theory. L4 , L6 include all foliation preserving diffeo
invariant scalars functions of ai , hij of 4th and 6th order derivatives.
Einstein-Aether theory. ܽ௜ = ߲௜ln ሺܰሻ

10. Simplest additional field. One scalar
• ܵ = ‫׬‬ሺ‫ܨ‬ሺܴ, ∎ܴ , ∎ଶ
ܴ … ߮ሻ − ݁/2߮,௔
߮,௔ + ‫ܮ‬ெሻܸ݀, [e.g.
DeLaurentis Phys Rep 509,2011]
• Ex1: ‫ܨ‬ = ‫ܨ‬ ߮ ܴ − ܸ ߮ ; ݁ = −1
– ‫ܨ‬ ‫ܩ‬௔௕ = −
ଵ
ଶ
ܶ௔௕
ఝ
− ݃௔௕∎߮ + ‫ܨ‬;௔௕
– ∎߮ = ܴ‫ܨ‬,ఝ − ܸ,ఝ
• Ex2: F=f(R), e=0
• ‫ܩ‬௔௕ = 1/݂ᇱሺ݂;௔௕
ᇱ
− ݃௔௕∎݂ᇱ +
௚ೌ್ ௙ି௙ᇲோ
ଶ
ሻ
(prime denotes differentiation wrt R)

11. How different?
• Recall, a conformal transformation ݃෤௔௕ = ݁ଶ௪݃௔௕
– (maintains angles, and relations between modules of vectors)
– Γ෨௔௕
௖
= Γ௔௕
௖
+ ݃ௗ௦ ‫ݓ‬,௔݃௕ௗ + ‫ݓ‬,௕݃௔ௗ − ‫ݓ‬,ௗ݃௔௕
• Thus
– ܴ෨௔௕ = ܴ௔௕ − 2‫ݓ‬;௔௕ + 2‫ݓ‬,௔‫ݓ‬,௕ − ݃௔௕∎‫ݓ‬ − 2݃௔௕‫ݓ‬,௖‫ݓ‬,௖
– ܴ෨ = ݁ିଶ௪
ܴ − 6∎‫ݓ‬ − 6‫ݓ‬,௖‫ݓ‬,௖
• Eg, for f(R), choose w=1/2 ln(f’), ߮ = ‫ݓ‬ 6 and action is
– ݃෤݂ ܴ = −݃෤ሺ−
ோ෨
ଶ
+
ଵ
ଶ
߮,௔߮,௕݃෤௔௕
− ܸሻ
So, equivalent to EEs coupled to a scalar field with a specific
potential standard GR results apply! (e.g. no hair!) . In
vacuum…coupled to matter fields is another story

12. • So much for a path to the unknown, let’s come back to
‘less speculative land for a while’. aLIGO/VIRGO, EHT, etc
are probing compact systems, what do we use to
explore them? [and draw lessons for exploring
elsewhere]
• Black holes: described in GR by the Kerr solution: unique
stationary, axisymmetric spacetime no hair theorem
– Particularly relevant here: existence of ISCO, light ring and
ergosphere
– Maximum amount of energy extractable 29%M (HW!)
– At the linear level, massless perturbations of BH are to a large
degree) described by QNMs, ℎ~݁௜௪௧
‫ݐ݅ݓ‬ℎ ‫ݓ‬ = ‫ݓ‬ோ + ݅‫ݓ‬ூ,
which depend on mass, spin and are functions of (l,m,n) [Berti
etal review]. Measurement of 1 w, fully determines all others!

13. • Black holes in astrophysics: as key ingredients of
AGNs, GRBs, modulating galaxy behavior….
• But much beyond… : holography (AdS/CFT),
CMT, quantum information, turbulence, chaos….

