Binary Systems as Test-beds of Gravity Theories
Binary Systems as Test-beds of Gravity Theories∗
Thibault Damour
Institut des Hautes Etudes Scientifiques, 35 route de Chartres,
F-91440 Bures-sur-Yvette, France
We review the general relativistic theory of the motion, and of the timing,
of binary systems containing compact objects (neutron stars or black holes).
Then we indicate the various ways one can use binary pulsar data to test the
strong-field and/or radiative aspects of General Relativity, and of general classes
of alternative theories of relativistic gravity.
1 Introduction
The discovery of binary pulsars in 1974 [1] opened up a new testing ground
for relativistic gravity. Before this discovery, the only available testing ground
for relativistic gravity was the solar system. As Einstein’s theory of General
Relativity (GR) is one of the basic pillars of modern science, it deserves to
be tested, with the highest possible accuracy, in all its aspects. In the solar
system, the gravitational field is slowly varying and represents only a very small
deformation of a flat spacetime. As a consequence, solar system tests can only
probe the quasi-stationary (non radiative) weak-field limit of relativistic gravity.
By contrast binary systems containing compact objects (neutron stars or black
holes) involve spacetime domains (inside and near the compact objects) where
the gravitational field is strong. Indeed, the surface relativistic gravitational
field h00 ≃ 2GM/c2R of a neutron star is of order 0.4, which is close to the one
of a black hole (2GM/c2R = 1) and much larger than the surface gravitational
fields of solar system bodies: (2GM/c2R)Sun ∼ 10−6, (2GM/c2R)Earth ∼ 10−9.
In addition, the high stability of “pulsar clocks” has made it possible to monitor
the dynamics of its orbital motion down to a precision allowing one to measure
the small (∼ (v/c)5) orbital effects linked to the propagation of the gravitational
field at the velocity of light between the pulsar and its companion.
The recent discovery of the remarkable double binary pulsar PSR J0737−
3039 [2, 3] (see also the contributions of M. Kramer and A. Possenti to these
∗Based on lectures given at the SIGRAV School “A Century from Einstein Relativity:
Probing Gravity Theories in Binary Systems”, Villa Olmo (Como Lake, Italy), 17-21 May
2005. To appear in the Proceedings, edited by M. Colpi et al. (to be published by Springer).
http://arxiv.org/abs/0704.0749v1
proceedings) has renewed the interest in the use of binary pulsars as test-beds
of gravity theories. The aim of these notes is to provide an introduction to the
theoretical frameworks needed for interpreting binary pulsar data as tests of GR
and alternative gravity theories.
2 Motion of binary pulsars in general relativity
The traditional (text book) approach to the problem of motion of N separate
bodies in GR consists of solving, by successive approximations, Einstein’s field
equations (we use the signature −+++)
Rµν −
Rgµν =
Tµν , (1)
together with their consequence
∇ν T µν = 0 . (2)
To do so, one assumes some specific matter model, say a perfect fluid,
T µν = (ε+ p)uµ uν + p gµν . (3)
One expands (say in powers of Newton’s constant)
gµν(x
λ) = ηµν + h
µν + h
µν + . . . , (4)
together with the use of the simplifications brought by the ‘Post-Newtonian’
approximation (∂0 hµν = c
−1 ∂t hµν ≪ ∂i hµν ; v/c ≪ 1, p ≪ ε). Then one
integrates the local material equation of motion (2) over the volume of each
separate body, labelled say by a = 1, 2, . . . , N . In so doing, one must define
some ‘center of mass’ zia of body a, as well as some (approximately conserved)
‘mass’ ma of body a, together with some corresponding ‘spin vector’ S
a and,
possibly, higher multipole moments.
An important feature of this traditional method is to use a unique coor-
dinate chart xµ to describe the full N -body system. For instance, the center
of mass, shape and spin of each body a are all described within this common
coordinate system xµ. This use of a single chart has several inconvenient as-
pects, even in the case of weakly self-gravitating bodies (as in the solar system
case). Indeed, it means for instance that a body which is, say, spherically sym-
metric in its own ‘rest frame’ Xα will appear as deformed into some kind of
ellipsoid in the common coordinate chart xµ. Moreover, it is not clear how
to construct ‘good definitions’ of the center of mass, spin vector, and higher
multipole moments of body a, when described in the common coordinate chart
xµ. In addition, as we are interested in the motion of strongly self-gravitating
bodies, it is not a priori justified to use a simple expansion of the type (4) be-
cause h
Gma/(c
2 |x − za|) will not be uniformly small in the common
coordinate system xµ. It will be small if one stays far away from each object a,
but, as recalled above, it will become of order unity on the surface of a compact
body.
These two shortcomings of the traditional ‘one-chart’ approach to the rela-
tivistic problem of motion can be cured by using a ‘multi-chart’ approach.The
multi-chart approach describes the motion of N (possibly, but not necessarily,
compact) bodies by using N+1 separate coordinate systems: (i) one global coor-
dinate chart xµ (µ = 0, 1, 2, 3) used to describe the spacetime outside N ‘tubes’,
each containing one body, and (ii) N local coordinate charts Xαa (α = 0, 1, 2, 3;
a = 1, 2, . . . , N) used to describe the spacetime in and around each body a.
The multi-chart approach was first used to discuss the motion of black holes
and other compact objects [4, 5, 6, 7, 8, 9, 10, 11]. Then it was also found to
be very convenient for describing, with the high-accuracy required for dealing
with modern technologies such as VLBI, systems of N weakly self-gravitating
bodies, such as the solar system [12, 13].
The essential idea of the multi-chart approach is to combine the information
contained in several expansions. One uses both a global expansion of the type
(4) and several local expansions of the type
Gαβ(X
a ) = G
a ;ma) +H
αβ (X
a ;ma,mb) + · · · , (5)
where G
αβ(X ;ma) denotes the (possibly strong-field) metric generated by an
isolated body of mass ma (possibly with the additional effect of spin).
The separate expansions (4) and (5) are then ‘matched’ in some overlapping
domain of common validity of the type Gma/c
2 . Ra ≪ |x−za| ≪ d ∼ |xa−xb|
(with b 6= a), where one can relate the different coordinate systems by expansions
of the form
xµ = zµa (Ta) + e
i (Ta)X
ij(Ta)X
a + · · · (6)
The multi-chart approach becomes simplified if one considers compact bodies
(of radius Ra comparable to 2Gma/c
2). In this case, it was shown [9], by
considering how the ‘internal expansion’ (5) propagates into the ‘external’ one
(4) via the matching (6), that, in General Relativity, the internal structure of
each compact body was effaced to a very high degree, when seen in the external
expansion (4). For instance, for non spinning bodies, the internal structure of
each body (notably the way it responds to an external tidal excitation) shows
up in the external problem of motion only at the fifth post-Newtonian (5PN)
approximation, i.e. in terms of order (v/c)10 in the equations of motion.
This ‘effacement of internal structure’ indicates that it should be possible
to simplify the rigorous multi-chart approach by skeletonizing each compact
body by means of some delta-function source. Mathematically, the use of dis-
tributional sources is delicate in a nonlinear theory such as GR. However, it
was found that one can reproduce the results of the more rigorous matched-
multi-chart approach by treating the divergent integrals generated by the use
of delta-function sources by means of (complex) analytic continuation [9]. The
most efficient method (especially to high PN orders) has been found to use
analytic continuation in the dimension of space d [14].
Finally, the most efficient way to derive the general relativistic equations of
motion of N compact bodies consists of solving the equations derived from the
action (where g ≡ − det(gµν))
dd+1 x
R(g)−
−gµν(zλa ) dz
a dzνa , (7)
formally using the standard weak-field expansion (4), but considering the space
dimension d as an arbitrary complex number which is sent to its physical value
d = 3 only at the end of the calculation.
Using this method1 one has derived the equations of motion of two compact
bodies at the 2.5PN (v5/c5) approximation level needed for describing binary
pulsars [15, 16, 9]:
d2 zia
= Aia0(za − zb) + c−2Aia2(za − zb,va,vb)
+ c−4Aia4(za − zb,va,vb,Sa,Sb)
+ c−5Aia5(za − zb,va − vb) +O(c−6) . (8)
Here Aia0 = −Gmb(zia − zib)/|za − zb|3 denotes the Newtonian acceleration, Aia2
its 1PN modification, Aia4 its 2PN modification (together with the spin-orbit
effects), and Aia5 the 2.5PN contribution of order v
5/c5. [See the references
above; or the review [17], for more references and the explicit expressions of A2,
A4 and A5.] It was verified that the term A
a5 has the effect of decreasing the
mechanical energy of the system by an amount equal (on average) to the energy
lost in the form of gravitational wave flux at infinity. Note, however, that here
Aia5 was derived, in the near zone of the system, as a direct consequence of the
general relativistic propagation of gravity, at the velocity c, between the two
bodies. This highlights the fact that binary pulsar tests of the existence of Aia5
are direct tests of the reality of gravitational radiation.
Recently, the equations of motion (8) have been computed to even higher
accuracy: 3PN ∼ v6/c6 [18, 19, 20, 21, 22] and 3.5PN ∼ v7/c7 [23, 24, 25]
(see also the review [26]). These refinements are, however, not (yet) needed for
interpreting binary pulsar data.
3 Timing of binary pulsars in general relativity
In order to extract observational effects from the equations of motion (8) one
needs to go through two steps: (i) to solve the equations of motion (8) so as to
1Or, more precisely, an essentially equivalent analytic continuation using the so-called
‘Riesz kernels’.
get the coordinate positions z1 and z2 as explicit functions of the coordinate
time t, and (ii) to relate the coordinate motion za(t) to the pulsar observables,
i.e. mainly to the times of arrival of electromagnetic pulses on Earth.
The first step has been accomplished, in a form particularly useful for dis-
cussing pulsar timing, in Ref. [27]. There (see also [28]) it was shown that,
when considering the full (periodic and secular) effects of the A2 ∼ v2/c2 terms
in Eq. (8), together with the secular effects of the A4 ∼ v4/c4 and A5 ∼ v5/c5
terms, the relativistic two-body motion could be written in a very simple ‘quasi-
Keplerian’ form (in polar coordinates), namely:
n dt+ σ = u− et sinu , (9)
θ − θ0 = (1 + k) 2 arctan
1 + eθ
1− eθ
, (10)
R ≡ rab = aR(1 − eR cosu) , (11)
ra ≡ |za − zCM | = ar(1− er cosu) , (12)
rb ≡ |zb − zCM | = ar′(1− er′ cosu) . (13)
Here n ≡ 2π/Pb denotes the orbital frequency, k = ∆θ/2π = 〈ω̇〉/n =
〈ω̇〉Pb/2π the fractional periastron advance per orbit, u an auxiliary angle (‘rel-
ativistic eccentric anomaly’), et, eθ, eR, er and er′ various ‘relativistic eccentric-
ities’ and aR, ar and ar′ some ‘relativistic semi-major axes’. See [27] for the
relations between these quantities, as well as their link to the relativistic energy
and angular momentum E, J . A direct study [28] of the dynamical effect of the
contribution A5 ∼ v5/c5 in the equations of motion (8) has checked that it led
to a secular increase of the orbital frequency n(t) ≃ n(0)+ ṅ(t−t0), and thereby
to a quadratic term in the ‘relativistic mean anomaly’ ℓ =
n dt+ σ appearing
on the left-hand side (L.H.S.) of Eq. (9):
ℓ ≃ σ0 + n0(t− t0) +
ṅ(t− t0)2 . (14)
As for the contribution A4 ∼ v4/c4 it induces several secular effects in the
orbital motion: various 2PN contributions to the dimensionless periastron pa-
rameter k (δ4 k ∼ v4/c4+ spin-orbit effects), and secular variations in the incli-
nation of the orbital plane (due to spin-orbit effects).
