1

21. Experimental Tests of Gravitational Theory

21. Experimental Tests of Gravitational Theory
Revised August 2023 by T. Damour (IHES, Bures-sur-Yvette).

21.1 General Relativity
Einstein’s theory of General Relativity (GR), the current “standard” theory of gravitation,
describes gravity as a universal deformation of the Minkowski metric:gµν (xλ ) = ηµν + hµν (xλ ) , where ηµν = diag(−1, +1, +1, +1) .

(21.1)

GR is classically defined by two postulates, embodied in the total action defining the theory:−1

Stot [gµν , ψ, Aµ , H] = cZ

d4 x(LEin + LSM ) .

(21.2)

The first postulate that states that the Lagrangian density describing the propagation and selfinteraction of the gravitational field isLEin [gαβ ] =c4 √ µν
gg Rµν (gαβ ) ,16πG

(21.3)

where G denotes Newton’s constant, g = − det(gµν ), g µν is the matrix inverse of gµν , and where
the Ricci tensor Rµν ≡ Rα µαν is the only independent trace of the curvature tensor
α
α
α
σ
α σ
Rα µβν = ∂β Γµν− ∂ν Γµβ
+ Γσβ
Γµν
− Γσν
Γµβ ,
1
λ
Γµν
= g λσ (∂µ gνσ + ∂ν gµσ − ∂σ gµν ) .
2(21.4)(21.5)The second postulate states that gµν (and its associated connection) couples universally, and minimally, to all the bosonic (respectively fermionic) fields of the Standard Model by replacing everywhere the Minkowski metric ηµν (respectively the flat Minkowski connection). Schematically(suppressing matrix indices and labels for the various gauge fields and fermions and for the Higgs
doublet),X√
1 X √ µα νβ a a1
LSM [ψ, Aµ , H, gµν ] = −gg g Fµν Fαβ −g ψ γ µ (Dµ + ωijµ γ ij )ψ
4
4
X √
√
1 √ µν
− 2 gg Dµ HDν H − g V (H) −λ g ψ Hψ .

(21.6)

a = ∂ Aa − ∂ Aa + g f a Ab Ac and the (representation-dependent) gauge-field covariant
Here Fµνµ ν
ν µAbc µ ν
derivative Dµ = ∂µ + gA Aaµ Tarep are defined as in Special Relativity, while the derivative of spin1
2 fermions also includes a coupling to the gravitational “spin-connection” ωijµ = −ωjiµ , via its
contraction with γ ij = 12 (γ i γ j − γ j γ i ), where i, j = 0, 1, 2, 3 and γ i = ei µ γ µ are usual (numerical)
Dirac matrices satisfying γ i γ j + γ j γ i = 2η ij . The connection components ωijµ are defined interms of the local orthonormal frame (vierbein) ei µ (such that gµν = ηij ei µ ej ν ) used to describe
the componentsof the variousfermions ψ, and of its inverse ei µ (such that ei µ ej µ = δij ), byωijµ = 12 Ci[jk] + Cj[ki] − Ck[ij] ekµ where Ci[jk] = ηis C s [jk] , with C i [jk] ≡ ∂µ ei ν − ∂ν ei µ ej µ ek ν .
From the total action follow Einstein’s field equations,Rµν − 21 Rgµν =

8πG
Tµν .
c4

(21.7)√
Here R = g µν Rµν is the scalar curvature, and Tµν ≡ gµα gνβ T αβ where T µν = (2/ g)δLSM /δgµν isthe (symmetric) energy-momentum tensor of the Standard Model matter. The theory is invariant
S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
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21. Experimental Tests of Gravitational Theory

under arbitrary coordinate transformations: x0µ = f µ (xν ) (as well as under arbitrary local SO(3,1)
rotations of the vierbein, e0i µ = Λi j (x)e0j µ ). To solve the field equations Eq. (21.7), one needs to
fix the coordinate gauge freedom, e.g., the “harmonic gauge” (which is the analogue of the Lorenz√
gauge, ∂µ Aµ = 0, in electromagnetism) corresponds to imposing the condition ∂ν ( gg µν ) = 0.
In this Review, we only consider the classical limit of gravitation (i.e. classical matter andclassical gravity). Quantum gravitational effects are expected (when considered at low energy) tocorrect the classical action Eq. (21.2) by additional terms involving quadratic and higher powers of
the curvature tensor. This suggests that the validityof classical gravity extends (at most) down top
3 ' 1.62 × 10−33 cm, i.e., up to energy scales
length scales of order the Planck length
L
=
~G/c
P
p
of order the Planck energy EP = ~c5 /G ' 1.22 × 1019 GeV. Considering quantum matter in aclassical gravitational background also poses interesting challenges, notably the possibility that thezero-point fluctuations of the matter fields generate a nonvanishing vacuum energy density ρvac ,√corresponding to a term − g ρvac in LSM [1]. This is equivalent to adding a “cosmological constant”
term +Λ gµν on the left-hand side of Einstein’s equations, Eq. (21.7), with Λ = 8πG ρvac /c4 . Recent
cosmological observations (see the following Reviews) suggest a positive value of Λ correspondingto ρvac ≈ (2.3 × 10−3 eV)4 . Such a small value has a negligible effect on the non-cosmological tests
discussed below.

21.2 Key features and predictions of GR
The definition of GR recalled above makes predictions both about the coupling of gravity tomatter, and about the structure of the gravitational field beyond its previously known Newtonian
aspects.
21.2.1 Equivalence Principle
First, the universal nature of the coupling between gµν and the Standard Model matter postulated in Eq. (21.6) entails many observable consequences that go under the generic name of“Equivalence Principle”.
A first aspect of the Equivalence Principle is that the outcome of a local non-gravitational
experiment, referred to local standards, should not depend on where, when, and in which locallyinertial frame, the experiment is performed. This means, for instance, that local experiments should
neither feel the cosmological evolution of the Universe (constancy of the “constants”), nor exhibit
preferred directions in spacetime (isotropy of space, local Lorentz invariance).
A second aspect of the Equivalence Principle is that the kinetic terms, g µν ∂µ φ∂ν φ or ψγ i ei µ ∂µ ψ,of all the fields of Nature (including the gravitational field itself) are universally coupled to the same
curved spacetime metric gµν (x) = ηij ei µ ej ν . This implies in particular that all massless fields should
propagate with the same speed.
A third aspect of the Equivalence Principle is that two (electrically neutral) test bodies dropped
at the same location and with the same velocity in an external gravitational field should fall inthe same way, independently of their masses and compositions (“universality of free fall” or “WeakEquivalence Principle”). In addition, the study (using the nonlinear structure of GR) of the motion,in an external gravitational field, of bodies having a non-negligible, or even strong, self-gravity (suchas planets, neutron stars, or black holes) has shown that the latter property of free-fall universality
holds equally well for self-gravitating bodies (“Strong Equivalence Principle”).
A last aspect of the Equivalence Principle concerns various universality features of the gravitational redshift of clock rates. GR predicts that, when intercomparing them by means of electromagnetic signals, two (non gravity-based) clocks located along two different spacetime worldlines should
exhibit a universal difference in clock rate that depends on their worldlines, but that is independentof their nature and constitution. For instance, two clocks located at two different positions in aPstatic external Newtonian potential U (x) = Gm/r should exhibit, when intercompared by elec31st May, 2024

21. Experimental Tests of Gravitational Theory3

tromagnetic signals, the difference in clock rate, τ1 /τ2 = ν2 /ν1 = 1 + [U (x 1 ) − U (x 2 )]/c2 + O(1/c4 ),(“universal gravitational redshift of clock rates”). Similarly, the comparison of atomic-transition
frequencies when observing on Earth a transition that took place on a far-away galaxy should
involve (at lowest order in cosmological perturbations) the universal cosmological redshift factor
1 + z = a(treception )/a(temission ) between the Friedmann scale factors a(t) (see below).
21.2.2 Quasi-stationary, weak-field (post-Newtonian) gravity
When applied to quasi-stationary, weak-field gravitational fields, Einstein equations, Eq. (21.7),
entail a spacetime structure which predicts deviations from Newtonian gravity of the first postNewtonian (1PN) order, i.e., fractionally smaller than Newtonian effects by a factor O(v 2 /c2 ) ∼O(GM/(c2 r)). The 1PN-accurate solution of Eq. (21.7) reads (in harmonic gauge)22
1
g00 = −1 + 2 V − 4 V 2 + O 6
c
c
c
 
4
1
g0i = − 3 Vi + O 5 ,
c
c


 
2
1
gij = δij 1 + 2 V + O 4 ,
c
c




,

(21.8)

where x0 = ct, i, j = 1, 2, 3, and where the scalar, V , and vector, Vi , (retarded) potentials are
00ii
0i
defined in terms of the sources σ = T c+T, σi = Tc by
2−1
V = −1
ret [−4πGσ] ; Vi = ret [−4πGσi ] .

(21.9)In GR the gravitational interaction of N moving point masses (labeled by A = 1, . . . , N ) is describedby a reduced (classical) action that admits a diagrammatic expansion:
Sreduced = S free + S tree−level + S one−loop + · · ·(21.10)where the free (special-relativistic) action reads
S

free

=−

XZ

q

mA c −ηµν dxµA dxνA

A

=−

XZ

q

2 /c2 ,
dtmA c2 1 − vA

(21.11)A

while the tree-level (one-graviton-exchange) interaction term reads
Stree−level

8πG
=− 4
c

Z

4

d xT

µν

−1



(Tµν − 21 T ηµν ) =Z

dtL(2) .