14. • perturbative approach (PN, PM, EOB…)
– Recast EEs with (v/c) and (M/L) as ‘smallness parameters’
– Internal structure effects ~ k (R/M)5 (v/c)10 [Damour]
– … BHs have k=0 [is this ‘special’ ? ]
– The above may change outside GR, e.g. ST (v/c)6 !
– Not good convergence properties as (v/c)~1, (M/L)~1…
resummation approaches help [Damour-Buonanno +..]
• Point-particle arguments:
– At m2/m1 0, test particle on a BH background [we know
this!, E, M + Carter constant determine orbits]
– Can ‘kludge’ waveforms [Hughes+], ‘adiabatically’ changing
parameters
– To leading order, a BH with mass m1 , spin a1 will have a
‘merger’ (or plunge) at higher freqn for higher spin!

15. consequences/observations…
• Light ring freqn associated to
frequency of QNM [but see Price-
Khanna ‘16]
• While it sounds ‘cute’… no real
‘hangup’ effect
• Different BHs (i.e. in different
theories) might have different
ISCO/LR properties, could be used
for testing them
• Measurement of spins!
• BH shadow, structure
of null geodesics and
accretion physics [D. Psaltis and A. Broderick.]

16. Predicting the ‘basic’ behavior
• In GR, simulations show a rather simple behavior
(and aLIGO supports this with whatever the true
gravitational theory is!)
– Are there any surprises?
– Can we map-out what to expect?
• Possible options:
– Run & run with different theories [but, this might not be
feasible, realistic, or even possible (TBD)]
– Fit & fit [direct fits [Campanelli et al], symmetry-based fits
[Boyle-Kesden-Nissanke], ‘semi-analytical’ fits [Rezzolla et al] ]
Anything else?

17. If simple behavior, and a simple way
to predict what to expect… then
there has to be a simple
explanation.
Which in turn might give us other
clues on the system…

18. Rules of the game…
• Initially, the system has much angular momentum
– Energy/Orbital ang momentum is radiated circularizing & tightening the orbit
– Keep individual angular momentum constant [PN backs this up]
• The ‘bare mass’ of the system is ~ M = m1 + m2 …
– Keep it constant till they get close enough
• What is close enough?
– For a particle, it makes sense to talk about the ISCO
– Push that concept to general cases. Where is the ISCO? determined by a
final Kerr black hole with spin aF
• Beyond ISCO, keep mass and ang. momentum constant , radiation is
just a small percentage (even when they’re the largest!)
• Put all together for angular momentum balance (related to
[Blandford-Hughes]):
Lf(aF,M) = Lorb(rISCO,aF,M) + L1(a1,m1) + L2(a2,m2)
[Buonanno,Kidder,LL 08,
also Kesden]

19. Max individual spin, arb mass; Arb spin; arb mass
No initial spin, arb mass (vs EOB, numeric) ; initial spin, equal mass (vs EOB, numerical)

20. Final 0 spin? Spins s1, s2=αs1 ; equal mass
Equal mass, arbitrary orientations

21. • Thus, a possible ‘recipe’ for exploring different theories
(while full solns are determined) is: for each theory…
– Are there BHs?
– If so, are they stable? [recall stability of Kerr not yet fully
established in GR!]
– What’s the ISCO? ( plunge frequency, final mass/spin)
– What’s the LR? ( final ringdown characteristics)
– Examples: Wei-Liu for MOG (’18); Jai-akson + for EMD (‘17)
(and with an eye to GW analysis, see Glampedakis+ ‘17)
Of course, this is for a ‘rough first pass’, but can we afford
anything else?