The second step in relating (8) to pulsar observations has been accomplished
through the derivation of a ‘relativistic timing formula’ [29, 30]. The ‘timing
formula’ of a binary pulsar is a multi-parameter mathematical function relating
the observed time of arrival (at the radio-telescope) of the center of the N th
pulse to the integer N . It involves many different physical effects: (i) dispersion
effects, (ii) travel time across the solar system, (iii) gravitational delay due to
the Sun and the planets, (iv) time dilation effects between the time measured
on the Earth and the solar-system-barycenter time, (v) variations in the travel
time between the binary pulsar and the solar-system barycenter (due to relative
accelerations, parallax and proper motion), (vi) time delays happening within
the binary system. We shall focus here on the time delays which take place
within the binary system (see the lectures of M. Kramer for a discussion of the
other effects).
For a proper derivation of the time delays occurring within the binary sys-
tem we need to use the multi-chart approach mentionned above. In the ‘rest
frame’ (X0a = c Ta, X
a) attached to the pulsar a, the pulsar phenomenon can be
modelled by the secularly changing rotation of a beam of radio waves:
Ωa(Ta) d Ta ≃ Ωa Ta +
Ω̇a T
Ω̈a T
a + · · · , (15)
where Φa is the longitude around the spin axis. [Depending on the precise defi-
nition of the rest-frame attached to the pulsar, the spin axis can either be fixed,
or be slowly evolving, see e.g. [13].] One must then relate the initial direction
(Θa,Φa), and proper time Ta, of emission of the pulsar beam to the coordinate
direction and coordinate time of the null geodesic representing the electromag-
netic beam in the ‘global’ coordinates xµ used to describe the dynamics of the
binary system [NB: the explicit orbital motion (9)–(13) refers to such global
coordinates x0 = ct, xi]. This is done by using the link (6) in which zia denotes
the global coordinates of the ‘center of mass’ of the pulsar, Ta the local (proper)
time of the pulsar frame, and where, for instance
e0i =
c2 rab
+ · · ·
+ · · · (16)
Using the link (6) (with expressions such as (16) for the coefficients e
i , . . .)
one finds, among other results, that a radio beam emitted in the proper direction
N i in the local frame appears to propagate, in the global frame, in the coordinate
direction ni where
ni = N i +
−N i N
. (17)
This is the well known ‘aberration effect’, which will then contribute to the
timing formula.
One must also write the link between the pulsar ‘proper time’ Ta and the
coordinate time t = x0/c = z0a/c used in the orbital motion (9)–(13). This reads
− c2 d T 2a = g̃µν(aλa) dzµa dzνa (18)
where the ‘tilde’ denotes the operation consisting (in the matching approach) in
discarding in gµν the ‘self contributions’ ∼ (Gma/Ra)n, while keeping the effect
of the companion (∼ Gmb/rab, etc. . .). One checks that this is equivalent (in
the dimensional-continuation approach) in taking xµ = zµa for sufficiently small
values of the real part of the dimension d. To lowest order this yields the link
1− 2Gmb
c2 rab
1− Gmb
c2 rab
which combines the special relativistic and general relativistic time dilation
effects. Hence, following [30] we can refer to them as the ‘Einstein time delay’.
Then, one must compute the (global) time taken by a light beam emitted
by the pulsar, at the proper time Ta (linked to temission by (19)), in the initial
global direction ni (see Eq. (17)), to reach the barycenter of the solar system.
This is done by writing that this light beam follows a null geodesic: in particular
0 = ds2 = gµν(x
λ) dxµ dxν ≃ −
1− 2U
c2 dt2 +
dx2 (20)
where U = Gma/|x−za|+Gmb/|x−zb| is the Newtonian potential within the
binary system. This yields (with te ≡ temission, ta ≡ tarrival)
ta − te =
dt ≃ 1
|dx|+ 2
|x− za|
|x− zb|
|dx| . (21)
The first term on the last RHS of Eq. (21) is the usual ‘light crossing time’
|zbarycenter(ta) − za(te)| between the pulsar and the solar barycenter. It con-
tains the ‘Roemer time delay’ due to the fact that za(te) moves on an orbit.
The second term on the last RHS of Eq. (21) is the ‘Shapiro time delay’ due to
the propagation of the beam in a curved spacetime (only the Gmb piece linked
to the companion is variable).
When inserting the ‘quasi-Keplerian’ form (9)–(13) of the relativistic motion
in the ‘Roemer’ term in (21), together with all other relativistic effects, one finds
that the final expression for the relativistic timing formula can be significantly
simplified by doing two mathematical transformations. One can redefine the
‘time eccentricity’ et appearing in the ‘Kepler equation’ (9), and one can define
a new ‘eccentric anomaly’ angle: u→ unew [we henceforth drop the superscript
‘new’ on u]. After these changes, the binary-system part of the general relativis-
tic timing formula [30] takes the form (we suppress the index a on the pulsar
proper time Ta)
tbarycenter − t0 = D−1[T +∆R(T ) + ∆E(T ) + ∆S(T ) + ∆A(T )] (22)
∆R = x sinω[cosu− e(1 + δr)] + x[1 − e2(1 + δθ)2]1/2 cosω sinu , (23)
∆E = γ sinu , (24)
∆S = −2r ln{1− e cosu− s[sinω(cosu− e) + (1− e2)1/2 cosω sinu]},(25)
∆A = A{sin[ω +Ae(u)] + e sinω}+B{cos[ω +Ae(u)] + e cosω} , (26)
where x = x0 + ẋ(T − T0) represents the projected light-crossing time (x =
apulsar sin i/c), e = e0 + ė(T − T0) a certain (relativistically-defined) ‘timing
eccentricity’, Ae(u) the function
Ae(u) ≡ 2 arctan
1 + e
, (27)
ω = ω0 + k Ae(u) the ‘argument of the periastron’, and where the (relativisti-
cally-defined) ‘eccentric anomaly’ u is the function of the ‘pulsar proper time’
T obtained by solving the Kepler equation
u− e sinu = 2π
T − T0
T − T0
. (28)
It is understood here that the pulsar proper time T corresponding to the N th
pulse is related to the integer N by an equation of the form
N = c0 + νp T +
ν̇p T
ν̈p T
3 . (29)
From these formulas, one sees that δθ (and δr) measure some relativistic distor-
tion of the pulsar orbit, γ the amplitude of the ‘Einstein time delay’2 ∆E , and
r and s the range and shape of the ‘Shapiro time delay’3 ∆S . Note also that
the dimensionless PPK parameter k measures the non-uniform advance of the
periastron. It is related to the often quoted secular rate of periastron advance
ω̇ ≡ 〈dω/dt〉 by the relation k = ω̇Pb/2π. It has been explicitly checked that
binary-pulsar observational data do indeed require to model the relativistic pe-
riastron advance by means of the non-uniform (and non-trivial) function of u
multiplying k on the R.H.S. of Eq. (27) [31]4. Finally, we see from Eq. (28) that
Pb represents the (periastron to periastron) orbital period at the fiducial epoch
T0, while the dimensionless parameter Ṗb represents the time derivative of Pb
(at T0).
Schematically, the structure of the DD timing formula (22) is
tbarycenter − t0 = F [TN ; {pK}; {pPK}; {qPK}] , (30)
where tbarycenter denotes the solar-system barycentric (infinite frequency) ar-
rival time of a pulse, T the pulsar emission proper time (corrected for aberra-
tion), {pK} = {Pb, T0, e0, ω0, x0} is the set of Keplerian parameters, {pPK =
k, γ, Ṗb, r, s, δθ, ė, ẋ} the set of separately measurable post-Keplerian parameters,
2The post-Keplerian timing parameter γ, first introduced in [29], has the dimension of
time, and should not be confused with the dimensionless post-Newtonian Eddington parameter
γPPN probed by solar-system experiments (see below).
3The dimensionless parameter s is numerically equal to the sine of the inclination angle i
of the orbital plane, but its real definition within the PPK formalism is the timing parameter
which determines the ‘shape’ of the logarithmic time delay ∆S(T ).
4Alas this function is theory-independent, so that the non-uniform aspect of the periastron
advance cannot be used to yield discriminating tests of relativistic gravity theories.
and {qPK} = {δr, A,B,D} the set of not separately measurable post-Keplerian
parameters [31]. [The parameter D is a ‘Doppler factor’ which enters as an
overall multiplicative factor D−1 on the right-hand side of Eq. (22).]
A further simplification of the DD timing formula was found possible. In-
deed, the fact that the parameters {qPK} = {δr, A,B,D} are not separately
measurable means that they can be absorbed in changes of the other param-
eters. The explicit formulas for doing that were given in [30] and [31]: they
consist in redefining e, x, Pb, δθ and δr. At the end of the day, it suffices to
consider a simplified timing formula where {δr, A,B,D} have been set to some
given fiducial values, e.g. {0, 0, 0, 1}, and where one only fits for the remaining
parameters {pK} and {pPK}.
Finally, let us mention that it is possible to extend the general parametrized
timing formula (30) by writing a similar parametrized formula describing the ef-
fect of the pulsar orbital motion on the directional spectral luminosity [d(energy)
/d(time) d(frequency) d(solid angle)] received by an observer. As discussed in
detail in [31] this introduces a new set of ‘pulse-structure post-Keplerian pa-
rameters’.
4 Phenomenological approach to testing rela-
tivistic gravity with binary pulsar data
As said in the Introduction, binary pulsars contain strong gravity domains and
should therefore allow one to test the strong-field aspects of relativistic gravity.
The question we face is then the following: How can one use binary pulsar data
to test strong-field (and radiative) gravity?
Two different types of answers can be given to this question: a phenomeno-
logical (or theory-independent) one, or various types of theory-dependent ap-
proaches. In this Section we shall consider the phenomenological approach.
The phenomenological approach to binary-pulsar tests of relativistic gravity
is called the parametrized post-Keplerian formalism [32, 31]. This approach
is based on the fact that the mathematical form of the multi-parameter DD
timing formula (30) was found to be applicable not only in General Relativity,
but also in a wide class of alternative theories of gravity. Indeed, any theory
in which gravity is mediated not only by a metric field gµν but by a general
combination of a metric field and of one or several scalar fields ϕ(a) will induce
relativistic timing effects in binary pulsars which can still be parametrized by the
formulas (22)–(29). Such general ‘tensor-multi-scalar’ theories of gravity contain
arbitrary functions of the scalar fields. They have been studied in full generality
in [33]. It was shown that, under certain conditions, such tensor-scalar gravity
theories could lead, because of strong-field effects, to very different predictions
from those of General Relativity in binary pulsar timing observations [34, 35, 36].
However, the point which is important for this Section, is that even when such
strong-field effects develop one can still use the universal DD timing formula
(30) to fit the observed pulsar times of arrival.
The basic idea of the phenomenological, parametrized post-Keplerian (PPK)
approach is then the following: By least-square fitting the observed sequence
of pulsar arrival times tN to the parametrized formula (30) (in which TN is
defined by Eq. (29) which introduces the further parameters νp, ν̇p, ν̈p) one can
phenomenologically extract from raw observational data the (best fit) values of
all the parameters entering Eqs. (29) and (30). In particular, one so determines
both the set of Keplerian parameters {pK} = {Pb, T0, e0, ω0, x0}, and the set of
post-Keplerian (PK) parameters {pPK} = {k, γ, Ṗb, r, s, δθ, ė, ẋ}. In extracting
these values, we did not have to assume any theory of gravity. However, each
specific theory of gravity will make specific predictions relating the PK param-
eters to the Keplerian ones, and to the two (a priori unknown) masses ma and
mb of the pulsar and its companion. [For certain PK parameters one must also
consider other variables related to the spin vectors of a and b.] In other words,
the measurement (in addition of the Keplerian parameters) of each PK param-
eter defines, for each given theory, a curve in the (ma,mb) mass plane. For any
given theory, the measurement of two PK parameters determines two curves
and thereby generically determines the values of the two masses ma and mb (as
the point of intersection of these two curves). Therefore, as soon as one mea-
sures three PK parameters one obtains a test of the considered gravity theory.