(21.12)

Corresponding to the 1PN-accurate metric of Eq. (21.8), the 1PN-accurate expansion of the latter
tree-level, two-body interaction Lagrangian L(2) reads (with rAB ≡ |xA − xB |, nAB ≡ (xA −xB )/rAB )L(2) = 21

X G mA mB 
A6=B

rAB

1+

3
7
1
1
(v 2A + v 2B ) − 2 (v A · v B ) − 2 (n AB · v A )(n AB · v B ) + O 4
2
2c
2c
2c
c




(21.13)
The two-body interactions, Eq. (21.13), exhibit v 2 /c2 corrections to Newton’s 1/r potential inducedby spin-2 exchange (“gravito-magnetism”). Consistency at the 1PN level, v 2 /c2 ∼ G m/rc2 , requires
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21. Experimental Tests of Gravitational Theory

4

that one also considers the three-body interactions contained in the one-loop contribution S one−loop ,corresponding to terms induced by some of the three-graviton vertices and other non-linearities(terms O(h2 ) and O(hT ) in Eq. (21.15) below), i.e., to the O(V 2 ) term in Eq. (21.8):(3)L

1 X G2 mA mB mC
1
=−
+O 4
2
2 B6=A6=C rAB rAC c
c




.

(21.14)21.2.3 Gravitational Waves in GR
The linearized approximation to Einstein’s field equations, Eq. (21.7), in harmonic gaugeν∂ (hµν − 12 hηµν ) = 0 (with h ≡ η µν hµν ), reads
hµν = −16πG(Tµν − 21 T ηµν ) + O(h2 ) + O(hT ) .
c4(21.15)

Outside of any source (i.e., when Tµν = 0), this yields hµν = 0, with ∂ ν (hµν − 21 hηµν ) = 0.
The generic linearized solution (modulo the diffeomorphism freedom) of the latter vacuum Einsteinequations can be written as (with k 2 = k · k = ηµν k µ k ν , k · x = kµ xµ )Zhµν (x) =d4 kδ(k 2 )µν (k)eik·x ,(21.16)

where the polarization tensor µν (k) must be transverse (µν k ν = 0) and traceless (η µν µν = 0).
In addition, µν (k) can be freely submitted to the gauge freedom 0µν = µν + ξµ kν + ξν kµ . Thisimplies that gravitational waves (GW) propagate with the speed of light, and (like electromagnetic
waves) have only two independent polarizations. In a frame where, say, k µ = (ck, 0, 0, k), the two
independent linear polarization tensors can be taken to have components only in the transverse
×
×
+
+
+
×
+
1-2 plane, of the following form: +
11 = −22 =  , with 12 = 21 = 0; or 12 = +21 =  , with××
11 = 22 = 0. Under a little-group rotation of angle θ in the 1-2 plane, the two circular polarization
amplitudes (±) = + ∓ i × vary as 0(±) = e±2iθ (±) , thereby characterizing the helicity-2 natureof GWs.
When solving the inhomogeneous equation Eq. (21.15), taking into account the nonlinear contributions O(h2 ) + O(hT ), one finds that, to lowest order, the GW amplitude emitted at large
distances by a matter distribution is given by the following “quadrupole formula”
hTT
ij (T, X) ≈2G TT
Q̈ab (T − R/c)
Pijab (N),4cR

(21.17)

where Qij (t) = d3 xσ(t, x)(xi xj − 31 δij x2 ) (a, b, i, j = 1, 2, 3) is the quadrupole moment of thesource, R = |X| the distance to the source, N = X/R the unit direction from the source to the
TT (N) = (δ − N N )(δ − N N ) − 1 (δ − N N )(δ − N N ) the transverseobserver, and Pijab
i j
a b
ia
i a
j b
jb
ab
2 ij
traceless projector onto the 2-plane orthogonal to N.
R

21.2.4 Strong gravitational fields: neutron stars and black holes
The nonlinear structure of Einstein’s equations implies many predictions for strong gravitational
fields that distinguish GR from Newtonian gravity. For instance, in Newtonian gravity, there is no
upper limit to the dimensionless gravitational potential U/c2 , with U satisfying Poisson’s equation∆U = −4πGρ, where ρ denotes the Newtonian mass density. By contrast, in GR, the dimensionlesssurface gravitational potential GM/(c2 R) of a spherically symmetric (perfect fluid) body cannot
exceed 49 [2].
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521. Experimental Tests of Gravitational Theory

Given an equation of state p = f (ρ) modeling the interior of a (cold) spherically symmetric body(say a non-rotating neutron star), Einstein equations, Eq. (21.7), with T µν = (ρ + p)uµ uν + pg µν ,
and


dr2
2
2
2
2
gµν dxµ dxν = −e2Φ(r) c2 dt2 +
+
r
dθ
+
sin
θdφ
,
(21.18)(r)
1 − 2GMc2 ryield the following Tolman-Oppenheimer-Volkoff radial equations:G(ρ + p/c2 )(M (r) + 4πr3 p/c2 );r2 (1 − 2GM (r)/(c2 r))
M 0 (r) = 4πr2 ρ ;p0 (r) = −Φ0 (r) =G(M (r) + 4πr3 p/c2 )r2 (1 − 2GM (r)/(c2 r)).

(21.19)(21.20)(21.21)In the exterior of the star (r ≥ R) , the metric takes the Schwarzschild formgµν dxµ dxν = − 1 −

2GM
c2 r



c2 dt2 +



dr2
2
2
2
2
+
r
dθ
+
sin
θdφ
,
1 − 2GMc2 r(21.22)where M ≡ M (R) is the total gravitational mass of the star. GR predicts, for any given p = f (ρ),
several (in principle) observable features of neutron stars, such as: (i) the maximum mass of a neutron star; (ii) the relation between the radius R and the total mass M ; (iii) theq dimensionless surface√
2
gravitational potential GM/(c R) (linked to the surface redshift −g00 = 1 − 2GM
measured by
c2 R
an observer at infinity); (iv) the moment of inertia; and (v) the Love number (tidal polarizability).
The current uncertainty on the equation of state of a neutron star yields the GR-predicted approximate range for the maximum mass of non-rotating neutron stars 1.5 M . Mmax . 2.5 M ,and the absolute upper bound Mmax < 3 M [3]. The surface gravitational potential of a typicalneutron star is GM/c2 RNS ' 0.17, which is a factor ∼ 108 higher than the surface potential of the
Earth, and a mere factor 3 below the black hole limit GM/c2 RBH = 21 to be discussed next.
The existence of a maximum mass for a neutron star led Oppenheimer and Snyder [4] topredict that the end point of stellar evolution for sufficiently heavy stars, after exhaustion of
all thermonuclear sources of energy, will be what are now called “black holes.” The latter are
solutions of Einstein’s equations whose past structure involves a gravitationally collapsing star, butwhose presently observable structure is essentially described (for non-rotating black holes) by thevacuum Schwarzschild solution Eq. (21.22). It took many years for theoretical (and mathematical)
physicists to understand that the apparent singularity of the Schwarzschild solution at r = 2GMc2
was a coordinate singularity and that the Schwarzschild spacetime was regular at the “black holehorizon”, RBH ≡ 2GM. The rotating analog of the Schwarzschild spacetime is the Kerr blackc2
hole [5].
Black holes are outstanding consequences of GR which enjoy many remarkable properties,
notably: (i) presence of a one-way surface (the horizon) for all waves and particles; (ii) absenceof “hair” (i.e., barring a possible electric charge, their structure is fully described by only two
parameters, total mass, M , and total angular momentum, J ≤ GM 2 /c); (iii) existence of a spectrumof damped quasi-normal vibrational modes; and (iv) a behavior under external perturbations similar
to ordinary physical objects satisfying the laws of (dissipative) thermodynamics. Moreover, though
no classical waves or particles can get out of the horizon, black holes are predicted to slowly
evaporate via quantum particle creation.
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21. Experimental Tests of Gravitational Theory

21.2.5 Cosmology
To complete our short tour of the main predictions of GR, let us mention that GR offers thecurrent standard framework for describing the large-scale structure of the Cosmos, from the nearlyhomogeneous Big Bang (and its plausible inflationary beginning) to the current inhomogeneous
Universe undergoing an accelerated expansion. The spacetime structure on large (temporal andspatial) scales is well described by a solution of Einstein’s equations of the form
ds2 = − (1 + 2Φ(t, x)) c2 dt2 + 2Wi (t, x)dtdxi + a2 (t) ((1 − 2Ψ(t, x)) δij + hij (t, x)) dxi dxj ,(21.23)where, after a suitable gauge-fixing [6], Wi (t, x) is transverse, while hij (t, x) is transverse andtraceless. The source T µν must involve a certain number of postulated ingredients: an inflaton
field; the matter of the Standard Model; a dark matter component; and a cosmological constantcontribution TΛµν = −ρvac g µν , with ρvac ≡ c4 Λ/(8πG). The scale factor a(t) of the Friedmannbackground metric ds20 = −c2 dt2 + a2 (t)δij dxi dxj satisfies the GR-predicted Friedmann equations(with vanishing spatial curvature k = 0),
 2
ȧ

8πG
ρtot ,
a
3


ä
4πG
3
=−
ρtot + 2 ptot ,
a
3
c
2

H ≡

=

(21.24)(21.25)

while the scalar (Φ(t, x), Ψ(t, x)), vector (Wi (t, x)), and tensor (hij (t, x)) inhomogeneous perturbations satisfy some GR-predicted propagation equations (coupled to matter perturbations); see [6]and the following Reviews. When the cosmic fluid is well approximated by a perfect fluid, Einstein’s
equations predict the following link between the scalar perturbations
Φ(t, x) = Ψ(t, x) .