22. Pressing to general case: Anatomy of a
binary merger (in GR)
4 stages: Newtonian, inspiral, plunge/merger, after-merger
Newtonian: tM < tH : other physics is needed to induce merger:
dynamical friction, n-body encounters, etc.
Inspiral: energy/ang. mom. Loss through GWs is the dominant
mechanism.
Perturbation techniques. Rely on: separation of scales! (v/c), M/R, etc

23. Perturbative to nonlinear and back
• During merger, v/c ~ 1 and objects have M/R ~ 1
Full solutions required, and in turn numerical
simulations
• Access the truly non-linear regime of GR
"Using a term like nonlinear science is like
referring to the bulk of zoology as the study of
non-elephant animals." (Stanislaw Ulam)

24. • Merger/plunge:
– 2 black holes merge into one if cosmic censorship holds.
– 2 NS will form another one which may collapse to a BH
– BH-NS. The BH will disrupt or swallow the NS depending on
typical radii involved
• After merger: use BH perturbation decaying
oscillations

25. Anatomy of ‘theoretical’ BBH signal
Energy radiated ~ 3- 12 % of total mass

26. What’s the possible outcome? (sGRB motivated)
Low spin/high mass,
small radius direct
plunge.
No sGRB, but could
still shine?
BHNS: High spin/low mass, large radius
disruption.
NSNS: Mtot > 1.3-1.5 Mmax
‘comfortable’ disk mass
GW: with a clear cutoff
NSNS: Mtot < 1.3-1.5 Mmax
GW: postmerger signal
sGRB from ‘sufficiently’
magnetized MNS?

27. ‘indirect impact’ on EM
[Palenzuela + ‘10]

28. But finally…
September 14, 2015

29. GW151226

30. Parameters inferred
Event Prob m1 (MO) m2 (MO) χχχχeff DL (Mpc) Mrad
(MO)
GW150914 > 5.1σ 365
4
(5,-4)
294
-4
(4,-4)
-0.06 0.17
-0.18
(0.17, -0.18)
410160
-180 3
LVT151012 2.1σ 23 13 0.00.3
-0.2 1100500
-500 2
GW151226 > 5σ 14.28.3
-3.7 7.52.5
-2.3 0.2 (…1 with spin) 440180
-190 1
GW170104 ~ 4.5σ 31.28.4
-6 19.45.3
-5.9 -0.120.21
-0.3 880450
-390 2
GW170608 SNR 13 127
-2 72
-2 0.070.23
−0.09 340140
-140 0.85
GW170814 SNR 18 30.55.7
-3 25.32.8
-4.2 0.060.12
-0.12 540130
-210 2.7
• Rate: ~ [12-213] Gpc-3 yr-1
• DM candidate? Still few to make an argument [peak in distribution?]
• Large masses in GW150914 not ‘first bet’ population implication?
• mg < 10-22 ev/c2

31. Black hole puzzles
• Spin?
(superradiance with
axions? [Arvanitaki
-Dimopoulos + ])
• Formation?
• Really a BH?
• Really as in GR/Kerr BH?
[Yang +]

32. How can one test GR?
• GR is a fully predictable theory, and BHs furnish the only
‘fully tractable’ astrophysical compact binary problem.
That is, given physical parameters (at far separations), GR
yields the full solution afterwards at any time (frequency)
• First target ideas
– Are the final spin/mass of BH as expected ?
– Is the inspiral stage behaving as expected from PN
(resummation) expressions?
– Is the final BH ringing down as advertised?
– (are they really BHs?!)
– What’s the extent to which values can be pinned down?
– How do we test for what we do not yet know?

33. Searching for deviations I (so far)
• Pameterize deviations of ‘closed form’ of gravitational waves in
IMR wrt effective parameters and allow them to vary (e.g. LSC
papers)
• Pameterized deviation from ‘stationary phase’ approximation –
informed by some available theories–
(for later: This assumes monotonic behavior…not necessarily true)
[figure: Yagi]

34. GW150914 [LVSC ’16]
• Separating in stages: IMR (stages
windows?)
– Results fully consistent with GR, but
systematics are still rather large
• GW150914 was very special! SNR in
inspiral phase ~ merger-ringdonw
phase
– Even with such a high SNR event, and
regime, final BH ringdown only enabling
fundamental QNM to be measured.
– Conditions for an ‘easier’ test? SNR and
higher strengths of other modes… but
must be aware ‘detector biases’