The test is passed only if the three curves meet at one point. More generally,
the measurement of n PK timing parameters yields n− 2 independent tests of
relativistic gravity. Any one of these tests, i.e. any simultaneous measurement
of three PK parameters can either confirm or put in doubt any given theory of
gravity.
As General Relativity is our current most successful theory of gravity, it is
clearly the prime target for these tests. We have seen above that the timing data
of each binary pulsar provides a maximum of 8 PK parameters: k, γ, Ṗb, r, s, δθ, ė
and ẋ. Here, we were talking about a normal ‘single line’ binary pulsar where,
among the two compact objects a and b only one of the two, say a is observed
as a pulsar. In this case, one binary system can provide up to 8− 2 = 6 tests of
GR. In practice, however, it has not yet been possible to measure the parameter
δθ (which measures a small relativistic deformation of the elliptical orbit), nor
the secular parameters ė and ẋ. The original Hulse-Taylor system PSR 1913+16
has allowed one to measure 3 PK parameters: k ≡ 〈ω̇〉Pb/2π, γ and Ṗb. The
two parameters k and γ involve (non radiative) strong-field effects, while, as
explained above, the orbital period derivative Ṗb is a direct consequence of the
term A5 ∼ v5/c5 in the binary-system equations of motion (5). The term A5 is
itself directly linked to the retarded propagation, at the velocity of light, of the
gravitational interaction between the two strongly self-gravitating bodies a and
b. Therefore, any test involving Ṗb will be a mixed radiative strong-field test.
Let us explain on this example what information one needs to implement a
phenomenological test such as the (k−γ−Ṗb)1913+16 one. First, we need to know
the predictions made by the considered target theory for the PK parameters k, γ
and Ṗb as functions of the two masses ma and mb. These predictions have been
worked out, for General Relativity, in Refs. [29, 28, 30]. Introducing the notation
(where n ≡ 2π/Pb)
M ≡ ma +mb (31)
Xa ≡ ma/M ; Xb ≡ mb/M ; Xa +Xb ≡ 1 (32)
βO(M) ≡
, (33)
they read
kGR(ma,mb) =
1− e2
β2O , (34)
γGR(ma,mb) =
Xb(1 +Xb)β
O , (35)
ṖGRb (ma,mb) = −
1 + 73
e2 + 37
(1− e2)7/2
XaXb β
O . (36)
However, if we use the three predictions (34)–(36), together with the best
current observed values of the PK parameters kobs, γobs, Ṗ obdb [37] we shall find
that the three curves kGR(ma,mb) = k
obs, γGR(ma,mb) = γ
obs, ṖGRb (ma,mb) =
Ṗ obsb in the (ma,mb) mass plane fail to meet at about the 13 σ level! Should
this put in doubt General Relativity? No, because Ref. [38] has shown that the
time variation (notably due to galactic acceleration effects) of the Doppler fac-
tor D entering Eq. (22) entailed an extra contribution to the ‘observed’ period
derivative Ṗ obsb . We need to subtract this non-GR contribution before drawing
the corresponding curve: ṖGRb (ma,mb) = Ṗ
b − Ṗ
galactic
b . Then one finds that
the three curves do meet within one σ. This yields a deep confirmation of Gen-
eral Relativity, and a direct observational proof of the reality of gravitational
radiation.
We said several times that this test is also a probe of the strong-field aspects
of GR. How can one see this? A look at the GR predictions (34)–(36) does not
exhibit explicit strong-field effects. Indeed, the derivation of Eqs. (34)–(36) used
in a crucial way the ‘effacement of internal structure’ that occurs in the general
relativistic dynamics of compact objects. This non trivial property is rather
specific of GR and means that, in this theory, all the strong-field effects can be
absorbed in the definition of the masses ma and mb. One can, however, verify
that strong-field effects do enter the observable PK parameters k, γ, Ṗb etc. . . by
considering how the theoretical predictions (34)–(36) get modified in alternative
theories of gravity. The presence of such strong-field effects in PK parameters
was first pointed out in Ref. [7] (see also [39]) for the Jordan-Fierz-Brans-Dicke
theory of gravity, and in Ref. [8] for Rosen’s bi-metric theory of gravity. A
detailed study of such strong-field deviations was then performed in [33, 34, 35]
for general tensor-(multi-)scalar theories of gravity. In the following Section
we shall exhibit how such strong-field effects enter the various post-Keplerian
parameters.
Continuing our historical review of phenomenological pulsar tests, let us
come to the binary system which was the first one to provide several ‘pure
strong-field tests’ of relativistic gravity, without mixing of radiative effects:
PSR 1534+12. In this system, it was possible to measure the four (non ra-
diative) PK parameters k, γ, r and s. [We see from Eq. (25) that r and s mea-
sure, respectively, the range and the shape of the ‘Shapiro time delay’ ∆S .] The
measurement of the 4 PK parameters k, γ, r, s define 4 curves in the (ma,mb)
mass plane, and thereby yield 2 strong-field tests of GR. It was found in [40]
that GR passes these two tests. For instance, the ratio between the measured
value sobs of the phenomenological parameter5 s and the value sGR[kobs, γobs]
predicted by GR on the basis of the measurements of the two PK parameters k
and γ (which determine, via Eqs. (34) , (35), the GR-predicted value of ma and
mb) was found to be s
obs/sGR[kobs, γobs] = 1.004± 0.007 [40]. The most recent
data [41] yield sobs/sGR[kobs, γobs] = 1.000± 0.007. We see that we have here a
confirmation of the strong-field regime of GR at the 1% level.
Another way to get phenomenological tests of the strong field aspects of
gravity concerns the possibility of a violation of the strong equivalence principle.
This is parametrized by phenomenologically assuming that the ratio between the
gravitational and the inertial mass of the pulsar differs from unity (which is its
value in GR): (mgrav/minert)a = 1+∆a. Similarly to what happens in the Earth-
Moon-Sun system [42], the three-body system made of a binary pulsar and of the
Galaxy exhibits a ‘polarization’ of the orbit which is proportional to ∆ ≡ ∆a −
∆b, and which can be constrained by considering certain quasi-circular neutron-
star-white-dwarf binary systems [43]. See [44] for recently published improved
limits6 on the phenomenological equivalence-principle violation parameter ∆.
The Parkes multibeam survey has recently discovered several new interesting
‘relativistic’ binary pulsars, thereby giving a huge increase in the number of
phenomenological tests of relativistic gravity. Among those new binary pulsar
systems, two stand out as superb testing grounds for relativistic gravity: (i)
PSR J1141−6545 [46, 47], and (ii) the remarkable double binary pulsar PSR
J0737−3039A and B [2, 3, 48, 49] (see also the lectures by M. Kramer and
A. Possenti).
The PSR J1141−6545 timing data have led to the measurement of 3 PK
parameters: k, γ, and Ṗb [47]. As in PSR 1913+16 this yields one mixed
radiative-strong-field test7.
5As already mentioned the dimensionless parameter s is numerically equal (in all theories)
to the sine of the inclination angle i of the orbital plane, but it is better thought, in the PPK
formalism, as a phenomenological timing parameter determining the ‘shape’ of the logarithmic
time delay ∆S(T ).
6Note, however, that these limits, as well as those previously obtained in [45], assume
that the (a priori pulsar-mass dependent) parameter ∆ ≃ ∆a is the same for all the analyzed
pulsars.
7In addition, scintillation data have led to an estimate of the sine of the orbital inclination,
sin i [50]. As said above, sin i numerically coincides with the PK parameter s measuring the
‘shape’ of the Shapiro time delay. Therefore, one could use the scintillation measurements as
an indirect determination of s, thereby obtaining two independent tests from PSR J1141−6545
data. A caveat, however, is that the extraction of sin i from scintillation measurements rests
on several simplifying assumptions whose validity is unclear. In fact, in the case of PSR
J0737−3039 the direct timing measurement of s disagrees with its estimate via scintillation
The timing data of the millisecond binary pulsar PSR J0737−3039A have
led to the direct measurement of 5 PK parameters: k, γ, r, s and Ṗb [3, 48,
49]. In addition, the ‘double line’ nature of this binary system (i.e. the fact
that one observes both components, A and B, as radio pulsars) allows one to
perform new phenomenological tests by using Keplerian parameters. Indeed, the
simultaneous measurement of the Keplerian parameters xa and xb representing
the projected light crossing times of both pulsars (A and B) gives access to the
combined Keplerian parameter
Robs ≡
xobsb
xobsa
. (37)
On the other hand, the general derivation of [30] (applicable to any Lorentz-
invariant theory of gravity, and notably to any tensor-scalar theory) shows that
the theoretical prediction for the the ratio R, considered as a function of the
masses ma and mb, is
Rtheory =
. (38)
The absence of any explicit strong-field-gravity effects in the theoretical predic-
tion (38) (to be contrasted, for instance, with the predictions for PK parameters
in tensor-scalar gravity discussed in the next Section) is mainly due to the con-
vention used in [30] and [31] for defining the masses ma and mb. These are
always defined so that the Lagrangian for two non interacting compact objects
reads L0 =
−ma c2(1− v2a/c2)1/2. In other words, ma c2 represents the total
energy of body a. This means that one has implicitly lumped in the definition
of ma many strong-self-gravity effects. [For instance, in tensor-scalar gravity
ma includes not only the usual Einsteinian gravitational binding energy due
to the self-gravitational field gµν(x), but also the extra binding energy linked
to the scalar field ϕ(x).] Anyway, what is important is that, when performing
a phenomenological test from the measurement of a triplet of parameters, e.g.
{k, γ,R}, at least one parameter among them be a priori sensitive to strong-
field effects. This is enough for guaranteeing that the crossing of the three
curves ktheory(ma,mb) = k
obs, γtheory(ma,mb) = γ
obs, Rtheory(ma,mb) = R
is really a probe of strong-field gravity.
In conclusion, the two recently discovered binary pulsars PSR J1141−6545
and PSR J0737−3039 have more than doubled the number of phenomenological
tests of (radiative and) strong-field gravity. Before their discovery, the ‘canoni-
cal’ relativistic binary pulsars PSR 1913+16 and PSR 1534+12 had given us four
data [49]. It is therefore safer not to use scintillation estimates of sin i on the same footing as
direct timing measurements of the PK parameter s. On the other hand, a safe way of obtaining
an s-related gravity test consists in using the necessary mathematical fact that s = sin i ≤ 1.
In GR the definition xa = aa sin i/c leads to sin i = nxa/(β0 Xb). Therefore we can write the
inequality nxa/(β0(M)Xb) ≤ 1 as a phenomenological test of GR.
s ≤ 1
0 0.5 1 1.5 2 2.5
PSR J1141−6545
intersection
0 0.5 1 1.5 2 2.5
PSR B1534+12
intersection
0 0.5 1 1.5 2 2.5
PSR J0737−3039
intersection
0 0.5 1 1.5 2 2.5
2.5 ω
s ≤ 1
PSR B1913+16
intersection
Figure 1: Phenomenological tests of General Relativity obtained from Keplerian
and post-Keplerian timing parameters of four relativistic pulsars. Figure taken
from [51].
such tests: one (k−γ−Ṗb) test from PSR 1913+16 and three (k−γ−r−s−Ṗb8)
tests from PSR 1534+12. The two new binary systems have given us five9 more
phenomenological tests: one (k−γ− Ṗb) (or two, k−γ− Ṗb−s) tests from PSR
J1141−6545 and four (k−γ− r−s− Ṗb−R) tests from PSR J0737−303910. As
illustrated in Figure 1, these nine phenomenological tests of strong-field (and
radiative) gravity are all in beautiful agreement with General Relativity.