(21.26)21.3 A roadmap of parametrizations of deviations from GR, and of modified
gravity
As will be discussed below, all currently performed gravitational experiments are compatible
with GR. However, similarly to what is done in discussions of precision electroweak experiments, it isuseful to quantify the significance of precision gravitational experiments by parameterizing possible
deviations from GR. One can distinguish two main approaches to considering, and parameterizing,
deviations from GR: (i) theory-agnostic phenomenological approaches; or, (ii) the study of thepredictions of specific classes of alternative theories of gravity. Both types have led to useful waysof discussing tests of gravity. Both types also have their limitations. Considering them together
leads to cross-fertilization.
21.3.1 Theory-agnostic phenomenological approaches to parameterizing deviations
from GRThe theory-agnostic phenomenological approach is the oldest, and, arguably, the most robust
one. It essentially consists in starting from specific observable predictions within the considered
standard theory, and of deforming them by introducing some free parameters measuring either
deviations from effects already present within the standard theory, or new effects absent from thestandard theory. A classic example is the periastron advance of Mercury (and the other planets).
When working within Newtonian gravity as a standard theory of gravity, the rate of periastronadvance of Mercury, ω̇, is (when neglecting the quadrupole moment of the Sun) a calculable functionof the masses and semi-major axes of the other planets of the solar system, say ω̇ Newton (mi , ai ).
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21. Experimental Tests of Gravitational Theory

However, ω̇ is also a directly observable quantity, so that one can parameterize the periastronadvance of Mercury by writingω̇ obs = ω̇ Newton (mi , ai ) + ∆ω̇ .

(21.27)

Using other observable data to determine some “observed” values of the mi ’s and ai ’s, one can thenmeasure the anomalous periastron precession ∆ω̇ and see whether it is compatible with zero, ornot. As is well-known, Leverrier used such a methodology and, in 1859, measured an anomalous
periastron precession of about ∆ω̇ ' 38 arcsec/century (later re-estimated at 43 arcsec/century),which was explained in 1915 as a GR prediction. Let us discuss further examples of the use of suchtheory-agnostic approaches for discussing deviations from GR.
21.3.2 Parameterized post-Newtonian (PPN) formalism.
When considering the weak-field slow-motion limit appropriate to describing gravitational experiments in the solar system, it has been traditional to parameterize possible (long-range) deviations from the GR-predicted 1PN metric by introducing extra dimensionless coefficients in thevarious terms of the metric of Eq. (21.8). The minimal version of the parameterized post-Newtonian(PPN) formalism (essentially due to Eddington) involves only two parameters β and γ, namely
22β1g00 = −1 + 2 V − 4 V 2 + O 6
c
c
c
 
2(γ + 1)1
g0i = −
Vi + O 5 ,
3
c
c

 

1
2γ
gij = δij 1 + 2 V + O 4 ,
c
c




,

(21.28)(21.29)(21.30)0iwith V and Vi defined by Eq. (21.9), with the same vectorial source σi = Tc , but a modified scalarsource



1
V
PPN
00
ii
σ
= 2 1 + (3γ − 2β − 1) 2 T + γ T.
(21.31)cc
In GR, β GR = 1 and γ GR = 1, so that deviations from GR are parameterized by β ≡ β − 1and γ ≡ γ − 1. Richer versions of the PPN formalism (involving up to ten parameters) weredeveloped in interaction with the study of classes of alternative theories of gravity [7,8]. This led toparameterizing new types of contributions to the 1PN metric that are absent in the GR framework.
When deriving the 1PN-accurate dynamics of N point masses predicted by the PPN-modified
metric, Eq. (21.28), one finds that the free Lagrangian is not modified (because we are considering here a Lorentz-invariant subclass of PPN metrics), while there are modifications of both thetwo-body Lagrangian, L(2) , Eq. (21.13), and the three-body one, L(3) , Eq. (21.14). More precisely, denoting η ≡ 4β − γ, the Newtonian interaction energy term in Eq. (21.13) is modified into
GAB mA mB /rAB , with a body-dependent gravitational “constant”
grav
grav
GAB = G[1 + η(EA
/mA c2 + EB
/mB c2 ) + O(1/c4 )] ,(21.32)grav
where EA
denotes the gravitational binding energy of body A. In addition, there is the additional
contribution + γ(v A − v B )2 /c2 in the brackets on the right-hand side of L(2) , Eq. (21.13). As forthe three-body interaction term L(3) , Eq. (21.14), it is modified by the overall factor 1 + 2β.
These results show how the introduction of the two minimal PPN deviation parameters β ≡ β−1
and γ ≡ γ − 1 suffices to introduce many different observable effects. Some of them (the ones linked
with γ) concern deviations at the linearized (one-graviton-exchange) level (and affect, for instance,light deflection and time-delay effects), while the deviation parameter β parameterizes effects linked
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21. Experimental Tests of Gravitational Theory

to the cubic vertex of Einstein’s gravity (and affects, for instance, periastron precession). Of
particular interest is the fact that Eq. (21.32) shows that the combination η ≡ 4β − γ parameterizes
a violation of the Strong Equivalence Principle, because the gravitational interaction between selfgravitating bodies is seen to be influenced by the gravitational binding energy of each body [9].
As stated above, this effect is absent in GR (where η GR = 0). This is an example where the factof contrasting GR with some deviations from it gives physical significance to a null effect in GR(namely the universality of free fall of self-gravitating bodies).
Finally, one can extend the PPN formalism by allowing for a slow, phenomenological time
variation of Newton’s constant:"G(t) = G0

#

Ġ0
1+
(t − t0 ) .
G0(21.33)Here, one assumes that there exist units in which the masses, mi , of elementary particles stay
constant, and that G is measured in such units. A possible time variation of G then corresponds
to a possible common variation of the dimensionless couplings Gm2i /(~c).
21.3.3 Parameterized post-Keplerian (PPK) formalism.
The discovery of pulsars (i.e., rotating neutron stars emitting a beam of radio noise) in gravitationally bound orbits [10, 11] has given us our first experimental handle on a regime of relativistic gravity going significantly beyond the uniformly weak-field, and quasi-stationary regimeof solar-system gravity. Binary pulsars allow us to probe some radiative effects, and also somestrong-gravitational-field effects. In these systems, the finite speed of propagation of the gravitational interaction between the pulsar and its companion generates damping-like terms at order(v/c)5 in the equations of motion [12]. These damping forces are the local counterparts of thegravitational radiation emitted at infinity by the system (“gravitational radiation reaction”). They
cause the binary orbit to shrink and its orbital period Pb to decrease. The remarkable stabilityof pulsar clocks has allowed one to measure the corresponding very small orbital period decayṖb ≡ dPb /dt ∼ −(v/c)5 ∼ −10−12 –10−14 in several binary systems, thereby giving us a direct
experimental handle on the propagation properties of the gravitational field. In addition, thelarge surface gravitational potential of a neutron star allows one to probe the quasi-static stronggravitational-field regime, as is discussed below.
It is possible to extract phenomenological (theory-independent) tests of gravity from binary
pulsar data by using the parameterized post-Keplerian (PPK) formalism [13]. The basis of thisformalism is the fact that, after correcting for the Earth’s motion around the Sun and for thedispersion due to propagation in the interstellar plasma, the time of arrival of the N th pulse tN
can be described by a generic, parameterized “timing formula” [13, 14], whose functional form iscommon to the whole class of tensor-scalar gravitation theories:tN − t0 = F [TN (νp , ν̇p , ν̈p ) ; {pK } ; {pP K }] .

