Binary Systems as Test-beds of Gravity Theories
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.
The discovery of binary pulsars in 1974  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 of a neutron star is of order , which is close to the one of a black hole () and much larger than the surface gravitational fields of solar system bodies: , . 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 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 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 separate bodies in GR consists of solving, by successive approximations, Einstein’s field equations (we use the signature )
together with their consequence
To do so, one assumes some specific matter model, say a perfect fluid,
One expands (say in powers of Newton’s constant)
together with the use of the simplifications brought by the ‘Post-Newtonian’ approximation (; , ). Then one integrates the local material equation of motion (2) over the volume of each separate body, labelled say by . In so doing, one must define some ‘center of mass’ of body , as well as some (approximately conserved) ‘mass’ of body , together with some corresponding ‘spin vector’ and, possibly, higher multipole moments.
An important feature of this traditional method is to use a unique coordinate chart to describe the full -body system. For instance, the center of mass, shape and spin of each body are all described within this common coordinate system . This use of a single chart has several inconvenient aspects, 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 symmetric in its own ‘rest frame’ will appear as deformed into some kind of ellipsoid in the common coordinate chart . Moreover, it is not clear how to construct ‘good definitions’ of the center of mass, spin vector, and higher multipole moments of body , when described in the common coordinate chart . 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) because will not be uniformly small in the common coordinate system . It will be small if one stays far away from each object , 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 relativistic problem of motion can be cured by using a ’multi-chart’ approach.The multi-chart approach describes the motion of (possibly, but not necessarily, compact) bodies by using separate coordinate systems: (i) one global coordinate chart () used to describe the spacetime outside ‘tubes’, each containing one body, and (ii) local coordinate charts (; ) used to describe the spacetime in and around each body . 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 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
where denotes the (possibly strong-field) metric generated by an isolated body of mass (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 (with ), where one can relate the different coordinate systems by expansions of the form
The multi-chart approach becomes simplified if one considers compact bodies (of radius comparable to ). In this case, it was shown , 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 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 distributional 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 . The most efficient method (especially to high PN orders) has been found to use analytic continuation in the dimension of space .
Finally, the most efficient way to derive the general relativistic equations of motion of compact bodies consists of solving the equations derived from the action (where )
formally using the standard weak-field expansion (4), but considering the space dimension as an arbitrary complex number which is sent to its physical value only at the end of the calculation.
Here denotes the Newtonian acceleration, its 1PN modification, its 2PN modification (together with the spin-orbit effects), and the PN contribution of order . [See the references above; or the review , for more references and the explicit expressions of , and .] It was verified that the term 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 was derived, in the near zone of the system, as a direct consequence of the general relativistic propagation of gravity, at the velocity , between the two bodies. This highlights the fact that binary pulsar tests of the existence of are direct tests of the reality of gravitational radiation.
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 get the coordinate positions and as explicit functions of the coordinate time , and (ii) to relate the coordinate motion 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 discussing pulsar timing, in Ref. . There (see also ) it was shown that, when considering the full (periodic and secular) effects of the terms in Eq. (8), together with the secular effects of the and terms, the relativistic two-body motion could be written in a very simple ‘quasi-Keplerian’ form (in polar coordinates), namely:
Here denotes the orbital frequency, the fractional periastron advance per orbit, an auxiliary angle (‘relativistic eccentric anomaly’), and various ‘relativistic eccentricities’ and and some ‘relativistic semi-major axes’. See  for the relations between these quantities, as well as their link to the relativistic energy and angular momentum . A direct study  of the dynamical effect of the contribution in the equations of motion (8) has checked that it led to a secular increase of the orbital frequency , and thereby to a quadratic term in the ‘relativistic mean anomaly’ appearing on the left-hand side (L.H.S.) of Eq. (9):
As for the contribution it induces several secular effects in the orbital motion: various 2PN contributions to the dimensionless periastron parameter ( spin-orbit effects), and secular variations in the inclination 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 pulse to the integer . 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 system we need to use the multi-chart approach mentionned above. In the ‘rest frame’ attached to the pulsar , the pulsar phenomenon can be modelled by the secularly changing rotation of a beam of radio waves:
where is the longitude around the spin axis. [Depending on the precise definition of the rest-frame attached to the pulsar, the spin axis can either be fixed, or be slowly evolving, see e.g. .] One must then relate the initial direction , and proper time , of emission of the pulsar beam to the coordinate direction and coordinate time of the null geodesic representing the electromagnetic beam in the ‘global’ coordinates used to describe the dynamics of the binary system [NB: the explicit orbital motion (9)–(13) refers to such global coordinates , ]. This is done by using the link (6) in which denotes the global coordinates of the ‘center of mass’ of the pulsar, the local (proper) time of the pulsar frame, and where, for instance
Using the link (6) (with expressions such as (16) for the coefficients ) one finds, among other results, that a radio beam emitted in the proper direction in the local frame appears to propagate, in the global frame, in the coordinate direction where
This is the well known ‘aberration effect’, which will then contribute to the timing formula.