In addition, let us recall that several quasi-circular wide binaries, made of
a neutron star and a white dwarf, have led to high-precision phenomenological
confirmations [44] (in strong-field conditions) of one of the deep predictions of
General Relativity: the ‘strong’ equivalence principle, i.e. the fact that var-
ious bodies fall with the same acceleration in an external gravitational field,
independently of the strength of their self-gravity.
Finally, let us mention that Ref. [31] has extended the philosophy of the
8The timing measurement of Ṗ obs
in PSR 1534+12 is even more strongly affected by
kinematic corrections (Ḋ terms) than in the PSR 1913+16 case. In absence of a precise,
independent measurement of the distance to PSR 1534+12, the k−γ− Ṗb test yields, at best,
a ∼ 15% test of GR.
9Or even six, if we use the scintillation determination of s in PSR J1141−6545.
10The companion pulsar 0737−3039B being non recycled, and being visible only during a
small part of its orbit, cannot be timed with sufficient accuracy to allow one to measure any
of its post-Keplerian parameters.
phenomenological (parametrized post-Keplerian) analysis of timing data, to a
similar phenomenological analysis of pulse-structure data. Ref. [31] showed that,
in principle, one could extract up to 11 ‘post-Keplerian pulse-structure param-
eters’. Together with the 8 post-Keplerian timing parameters of a (single-line)
binary pulsar, this makes a total of 19 phenomenological PK parameters. As
these parameters depend not only on the two massesma,mb but also on the two
angles λ, η determining the direction of the spin axis of the pulsar, the maximum
number of tests one might hope to extract from one (single-line) binary pulsar
is 19 − 4 = 15. However, the present accuracy with which one can model and
measure the pulse structure of the known pulsars has not yet allowed one to
measure any of these new pulse-structure parameters in a theory-independent
and model-independent way.
Nonetheless, it has been possible to confirm the reality (and order of mag-
nitude) of the spin-orbit coupling in GR which was pointed out [52, 53] to be
observable via a secular change of the intensity profile of a pulsar signal. Confir-
mations of general relativistic spin-orbit effects in the evolution of pulsar profiles
were obained in several pulsars: PSR 1913+16 [54, 55], PSR B1534+12 [56] and
PSR J1141−6545 [57]. In this respect, let us mention that the spin-orbit interac-
tion affects also several PK parameters, either by inducing a secular evolution in
some of them (see [31]) or by contributing to their value. For instance, the spin-
orbit interaction contributes to the observed value of the periastron advance
parameter k an amount which is significant for the pulsars (such as 1913+16
and 0737−3039) where k is measured with high-accuracy. It was then pointed
out [58] that this gives, in principle, and indirect way of measuring the moment
of inertia of neutron stars (a useful quantity for probing the equation of state
of nuclear matter [59, 60]). However, this can be done only if one measures,
besides k, two other PK parameters with 10−5 accuracy. A rather tall order
which will be a challenge to meet.
The phenomenological approach to pulsar tests has the advantage that it can
confirm or invalidate a specific theory of gravity without making assumptions
about other theories. Moreover, as General Relativity has no free parameters,
any test of its predictions is a potentially lethal test. From this point of view, it is
remarkable that GR has passed with flying colours all the pulsar tests if has been
submitted to. [See, notably, Fig. 1.] As argued above, these tests have probed
strong-field aspects of gravity which had not been probed by solar-system (or
cosmological) tests. On the other hand, a disadvantage of the phenomenological
tests is that they do not tell us in any precise way which strong-field structures,
have been actually tested. For instance, let us imagine that one day one specific
PPK test fails to be satisfied by GR, while the others are OK. This leaves us in a
quandary: If we trust the problematic test, we must conclude that GR is wrong.
However, the other tests say that GR is OK. This example shows that we would
like to have some idea of what physical effects, linked to strong-field gravity,
enter in each test, or even better in each PK parameter. The ‘effacement of
internal structure’ which takes place in GR does not allow one to discuss this
issue. This gives us a motivation for going beyond the phenomenological PPK
approach by considering theory-dependent formalisms in which one embeds GR
within a space of alternative gravity theories.
5 Theory-space approach to testing relativistic
gravity with binary pulsar data
A complementary approach to testing gravity with binary pulsar data consists
in embedding General Relativity within a multi-parameter space of alternative
theories of gravity. In other words, we want to contrast the predictions of GR
with the predictions of continuous families of alternative theories. In so doing we
hope to learn more about which structures of GR are actually being probed in
binary pulsar tests. This is a bit similar to the well-known psycho-physiological
fact that the best way to appreciate a nuance of colour is to surround a given
patch of colour by other patches with slightly different colours. This makes it
much easier to detect subtle differences in colour. In the same way, we hope to
learn about the probing power of pulsar tests by seeing how the phenomeno-
logical tests summarized in Fig. 1 fail (or continue) to be satisfied when one
continuously deform, away from GR, the gravity theory which is being tested.
Let us first recall the various ways in which this theory-space approach has
been used in the context of the solar-system tests of relativistic gravity.
5.1 Theory-space approaches to solar-system tests of rel-
ativistic gravity
In the quasi-stationary weak-field context of the solar-system, this theory-space
approach has been implemented in two different ways. First, the parametrized
post-Newtonian (PPN) formalism [61, 62, 63, 42, 64, 65, 11, 66] describes
many ‘directions’ in which generic alternative theories of gravity might dif-
fer in their weak-field predictions from GR. In its most general versions the
PPN formalism contains 10 ‘post-Einstein’ PPN parameters, γ̄ ≡ γPPN − 111,
β̄ ≡ βPPN−1, ξ, α1, α2, α3, ζ1, ζ2, ζ3, ζ4. Each one of these dimensionless quanti-
ties parametrizes a certain class of slow-motion, weak-field gravitational effects
which deviate from corresponding GR predictions. For instance, γ̄ parametrizes
modifications both of the effect of a massive body (say, the Sun) on the light
passing near it, and of the terms in the two-body gravitational Lagrangian which
are proportional to (Gmamb/rab) · (va − vb)2/c2.
A second way of implementing the theory-space philosophy consists in con-
sidering some explicit, parameter-dependent family of alternative relativistic
theories of gravity. For instance, the simplest tensor-scalar theory of gravity
11The PPN parameter γPPN is usually denoted simply as γ. To distinguish it from the
Einstein-time-delay PPK timing parameter γ used above we add the superscript PPN. In
addition, as the value of γPPN in GR is 1, we prefer to work with the parameter γ̄ ≡ γPPN−1
which vanishes in GR, and therefore measures a ‘deviation’ from GR in a certain ‘direction’
in theory-space. Similarly with β̄ ≡ βPPN − 1.
put forward by Jordan [67], Fierz [68] and Brans and Dicke [69] has a unique
free parameter, say α20 = (2ωBD + 3)
−1. When α20 → 0, this theory reduces
to GR, so that α20 (or 1/ωBD) measures all the deviations from GR. When
considering the weak-field limit of the Jordan-Fierz-Brans-Dicke (JFBD) the-
ory, one finds that it can be described within the PPN formalism by choosing
γ̄ = −2α20(1 + α20)−1, β̄ = 0 and ξ = αi = ζj = 0.
Having briefly recalled the two types of theory-space approaches used to
discuss solar-system tests, let us now consider the case of binary-pulsar tests.
5.2 Theory-space approaches to binary-pulsar tests of rel-
ativistic gravity
There exist generalizations of these two different theory-space approaches to the
context of strong-field gravity and binary pulsar tests. First, the PPN formalism
has been (partially) extended beyond the ‘first post-Newtonian’ (1PN) order
deviations from GR (∼ v2/c2+Gm/c2 r) to describe 2PN order deviations from
[70]. Remarkably, there appear only two new parameters
at the 2PN level12: ǫ and ζ. Also, by expanding in powers of the self-gravity
parameters of body a and b the predictions for the PPK timing parameters
in generic tensor-multi-scalar theories, one has shown that these predictions
depended on several ‘layers’ of new dimensionless parameters [33]. Early among
these parameters one finds, the 1PN parameters β̄, γ̄ and then the basic 2PN
parameters ǫ and ζ, but one also finds further parameters β3, (ββ
′), β′′, . . .
which would not enter usual 2PN effects. The two approaches that we have just
mentionned can be viewed as generalizations of the PPN formalism.
There exist also useful generalizations to the strong-field context of the idea
of considering some explicit parameter-dependent family of alternative theo-
ries of relativistic gravity. Early studies [7, 8, 39] focussed either on the one-
parameter JFBD tensor-scalar theory, or on some theories which are not con-
tinuously connected to GR, such as Rosen’s bimetric theory of gravity. Though
the JFBD theory exhibits a marked difference from GR in that it predicts the
existence of dipole radiation, it has the disadvantage that the weak field, solar-
system constraints on its unique parameter α20 are so strong that they drastically
constrain (and essentially forbid) the presence of any non-radiative, strong-field
deviations from GR. In view of this, it is useful to consider other ‘mini-spaces’
of alternative theories.
A two-parameter mini-space of theories, that we shall denote13 here as
′, β′′), was introduced in [33]. This two-parameter family of tensor-bi-scalar
theories was constructed so as to have exactly the same first post-Newtonian
limit as GR (i.e. γ̄ = β̄ = · · · = 0), but to differ from GR in its predictions
12When restricting oneself to the general class of tensor-multi-scalar theories. At the 1PN
level, this restriction would imply that only the ‘directions’ γ̄ and β̄ are allowed.
13We add here an index 2 to T as a reminder that this is a class of tensor-bi-scalar theories,
i.e. that they contain two independent scalar fields ϕ1, ϕ2 besides a dynamical metric gµν .
for the various observables that can be extracted from binary pulsar data. Let
us give one example of this behaviour of the T2(β
′, β′′) class of theories. For
a general theory of gravity we expect to have violations of the strong equiva-
lence principle in the sense that the ratio between the gravitational mass of a
self-gravitating body to its inertial mass will admit an expansion of the type
mgrava
minerta
≡ 1 + ∆a = 1−
η1 ca + η2 c
a + . . . (39)
where ca ≡ −2 ∂ lnma∂ lnG measures the ‘gravitational compactness’ (or fractional
gravitational binding energy, ca ≃ −2Egrava /ma c2) of body a. The numerical
coefficient η1 of the contribution linear in ca is a combination of the first post-
Newtonian order PPN parameters, namely η1 = 4 β̄ − γ̄ [42]. The numerical
coefficient η2 of the term quadratic in ca is a combination of the 1PN and 2PN
parameters. When working in the context of the T2(β
′, β′′) theories, the 1PN
parameters vanish exactly (β̄ = 0 = γ̄) and the coefficient of the quadratic term
becomes simply proportional to the theory parameter β′ : η2 =
Bβ′, where
B ≈ 1.026. This example shows explicitly how binary pulsar data (here the data
constraining the equivalence principle violation parameter ∆ = ∆a − ∆b, see
above) can go beyond solar-system experiments in probing certain strong-self-
gravity effects. Indeed, solar-system experiments are totally insensitive to 2PN
parameters because of the smallness of ca ∼ Gma/c2Ra and of the structure
of 2PN effects [70]. By contrast, the ‘compactness’ of neutron stars is of order
ca ∼ 0.21ma/M⊙ ∼ 0.3 [33] so that the pulsar limit |∆| < 5.5×10−3 [44] yields,
within the T2(β
′, β′′) framework, a significant limit on the dimensionless (2PN
order) parameter β′ : |β′| < 0.12.
Ref. [35] introduced a new two-parameter mini-space of gravity theories,
denoted here as T1(α0, β0), which, from the point of view of theoretical physics,
has several advantages over the T2(β
′, β′′) mini-space mentionned above. First,
it is technically simpler in that it contains only one scalar field ϕ besides the
metric gµν (hence the index 1 on T1(α0, β0)). Second, it contains only positive-
energy excitations (while one combination of the two scalar fields of T2(β
′, β′′)
carried negative-energy waves). Third, it is the minimal way to parametrize
the huge class of tensor-mono-scalar theories with a ‘coupling function’ a(ϕ)
satisfying some very general requirements (see below).