(21.34)Here, TN is the pulsar proper time corresponding to the N th turn given by N/2π = νp TN +1ν̇ T 2 + 61 ν̈p TN3 (with νp ≡ 1/Pp the spin frequency of the pulsar, etc.), {pK } = {Pb , T0 , e, ω0 , x} is
2 p N
the set of “Keplerian” parameters (notably, orbital period Pb , eccentricity e, periastron longitude ω0
and projected semi-major axis x = a sin i/c), and {pP K } = {k, γtiming , Ṗb , r, s, δθ , ė, ẋ} denotes theset of (separately measurable) “post-Keplerian” parameters. Most important among these are: thefractional periastron advance per orbit k ≡ ω̇Pb /2π; a dimensionful time-dilation parameter γtiming ;the orbital period derivative Ṗb ; and the “range” and “shape” parameters of the gravitational timedelay caused by the companion, r and s.
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921. Experimental Tests of Gravitational Theory

Without assuming any specific theory of gravity, one can phenomenologically analyze the datafrom any binary pulsar by least-squares fitting the observed sequence of pulse arrival times to thetiming formula of Eq. (21.34). This fit yields the “measured” values of the parameters {νp , ν̇p , ν̈p },{pK }, {pP K }. Now, each specific relativistic theory of gravity predicts that, for instance, k, γtiming ,Ṗb , r, and s (to quote parameters that have been successfully measured from some binary pulsar
data) are some theory-dependent functions of the Keplerian parameters and of the (unknown)masses m1 , m2 of the pulsar and its companion. For instance, in GR, one finds (with M ≡ m1 +m2 ,n ≡ 2π/Pb ),k GR (m1 , m2 ) =3(1 − e2 )−1 (GM n/c3 )2/3 ,GRγtiming(m1 , m2 ) =en−1 (GM n/c3 )2/3 m2 (m1 + 2m2 )/M 2 ,



73 2
37 4
ṖbGR (m1 , m2 ) = − (192π/5)(1 − e2 )−7/2 1 + 24
e + 96e× (GM n/c3 )5/3 m1 m2 /M 2 ,rGR (m1 , m2 ) =Gm2 /c3 ,sGR (m1 , m2 ) =nx(GM n/c3 )−1/3 M/m2 .
(21.35)
theory(m1 , m2 ), Ṗbtheory(m1 , m2 ),In alternative gravity theories each of the functions k theory(m1 , m2 ), γtiming
etc., is modified by quasi-static strong field effects (associated with the self-gravities of the pulsar
and its companion), while the particular function Ṗbtheory (m1 , m2 ) is further modified by radiative
effects [15–18]. If one measures N > 2 PPK parameters from the data of a specific binary pulsar,
these N measurements determine, for each given theory, N curves (defined by the N equations
kitheory (m1 , m2 ) = kiobs ) in the two-dimensional mass plane (m1 , m2 ). This yields N − 2 tests of thespecified theory, according to whether the N curves (or strips) have one point in common, as theyshould.

21.3.4 Parameterized-post-Friedmannian (PPF) formalisms.
We have recalled above that, in GR, the two functions, Φ(t, x), and Ψ(t, x), parameterizing (inthe “longitudinal gauge”) the scalar perturbations of the background Friedmann metric are related(in absence of anisotropic stresses) by Eq. (21.26). Several authors [19–28] have defined various
types of parameterized-post-Friedmannian (PPF) formalisms involving (generally space and timedependent) phenomenological parameters. The simplest versions of these formalisms involve two
phenomenological parameters measuring: (i) the ratio between Φ(t, x), and Ψ(t, x), say (using aparametrization which parallels the usual PPN parametrization)
Ψ(t, x) = γcosmo (t, x)Φ(t, x) ;(21.36)and (ii) the effective gravitational constant entering the Poisson equation for Φ(t, x), say∆Φ(t, x) = 4πGΦ (t, x)δρ(t, x) .

(21.37)

However, the peculiarities of cosmological observables limit the domain of applicability of suchphenomenological approaches [26] (notably because the strong dependence of cosmological probes
on epochs and scales obliges one to rely on specific parameterizations of the functions γcosmo (t, x)
and GΦ (t, x), e.g., [25, 28]). Approaches based on specific classes of modified-gravity theories
allow for a more complete treatment involving, in principle, all existing cosmological observables:
Big Bang nucleosynthesis, cosmic microwave background, large-scale structure, Hubble diagram,
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1021. Experimental Tests of Gravitational Theory

weak lensing, etc. Discussing the current cosmological tests using either such PPF formalisms, orcomparisons with the predictions of modified-gravity theories, is beyond the scope of this review.
See [29] for a comprehensive recent discussion. The bottom line is that all present cosmological
data have been found to be compatible with GR (within the Friedmann-Lemaître-based ΛCDM
model). Beyond the quantitative limits on various parameterized theoretical models [29], one should
remember the striking (strong-field-type) qualitative verification of GR embodied in the fact that
relativistic cosmological models give an accurate picture of the Universe over a period during which
the spatial metric has been blown up by a gigantic factor, say (1 + z)2 ∼ 1019 between Big Bang
nucleosynthesis and now.
21.3.5 Various phenomenological tests of GR from gravitational wave (GW) data
The observations by the US-based Laser Interferometer Gravitational-wave Observatory (LIGO),
later joined by the Europe-based Virgo detector, of gravitational-wave (GW) signals [30–34], haveopened up a novel testing ground for relativistic gravity. The first three observing runs (O1, O2,
O3a and O3b) of the LIGO-Virgo collaboration, recently joined by the Japan-based KAGRA detector (LVK collaboration), have led to the confident detection of 93 GW signals, most of which comefrom binary black-hole coalescences. These observations are summarized in three GW transient
catalogs: GWTC-1 [35], GWTC-2 [36] and GWTC-3 [37].
Several approaches have been used to either test consistency with GR, or to look for special
types of possible deviations. Making accurate predictions for GW signals from coalescing black
holes within GR took years of both analytical [38, 39] and numerical [40] work. Some works havestarted (both analytically [15, 41–45] and numerically [46–48]) to derive the corresponding predictions within some modified-gravity theories. Phenomenological approaches are very useful forparameterizing general, conceivable deviations from GR when analyzing the GW signals emitted
by coalescing black holes or neutron stars.
A first phenomenological, global consistency test simply consists of measuring the noise-weighted
correlation C between each detected strain signal and the corresponding best-fit GR-predictedwaveform. C obs should be equal to 1, modulo statistical (and/or systematic) errors.
Various other phenomenological tests of the structure of the GR-predicted waveforms emitted bycoalescing compact binaries have been suggested. One general idea [49–51] (dubbed “parameterized
post-Einsteinian formalism” in [52]) is to modify the GR-predicted Fourier-domain value ψ(f ) of
the phase of black-hole coalescence GW signals h(f ) = A(f )eiψ(f ) by introducing GR-deviation
parameters, say
ψ(f ) =

X h GR,NSpii(m1 , m2 )(1 + δ p̂i ) + piGR,S (m1 , m2 , S1 , S2 ) ui (f ).

(21.38)iHere, the ui (f )’s define a basis of functions of the GW frequency f , and the superscript NS
refers to the nonspinning contribution, while the superscript S refers to the spinning one. Such a
GR-modification directly applies to the phenomenological representation [53] of ψ(f ), and can bePgeneralized to any waveform model by adding the non-GR phase term i pGR,NS(m1 , m2 )δ p̂i ui (f )ito the corresponding GR-predicted Fourier-domain phase ψ(f ) [54]. For instance, the leadingorder (LO), quadrupolar term in the GR phase evolution during the early inspiral corresponds2
1 +m2 )
3 −5/3 , while the next-to-leading-order
to u0 (f ) = f −5/3 and pGR= 3(m0128m1 m2 (πG(m1 + m2 )/c )22GR(NLO) term is a O(v /c ) correction p2 u2 (f ) with u2 (f ) = f −1 . In the phenomenological model[53] these terms (as well as the other inspiral contributions) are cut-off beyond the frequency
G(m1 + m2 )f /c3 = 0.018.
Each dimensionless parameter δ p̂i introduces a fractional deviation from the corresponding individual phasing GR effect having the frequency dependence ui (f ), and can, in principle, be extracted
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1121. Experimental Tests of Gravitational Theory

by fitting the inspiral part of the observed waveform to the deformed template of Eq. (21.38).
However, one must also use this deformed template for simultaneously extracting the values of
m1 , m2 , S1 , and S2 . Together with signal-to-noise ratio (SNR) considerations, and parametercorrelation issues, this limits the applicability of such a test to introducing only one deformation
parameter δ p̂i at a time. A particularly meaningful test [49] is to leave undeformed the LO andGR
NLO terms pGR0 u0 (f ) + p2 u2 (f ) and to vary the third coefficient p3 parameterizing the next,
3
3“GW tail”-related O(v /c ) correction, with u3 (f ) = f −2/3 . Another well-motivated test [52] is
to introduce a new coefficient δ p̂−2 , which is absent in GR, and which parameterizes an O(( vc )−2 )fractional correction to the LO, quadrupolar term, thereby allowing for a possible dipolar GW flux
(indeed, dipolar GW radiation generally exists in theories containing scalar excitations). As pGR−2vanishes, δ p̂−2 is added as an absolute deviation, scaled by the LO term pGR
0 .
The coalescence of two black holes, or of a black hole and a neutron star (or of two heavy-enoughneutron stars) leads to the formation of a black hole that is initially formed in a perturbed state.
The relaxation of the latter perturbed black hole into its stationary, equilibrium state leads to theemission of characteristic (rapidly decaying) ringing GW modes (a.k.a. quasi-normal modes) [55,56],
whose frequencies and decay times are functions of the mass (Mf ) and spin (Sf ≡ GMf2 af /c) of
the final black hole, sayωa = (c3 /GMf )[2π fˆaQNM (af ) − i/τ̂aQNM (af )] ,(21.39)where a = 1, 2, . . . labels the various ringing modes, starting from the least-damped one. In principle, if the SNR is large enough, one can directly test for the presence of one or several of thesemodes in the post-merger signal, and measure both Re(ωa ) and Im(ωa ) in a theory-independentway. These phenomenological measurements then lead to null tests of GR, from which one canextract theoretical information about eventual deviations from GR [57, 58].
As recalled above, GR predicts that GWs propagate (in vacuum) at exactly the same speed
as light (i.e., they have the same dispersion law g µν kµ kν = 0 in curved spacetime). Deviations
from such a universal, scale-free dispersion law can be phenomenologically parameterized in several
ways. If one phenomenologically assumes that the graviton dispersion law includes a mass term,say g µν kµ kν + m2g /~2 = 0, or some more general type of frequency-dependent modification, suchchanges affect the phasing of the inspiral GW signal and can be directly tested [59]. When one
observes both GWs and electromagnetic waves emitted by the same system, one can also directly
test whether both types of waves propagate in the same way.
Let us now present some examples of theory-dependent discussions of experimental tests basedon considering specific classes of alternative theories. The most conservative deviations from Einstein’s pure spin-2 theory are defined by adding new, bosonic, light or massless, macroscopicallycoupled fields.
21.3.6 Gravity tests within classes of tensor-scalar theories of gravity
The possible existence of new gravitational-strength couplings leading to deviations from Einsteinian (and Newtonian) gravity has been suggested by many natural extensions of GR, startingwith the classic Kaluza-Klein idea, and continuing up to now with the study of extended supergravity theories, and of (super-)string theory. In particular, a recurrent suggestion of such theories(which dates back to pioneering work by Jordan, and by Fierz [60]) is the existence of a scalar fielda 2 gauge-field actions. Such fields
ϕ coupled both to the scalar curvature R and to the various Fµν(“dilaton” or “moduli”) generically appear in string theory and are massless at the tree-level, but
could acquire a self-interaction potential V (ϕ) beyond the tree-level.
The exchange of such a dilaton-like field leads to several types of observational deviations fromGR. For experimental limits on the gravitational inverse-square-law (down to the micrometer range)
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1221. Experimental Tests of Gravitational Theory