where the ‘tilde’ denotes the operation consisting (in the matching approach) in discarding in the ‘self contributions’ , while keeping the effect of the companion (, etc). One checks that this is equivalent (in the dimensional-continuation approach) in taking for sufficiently small values of the real part of the dimension . To lowest order this yields the link
which combines the special relativistic and general relativistic time dilation effects. Hence, following  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 (linked to by (19)), in the initial global direction (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
where is the Newtonian potential within the binary system. This yields (with , )
The first term on the last RHS of Eq. (21) is the usual ‘light crossing time’ between the pulsar and the solar barycenter. It contains the ‘Roemer time delay’ due to the fact that 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 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’ appearing in the ‘Kepler equation’ (9), and one can define a new ‘eccentric anomaly’ angle: [we henceforth drop the superscript ‘new’ on ]. After these changes, the binary-system part of the general relativistic timing formula  takes the form (we suppress the index on the pulsar proper time )
where represents the projected light-crossing time (), a certain (relativistically-defined) ‘timing eccentricity’, the function
the ‘argument of the periastron’, and where the (relativistically-defined) ‘eccentric anomaly’ is the function of the ‘pulsar proper time’ obtained by solving the Kepler equation
It is understood here that the pulsar proper time corresponding to the pulse is related to the integer by an equation of the form
From these formulas, one sees that (and ) measure some relativistic distortion of the pulsar orbit, the amplitude of the ‘Einstein time delay’
Schematically, the structure of the DD timing formula (22) is
where denotes the solar-system barycentric (infinite frequency) arrival time of a pulse, the pulsar emission proper time (corrected for aberration), is the set of Keplerian parameters, the set of separately measurable post-Keplerian parameters, and the set of not separately measurable post-Keplerian parameters . [The parameter is a ‘Doppler factor’ which enters as an overall multiplicative factor on the right-hand side of Eq. (22).]
A further simplification of the DD timing formula was found possible. Indeed, the fact that the parameters are not separately measurable means that they can be absorbed in changes of the other parameters. The explicit formulas for doing that were given in  and : they consist in redefining and . At the end of the day, it suffices to consider a simplified timing formula where have been set to some given fiducial values, e.g. , and where one only fits for the remaining parameters and .
Finally, let us mention that it is possible to extend the general parametrized timing formula (30) by writing a similar parametrized formula describing the effect of the pulsar orbital motion on the directional spectral luminosity received by an observer. As discussed in detail in  this introduces a new set of ‘pulse-structure post-Keplerian parameters’.