36. • Further : (i) no ‘weird’ behavior 2 disjoint regions
merged into one [BHs can not bifurcate!]; (ii) Penrose
‘end state conjecture’ is supported; (iii) relaxation is very
short
• Observations can bound ‘tidal defformability’ parameters
up to some value but not yet concluding is zero. could
be a non-black hole that is sufficiently compact? Yes! but
a very constrained object. Suppose it’s a ‘blob’ of
something. We see it relaxing (from GWs) viscosity in
its description values derived at much too high to be :
neutron stars or boson stars [Yunes + ‘16]

37. Continuing…challenges?
• GW150914 was great (high SNR), GW151226 (long!), the
others somewhere in between. We’ve gotten a glimpse
of what can be done per signal. ALIGO/VIRGO will obtain
many events. What else can be done with them?
• Are we searching for deviations in the best possible
way?
• If we detect a deviation? What would it mean?

38. BH? Digging deeper
• Beyond the fundamental mode? –Test of gravity
– Wait for a sufficiently high SNR event. SNR in
postmerger phase for GW150914 ~ 8 unlikely in
ALIGO [Sesana+], realistic (but hard) in ET/CE.
– Combine multiple events
• Posterior combination. SNR ~ N1/4 [Meidam+ ‘14]
• Coherent stacking. SNR ~ N1/2 [Yang + ‘17]

39. Example
• Let’s concentrate on the ‘after-merger’ regime. GR predicts
the signal is dominated by QNMs. For simplicity let’s focus on
the leading (2-2) and one of the sub-leading modes (3-3). The
signal at the detector is:
A key observation is that A, ω and φ are (in theory) known from
the model

40. • Thus, we can consider adding coherently different signals targeting
specific modes/behavior with N events:
1. Pick any given event and define ω33,1 =: ω33 & φ33,1 =: φ33
2. Define aj = ω33,j / ω33 & ∆j =(φ33,j − φ33)/ ω33,j
3. Shift/rescale each sj(t) = sj(t/aj + ∆j)
4. Add them!, s = Σ cj sj
The resulting sum contains a single oscillating frequency ω33 and a
collection of rescaled ω22’s
The rest… are details of Bayes analysis and facing the fact that
parameters have uncertainties.
[Yang,Yagi,Blackman,LL,Pascalidis,Pretorius,Yunes ‘17]

41. Hypothesis testing
• For simplicity work in freqn domain and consider N events
• Hypothesis:
H1 : y:= s-h22 = n + A h33; H2 : y:= s-h22 = n
• PA ~ exp( - Πf 2 |y-Ah33|/Sn |2 ) (Probability function for the 2nd mode to be
present)
• With PA perform a Generalized Likelihood Ratio Test (<h33,y>/|h33|>
γ), obtaining the requirement to favour H1 over H2 & H33 ~ <h33>
(1+offsets)
• With ρcrit related to the false alarm and detection rates and σp the
variance of distribution. For 0.01 and 0.99 respectively ρcrit = 4.65.

42. • Assume uniform merger rate (40 Gpc-3 yr-1)
• For simplicity no spins in individual BHs (in the binary)
and masses in [10-50]MO
• Adopt zero-detuned, high-power noise spectral density
for aLIGO at design sensitivity.
• Distribute events up to z=1
• Use MC for sampling
• 40-65 events with ρ22 > 8
• Without ‘stacking’ 28% chance
• With ‘staking’ 97% chance of detecting 33 mode in 1yr of
observation. [if rate is 13 Gpc-3 yr-1, 12% and 50% instead]

43. • Of course, the idea is more general than this
application
– Dig main mode in low SNR/low mass events
– Dig pre-merger modulations
– Etc.
• Also, if full waveforms are unknown, but
particular features are: ‘incoherent’ stacking
[e.g. without known phase] can still be
implemented [Yang+ ‘18]) ; e.g
– Post-merger oscillations in BNS
– QNMs in extensions to GR
– etc