Let us now motivate the use of tensor-scalar theories of gravity as alternatives
to general relativity.
5.3 Tensor-scalar theories of gravity
Let us start by recalling (essentially from [35]) why tensor-(mono)-scalar theories
define a natural class of alternatives to GR. First, and foremost, the existence
of scalar partners to the graviton is a simple theoretical possibility which has
surfaced many times in the development of unified theories, from Kaluza-Klein
to superstring theory. Second, they are general enough to describe many inter-
esting deviations from GR (both in weak-field and in strong field conditions),
but simple enough to allow one to work out their predictions in full detail.
Let us therefore consider a general tensor-scalar action involving a metric
g̃µν (with signature ‘mostly plus’), a scalar field Φ, and some matter variables
ψm (including gauge bosons):
16πG∗
g̃1/2
F (Φ)R̃ − Z(Φ)g̃µν∂µΦ ∂νΦ− U(Φ)
+ Sm[ψm; g̃µν ] .
For simplicity, we assume here that the weak equivalence principle is satisfied,
i.e., that the matter variables ψm are all coupled to the same ‘physical metric’
g̃µν . The general model (40) involves three arbitrary functions: a function F (Φ)
coupling the scalar Φ to the Ricci scalar of g̃µν , R̃ ≡ R(g̃µν), a function Z(Φ)
renormalizing the kinetic term of Φ, and a potential function U(Φ). As we
have the freedom of arbitrary redefinitions of the scalar field, Φ → Φ′ = f(Φ),
only two functions among F , Z and U are independent. It is often convenient
to rewrite (40) in a canonical form, obtained by redefining both Φ and g̃µν
according to
g∗µν = F (Φ) g̃µν , (41)
ϕ = ±
F ′2(Φ)
F 2(Φ)
F (Φ)
. (42)
This yields
16πG∗
∗ [R∗ − 2gµν∗ ∂µϕ∂νϕ− V (ϕ)] + Sm
2(ϕ) g∗µν
where R∗ ≡ R(g∗µν), where the potential
V (ϕ) = F−2(Φ)U(Φ) , (44)
and where the conformal coupling function A(ϕ) is given by
A(ϕ) = F−1/2(Φ) , (45)
with Φ(ϕ) obtained by inverting the integral (42).
The two arbitrary functions entering the canonical form (43) are: (i) the con-
formal coupling function A(ϕ), and (ii) the potential function V (ϕ). Note that
the ‘physical metric’ g̃µν (the one measured by laboratory clocks and rods) is
conformally related to the ‘Einstein metric’ g∗µν , being given by g̃µν = A
2(ϕ) g∗µν .
The canonical representation is technically useful because it decouples the two
14Actually, most unified models suggest that there are violations of the weak equivalence
principle. However, the study of general string-inspired tensor-scalar models [71] has found
that the composition-dependent effects would be negligible in the gravitational physics of
neutron stars that we consider here. The experimental limits on tests of the equivalence
principle would, however, bring a strong additional constraint of order 10−5 α2
∼ ∆a/a .
10−12. As this constraint is strongly model-dependent, we will not use it in our exclusion
plots below. One should, however, keep in mind that a limit on the scalar coupling strength
of order α2
. 10−7 [71, 72] is likely to exist in many, physically-motivated, tensor-scalar
models.
irreducible propagating excitations: the spin-0 excitations are described by ϕ,
while the pure spin-2 excitations are described by the Einstein metric g∗µν (with
kinetic term the usual Einstein-Hilbert action ∝ R(g∗µν)).
In many technical developments it is useful to work with the logarithmic
coupling function a(ϕ) such that:
a(ϕ) ≡ lnA(ϕ) ; A(ϕ) ≡ ea(ϕ) . (46)
In the case of the general model (40) this logarithmic15 coupling function is
given by
a(ϕ) = −
lnF (Φ) ,
where Φ(ϕ) must be obtained from (42).
In the following, we shall assume that the potential V (ϕ) is a slowly varying
function of ϕ which, in the domain of variation we shall explore, is roughly
equivalent to a very small mass term V (ϕ) ∼ 2m2ϕ(ϕ−ϕ0)2 with m2ϕ of cosmo-
logical order of magnitude m2ϕ = O(H20 ), or, at least, with a range λϕ = m−1ϕ
much larger than the typical length scales that we shall consider (such as the
size of the binary orbit, or the size of the Galaxy when considering violations of
the strong equivalence principle). Under this assumption16 the potential func-
tion V (ϕ) will only serve the role of fixing the value of ϕ far from the system
(to ϕ(r = ∞) = ϕ0), and its effect on the propagation of ϕ within the system
will be negligible. In the end, the tensor-scalar phenomenology that we shall
explore only depends on one function: the coupling function a(ϕ).
Let us consider some examples to see what kind of coupling functions might
naturally arise. First, the simplest case is the Jordan-Fierz-Brans-Dicke action,
which is of the general type (40) with
F (Φ) = Φ (47)
Z(Φ) = ωBD Φ
−1 , (48)
where ωBD is an arbitrary constant. Using Eqs. (42), (45) above, one finds that
− 2α0 ϕ = lnΦ and that the (logarithmic) coupling function is simply
a(ϕ) = α0 ϕ+ const. , (49)
where α0 = ∓(2ωBD + 3)−1/2, depending on the sign chosen in Eq. (42). Inde-
pendently of this sign, one has the link
α20 =
2ωBD + 3
. (50)
15As we shall mostly work with a(ϕ) below, we shall henceforth drop the adjective ‘loga-
rithmic’.
16Note, however, that, as was recently explored in [73, 74, 75], a sufficiently fast varying
potential V (ϕ) can change the tensor-scalar phenomenology by endowing ϕ with a mass term
m2ϕ =
∂2V/∂ϕ2 which strongly depends on the local value of ϕ and, thereby can get large
in sufficiently dense environments.
Note that 2ωBD + 3 must be positive for the spin-0 excitations to have the
correct (non ghost) sign.
Let us now discuss the often considered case of a massive scalar field having
a nonminimal coupling to curvature
16πG∗
g̃1/2
R̃− g̃µν∂µΦ ∂νΦ−m2ΦΦ2 + ξR̃Φ2
+ Sm[ψm; g̃µν ] .
This is of the form (40) with
F (Φ) = 1 + ξΦ2 , Z(Φ) = 1 , U(Φ) = m2ΦΦ
2 . (52)
The case ξ = − 1
is usually referred to as that of ‘conformal coupling’. With the
variables (51) the theory is ghost-free only if 2 (1+ ξΦ2)2 (dϕ/dΦ)2 = 1+ ξ(1 +
6 ξ)Φ2 is everywhere positive. If we do not wish to restrict the initial values of
Φ, we must have ξ(1+6 ξ) > 0. Introducing then the notation χ ≡
ξ(1 + 6 ξ),
we get the following link between Φ and ϕ:
1 + 2χΦ
1 + χ2 Φ2 + χΦ
1 + χ2 Φ2 −
1 + ξΦ2
. (53)
For small values of Φ, this yields ϕ = Φ/
2 + O(Φ3). The potential and the
coupling functions are given by
V (ϕ) =
1 + ξΦ2
, (54)
a(ϕ) = −1
ln(1 + ξΦ2) . (55)
These functions have singularities when 1+ ξΦ2 vanishes. If we do not wish
to restrict the initial value of Φ we must assume ξ > 0 (which then implies our
previous assumption ξ(1+6 ξ) > 0). Then there is a one-to-one relation between
Φ and ϕ over the entire real line. Small values of Φ correspond to small values
of ϕ and to a coupling function
a(ϕ) = − ξ ϕ2 +O(ϕ4) . (56)
On the other hand, large values of |Φ| correspond to large values of |ϕ|, and to
a coupling function of the asymptotic form
a(ϕ) ≃ −
|ϕ|+ const. (57)
The potential V (ϕ) has a minimum at ϕ = 0, as well as other minima at
ϕ → ±∞. If we assume, for instance, that m2Φ and the cosmological dynamics
are such that the cosmological value of ϕ is currently attracted towards zero,
the value of ϕ at large distances from the local gravitating systems we shall
consider will be ϕ0 ≪ 1.
As a final example of a possible tensor-scalar gravity theory, let us discuss
the string-motivated dilaton-runaway scenario considered in [76]. The starting
action (a functional of ḡµν and Φ) was taken of the general form
Bg(Φ)
BΦ(Φ)
[2 �̄Φ
−(∇̄Φ)2]− 1
BF (Φ)F̄
2 − V (Φ) + · · ·
and it was assumed that all the functions Bi(Φ) have a regular asymptotic be-
havior when Φ → +∞ of the form Bi(Φ) = Ci+O(e−Φ). Under this assumption
the early cosmological evolution can push Φ towards +∞ (hence the name ‘run-
away dilaton’). In the canonical, ‘Einstein frame’ representation (43), one has,
for large values of Φ, Φ ≃ c ϕ, where c is a numerical constant, and the coupling
function to hadronic matter is given by
ea(ϕ) ∝ ΛQCD(ϕ) ∝ B−1/2g (ϕ) exp[−8π2 b−13 BF (ϕ)]
where b3 is the one-loop rational coefficient entering the renormalization-group
running of the gauge field coupling g2F . This finally yields a coupling function
of the approximate form (for large values of ϕ):
a(ϕ) ≃ k e−cϕ + const. ,
where the dimensionless constants k and c are both expected to be of order unity.
[The constant c must be positive, but the sign of k is not a priori restricted.]
Summarizing: the JFBD model yields a coupling function which is a linear
function of ϕ, Eq. (49), a nonminimally coupled scalar yields a coupling function
which interpolates between a quadratic function of ϕ, Eq. (56), and a linear one,
Eq. (57), and the dilaton-runaway scenario of Ref. [76] yields a coupling function
of a decaying exponential type.
5.4 The role of the coupling function a(ϕ); definition of the
two-dimensional space of tensor-scalar gravity theo-
ries T1(α0, β0)
Let us now discuss how the coupling function a(ϕ) enters the observable pre-
dictions of tensor-scalar gravity at the first post-Newtonian (1PN) level, i.e.,
in the weak-field conditions appropriate to solar-system tests. It was shown in
previous work that, if one uses appropriate units in the asymptotic region far
from the system, namely units such that the asymptotic value a(ϕ0) of a(ϕ)
vanishes17, all observable quantities at the 1PN level depend only on the values
17In these units the Einstein metric g∗µν and the physical metric g̃µν asymptotically coin-
cide.
of the first two derivatives of the a(ϕ) at ϕ = ϕ0. More precisely, if one defines
α(ϕ) ≡ ∂ a(ϕ)
; β(ϕ) ≡ ∂ α(ϕ)
∂2 a(ϕ)
, (58)
and denotes by α0 ≡ α(ϕ0), β0 ≡ β(ϕ0) their asymptotic values, one finds
(see, e.g., [33]) that the effective gravitational constant between two bodies (as
measured by a Cavendish experiment) is given by
G = G∗(1 + α
0) , (59)
while, among the PPN parameters, only the two basic Eddington ones, γ̄ ≡
γPPN − 1, and β̄ ≡ βPPN − 1, do not vanish, and are given by
γ̄ ≡ γPPN − 1 = −2 α
1 + α20
, (60)
β̄ ≡ βPPN − 1 = 1
α0 β0 α0
(1 + α20)
. (61)
The structure of the results (60) and (61) can be transparently expressed by
means of simple (Feynman-like) diagrams (see, e.g., [77]). Eqs. (59) and (60)
correspond to diagrams where the interaction between two worldlines (repre-
senting two massive bodies) is mediated by the sum of the exchange of one
graviton and one scalar particle. The scalar couples to matter with strength
G∗. The exchange of a scalar excitation then leads to a term ∝ α20. On
the other hand, Eq. (61) corresponds to a nonlinear interaction between three
worldlines involving: (i) the ‘generation’ of a scalar excitation on a first world-
line (factor α0), (ii) a nonlinear vertex on a second worldline associated to the
quadratic piece of a(ϕ) (aquad(ϕ) =
β0(ϕ−ϕ0)2; so that one gets a factor β0),
and (iii) the final ‘absorption’ of a scalar excitation on a third worldline (second
factor α0).