4 Phenomenological approach to testing relativistic 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 phenomenological (or theory-independent) one, or various types of theory-dependent approaches. 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 but by a general combination of a metric field and of one or several scalar fields 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 . 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 to the parametrized formula (30) (in which is defined by Eq. (29) which introduces the further parameters ) 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 , and the set of post-Keplerian (PK) parameters . 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 parameters to the Keplerian ones, and to the two (a priori unknown) masses and of the pulsar and its companion. [For certain PK parameters one must also consider other variables related to the spin vectors of and .] In other words, the measurement (in addition of the Keplerian parameters) of each PK parameter defines, for each given theory, a curve in the 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 and (as the point of intersection of these two curves). Therefore, as soon as one measures 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 PK timing parameters yields 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: and . Here, we were talking about a normal ‘single line’ binary pulsar where, among the two compact objects and only one of the two, say is observed as a pulsar. In this case, one binary system can provide up to tests of GR. In practice, however, it has not yet been possible to measure the parameter (which measures a small relativistic deformation of the elliptical orbit), nor the secular parameters and . The original Hulse-Taylor system PSR 191316 has allowed one to measure 3 PK parameters: , and . The two parameters and involve (non radiative) strong-field effects, while, as explained above, the orbital period derivative is a direct consequence of the term in the binary-system equations of motion (5). The term is itself directly linked to the retarded propagation, at the velocity of light, of the gravitational interaction between the two strongly self-gravitating bodies and . Therefore, any test involving 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 one. First, we need to know the predictions made by the considered target theory for the PK parameters and as functions of the two masses and . These predictions have been worked out, for General Relativity, in Refs. [29, 28, 30]. Introducing the notation (where )
However, if we use the three predictions (34)–(36), together with the best current observed values of the PK parameters  we shall find that the three curves , , in the mass plane fail to meet at about the level! Should this put in doubt General Relativity? No, because Ref.  has shown that the time variation (notably due to galactic acceleration effects) of the Doppler factor entering Eq. (22) entailed an extra contribution to the ‘observed’ period derivative . We need to subtract this non-GR contribution before drawing the corresponding curve: . Then one finds that the three curves do meet within one . This yields a deep confirmation of General 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 and . One can, however, verify that strong-field effects do enter the observable PK parameters 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.  (see also ) for the Jordan-Fierz-Brans-Dicke theory of gravity, and in Ref.  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 153412. In this system, it was possible to measure the four (non radiative) PK parameters and . [We see from Eq. (25) that and measure, respectively, the range and the shape of the ‘Shapiro time delay’ .] The measurement of the 4 PK parameters define 4 curves in the mass plane, and thereby yield 2 strong-field tests of GR. It was found in  that GR passes these two tests. For instance, the ratio between the measured value of the phenomenological parameter
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): . Similarly to what happens in the Earth-Moon-Sun system , the three-body system made of a binary pulsar and of the Galaxy exhibits a ‘polarization’ of the orbit which is proportional to , and which can be constrained by considering certain quasi-circular neutron-star-white-dwarf binary systems . See  for recently published improved limits
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 J11416545 [46, 47], and (ii) the remarkable double binary pulsar PSR J07373039A and B [2, 3, 48, 49] (see also the lectures by M. Kramer and A. Possenti).
The PSR J11416545 timing data have led to the measurement of 3 PK parameters: , , and . As in PSR 191316 this yields one mixed radiative-strong-field test
The timing data of the millisecond binary pulsar PSR J07373039A have led to the direct measurement of 5 PK parameters: , , , and [3, 48, 49]. In addition, the ‘double line’ nature of this binary system (i.e. the fact that one observes both components, and , as radio pulsars) allows one to perform new phenomenological tests by using Keplerian parameters. Indeed, the simultaneous measurement of the Keplerian parameters and representing the projected light crossing times of both pulsars ( and ) gives access to the combined Keplerian parameter
On the other hand, the general derivation of  (applicable to any Lorentz-invariant theory of gravity, and notably to any tensor-scalar theory) shows that the theoretical prediction for the the ratio , considered as a function of the masses and , is
The absence of any explicit strong-field-gravity effects in the theoretical prediction (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 convention used in  and  for defining the masses and . These are always defined so that the Lagrangian for two non interacting compact objects reads . In other words, represents the total energy of body . This means that one has implicitly lumped in the definition of many strong-self-gravity effects. [For instance, in tensor-scalar gravity includes not only the usual Einsteinian gravitational binding energy due to the self-gravitational field , but also the extra binding energy linked to the scalar field .] Anyway, what is important is that, when performing a phenomenological test from the measurement of a triplet of parameters, e.g. , 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 , , is really a probe of strong-field gravity.