44. Comments
• We have seen:
– Significant info can be/is obtained with GWs, and
combination of I-M-R is a powerful discerning tool. However,
unless right masses, and right SNR analysis per-event can only
give so much info until LISA, 3G detectors (ET, CE, Voyager…)
– Further info can be ‘squeezed’ from multiple events jointly
analysed. Posterior-distribution multiplication (~no assuming
coherent emission) SNR ~ N1/4 . If only 1 unknown or ‘stacked’
SNR ~ N1/2 if Hypothesis of GR is correct
– Beyond GR? Analysis employing knowledge of expected
deviations can help tremendously (e.g. matched filtering vs
burst search). Does e.g. : PPE; modified parameterization do
the job? Perhaps but it would be nice to know how binaries
should behave beyond GR

45. Where were we?
• Reviewed (some!) options to look beyond GR
• Discussed some basic arguments to gain a coarse
qualitative and (to a degree) some quantitative
understanding of what to expect in some extensions
• Covered some particular features of detections & some
messages/opportunities ahead
• Recalled the importance of templates (or at least info
that can guide data analysis) & some current ideas for
how to check for deviations
• But… we have yet to look into things in detail…

46. Back to merger anatomy
(2000’s déjà vu)
INSPIRAL MERGER POST-MERGER
With the benefits of knowing detectors *do* work, gravitational
waves are ‘catchable’ and, at least so far, no significant deviations
from GR.
RESTRICTION: we’ll discuss theories that have been used to study
compact binaries in full

47. Beyond GR I?
• Restricting to theories known to allow for well-posed problems.
I.e. those where one can show ‫ݑ‬ ܶ ൑ ܽ݁௕௧
|‫ݑ‬ 0 |
• Few options known to be amenable to well defined initial
(boundary) value problems. Examples: Scalar-Vector-Tensor
theories.
Scalar-Tensor (ST) {many incarnations}
Scalar-Vector-Tensor (EMD)

48. • Scalar tensor

49. What’s new? : ‘phase’ transitions
• A non-trivial solution ߮ ് 0 can develop for sufficiently
dense objects and the asymptotic boundary condition on
߮ [Damour-Esposito-Farese ‘96]
• In ‘gravitational terms’ the scalar field endows stars with
a ‘scalar charge’ that introduces several effects:
– Newtonian constant is renormalized Ge ~ G (1+a1 a2)
– Dipolar radiation

50. First GR : NSNS
• No-rescaling of mass possible, though constrained masses
• Recall tidal effects at 5PN (in GR)

51. But now ST
• Dynamical/induced scalarization non-
monotonicity of scalar charges! –pressure on
parameterized deviations--

52. • Differences can be significant but with low SNR and even could be
‘degenerate’ with equation of state variations [but recall PN tests!]
• EM counterparts can be significant aids
[Barausse,Palenzuela,Ponce,LL][Sampson etal]

53. F(R)=R+aR2 (‘turn a potential on’)
[15.5-24] [17.5-24]
[19-24]
[Sagunski,Zhang,+ ‘17]

54. Scalar tensor vector. EMD
• Natural ‘low energy limit’ of string theory (to leading order).
In the Einstein frame, the equations are:
• As opposed to ST case, scalarization can take place even in
the absence of matter (gauge field triggers 2 types of
instabilities ‘charged BH’) [Hirschmann + ‘17]
– Tachyon ‘like’ or superradiance like
– Natural way to give ‘hair’ to the BH
– Scalar field behavior can ‘screen’ BH charge away from BH

55. Black holes (EMD)
• In the absence of matter, scalar
charge ‘induced’ through
coupling with vector field [or
time dependent cosmological
constant]
• Charge largely independent of
asymptotic value of scalar field.
Proportional to α0(Q/M)2 or
Q/M [for small/large coupling]
--behavior interpolates KN to Kerr!
• For small values only subtle
differences in dynamics and
radiation characteristics
[Hirschman,LL,Liebling,Palenzuela, ‘17]

56. [equal mass case]
[unequal mass case q=2/3]