see Refs. [61–65]. If the potential V (ϕ) is zero or negligible for the considered range, the couplinga 2 leads to apparent violations of the weak equivalence principle, with rather specificof ϕ to Fµν
composition-dependence [66]. Next, when neglecting the fractionally small composition-dependent
effects, such a field approximately couples to the trace of the energy-momentum tensor T = gµν T µν .
The most general such theory contains (after suitable field redefinitions) two arbitrary functions of
the scalar field, namely the self-interaction potential V (ϕ), and a matter-coupling function a(ϕ):
c4 √Ltot [gµν , ϕ, ψ, Aµ , H] =g(R(gµν ) − 2g µν ∂µ ϕ∂ν ϕ)
16πG∗√− gV (ϕ) + LSM [ψ, Aµ , H, geµν ] .
(21.40)Here G∗ is a “bare” Newton constant, and the Standard Model matter is coupled not to the“Einstein” (pure spin-2) metric gµν , but to the conformally related (“Jordan-Fierz”) metricgeµν = exp(2a(ϕ))gµν .

(21.41)The scalar field equation
4πG∂V (ϕ)
g ϕ = 4−α(ϕ)T +
c
∂ϕ




,

(21.42)

featuresα(ϕ) ≡ ∂a(ϕ)/∂ϕ ,(21.43)as the basic (field-dependent) coupling between ϕ and matter [15,67]. The best-known, special caseof these theories is the one-parameter (ω) Jordan-Fierz-Brans-Dicke theory [68], with V (ϕ) = 0
and a(ϕ) = α0 ϕ, leading to a field-independent coupling α(ϕ) = α0 (with α0 2 = 1/(2ω + 3)). Moregenerally, if we consider the massless theories (V (ϕ) = 0) with arbitrary (non-linear) coupling
function a(ϕ), they modify Einstein’s predictions in the weak-field slow-motion limit appropriate
to describing gravitational experiments in the solar system (1PN approximation) only throughthe appearance of exactly the same two “post-Einstein” dimensionless parameters γ = γ − 1 andβ = β − 1 that entered the minimal (Eddington) PPN formalism presented above. However, we
now have the following theoretical expressions relating the latter phenomenological parameters to
the coupling functions entering the tensor-scalar action Eq. (21.40):
α02;
1 + α02
β0 α02β = + 21.
(1 + α02 )2
γ = −2(21.44)(21.45)Here α0 ≡ α(ϕ0 ), and β0 ≡ ∂α(ϕ0 )/∂ϕ0 , with ϕ0 denoting the vacuum expectation value (VEV)
of ϕ around the solar system. In addition, the observable value Gobs of the gravitational constantis found to be field-dependent and given (at a place where ϕ = ϕ0 ) by
Gobs = G(ϕ0 ) ≡ G∗ exp[2a(ϕ0 )](1 + α02 ) .

(21.46)This makes it clear that the parameter γ is the basic post-Einstein parameter, which measures
the admixture of an additional field (here a spin-0 field) to the pure spin-2 GR. One also sees
how the parameter β is linked to non-linear effects (here coupling terms β0 (ϕ − ϕ0 )2 T in theaction), and how the Nordtvedt parameter η ≡ 4β − γ is related to the field-dependence of Gobs
(η = (α0 /(1 + α02 ))∂ ln G(ϕ0 )/∂ϕ0 ).
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21. Experimental Tests of Gravitational Theory

The advantage of a theory-dependent approach, such as Eq. (21.40), over the phenomenologicalminimal PPN approach of Eq. (21.28), is that it allows one to consistently predict the observational deviations from GR in all possible gravity regimes: the quasi-stationary weak-field regime;
the wavelike weak-field regime; the strong-field regime; the cosmological regime, etc. All such observational deviations can be consistently worked out once one chooses specific forms of the coupling
function a(ϕ), and of V (ϕ). The simple choice of a two-parameter quadratic coupling function, saya(ϕ) = α0 (ϕ − ϕ0 ) + 12 β0 (ϕ − ϕ0 )2 , has been found useful for describing many possible observabledeviations from GR.
The observable consequences for binary pulsar observations of the strong-field and radiative
effects linked to the coupling to ϕ have been explicitly worked out in Refs. [15,44] in the case where
ϕ is massless (see Ref. [69] for the case where ϕ is massive). In particular, the strong-field natureof the pulsar tests is demonstrated by the fact that some tensor-scalar theories can be as close as
desired to GR in the weak-field regime of the solar-system (i.e., γ and β can be as small as desired,or even exactly zero), while developing (via a “spontaneous scalarization” mechanism) differencesof order unity with GR in binary pulsar experiments [17, 18].
21.3.7 Attractor and screening mechanisms in modified gravity
As will follow from the discussion of experimental data below, the comparison between thepredictions of general massless tensor-scalar theories and current data shows that the basic coupling
parameter α0 must be tuned to a small value (especially when allowing for composition-dependent
effects). This raises the issue of the naturalness of such small coupling parameters. It has beenshown in this respect that, in many tensor-scalar theories, there is an attractor mechanism by which
the cosmological evolution naturally drives the VEV ϕ0 (t) towards a value for which the coupling
parameter α0 = α(ϕ0 ) vanishes, thereby making it natural to expect only small deviations fromGR (at least for the weak-field regime) at our current cosmological epoch [70, 71].
There are other theoretical mechanisms (generically called “screening mechanisms”) that could
explain why a theory of gravity whose theoretical content significantly differs from that of GR
could naturally pass all the stringent, GR-compatible experimental limits that will be discussed
below. In particular, when considering a self-interacting scalar field (V (ϕ) 6= 0), the interplay
between the two terms on the right-hand side of Eq. (21.42) tends to drive the local VEV ϕ0 of
ϕ to a density-dependent
value. In turn, this leads to a corresponding density-dependent effectiveq
2
mass m0 (ϕ0 ) = 4πG∂ V (ϕ0 )/∂ϕ20 of the ϕ field, and to density-dependent matter couplings [72].
Various choices of the functions V (ϕ) and a(ϕ) can then reduce the ϕ-induced deviations from GR
in dense environments while still allowing for significant deviations in different (e.g., cosmological)
regimes [73–77].
Other screening mechanisms have been invoked, based on an environment dependence mediatedby (first or second) derivatives of a scalar degree of freedom. Roughly speaking, such mechanisms
involve a (possibly effective) scalar degree of freedom ϕ that satisfies a field equation that is moregeneral than Eq. (21.42) in that the left-hand side, g ϕ, is replaced by a non-linear function of ϕ,∂ϕ and ∂ 2 ϕ. The presence of non-linear derivative self-interactions of ϕ can weaken the effective
coupling of ϕ to matter. A simple toy-model showing this weakening would be to replace Eq. (21.42)
by an equation of the form
∂V (ϕ)
4πG
Z(ϕ, ∂ϕ, ∂ ϕ)g ϕ = 4−α(ϕ)T +
c
∂ϕ
2





.

(21.47)

Such an equation is equivalent, at a first level of approximation, to replacing the gravitational
constant G entering Eq. (21.42) by Geff (ϕ0 , ∂ϕ0 , ∂ 2 ϕ0 ) ≡ G/Z0 , where Z0 ≡ Z(ϕ0 , ∂ϕ0 , ∂ 2 ϕ0 ). This
has a screening effect if Z0  1. Indeed, the replacement G → Geff diminishes the strength of the
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1421. Experimental Tests of Gravitational Theory

interaction potential due to q
ϕ exchange by a factor of 1/Z0 . In addition, theqrange of this interactionis also affected: m0 (ϕ0 ) =4πG∂ 2 V (ϕ0 )/∂ϕ20 → m0eff (ϕ0 , ∂ϕ0 , ∂ 2 ϕ0 ) =4πGeff ∂ 2 V (ϕ0 )/∂ϕ20 =

−1/2

Z0

m0 .
Screening mechanisms based on such non-linear derivative self-interactions are often referred toas being “Vainshtein-like” because a similar mechanism was first invoked in Ref. [78] as a conjectural
way to ensure that the extra degrees of freedom associated with a massive (rather than massless)graviton become effectively weakly coupled to matter within a large domain around gravitationalsources. Here, one is considering massive deformations of the massless spin-2 metric field of GR
by a very small mass, possibly of cosmological scale: mg ∼ ~H0 ∼ 10−33 eV. The constructionof ghost-free potential terms for a spin-2 field has turned out to be a delicate matter [79]. The
phenomenology of a very-low-mass graviton is still partly uncontrolled, both because of the unknown
extent to which the Vainshtein screening is really active, and because of subtle constraints linked toan eventual UV completion of the theory beyond the unusually low energy scale where it becomesstrongly coupled:Λstrong coupling ∼ (MPlanck m20 )1/3 ∼ 10−13



m0
~H0

2/3

eV .