Eqs. (60) and (61) can be summarized by saying that the first two coefficients
in the Taylor expansion of the coupling function a(ϕ) around ϕ = ϕ0 (after
setting a(ϕ0) = 0)
a(ϕ) = α0(ϕ− ϕ0) +
β0(ϕ − ϕ0)2 + · · · (62)
suffice to determine the quasi-stationary, weak-field (1PN) predictions of any
tensor-scalar theory. In other words, the solar-system tests only explore the
‘osculating approximation’ (62) (slope and local curvature) to the function a(ϕ).
Note that GR corresponds to a vanishing coupling function a(ϕ) = 0 (so that
α0 = β0 = · · · = 0), the JFBD model corresponds to keeping only the first term
on the R.H.S. of (62), while, for instance, the nonminimally coupled scalar field
(with asymptotic value ϕ0 ≪ 1) does indeed lead to nonzero values for both α0
and β0, namely
α0 ≃ − 2 ξ ϕ0 ; β0 ≃ − 2 ξ . (63)
Finally the dilaton-runaway scenario considered above leads also to non zero
values for both α0 and β0, namely
α0 ≃ − k c e−cϕ0 ; β0 ≃ + k c2 e−cϕ0 , (64)
for a largish value of ϕ0. Note that the dilaton-runaway model naturally predicts
that α0 ≪ 1, and that β0 is of the same order of magnitude as α0 : β0 ≃ − c α0
with c being (positive and) of order unity. The interesting outcome is that such
a model is well approximated by the usual JFBD model (with β0 = 0). This
shows that a JFBD-like theory could come out from a model which is initially
quite different from the usual exact JFBD theory.
As we shall discuss in detail below, solar-system tests constrain α20 and α
0 |β0|
to be both small. This immediately implies that |α0| must be small, i.e., that
the scalar field is linearly weakly coupled to matter. On the other hand, the
quadratic coupling parameter β0 is not directly constrained. Both its magnitude
and its sign can be more or less arbitrary. Note that there are no a priori sign
restrictions on β0. The conformal factor A
2(ϕ) = exp(2 a(ϕ)) entering Eq. (43)
had to be positive, but this leads to no restrictions on the sign of a(ϕ) and of
its various derivatives18. For instance, in the nonminimally coupled scalar field
case, it seemed more natural to require ξ > 0, which leads to a negative β0 in
view of Eq. (63).
Let us summarize the results above: (i) the most general tensor-scalar the-
ory19 is described by one arbitrary function a(ϕ); and (ii) weak-field tests depend
only on the first two terms, parametrized by α0 and β0, in the Taylor expansion
(62) of a(ϕ) around its asymptotic value ϕ0.
From this follows a rather natural way to define a simplemini space of tensor-
scalar theories. It suffices to consider the two-dimensional space of theories, say
T1(α0, β0), defined by the coupling function which is a quadratic polynomial in
ϕ [34, 35], say
aα0,β0(ϕ) = α0(ϕ− ϕ0) +
β0(ϕ− ϕ0)2 . (65)
As indicated, this class of theories depends only on two parameters: α0 and
β0. The asymptotic value ϕ0 of ϕ does not count as a third parameter (when
using the form (65)) because one can always work with the shifted field ϕ̄ ≡
ϕ−ϕ0, with asymptotic value ϕ̄0 = 0 and coupling function aα0,β0(ϕ̄) = α0 ϕ̄+
β0 ϕ̄
2. Moreover, as already said, the asymptotic value a(ϕ0) of a(ϕ) has also
no physical meaning, because one can always use units such that it vanishes (as
done in (65)).
18As explained above, we assume here the presence of a potential term V (ϕ) to fix the
asymptotic value ϕ0 of ϕ. If the potential V (ϕ) is absent (or negligible), the ‘attractor
mechanism’ of Refs. [78, 71] would attract ϕ to a minimum of the coupling function a(ϕ),
thereby favoring a positive value of β0.
19Under the assumption that the potential V (ϕ) is a slowly-varying function of ϕ, which
modifies the propagation of ϕ only on very large scales.
Note also that an alternative way to represent the same class of theories is
to use a coupling function of the very simple form
aβ(ϕ) =
β ϕ2 , (66)
but to keep the asymptotic value ϕ0 as an independent parameter. This class
of theories is clearly equivalent to T1(α0, β0), Eq. (65), with the dictionary:
α0 = β ϕ0, β0 = β.
5.5 Tensor-scalar gravity, strong-field effects, and binary-
pulsar observables
Having chosen some mini-space of gravity theories, we now wish to derive what
predictions these theories make for the timing observables of binary pulsars. To
do this we need to generalize the general relativistic treatment of the motion and
timing of binary systems comprising strongly self-gravitating bodies summarized
above. Let us recall that this treatment was based on a multi-chart method,
using a matching between two separate problems: (i) the ‘internal problem’
considers each strongly self-gravitating body in a suitable approximately freely
falling frame where the influence of its companion is small, and (ii) the ‘external
problem’ where the two bodies are described as effective point masses which
interact via the various fields they are coupled to. Let us first consider the
internal problem, i.e., the description of a neutron star in an approximately
freely falling frame where the influence of the companion is reduced to imposing
some boundary conditions on the tensor and scalar fields with which it interacts
[7, 8, 33, 34, 35]. The field equations of a general tensor-scalar theory, as derived
from the canonical action (43) (neglecting the effect of V (ϕ)) read
R∗µν = 2 ∂µϕ∂νϕ+ 8πG∗
T ∗µν −
T ∗g∗µν
, (67)
�g∗ ϕ = − 4πG∗ α(ϕ)T∗ , (68)
where T
∗ ≡ 2 c (g∗)−1/2 δSm/δg∗µν denotes the material stress-energy tensor in
‘Einstein units’, and α(ϕ) the ϕ-derivative of the coupling function, see Eq. (58).
All tensorial operations in Eqs. (67) and (68) are performed by using the Einstein
metric g∗µν .
Explicitly writing the field equations (67) and (68) for a slowly rotating
(stationary, axisymmetric) neutron star, labelled20 A, leads to a coupled set of
ordinary differential equations constraining the radial dependence of g∗µν and
ϕ [35, 79]. Imposing the boundary conditions g∗µν → ηµν , ϕ → ϕa at large
radial distances, finally determines the crucial ‘form factors’ (in Einstein units)
describing the effective coupling between the neutron star A and the fields to
20We henceforth use the labels A and B for the (recycled) pulsar and its companion, instead
of the labels a and b used above. We henceforth use the label a to denote the asymptotic
value of some quantity (at large radial distances within the local frame, Xi
or Xi
, of the
considered neutron star A or B).
which it is sensitive: total mass mA(ϕa), total scalar charge ωA(ϕa), and inertia
moment IA(ϕa). As indicated, these quantities are functions of the asymptotic
value ϕa of ϕ felt by the considered neutron star
21. They satisfy the relation
ωA(ϕa) = −∂ mA(ϕa)/∂ ϕa. From them, one defines other quantities that play
an important role in binary pulsar physics, notably
αA(ϕa) ≡ −
≡ ∂ lnmA
, (69)
βA(ϕa) ≡
, (70)
as well as
kA(ϕa) ≡ −
∂ ln IA
. (71)
The quantity αA, Eq. (69), plays a crucial role. It measures the effective coupling
strength between the neutron star and the ambient scalar field. If we formally let
the self-gravity of the neutron A tend toward zero (i.e., if we consider a weakly
self-gravitating object), the function αA(ϕa) becomes replaced by α(ϕa) where
α(ϕ) ≡ ∂ a(ϕ)/∂ ϕ is the coupling strength appearing in the R.H.S. of Eq. (68).
Roughly speaking, we can think of αA(ϕa) as a (suitable defined) average value
of the local coupling strength α(ϕ(r)) over the radial profile of the neutron star
0.5 1 1.5 2 2.5 3
critical
maximum
maximum
mass in GR
scalar charge
baryonic mass
neutron star
Figure 2: Dependence upon the baryonic mass m̄A of the coupling parameter
αA in the theory T1(α0, β0) with α0 = −0.014, β0 = −6. Figure taken from
[80].
It was pointed out in Refs. [34, 35] that the strong self-gravity of a neutron
star can cause the effective coupling strength αA(ϕa) to become of order unity,
21This ϕa is a combination of the cosmological background value ϕ0 and of the scalar
influence of the companion of the considered neutron star. It varies with the orbital period and
is determined as part of the ‘external problem’ discussed below. Note that, strictly speaking,
the label a (for asymptotic) should be indexed by the label of the considered neutron star: i.e.
one should use a label aA (and a locally asymptotic value ϕaA) when considering the neutron
star A, and a label aB (with a corresponding ϕaB ) when considering the neutron star B.
even when its weak-field counterpart α0 = α(ϕa) is extremely small (as is im-
plied by solar-system tests that put strong constraints on the PPN combination
γ̄ = −2α20/(1+α20)). This is illustrated, in the minimal context of the T1(α0, β0)
class of theories, in Figure 2.
Note that when the baryonic mass m̄A of the neutron star is smaller than
the critical mass m̄cr ≃ 1.24M⊙ the effective scalar coupling strength αA of
the star is quite small (because it is proportional to its weak-field limit α0 =
α(ϕa)). By contrast, when m̄A > m̄cr, |αA| becomes of order unity, nearly
independently of the externally imposed α0 = αa = α(ϕa). This interesting
non-perturbative behaviour was related in [34, 35] to a mechanism of spontaneous
scalarization, akin to the well-known mechanism of spontaneous magnetization
of ferromagnets. See also [51] for a simple analytical description of the behaviour
of αA.
Let us also mention in passing that, in the case where A is a black hole, the
effective coupling strength αA actually vanishes [33]. This result is related to
the impossibility of having (regular) ‘scalar hair’ on a black hole.
We have sketched above the first part of the matching approach to the mo-
tion and timing of strongly self-gravitating bodies: the ‘internal problem’. It
remains to describe the remaining ‘external problem’. As already mentionned
(and emphasized, in the present context, by Eardley [7, 11]), the most efficient
way to describe the external problem is, instead of matching in detail the exter-
nal fields (g∗µν , ϕ) to the fields generated by each body in its comoving frame, to
‘skeletonize’ the bodies by point masses. Technically this means working with
the action
16πG∗
∗ [R∗ − 2 gµν∗ ∂µϕ∂νϕ]
mA(ϕ(zA))(−g∗µν(zA) dz
1/2 , (72)
where the function mA(ϕ) in the last term on the R.H.S. is the function mA(ϕa)
obtained above by solving the internal problem. Eq. (72) indicates that the ar-
gument of this function is taken to be ϕa = ϕ(zA), i.e., the value that the scalar
field (as viewed in the external problem) takes at the location z
A of the center of
mass of body A. However, as body A is described, in the external problem, as a
point mass this causes a technical difficulty: the externally determined field ϕ(x)
becomes formally singular at the location of the point sources, so that ϕ(zA) is a
priori undefined. One can either deal with this problem by coming back to the
physically well-defined matching approach (which shows that ϕ(zA) should be
replaced by ϕa, the value of ϕ in an intermediate domain RA ≪ r ≪ |zA−zB |),
or use the efficient technique of dimensional regularization. This means that the
spacetime dimension D in Eq. (72) is first taken to have a complex value such
that ϕ(zA) is finite, before being analytically continued to its physical value
D = 4.