57. • However, Julie ‘18
[Ferrari + ’00]

58. What if one changes the ‘BH’ model?
• We need to understand how ‘alternative to BHs’ would
behave (don’t want to mistake beyond GR with beyond
BHs… can we avoid it?)
• As with extensions to GR many suggestions exist: e.g.
boson stars, fuzzballs, gravastars, …[unless sufficiently
defined to study their nonlinear behavior, let’s affectively
call them:``crapastars’’? J ] [Boson stars are not]
• Boson stars, fully determined model, amenable to study
[Palenzuela + .. ’06, ‘07, ‘17, ‘18]

59. • ߣ = ߪ଴ 8ߨ

61. dynamics
• Merger of boson stars that do not yield a BH gravitational wave
corresponding to fundamental modes of a BS without angular momentum!
Large amounts of ang momentum radiated!
• For boson stars collapsing promptly to a BH gravitational wave
corresponding to QNM of a BH relaxing to equilibrium. Mass/spin are
different from BHBH case if stars are not initially compact enough.
Otherwise inspiral & aftermerger can be the same as in BHBH analog case
• To date, such compactions not achieved, further they are distinguishable
from NS [Sennett+’17]… however this need not be the case for all
potentials…
[Palenzuela + ‘18]

62. What else?
• Beyond ScalarTensor, and a EMD theory, no full IMR for
compact binaries done [extensions to GR]
• Beyond boson stars (and neutron stars) no full IMR
compact binaries done [beyond BHs]
Is this really all?
• Slowly rotating BHs in
EinsteinAether theory
(Barausse-Sotiriou+)
Lorentz violating theory! A further ‘specialized’ field to
‘restore’ gauge invariance

63. • BHs in some quadratic gravity…
EoMs in : Einstein-Dilaton-Gauss-Bonet [Let’s count
derivatives!]

64. • Dynamical Chern-Simons
(Yunes, Yagi, Stein, Pretorius +)

65. • And may be a few others…. But
– Not known if they are stable [in spite of claims to the contrary!]
– unknown QNM spectra
– Traditionally hoped that full dynamical behavior could be
worked out [HOPE is the traditional word here]

66. Why do I worry? PDE theory
• ߲ଶ
‫ݑ‬ + ߲‫ݒ‬ … & ߲ଶ
‫ݒ‬ + ߲‫ݑ‬ {u,v} perturbations propagate
independently
• ߲ଶ
‫ݑ‬ + ߲ଶ
‫ ݒ‬ … & ߲ଶ
‫ݒ‬ + ߲?
‫ ݒ‬ {u,v} affect each other propagation
Possible dispersion relation change, speed changes, etc
• ‫ݑ߲߲ ݑ‬ + ܰ‫ܮ‬ … ‘linearly degenerate’, you’re golden
• ߲‫ݑ߲߲ ݑ‬ + ܰ‫ܮ‬ … ‘truly nonlinear’, shocks! Do worry!
• ߲߲‫ݑ߲߲ ݑ‬ + ܰ‫ܮ‬ … ‘you are on your own…(and probably
crazy)’
• ߲߲‫ݑ‬ + ߲߲߲‫ ݑ‬ … ‘you are definitively crazy’
• ߲߲‫ݑ‬ + ߲߲߲߲‫ ݑ‬ … ‘you are playing with fire’ [e.g Cayuso+’17]

67. Random thoughts…
(what I think I think...at least today…)
• We have some theories, motivated by different fronts…
do we know any is strongly preferred?
– Even if so… one could argue chances aren’t great it is *the
theory* to consider
– Short of having a QG theory that guide corrections to consider,
we are left with biased views
– We’ve done reasonably well with effective field theory
philosophy, so considering EEs as the leading term in an
expansion (on what? curvature seems natural…but other
options could arise). Must learn to live in this muck!