In conclusion, the two recently discovered binary pulsars PSR J11416545 and PSR J07373039 have more than doubled the number of phenomenological tests of (radiative and) strong-field gravity. Before their discovery, the ‘canonical’ relativistic binary pulsars PSR 191316 and PSR 153412 had given us four such tests: one test from PSR 191316 and three (
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  (in strong-field conditions) of one of the deep predictions of General Relativity: the ‘strong’ equivalence principle, i.e. the fact that various 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.  has extended the philosophy of the phenomenological (parametrized post-Keplerian) analysis of timing data, to a similar phenomenological analysis of pulse-structure data. Ref.  showed that, in principle, one could extract up to 11 ‘post-Keplerian pulse-structure parameters’. Together with the post-Keplerian timing parameters of a (single-line) binary pulsar, this makes a total of phenomenological PK parameters. As these parameters depend not only on the two masses but also on the two angles 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 . 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 magnitude) 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. Confirmations of general relativistic spin-orbit effects in the evolution of pulsar profiles were obained in several pulsars: PSR 191316 [54, 55], PSR B153412  and PSR J11416545 . In this respect, let us mention that the spin-orbit interaction affects also several PK parameters, either by inducing a secular evolution in some of them (see ) or by contributing to their value. For instance, the spin-orbit interaction contributes to the observed value of the periastron advance parameter an amount which is significant for the pulsars (such as 191316 and 07373039) where is measured with high-accuracy. It was then pointed out  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 , two other PK parameters with 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 phenomenological 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 relativistic 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 differ in their weak-field predictions from GR. In its most general versions the PPN formalism contains ‘post-Einstein’ PPN parameters,
A second way of implementing the theory-space philosophy consists in considering some explicit, parameter-dependent family of alternative relativistic theories of gravity. For instance, the simplest tensor-scalar theory of gravity put forward by Jordan , Fierz  and Brans and Dicke  has a unique free parameter, say . When , this theory reduces to GR, so that (or ) measures all the deviations from GR. When considering the weak-field limit of the Jordan-Fierz-Brans-Dicke (JFBD) theory, one finds that it can be described within the PPN formalism by choosing , and .
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 relativistic 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 () to describe 2PN order deviations from GR . Remarkably, there appear only two new parameters at the 2PN level
There exist also useful generalizations to the strong-field context of the idea of considering some explicit parameter-dependent family of alternative theories 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 continuously 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 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 denote
where measures the ‘gravitational compactness’ (or fractional gravitational binding energy, ) of body . The numerical coefficient of the contribution linear in is a combination of the first post-Newtonian order PPN parameters, namely . The numerical coefficient of the term quadratic in is a combination of the 1PN and 2PN parameters. When working in the context of the theories, the 1PN parameters vanish exactly and the coefficient of the quadratic term becomes simply proportional to the theory parameter , where . This example shows explicitly how binary pulsar data (here the data constraining the equivalence principle violation parameter , 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 and of the structure of 2PN effects . By contrast, the ‘compactness’ of neutron stars is of order  so that the pulsar limit  yields, within the framework, a significant limit on the dimensionless (2PN order) parameter .
Ref.  introduced a new two-parameter mini-space of gravity theories, denoted here as , which, from the point of view of theoretical physics, has several advantages over the mini-space mentionned above. First, it is technically simpler in that it contains only one scalar field besides the metric (hence the index on ). Second, it contains only positive-energy excitations (while one combination of the two scalar fields of carried negative-energy waves). Third, it is the minimal way to parametrize the huge class of tensor-mono-scalar theories with a ‘coupling function’ 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 ) 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 interesting 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 (with signature ‘mostly plus’), a scalar field , and some matter variables (including gauge bosons):
For simplicity, we assume here that the weak equivalence principle is satisfied, i.e., that the matter variables are all coupled to the same ‘physical metric’
where , where the potential
and where the conformal coupling function is given by