One then derives from the action (72) two important consequences for the
motion and timing of binary pulsars. First, one derives the Lagrangian de-
scribing the relativistic interaction between N strongly self-gravitating bodies
(including orbital ∼ (v/c)2 effects, and neglecting O(v4/c4) ones) [11, 7, 33, 39].
It is the sum of one-body, two-body and three-body terms.
The one-body action has the usual form of the sum (over the label A) of the
kinetic term of each point mass:
one-body
A = −mA c
1− v2A/c2
= −mA c2 +
(v2A)
. (73)
Here, we use Einstein units, and the inertial mass mA entering Eq. (73) is mA ≡
mA(ϕ0), where ϕ0 is the asymptotic value of ϕ far away from the considered
N -body system.
The two-body action is a sum over the pairs A,B of a term L
2-body
AB which
differs from the GR-predicted 2-body Lagrangian in two ways: (i) the usual
gravitational constant G appearing as an overall factor in L
2-body
AB must be re-
placed by an effective (body-dependent) gravitational constant (in the appro-
priate units mentioned above) given by
GAB = G∗(1 + αA αB) , (74)
and (ii) the relativistic (O(v2/c2)) terms in L2-bodyAB contain, in addition to those
predicted by GR, new velocity-dependent terms of the form
2-body
AB = (γ̄AB)
GAB mAmB
(vA − vB)2
, (75)
γ̄AB ≡ γAB − 1 = − 2
αA αB
1 + αA αB
. (76)
In these expressions αA ≡ αA(ϕ0) ≡ ∂ lnmA(ϕ0)/∂ϕ0 (see Eq. (69) with ϕa →
Finally, the 3-body action is a sum over the pairs B,C and over A (with
A 6= B, A 6= C, but the possibility of having B = C) of
3-body
ABC = −(1 + 2 β̄
GAB GAC mAmB mC
c2 rAB rAC
where
β̄ABC ≡ βABC − 1 =
αB βA αC
(1 + αA αB)(1 + αA αC)
, (78)
with βA = ∂αA(ϕ0)/∂ϕ0 (see Eq. (70) with ϕa → ϕ0).
When comparing the strong-field results (74), (76), (78) to their weak-field
counterparts (59), (60), (61) one sees that the body-dependent quantity αA
replaces the weak-field coupling strength α0 in all quantities which are linked
to a scalar effect generated by body A. Note also that, in keeping with the
‘3-body’ nature of Eq. (77), the quantity βABC −1 is linked to scalar interactions
which are generated in bodies B and C and which nonlinearly interact on body
A. The notation used above has been chosen to emphasize that γAB and β
are strong-field analogs of the usual Eddington parameters γPPN, βPPN, so that
γ̄AB and β̄
BC are strong-field analogs of the ‘post-Einstein’ 1PN parameters γ̄
and β̄ (which vanish in GR). Indeed the usual PPN results for the post-Einstein
terms in the O(1/c2) 2-body and 3-body Lagrangians are obtained by replacing
in Eqs. (75) and (77) γ̄AB → γ̄, β̄ABC → β̄ and GAB → G.
The non-perturbative strong-field effects discussed above show that the strong
self-gravity of neutron stars can cause γAB and β
BC to be significantly different
from their GR values γGR = 1, βGR = 1, in some scalar-tensor theories having
a small value of the basic coupling parameter α0 (so that γ
PPN − 1 ∝ α20 and
βPPN − 1 ∝ β0 α20 are both small). For instance, Fig. 2 shows that it is possible
to have αA ∼ αB ∼ ± 0.6 which implies γAB − 1 ∼ − 0.53, i.e., a 50% deviation
from GR! Even larger effects can arise in βABC − 1 because of the large values
that βA = ∂αA/∂ϕ0 can reach near the spontaneous scalarization transition
[35].
Those possible strong-field modifications of the effective Eddington param-
eters γAB, β
BC , which parametrize the ‘first post-Keplerian’ (1PK) effects
(i.e., the orbital effects ∼ v2/c2 smaller than those entailed by the Lagrangian
A 6=B
GAB mAmB/rAB), can then significantly modify the usual
GR predictions relating the directly observable parametrized post-Keplerian
(PPK) parameters to the values of the masses of the pulsar and its compan-
ion. As worked out in Refs. [11, 31, 33, 35] one finds the following modified
predictions for the PPK parameters k ≡ 〈ω̇〉/n, r and s:
kth(mA,mB) =
1− e2
GAB(mA +mB)n
αA αB
1 + αA αB
− XA βB α
A +XB βA α
6 (1 + αA αB)2
, (79)
rth(mA,mB) = G0B mB , (80)
sth(mA,mB) =
GAB(mA +mB)n
]−1/3
. (81)
Here, the label A refers to the object which is timed (‘the pulsar’22), the label B
refers to its companion, xA = aA sin i/c denotes the projected semi-major axis
of the orbit of A (in light seconds), XA ≡ mA/(mA+mB) andXB ≡ mB/(mA+
mB) = 1 − XA the mass ratios, n ≡ 2π/Pb the orbital frequency and G0B =
G∗(1 + α0 αB) the effective gravitational constant measuring the interaction
22In the double binary pulsar, both the first discovered pulsar and its companion are pulsars.
However, the companion B is a non recycled, slow pulsar whose motion is well described by
Keplerian parameters only.
between B and a test object (namely electromagnetic waves on their way from
the pulsar toward the Earth). In addition one must replace the unknown bare
Newtonian G∗ by its expression in terms of the one measured in Cavendish
experiments, i.e., G∗ = G/(1 + α
0) as deduced from Eq. (59).
The modified theoretical prediction for the PPK parameter γ entering the
‘Einstein time delay’ ∆E , Eq. (24), is more complicated to derive because one
must take into account the modulation of the proper spin period of the pulsar
caused by the variation of its moment of inertia IA under the (scalar) influence
of its companion [11, 7, 35]. This leads to
γth(mA,mB) =
1 + αA αB
GAB(mA +mB)n
[XB(1 + αA αB) + 1 + kA αB ] , (82)
where kA(ϕ0) = −∂ ln IA(ϕ0)/∂ϕ0 (see Eq. (71) with ϕa → ϕ0). Numerical
studies [35] show that kA can take quite large values. Actually, the quantity
kA αB entering (82) blows up near the scalarization transition when α0 → 0
(keeping β0 < 0 fixed). In other words a theory which is closer to GR in weak-
field conditions predicts larger deviations in the strong-field regime.
The structure dependence of the effective gravitational constantGAB , Eq. (74),
has also the consequence that the object A does not fall in the same way
as B in the gravitational field of the Galaxy. As most of the mass of the
Galaxy is made of non strongly-self-gravitating bodies, A will fall toward the
Galaxy with an acceleration ∝ GA0, while B will fall with an acceleration
∝ GB0. Here, as above, GA0 = G0A = G∗(1 + α0 αA) is the effective gravi-
tational constant between A and any weakly self-gravitating body. As pointed
out in Ref. [43] this possible violation of the universality of free fall of self-
gravitating bodies can be constrained by using observational data on the class
of small-eccentricity long-orbital-period binary pulsars. More precisely, the
quantity which can be observationally constrained is not exactly the violation
∆AB = (G0A −G0B)/G = (1 +α20)−1(α0 αA −α0 αB) of the strong equivalence
principle [which simplifies to ∆A0 = (G0A − G)/G = (1 + α20)−1(α0 αA − α20)
in the case of observational relevance where one neglects the self-gravity of the
white-dwarf companion] but rather23 [33]
∆effective ≡
2 γAB − (XA βBAA +XB βABB) + 2
(1 + αA αB)
−3/2(1 + α20)
−1(α0 αA − α0 αB) . (83)
Here, the index B (= white-dwarf companion) can be replaced by 0 (weakly self-
gravitating body) so that, for instance, γAB = γA0 = 1− 2αAα0/(1+αA α0) =
(1− αA α0)/(1 + αA α0), as deduced from Eq. (76).
23This refinement is given here for pedagogical completeness. However, in practice, the
lowest-order result ∆ ≃ (1 + α2
)−1(α0 αA − α
) ≃ α0 αA − α
is accurate enough.
It remains to discuss the possible strong-field modifications of the theoretical
prediction for the orbital period derivative Ṗb = Ṗ
b (mA,mB). This is obtained
by deriving from the effective action (72) the energy lost by the binary system
in the form of fluxes of spin-2 and spin-0 waves at infinity. The needed results
in a generic tensor-scalar theory were derived in Refs. [33, 39] (in addition one
must take into account the tensor-scalar modification of the additional ‘varying-
Doppler’ contribution to the observed Ṗb due to the Galactic acceleration [38]).
The final result for Ṗb is of the form
Ṗ thb (mA,mB) = Ṗ
monopole
bϕ + Ṗ
dipole
bϕ + Ṗ
quadrupole
bϕ + Ṗ
quadrupole
galagtic
bGR + δ
th Ṗ
galactic
b , (84)
where, for instance, Ṗ
monopole
bϕ is (heuristically
24) related to the monopolar flux
of spin-0 waves at infinity. The term Ṗ
quadrupole
bg∗ corresponds to the usual
quadrupolar flux of spin-2 waves at infinity. It reads:
quadrupole
bg∗ (mA,mB) = −
5(1 + αA αB)
(mA +mB)2
GAB(mA +mB)n
1 + 73 e2/24 + 37 e4/96
(1− e2)7/2
with GAB = G∗(1 + αA αB) = G(1 + αA αB)/(1 + α
0), where G∗ is the ‘bare’
gravitational constant appearing in the action, while G is the gravitational con-
stant measured in Cavendish experiments. The flux (85) is the only one which
survives in GR (although without any αA-related modifications). Among the
several other contributions which arise in tensor-scalar theories, let us only write
down the explicit expression of the contribution to (84) coming from the dipolar
flux of scalar waves. Indeed, this contribution is, in most cases, the dominant
one [7] because it scales as (v/c)3, while the monopolar and quadrupolar con-
tributions scale as (v/c)5. It reads
dipole
bϕ (mA,mB) = −2π
G∗mAmB n
c3(mA +mB)
1 + e2/2
(1 − e2)5/2
(αA − αB)2 . (86)
Note that the dipolar effect (86) vanishes when αA = αB . Indeed, a binary
system made of two identical objects (A = B) cannot select a preferred direction
for a dipole vector, and cannot therefore emit any dipolar radiation. This also
implies that double neutron star systems (which tend to have mA ≈ mB ∼
1.35M⊙) will be rather poor emitters of dipolar radiation (though (86) still tends
to dominate over the other terms in (84), because of the remaining difference
(mA − mB)/(mA + mB) 6= 0). By contrast, very dissymmetric systems such
24Contrary to the GR case where a lot of effort was spent to show how the observed Ṗb
was directly related to the GR predictions for the (v/c)5-accurate orbital equations of motion
of a binary system [9], we use here the indirect and less rigorous argument that the energy
flux at infinity should be balanced by a corresponding decrease of the mechanical energy of
the binary system.
as a neutron-star and a white-dwarf (or a neutron-star and a black hole) will
be very efficient emitters of dipolar radiation, and will potentially lead to very
strong constraints on tensor-scalar theories. See below.
5.6 Theory-space analyses of binary pulsar data
Having reviewed the theoretical results needed to discuss the predictions of
alternative gravity theories, let us end by summarizing the results of various
theory-space analyses of binary pulsar data.
Let us first recall what are the best, current solar-system limits on the two
1PN ‘post-Einstein’ parameters γ̄ ≡ γPPN − 1 and β̄ ≡ βPPN − 1. They are:
γ̄ = (2.1± 2.3)× 10−5 , (87)
from frequency shift measurements made with the Cassini spacecraft [81], which
supersedes the constraint
γ̄ = (−1.7± 4.5)× 10−4 (88)
from VLBI measurements [82],
|2 γ̄ − β̄| < 3× 10−3 , (89)
from Mercury’s perihelion shift [66, 83], and
4 β̄ − γ̄ = (4.4± 4.5)× 10−4 , (90)
from Lunar laser ranging measurements [84].