(21.48)The search for modified gravity theories incorporating an extra scalar degree of freedom potentially able to yield a Vainshtein-like screening led to writing down the following general class of
tensor-scalar Lagrangian [80, 81]:
Ltot [gµν , ϕ, ψ] =G2 (ϕ, X) − G3 (ϕ, X)g ϕ + G4 (ϕ, X)R+G4X (ϕ, X)[(g ϕ)2 − ϕµν ϕµν ]1+G5 (ϕ, X)Gµν ϕµν − G5X (ϕ, X)[(g ϕ)36−3g ϕϕµν + 2ϕµν ϕµλ ϕνλ ] + Lmatter [gµν , ψ] .
(21.49)Here gµν denotes the matter-coupled metric, X ≡ − 21 g µν ∂µ ϕ∂ν ϕ, ϕµν ≡ ∇µ ∇ν ϕ, Gµν ≡ Rµν −1µν2 Rg , and the various coefficients Gn (ϕ, X) are arbitrary functions of two variables (with GnX ≡∂Gn /∂X). The field equations derived from the Lagrangian of Eq. (21.49) are only of second orderin derivatives in spite of the non-linear structure of Ltot . This implies that the tensor-scalar theories
defined by Eq. (21.49) feature three degrees of freedom, corresponding to a massless spin-2 excitation(GW) and a spin-0 excitation. Contrary to the simpler tensor-scalar theories of Eq. (21.40), it isfound that the speed of propagation of GWs implied by Eq. (21.49) is generically different from the
speed of light:
c2GW
G4 − X(ϕ̈G5X + G5ϕ )=.
(21.50)c2
G4 − 2XG4X − X(H ϕ̇G5X − G5ϕ )More general modified gravity models have been proposed (see, e.g. Refs. [82, 83]). Apartfrom the simplest of them, most of these models have a rather artificial flavor, and do not leadto convincing alternative explanations either of dark matter or of dark energy. In addition, manyof them do not lead (contrary to GR) to mathematically “well-posed” evolution problems [84–86].
This entails a serious challenge to deriving strong-field predictions for such models. It has beenargued that many of these (dark-energy motivated) models should be viewed as effective field
theory (EFT) approximations that need some sort of UV completion at an unusually low frequency
scale [87]. In spite of these shortcomings, such models are conceptually interesting because they
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1521. Experimental Tests of Gravitational Theory

give examples of deviations for various predictions of GR, existing independently from each other,in various regimes. For instance, some special tensor-scalar models lead to black hole solutions
modified by scalar-hair [88–90]. For other types of black holes with scalar-hair, see Ref. [91]. Thisshows the interest of phenomenologically testing, in a democratic and agnostic way, all conceivabledeviations from GR.
Let us now turn to briefly presenting current experimental results of various phenomenologicaltests of the main GR predictions recalled in Section 21.2 above.

21.4 Experimental tests of the Equivalence Principle ( i.e., of the matter-gravity
coupling)
21.4.1 Tests of the constancy of constants
Stringent limits on a possible time variation of the basic coupling constants have been obtainedby analyzing a natural fission reactor phenomenon that took place at Oklo, Gabon, two billion years
ago [92, 93]. These limits are at the 1 × 10−8 level for the fractional variation of the fine-structure
constant αem [93], and at the 4 × 10−9 level for the fractional variation of the ratio mq /ΛQCD
between the light quark masses and ΛQCD [94]. The determination of the lifetime of Rhenium 187
from isotopic measurements of some meteorites dating back to the formation of the solar system(about 4.6 Gyr ago) yields comparably strong limits [95]. Measurements of absorption lines inastronomical spectra also give stringent limits on the variability of both αem and µ = mp /me atcosmological redshifts, e.g.,∆αem /αem = (1.2 ± 1.7stat ± 0.9sys ) × 10−6 ,(21.51)at redshifts z = 1.0–2.4 [96], and|∆µ/µ| < 4 × 10−7 (95% C.L.) ,(21.52)

at a redshift z = 0.88582 [97]. There are also significant limits on the variation of αem andµ = mp /me at redshift z ∼ 103 from cosmic microwave background data, e.g., ∆αem /αem =(3.6 ± 3.7) × 10−3 [98]. Direct laboratory limits (based on monitoring the frequency ratio of several
different atomic clocks) on the present time variation of αem , and µ = mp /me have reached thelevels [99, 100]d ln(αem )/dt = (1.8 ± 2.5) × 10−19 yr−1 ,d ln(µ)/dt = (−8 ± 36) × 10−18 yr−1 .

(21.53)

There are also experimental limits on a possible dependence of coupling constants on the gravitational potential [99, 100].
Experimental limits on the present time variation of the gravitational constant, Eq. (21.33),
have been derived from planetary ephemerides [101], lunar laser ranging [102], and binary-pulsar
data [103, 104]. The most stringent limits come from lunar-laser-ranging data [102]:Ġ0= (7.1 ± 7.6) × 10−14 yr−1 .
G0(21.54)21.4.2 Tests of the isotropy of space and of Local Lorentz invariance
The highest precision tests of the isotropy of space have been performed by looking for possible
quadrupolar shifts of nuclear energy levels [105]. The (null) results can be interpreted as testing
the fact that the various pieces in the matter Lagrangian, Eq. (21.6), are indeed coupled to thesame external metric gµν to the 10−29 level.
Stringent tests of possible violations of local Lorentz invariance in gravitational interactions
have been obtained both from solar-system data [8] and pulsar data [106, 107]. For astrophysicalconstraints on possible Planck-scale violations of Lorentz invariance, see Ref. [108].
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21. Experimental Tests of Gravitational Theory

21.4.3 Tests of the universality of free fall (weak, and strong equivalence principles)
The universality of the acceleration of free fall has been verified, for laboratory bodies, both on
the ground [109, 110] (at the 10−13 level), and in space [111, 112] (at the 10−15 level):(∆a/a)BeTi = (0.3 ± 1.8) × 10−13 ;(∆a/a)BeAl = (−0.7 ± 1.3) × 10−13 ;(∆a/a)TiPt = (−1.5 ± 2.3(stat) ± 1.5(syst)) × 10−15 .

(21.55)The universality of free fall has also been verified when comparing the fall of classical and quantum
objects (at the 6 × 10−9 level [113]), or of two quantum objects (at the (0.3 ± 5.4) × 10−7 [114], and(1 ± 1.4) × 10−9 , levels [115]).
The universality of free fall of self-gravitating bodies (strong equivalence principle) has beenverified in both the weak-gravity, and the strong-gravity regimes. The gravitational accelerationsof the Earth and the Moon toward the Sun have been checked to agree at the 10−13 level [102](∆a/a)EarthMoon = (−3 ± 5) × 10−14 .

(21.56)The latter result constrains the Nordtvedt PPN parameter [9] η ≡ 4β − γ to the 10−4 level:
η = (−0.2 ± 1.1) × 10−4 .

(21.57)

See below for strong-field tests of the strong equivalence principle.
Finally, the universality of the gravitational redshift of clock rates has been verified at the 10−4level by comparing a hydrogen-maser clock flying on a rocket up to an altitude of about 10, 000 km
to a similar clock on the ground [116]. The redshift due to a height change of only 33 cm has been
detected by comparing two optical clocks based on 27 Al+ ions [117]. The gravitational redshift
has also been detected in the orbit of a star near the supermassive black hole at the center of our
Galaxy [118, 119], and its universality has been verified at the 5% level [120].

21.5 Tests of quasi-stationary, weak-field gravity
All currently performed gravitational experiments in the solar system, including perihelion
advances of planetary orbits, the bending and delay of electromagnetic signals passing near theSun, and very accurate ranging data to the Moon obtained by laser echoes, are compatible withthe post-Newtonian results of Eq. (21.15), Eq. (21.13), and Eq. (21.14). The “gravito-magnetic”
interactions ∝ vA vB contained in Eq. (21.13) are involved in many of these experimental tests. They
have been particularly tested in lunar-laser-ranging data [121], in the combined LAGEOS-LARES
satellite data [122, 123], and in the dedicated Gravity Probe B mission [124].
To assess in a quantitative manner the results of the various solar-system tests of gravity it isconvenient to express them in terms of the PPN parameters defined above. The best current limiton the post-Einstein parameter γ ≡ γ − 1 isγ = (2.1 ± 2.3) × 10−5 ,(21.58)as deduced from the additional Doppler shift experienced by radio-wave beams connecting the
Earth to the Cassini spacecraft when they passed near the Sun [125].
The (cubic-vertex-related) post-Einstein parameter β ≡ β − 1 is constrained at the 10−4 level
both from a study of the global sensitivity of planetary ephemerides to post-Einstein parameters[101],(21.59)|β| < 7 × 10−5 ,31st May, 2024

17

21. Experimental Tests of Gravitational Theory

and from lunar-laser-ranging data [102]β = (−4.5 ± 5.6) × 10−5 .