68. To begin answer possible questions, we need to reconcile
the following picture
‘Fixed’ version
Truncated
theory

69. Beyond GR II?
• Many extensions contemplate: R + k (higher order curv)
– Nicely/loftily motivated by QG, Λ/DM considerations, EFT, etc
– Some degree of testing in weak-field scenarios over very
specialized backgrounds
– Problems: higher derivatives, very dubious character (or
unknown character) of resulting equations of motion, possible
runaway of energy to UV, etc. [ill-posednes… Hadamard would
even say throw it away!]
– How to ask nonlinear qns from potentially
(arguably/bettably/swearably) sick theories? Need to
understand non-linear regime and what to expect, or hope!,
will happen with higher order curvature corrections

70. A bit more ‘in detail’
• Many alternatives motivated by: further physics, EFT
considerations, quantum gravity arguments, [often
involving curvature corrections with/without extra
d.o.f, nonlocalities, etc]
‫ܩ‬௔௕ = ܱ݇ሺ݃௔௕, ߠሻ ; ‫ܮ‬ ߠ = Ωሺ݃௔௕, ߠሻ
• Do BHs differ from those in GR ?
• Are they stable?
• What’s their dynamical behavior?
• Relaxation to equilibrium? (what’s the final state?)
• Interaction with further fields? (e.g. matter)
How to probe what could take place?

71. • What to do in non-linear regimes?
– Option 1: ``reduction of order’’: ‫ܮ‬ ∅ = ܵሺ∅ሻ. Treat
S ‘iteratively’ keeping L a well defined hyperbolic
operator. Depending on the scheme can render the
problem seemingly well posed, but is it physically
doing the right job?
– Example: for dCS [Okounkova+ ‘17], perturbative expansion
to apply above scheme. So far carried to linear order (Chern-
Simons field on a BBH background)

72. Option 2
• Modify the system of eqns, in an ad-hoc manner to
control higher gradients and prevent wild runaway to
the UV
• E.g. Israel-Stewart formultion of viscous relativistic
hydrodynamics: T = Tpf + gradient terms
– Define Π = (shear/bulk)ab + Grad(shear/bulk..)ab as new and
independent variable
– Force an eqn on Π such that Π ∼ (shear/bulk)ab to leading
order always
– τ Π,t = - Π + (shear/bulk)ab …. [Geroch, details shouldn’t matter]
So, mathematically all now correct. How about physically?
Without a complete problem it is hard to tell

73. Example!!
• Burgess-Williams ’14
• Integrating out ߷ →
• and EOM

74. Further freedom!
[Solomon-Trodden ’18]

75. To fix ideas, keep to O(1/M4)
• Option 0: integrate through
• Option 1:
Reduction of order
• Option 2: ‘fix equations’ [Cayuso +’17, Allwright ‘18]

79. • Option 0: ‘impossible & bad’. Option 1: delicate &
uncertain?. Option 2: Can be justified? One could argue
yes in 3+1 dimensions but not above
– (drawing from LIGO/VIRGO, fluid-gravity correspondence and
specific ‘2nd order’ BH perturbation calculations).
There seems to be a way to avoid ‘not going to non-linear-
land’ with (many) GR alternatives and face upcoming data
[Cayuso,Ortiz,LL ‘17]
‘Fixed’
version
Extension to
GR
(truncated
version)

80. Final words
• For beyond GR question, the good news is that ‘everything is an
opportunity’, the bad news is that it is yet unclear (to me!)
whether there is any preferable option
– Short of knowing which one, exploration of some might help be ready for
specific phenomenology
– New ideas are fueling some hopes to try and study what could happen in
the ‘extreme’ regimes of compact binary mergers
– Some basic suggestions to explore possible behavior
– ‘Numerical Relativity’ codes mature enough for ‘outsiders’ to take and use
• However, I do worry about ‘theoretical biases’, ultimately the true
guide will come from data, and ‘residual analysis’ wrt GR
templates might be the most solid path forward. Nevertheless if a
non-trivial residual is obtained, we will need insights to
understand what is going on