"The Confrontation between General Relativity and Experiment"
Clifford M. Will 

6 Stellar System Tests of Gravitational Theory

6.1 The binary pulsar and general relativity

The 1974 discovery of the binary pulsar B1913+16 by Joseph Taylor and Russell Hulse during a routine search for new pulsars provided the first possibility of probing new aspects of gravitational theory: the effects of strong relativistic internal gravitational fields on orbital dynamics, and the effects of gravitational radiation reaction. For reviews of the discovery, see the published Nobel Prize lectures by Hulse and Taylor [195, 385]. For reviews of the current status of testing general relativity with pulsars, including binary and millisecond pulsars, see [261, 374, 412]; specific details on every pulsar discovered to date, along with orbit elements of pulsars in binary systems, can be found at the Australia Telescope National Facility (ATNF) online pulsar catalogue [28]. Table 7 lists the current values of the key orbital and relativistic parameters for B1913+16, from analysis of data through 2006 [409].

Table 7: Parameters of the binary pulsar B1913+16. The numbers in parentheses denote errors in the last digit. Data taken from [409]. Note that γ ′ is not the same as the PPN parameter γ [see Eqs. (108*)].
(i) Astrometric and spin parameters:
Right Ascension α 19h15m27.s99928(9)
Declination δ 16∘06′27.′′3871(13)
Pulsar period Pp (ms) 59.0299983444181 (5)
Derivative of period P˙p 8.62713 (8) × 10−18
(ii) “Keplerian” parameters:
Projected semimajor axis ap sin i (s) 2.341782 (3)
Eccentricity e 0.6171334 (5 )
Orbital period Pb (day) 0.322997448911 (4)
Longitude of periastron ω0 (∘) 292.54472 (6 )
Julian date of periastron T0 (MJD) 52144.90097841(4)
(iii) “Post-Keplerian” parameters:
Mean rate of periastron advance ⟨ω˙⟩ (∘ − 1 yr) 4.226598 (5)
Redshift/time dilation ′ γ (ms) 4.2992 (8)
Orbital period derivative ˙ Pb (− 12 10) − 2.423(1)

The system consists of a pulsar of nominal period 59 ms in a close binary orbit with an unseen companion. The orbital period is about 7.75 hours, and the eccentricity is 0.617. From detailed analyses of the arrival times of pulses (which amounts to an integrated version of the Doppler-shift methods used in spectroscopic binary systems), extremely accurate orbital and physical parameters for the system have been obtained (see Table 7). Because the orbit is so close (≈ 1R ⊙) and because there is no evidence of an eclipse of the pulsar signal or of mass transfer from the companion, it is generally agreed that the companion is compact. Evolutionary arguments suggest that it is most likely a dead pulsar, while B1913+16 is a “recycled” pulsar. Thus the orbital motion is very clean, free from tidal or other complicating effects. Furthermore, the data acquisition is “clean” in the sense that by exploiting the intrinsic stability of the pulsar clock combined with the ability to maintain and transfer atomic time accurately using GPS, the observers can keep track of pulse time-of-arrival with an accuracy of 13 μs, despite extended gaps between observing sessions (including a several-year gap in the middle 1990s for an upgrade of the Arecibo radio telescope). The pulsar has experienced only one small “glitch” in its pulse period, in May 2003.

Three factors made this system an arena where relativistic celestial mechanics must be used: the relatively large size of relativistic effects [vorbit ≈ (m ∕r)1∕2 ≈ 10−3], a factor of 10 larger than the corresponding values for solar-system orbits; the short orbital period, allowing secular effects to build up rapidly; and the cleanliness of the system, allowing accurate determinations of small effects. Because the orbital separation is large compared to the neutron stars’ compact size, tidal effects can be ignored. Just as Newtonian gravity is used as a tool for measuring astrophysical parameters of ordinary binary systems, so GR is used as a tool for measuring astrophysical parameters in the binary pulsar.

The observational parameters that are obtained from a least-squares solution of the arrival-time data fall into three groups:

  1. non-orbital parameters, such as the pulsar period and its rate of change (defined at a given epoch), and the position of the pulsar on the sky;
  2. five “Keplerian” parameters, most closely related to those appropriate for standard Newtonian binary systems, such as the eccentricity e, the orbital period Pb, and the semi-major axis of the pulsar projected along the line of sight, apsini; and
  3. five “post-Keplerian” parameters.