(21.60)The periastron advance of the star S2 around the Galactic center massive black hole has been
observed to agree with GR within 20% [126]. More stringent limits on γ (i.e. the coupling of
ϕ to matter) are obtained in dilaton-like models where scalar couplings violate the Equivalence
Principle [127].

21.6 Tests of strong-field gravity (neutron stars and black holes)
Experimental tests of strong-field gravity have been obtained in various physical systems, notably binary pulsars and coalescing binary black holes.
It is convenient to quantitatively express binary-pulsar tests of strong-field gravity by using
the PPK formalism defined above. We recall that the measurement of N phenomenological PPK
parameters leads to N − 2 tests of strong-field gravity. In all, thirteen tests of strong-field and/or
radiative gravity have been obtained in the four different (double neutron-star) binary pulsar systems PSR1913+16 [10, 11, 128], PSR1534+12 [129–131], PSR J1141−6545 [132–135], and PSR
J0737−3039 A,B [136–140]. These consist of N − 2 = 5 − 2 = 3 tests from PSR1913+16 ; 5 − 2 = 3
tests from PSR1534+12; 4 − 2 = 2 tests from PSR J1141−6545; and 7 − 2 = 5 tests from PSR
J0737−3039 (see, also, Ref. [141] for additional, less accurate tests of relativistic gravity). Amongthese tests, four of them (those involving the measurement of the PPK parameter Ṗb ) probe radiative effects, and will be discussed in the following section. The four binary pulsar systems
PSR1913+16, PSR1534+12, PSR J1141−6545, and PSR J0737−3039 A,B have given nine tests ofquasi-static, strong-field gravity. GR passes all these tests within the measurement accuracy. Let
us only highlight here some of the most accurate strong-field tests.
In the binary pulsar PSR 1534+12 [129] one has measured five post-Keplerian parameters: k,γtiming , r, s, and (with less accuracy) Ṗb [130, 131]. This yields three tests of relativistic gravity.
Among these tests, the two involving the measurements of k, γtiming , r, and s accurately probe
strong field gravity, without mixing of radiative effects [130]. The most precise (10−3 level) of thesepure strong-field tests is the one obtained by combining the measurements of k, γtiming , and s;
namely, [131],"#sobs= 1.002 ± 0.002 .
(21.61)obs ]sGR [k obs , γtiming
1534+12The discovery of the remarkable double binary pulsar PSR J0737−3039 A and B [136, 137]
has led to the measurement of seven independent parameters [138–140]: five of them are thepost-Keplerian parameters k, γtiming , r, s, and Ṗb entering the relativistic timing formula of the
fast-spinning pulsar PSR J0737−3039 A; a sixth is the ratio R = xB /xA between the projectedsemi-major axis of the more slowly spinning companion pulsar PSR J0737−3039 B, and that of PSR
J0737−3039 A (the theoretical prediction for the ratio R = xB /xA , considered as a function of the(inertial) masses m1 = mA and m2 = mB , is Rtheory = m1 /m2 + O((v/c)4 ) [13, 14], independentlyof the gravitational theory considered). Finally, the seventh parameter ΩSO,B is the angular rate of(spin-orbit) precession of PSR J0737−3039 B around the total angular momentum vector [139,140].
These seven measurements give us five tests of relativistic gravity [138, 142, 143], four of which are
quasi-static, strong-field tests. GR passes all those tests with flying colors [144]. The most accurate
is at the 2 × 10−4 level:
"

#

sobs
= 1.00009 ± 0.00018 .
sGR [k obs , γtiming obs ] 0737−3039
31st May, 2024

(21.62)

18

21. Experimental Tests of Gravitational Theory

Binary pulsar data on other types of pulsar systems can be used to test strong-field aspectsof the “strong equivalence principle,” namely the GR prediction that strong-self-gravity objects(such as neutron stars) should fall with the same acceleration as weak-self-gravity objects (such as
white-dwarfs) in the (external) gravitational field created by other objects (such as the Galaxy, or
another white dwarf). The first binary-pulsar tests of this property have been obtained in nearlycircular binary systems (made of a neutron star and a white dwarf) falling in the field of the
Galaxy, and have led to strong-field confirmations (at the 2 × 10−3 level) of the strong equivalenceprinciple [104, 145–147]. The remarkable discovery of the pulsar PSR J0337+1715 in a hierarchicaltriple system [148] has allowed one to derive a much more accurate test of the strong equivalenceprinciple because the inner binary (comprising a pulsar and a close white-dwarf companion) falls
toward the outer white-dwarf companion with an acceleration that is 108 times larger than the
Galactic acceleration. This leads to a 95% confidence level limit on a possible fractional differencein free-fall acceleration of the pulsar and its close companion of [149, 150]|∆a/a| < 2.05 × 10−6 (95% C.L.) .

(21.63)This limit yields strong constraints on tensor-scalar gravity models.
Measurements over several years of the pulse profiles of various pulsars have detected secularchanges compatible with the prediction [151] that the general relativistic spin-orbit coupling shouldcause a secular change in the orientation of the pulsar beam with respect to the line of sight(“geodetic precession”). Such confirmations of general-relativistic spin-orbit effects were obtainedin PSR 1913+16 [152], PSR B1534+12 [131], PSR J1141−6545 [153], PSR J0737−3039 [139, 140],
and PSR J1906+0746 [154, 155]. In some cases (notably PSR 1913+16 and PSR J1906+0746) the
secular change in the orientation of the pulsar beam is expected to lead to the disappearance of thebeam (as seen on the Earth) on a human time scale (the second pulsar in the double system PSR
J0737−3039 already disappeared in March 2008 and is expected to reappear around 2035 [140]).
Recently, the ultimate strong-field regime of black holes has started to be quantitatively probed
via GW observations. The LIGO-Virgo(-Kagra) collaboration has detected (starting in September2015) GW signals [35], which, besides testing the radiative structure of gravity (see next section),
are in excellent qualitative and quantitative agreement with the structure and dynamics of blackhole horizons in GR. Because of the mixing of strong-field effects with radiative effects during thecoalescence of two black holes, and because of the lack of detailed alternative-theory predictions
for this process (see, however, Refs. [46–48]), it is not easy to set quantitative limits on possible
strong-field deviations from GR, independently of radiative effects. Direct tests of the existenceof black-hole horizons are scarce (see, however, Sec. VIIIB of [156] which reports the lack of any
statistical evidence for GW echoes).
Let us also mention that the Event Horizon Telescope collaboration has obtained event-horizonscale images of the supermassive black hole candidates in the center of the giant elliptical galaxy
M87, and in the center of our Galaxy. These images are consistent with GR models of accreting
Kerr black holes [157, 158]. For discussions of the corresponding constraints on the black holegeometry in the vicinity of the light ring see Refs. [159–161].

21.7 Tests of radiative gravity (both in binary-pulsar data and in GW data)
Experimental confirmations of the GR predictions for the radiative structure of gravity havebeen obtained both in binary-pulsar data and in the observation of GW signals from coalescing
compact binaries (binary black holes and binary neutron stars).
Binary-pulsar observations involving the measurement of the orbital period derivative Ṗb give
direct experimental tests of the reality of gravitational radiation, and, in particular, an experimental
confirmation that the speed of propagation of gravity cg is equal to the speed of light c (indeed, as
31st May, 2024

1921. Experimental Tests of Gravitational Theory

recalled above, Ṗb is a consequence of the propagation of the gravitational interaction between thetwo neutron stars [12]). Even in the presence of screening mechanisms within the binary system,
the value of Ṗb yields a measurement of the speed of propagation of GWs at the 10−2 level [162].
The currently most accurate binary-pulsar tests of the radiative properties of gravity come fromthe binary neutron-star systems PSR1913+16 and PSR J0737−3039 A,B, as well as from several
neutron-star-white-dwarf systems, notably PSR J1738+0333.
After subtracting a small (∼ 10−14 level in Ṗbobs = (−2.423 ± 0.001) × 10−12 ), but significant,“Galactic” perturbing effect (linked to Galactic accelerations and to the pulsar proper motion) [163],
one finds that the phenomenological test obtained by combining the measurements of the three PPKparameters (k − γtiming − Ṗb )1913+16 is passed by GR with complete success [128]:"

Ṗbobs − Ṗbgal= 0.9983 ± 0.0016 .
obs ]ṖbGR [k obs , γtiming
1913+16#(21.64)obs ] is the result of inserting in Ṗ GR (m , m ) the values of the masses predicted
Here ṖbGR [k obs , γtiming
1
2
b
obs
GR
obs
GR
by the two equations k = k (m1 , m2 ), and γtiming = γtiming (m1 , m2 ). This yields experimental
evidence for the reality of gravitational radiation damping forces at the (−1.7 ± 1.6) × 10−3 level.
An even better experimental test of the radiative structure of gravity (6 × 10−5 level) has been
recently obtained from the combined measurement in PSR J0737−3039 A,B of the three parametersk, s, and Ṗb [144]:#"Ṗbobs − Ṗbgal= 0.999963 ± 0.000063 .
(21.65)ṖbGR [k obs , sobs ] 0737−3039