Concerning binary pulsar data, we can make use of the published measure-
ments of various Keplerian and post-Keplerian timing parameters in the binary
pulsars: PSR 1913+16 [37], PSR B1534+12 [41], PSR J1141−6545 [47] and
PSR J0737−3039A+B [3, 48, 49]. In addition, we can use25 the recently up-
dated limit on the parameter ∆ measuring a possible violation of the strong
equivalence principle (SEP), namely |∆| < 5.5 × 10−3 at the 95% confidence
level [44].
This ensemble of solar-system and binary-pulsar data can then be analyzed
within any given parametrized theoretical framework. For instance, one might
work within
(i) the 4-parameter framework T0(γ̄, β̄; ǫ, ζ) [70] which defines the 2PN exten-
sion of the original (Eddington) PPN framework T0(γ̄, β̄); or
(ii) the 2-parameter class of tensor-mono-scalar theories T1(α0, β0) [34]; or
25There is, however, a caveat in the theoretical use one can make of the phenomenological
limits on ∆. Indeed, in the small-eccentricity long-orbital-period binary pulsar systems used
to constrain ∆ one does not have access to enough PK parameters to measure the pulsar mass
mA directly. As the theoretical expression of ∆ ≃ α0 αA −α
depends on mA (through αA),
one needs to assume some fiducial value of mA (say mA ≃ 1.35M⊙).
(iii) the 2-parameter class of tensor-bi-scalar theories T2(β
′, β′′) [33].
Here, the index 0 on T0(γ̄, β̄; ǫ, ζ) is a reminder of the fact that this framework
is not a family of specific theories (it contains zero explicit dynamical fields),
but is a parametrization of 2PN deviations from GR. As a consequence, its use
for analyzing binary pulsar data is somewhat ill-defined because one needs to
truncate the various timing observables (which are functions of the compactness
of the two bodies A and B, say PPK = f(cA, cB)) at the 2PN order (i.e. es-
sentially at the quadratic order in cA and/or cB). For some observables (or for
product of observables) there might be several ways of defining this truncation.
In spite of this slight inconvenience, the use of the T0(γ̄, β̄; ǫ, ζ) framework is
conceptually useful because it shows very clearly why and how binary-pulsar
data can probe the behaviour of gravitational theories beyond the usual 1PN
regime probed by solar-system tests.
For instance, the parameter ∆A ≡ mgravA /minertA − 1 measuring the strong
equivalence principle (SEP) violation in a neutron star has, within the T0(γ̄, β̄; ǫ, ζ)
framework, a 2PN-order expansion of the form [33, 70]
∆A = −
(4 β̄ − γ̄) cA +
+ ζ +O(β̄)
bA , (91)
where cA = −2 ∂ lnmA∂ lnG ≃
〈U〉A, bA = 1c4 〈U
2〉A ≃ B c2A, with B ≃ 1.026 and
cA ≃ kmA/M⊙ with k ∼ 0.21. The general result (91) is compatible with the
result quoted in subsection 5.2 within the context of the theory T2(β
′, β′′) when
taking into account the fact that, within T2(β
′, β′′), one has β̄ = γ̄ = 0, ǫ = β′
and ζ = 0 [and that β′′ parametrizes some effects beyond the 2PN level].
On the example of Eq. (91) one sees that, after having used solar-system tests
to constrain the first contribution on the RHS to a very small value, one can
use binary-pulsar tests of the SEP to set a significant limit on the combination
ǫ + ζ of 2PN parameters. Other pulsar data then yield significant limits on
other combinations of the two 2PN parameters ǫ and ζ. The final conclusion
is that binary-pulsar data allow one to set significant limits (around or better
than the 1% level) on the possible 2PN deviations from GR (in contrast to
solar-system tests which are unable to yield any limit on ǫ and ζ) [70]. For a
recent update of the limits on ǫ and ζ, which makes use of recent pulsar data
see [51].
Let us now briefly discuss the use of mini-space of theories, such as T1(α0, β0)
or T2(β
′, β′′), for analyzing solar-system and binary-pulsar data. The basic
methodology is to compute, for each given theory (e.g. for each given values of
α0 and β0 if one chooses to work in the T1(α0, β0) theory space) a goodness-
of-fit statistics χ2(α0, β0) measuring the quality of the agreement between the
experimental data and the considered theory. For instance, when considering
the timing data of a particular pulsar, for which one has measured several PK
parameters pi (i = 1, . . . , n) with some standard deviations σ
, one defines,
for this pulsar
χ2(α0, β0) = min
mA,mB
(σobspi )
theory
i (α0, β0;mA,mB)− p
2 , (92)
where ‘min’ denotes the result of minimizing over the unknown masses mA,mB
and where p
theory
i (α0, β0;mA,mB) denotes the theoretical prediction (within
T1(α0, β0)) for the PK observable pi (given also the observed values of the
Keplerian parameters).
The goodness-of-fit quantity χ2(α0, β0) will reach its minimum χ
min for some
values, say αmin0 , β
0 , of α0 and β0. Then, one focusses, for each pulsar, on the
level contours of the function
∆χ2(α0, β0) ≡ χ2(α0, β0)− χ2min . (93)
Each choice of level contour (e.g. ∆χ2 = 1 or ∆χ2 = 2.3) defines a certain
region in theory space, which contains, with a certain corresponding ‘confidence
level’, the ‘correct’ theory of gravity (if it belongs to the considered mini-space of
theories). When combining together several independent data sets (e.g. solar-
system data, and different pulsar data) we can define a total goodness-of-fit
statistics χ2tot(α0, β0), by adding together the various individual χ
2(α0, β0). This
leads to a corresponding combined contour ∆χ2tot(α0, β0).
Let us end by briefly summarizing the results of the theory-space approach
to relativistic gravity tests. For detailed discussions the reader should consult
Refs. [33, 40, 35, 36, 80], and especially the recent update [51] which uses the
latest binary-pulsar data.
Regarding the two-parameter class of tensor-bi-scalar theories T2(β
′, β′′) the
recent analysis [51] has shown that the ∆χ2(β′, β′′) corresponding to the double
binary pulsar PSR J0737−3039 was defining quite a small elliptical allowed
region in the (β′, β′′) plane. By contrast the other pulsar data define much wider
allowed regions, while the strong equivalence principle tests define (in view of
the theoretical result ∆ ≃ 1 + 1
Bβ′(c2A − c2B)) a thin, but infinitely long, strip
|β′| < cst. in the (β′, β′′) plane. This highlights the power of the double binary
pulsar in probing certain specific strong-field deviations from GR.
Contrary to the T2(β
′, β′′) tensor-bi-scalar theories, which were constructed
to have exactly the same first post-Newtonian limit as GR26 (so that solar-
system tests put no constraints on β′ and β′′), the class of tensor-mono-scalar
theories T1(α0, β0) is such that its parameters α0 and β0 parametrize both the
weak-field 1PN regime (see Eqs. (60) and (61) above) and the strong-field regime
(which plays an important role in compact binaries). This means that each class
of solar-system data (see Eqs. (87)–(90) above) will define, via a corresponding
goodness-of-fit statistics of the type, say
χ2Cassini(α0, β0) = (σ
Cassini
−2 (γ̄theory(α0, β0)− γ̄Cassini)2
26However, this could be achieved only at the cost of allowing some combination of the two
scalar fields to carry a negative energy flux.
a certain allowed region27 in the (α0, β0) plane. As a consequence, the analysis
in the framework of the T1(α0, β0) space of theories allows one to compare
and contrast the probing powers of solar-system tests versus binary-pulsar tests
(while comparing also solar-system tests among themselves and binary-pulsar
ones among themselves). The result of the recent analysis [51] is shown in
Figure 3.
general relativity
B1534+12
J1141–6545
J0737–3039
B1913+16
−6 −4 −2 0 2 4 6
0.025
matter
matter
0.175
0.075
0.125
solar
system
Figure 3: Solar-system and binary-pulsar constraints on the two-parameter fam-
ily of tensor-mono-scalar theories T1(α0, β0). Figure taken from [51].
In Fig. 3, the various solar-system constraints (87)–(90) are concentrated
around the horizontal β0 axis. In particular, the high-precision Cassini con-
straint is the lower small grey strip. The various pulsar constraints are labelled
by the name of the pulsar, except for the strong equivalence principle constraint
which is labelled SEP. Note that General Relativity corresponds to the origin
of the (α0, β0) plane, and is compatible with all existing tests.
The global constraint obtained by combining all the pulsar tests would, to a
good accuracy, be obtained by intersecting the various pulsar-allowed regions.
One can then see on Fig. 3 that it would be comparable to the pre-Cassini
solar-system constraints and that its boundaries would be defined successively
(starting from the left) by 1913+16, 1141−6545, 0737−3039, 1913+16 again
and 1141−6545 again.
A first conclusion is therefore that, at the quantitative level, binary-pulsar
tests constrain tensor-scalar gravity theories as strongly as most solar-system
27Actually, in the case of the Cassini data, as it is quite plausible that the positive value of
the published central value γ̄Cassini = +2.1× 10−5 is due to unsubtracted systematic effects,
we use σCassiniγ = 2.3× 10
−5 but γ̄Cassini = 0. Otherwise, we would get unreasonably strong
1σ limits on α2
because tensor-scalar theories predict that γ̄ must be negative, see Eqs. (60)
and (61).
tests (excluding the exceptionally accurate Cassini result which constrains α20
to be smaller than 1.15 × 10−5, i.e. |α0| < 3.4 × 10−3). A second conclusion
is obtained by comparing the behaviour of the solar-system exclusion plots and
of the binary-pulsar ones around the negative β0 axis. One sees that binary-
pulsar tests exclude a whole domain of the theory space (located on the left
of β0 < −4) which is compatible with all solar-system experiments (even when
including the very tight Cassini constraint). This remarkable qualitative feature
of pulsar tests is a direct consequence of the existence of (non-perturbative)
strong-field effects which start developing when the product −β0 cA (with cA
denoting, as above, the compactness of the pulsar) becomes of order unity.
6 Conclusion
In conclusion, we hope to have convinced the reader of the superb opportunities
that binary pulsar data offer for testing gravity theories. In particular, they
have been able to go qualitatively beyond solar-system experiments in probing
two physically important regimes of relativistic gravity: the radiative regime
and the strong-field one. Up to now, General Relativity has passed with flying
colours all the radiative and strong-field tests provided by pulsar data. However,
it is important to continue testing General Relativity in all its aspects (weak-
field, radiative and strong-field). Indeed, history has taught us that physical
theories have a limited range of validity, and that it is quite difficult to predict
in which regime a theory will cease to be an accurate description of nature. Let
us look forward to new results, and possibly interesting surprises, from binary
pulsar data.
Acknowledgments
It is a pleasure to thank my long-term collaborator Gilles Esposito-Farèse for his
useful remarks on the text, and for providing the figures. I wish also to thank
the organizers of the 2005 Sigrav School, and notably Monica Colpi and Ugo
Moschella, for organizing a warm and intellectually stimulating meeting. This
work was partly supported by the European Research and Training Network
“Forces Universe” (contract number MRTN-CT-2004-005104).
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	Introduction
	Motion of binary pulsars in general relativity
	Timing of binary pulsars in general relativity
	Phenomenological approach to testing relativistic gravity with binary pulsar data
	Theory-space approach to testing relativistic gravity with binary pulsar data
	Theory-space approaches to solar-system tests of relativistic gravity
	Theory-space approaches to binary-pulsar tests of relativistic gravity
	Tensor-scalar theories of gravity
	The role of the coupling function a(); definition of the two-dimensional space of tensor-scalar gravity theories T1 (0 , 0)
	Tensor-scalar gravity, strong-field effects, and binary-pulsar observables
	Theory-space analyses of binary pulsar data
	Conclusion