In addition to the above tests, further very stringent tests of radiative gravity follow from themeasurement of the orbital period decay Ṗb of low-eccentricity pulsar-white dwarf systems. Notably,
the system PSR J1738+0333 yields an intrinsic orbital decay of [164]hṖbobs − Ṗbgali1738+0333= (−25.9 ± 3.2) × 10−15 ,(21.66)

to be compared to
h

ṖbGR

i
1738+0333−15= (−27.7+1.5.
−1.9 ) × 10

(21.67)

The fractional agreement between the (corrected) observed period decay and the GR-predicted one
seems to be quantitatively less impressive than the double-neutron-star results cited above, but
the crucial point is that asymmetric binary systems (such as neutron-star-white-dwarf ones) arestrong emitters of dipolar gravitational radiation in tensor-scalar theories, with Ṗb scaling (modulo
matter-scalar couplings) like m1 m2 /(m1 + m2 )2 (v/c)3 , instead of the parametrically smaller GRpredicted quadrupolar radiation Ṗb ∼ (v/c)5 [7, 15]. In view of the very small absolute value ofṖb , this makes such systems (and notably PSR J1738+0333) very sensitive probes of tensor-scalar
gravity [103,164–168]. It is then useful to turn to a theory-dependent analysis of pulsar data. Suchan analysis (see, e.g., [17, 130, 164, 167, 168]) leads to excluding a large portion of the parameter
space of tensor-scalar gravity allowed by solar-system tests. As a result, the basic matter-scalar
coupling α02 is more strongly constrained, over most of the parameter space, than the best current
solar-system limits of Eq. (21.58) (namely below the 10−5 level) [164, 167].
We now turn to the tests of radiative gravity that can be deduced from the GW data gatheredfrom the first three observing runs of the LIGO-Virgo-Kagra collaboration. All currently detected
GW signals are consistent with GR predictions. Several phenomenological approaches were used
and led to setting limits on possible deviations from GR [51, 156, 169, 170].
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20

21. Experimental Tests of Gravitational Theory

A theory-agnostic quantitative assessment on possible deviations from GR is given by measuring
the agreement between the full observed GW signal of coalescing binary black holes, and the GRpredicted one. One of the strongest results was obtained with the first event: GW150914, which
had an SNR of 24. The noise-weighted correlation between the GW150914 signal and the best-fit
GR-predicted waveform was found to be ≥ 97% [51, 169]. In other words, GR-violation effects
that cannot be reabsorbed in a redefinition of physical parameters are limited (in a noise-weighted
sense) to less than 3%.
Besides checking the agreement between the full observed GW signals and the corresponding
best-fit full signals predicted by GR, one also tested the consistency between separate parts of thesignals. A first approach [156,169,170] separates: (i) the lower-frequency signal emitted during theinspiral phase (considered up to the innermost stable circular orbit); and (ii) the higher-frequency
remaining signal emitted during the postinspiral phase, comprising the late-inspiral, the merger,
and the ringdown. Separately fitting each of these partial signals to GR-based templates thenyields separate estimates of the binary’s parameters, leading to separate estimates of the mass Mf
and dimensionless spin parameter af = Jf /(GMf2 ) of the final black hole that would be formed(in GR) by the coalescence of the two initial black holes. The consistency with GR then consists
in testing whether the two estimates (Mf , af )insp and (Mf , af )postinsp are compatible with eachother. They were found to be compatible for all events whose corresponding separate SNRs madesuch an analysis meaningful (see Figs. 3 and 4 in [170] and Fig.3 in [156]). Quantitatively, the(population-marginalized) fractional differences
Mfinsp − Mfpostinsp∆Mf= 2 insp,M̄f
Mf + Mfpostinsp
ainsp− apostinsp
∆af
f
f
,
= 2 inspāf
af + apostinspf(21.68)

between the two estimates were found to be consistent with zero (i.e. with GR) [156]∆Mf= −0.02+0.07−0.06 ,
M̄f
∆af
= −0.06+0.10−0.07 .
āf

(21.69)A second approach studies the consistency of the ringdown GW signal (emitted by the final, vibrating black hole) with GR predictions. See, in particular, section VIIIA, and Figs. 13 and 14,of [156]. The results are rather sensitive to various data analysis assumptions (notably the useof complete waveform models versus an analysis using only the ringdown signal). Though there
is no clear sign of any deviations from GR, the current SNRs do not yield strong theory-agnostic
evidence for the presence of ringing overtones.
The parametrization of Eq. (21.38) for possible deviations in the frequency dependence of theFourier-domain phase ψ(f ) of the black hole coalescence GW signal was used to measure best-fit
values for each fractional deviation parameter δ p̂i , considered separately (the other ones being set
to zero). In all cases, the posterior distribution for each δ p̂i is consistent with the GR value, i.e.,δ p̂GR= 0 (see Fig. 6 in [156]). The current limits on δ p̂i are (roughly) of order unity, except forithe two parameters highlighted above: δ p̂3 (parameterizing the O(( vc )3 ) fractional correction to theLO, quadrupolar term); and δ p̂−2 (parameterizing a possible dipolar-radiation-related O(( vc )−2 )fractional correction to the LO, quadrupolar term). Ref. [170] gives 90%-credible intervals for δ p̂3+0.10
of −0.02+0.11−0.10 (when using a phenomenological model [53]), and −0.01−0.11 (when using an effective
31st May, 2024

2121. Experimental Tests of Gravitational Theory

one-body model [38]). Ref. [156] gives a 90%-credible upper bound for a possible dipolar term of|δ p̂−2 | ≤ 7.3 × 10−4 .
As recalled above, GR predicts that the polarization content of GWs is pure helicity-2, i.e.
described by the two independent components of a traceless tensor transverse to the propagation
direction. A (massless) scalar excitation would add a pure-trace “breathing mode” in the planetransverse to the propagation direction. A phenomenological approach to generic metric theories ofgravity would allow for up to six polarizations for a GW [171], namely two tensor, two vector andtwo scalar modes. The LVK collaboration tested possible polarization deviations from GR in various
ways [33,156,169]. Recent results are given in section VII of [156]. Though pure-scalar, pure-vector
and vector-scalar hypotheses are significantly disfavored, any mixed hypothesis involving tensor
modes (i.e., tensor-scalar, tensor-vector, and tensor-vector-scalar) cannot be ruled out conclusively.
GR also predicts that GWs are non dispersive, and propagate at the same speed as light. One
can phenomenologically modify the GR-predicted GW phase evolution by adding the putative effectof an anomalous dispersion relation of the form E 2 = p2 c2 + Apα cα . GW data have been used to
set bounds on the anomalous coefficient A for various values of the exponent α. The case α = 0
is equivalent to assuming that gravitons disperse as a massive particle [59]. The current (90%credibility) phenomenological GW limit on the graviton mass is mg ≤ 1.27 × 10−23 eV/c2 [156].
This is 2.5 times more stringent than the Solar System bound of 3.16 × 10−23 eV/c2 [65].
Finally, a very constraining bound on the speed of propagation of gravity cGW was derived fromthe observed time delay of 1.7 s between GW170817 and the associated γ-ray burst. Namely, thefractional difference between cGW and clight ≡ c is constrained to be [172]− 3 × 10−15 <cGW − c< +7 × 10−16 .
c(21.70)

When comparing the latter bound to the prediction Eq. (21.50) from general second-order tensorscalar theories, Eq. (21.49), one is led to conclude that the coupling function G5 (ϕ, X) has to beignored and that the coupling function G4 (ϕ, X) has to be restricted to depend only on ϕ. Thisdrastically reduces the viable tensor-scalar modified-gravity models [173–176].
Let us finally mention that four pulsar timing arrays have recently given tantalizing evidence forthe existence of a stochastic background of gravitational waves with frequency ∼ 10−9 Hz [177–180].

21.8 Conclusions
All present experimental tests are compatible with the predictions of the current “standard”
theory of gravitation, Einstein’s General Relativity. Let us recap the main tests. The universalityof the coupling between matter and gravity (Equivalence Principle) has been verified at around the
10−15 level. Solar system experiments have tested the weak-field predictions of Einstein’s theory at
the few times 10−5 level. The propagation properties (in the near zone) of relativistic gravity, as well
as several of its static strong-field aspects, have been verified at the 10−4 level (or better) in several
binary pulsar experiments. Interferometric detectors of gravitational radiation have given direct
observational proofs of the existence, and properties, of gravitational waves (in the wave zone),and of the existence of coalescing black holes, and they have already set strong limits on possible
deviations; in particular: an upper bound |δ p̂−2 | < 7.3 × 10−4 on a possible dipolar contribution
to the GW flux; the O(10−15 ) bound of Eq. (21.70) on the speed of gravity; and confirmation of
the tensor polarization structure of gravitational waves. In addition, laboratory experiments haveset strong constraints on sub-millimeter modifications of Newtonian gravity, while many different
cosmological data sets have been used to set limits on possible GR deviations on cosmologicalscales [29]. In spite of the uneasiness of having to assume the existence of dark matter, and thepresence of an unnaturally small cosmological constant (as dark energy), General Relativity stands
31st May, 2024

2221. Experimental Tests of Gravitational Theory

out as a uniquely successful description of gravity on all the scales that have been explored so far.
There are no modified-gravity models which naturally pass all existing experimental tests, while
either explaining away the need for dark matter or for dark energy.
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