The five post-Keplerian parameters are: ⟨ω˙⟩, the average rate of periastron advance; γ′, the amplitude of delays in arrival of pulses caused by the varying effects of the gravitational redshift and time dilation as the pulsar moves in its elliptical orbit at varying distances from the companion and with varying speeds; ˙ Pb, the rate of change of orbital period, caused predominantly by gravitational radiation damping; and r and s = sini, respectively the “range” and “shape” of the Shapiro time delay of the pulsar signal as it propagates through the curved spacetime region near the companion, where i is the angle of inclination of the orbit relative to the plane of the sky. An additional 14 relativistic parameters are measurable in principle [119].

In GR, the five post-Keplerian parameters can be related to the masses of the two bodies and to measured Keplerian parameters by the equations (TEGP 12.1, 14.6 (a) [420*])

⟨ω˙⟩ = 6πfb(2πmfb )2∕3(1 − e2)−1, m ( m ) γ′ = e(2πfb)−1(2πmfb )2∕3--2 1 + --2 , m m P˙b = − 192π-(2 πℳfb )5∕3F (e), (108 ) 5 r = m2, s = sin i,
where m1 and m2 denote the pulsar and companion masses, respectively. The formula for ⟨˙ω⟩ ignores possible non-relativistic contributions to the periastron shift, such as tidally or rotationally induced effects caused by the companion (for discussion of these effects, see TEGP 12.1 (c) [420*]). The formula for P˙b includes only quadrupole gravitational radiation; it ignores other sources of energy loss, such as tidal dissipation (TEGP 12.1 (f) [420*]). Notice that, by virtue of Kepler’s third law, (2πfb )2 = m ∕a3, (2πmfb )2∕3 = m ∕a ∼ 𝜖, thus the first two post-Keplerian parameters can be seen as 𝒪 (𝜖), or 1PN corrections to the underlying variable, while the third is an 5∕2 𝒪 (𝜖 ), or 2.5PN correction. The current observed values for the Keplerian and post-Keplerian parameters are shown in Table 7. The parameters r and s are not separately measurable with interesting accuracy for B1913+16 because the orbit’s 47∘ inclination does not lead to a substantial Shapiro delay. However they are measurable in the double pulsar, for example.

Because fb and e are separately measured parameters, the measurement of the three post-Keplerian parameters provides three constraints on the two unknown masses. The periastron shift measures the total mass of the system, P˙b measures the chirp mass, and γ′ measures a complicated function of the masses. GR passes the test if it provides a consistent solution to these constraints, within the measurement errors.

View Image
Figure 6: Constraints on masses of the pulsar and its companion from data on B1913+16, assuming GR to be valid. The width of each strip in the plane reflects observational accuracy, shown as a percentage. An inset shows the three constraints on the full mass plane; the intersection region (a) has been magnified 400 times for the full figure.

From the intersection of the ⟨ω˙⟩ and ′ γ constraints we obtain the values m1 = 1.4398 ± 0.0002M ⊙ and m2 = 1.3886 ± 0.0002M ⊙. The third of Eqs. (108*) then predicts the value P˙b = − 2.402531 ± 0.000014 × 10−12. In order to compare the predicted value for P˙b with the observed value of Table 7, it is necessary to take into account the small kinematic effect of a relative acceleration between the binary pulsar system and the solar system caused by the differential rotation of the galaxy. Using data on the location and proper motion of the pulsar, combined with the best information available on galactic rotation; the current value of this effect is gal P˙b ≃ − (0.027 ± 0.005) × 10−12. Subtracting this from the observed P˙b (see Table 7) gives the corrected P˙bcorr= − (2.396 ± 0.005) × 10−12, which agrees with the prediction within the errors. In other words,

P˙cborr -˙GR- = 0.997 ± 0.002. (109 ) Pb
The consistency among the measurements is displayed in Figure 6*, in which the regions allowed by the three most precise constraints have a single common overlap. Uncertainties in the parameters that go into the galactic correction are now the limiting factor in the accuracy of the test of gravitational damping.
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Figure 7: Plot of the cumulative shift of the periastron time from 1975 – 2005. The points are data, the curve is the GR prediction. The gap during the middle 1990s was caused by a closure of Arecibo for upgrading. Image reproduced with permission from [409], copyright by AAS.

A third way to display the agreement with GR is by comparing the observed phase of the orbit with a theoretical template phase as a function of time. If fb varies slowly in time, then to first order in a Taylor expansion, the orbital phase is given by Φb(t) = 2πfb0t + πf˙b0t2. The time of periastron passage tP is given by Φ (tP ) = 2πN, where N is an integer, and consequently, the periastron time will not grow linearly with N. Thus the cumulative difference between periastron time tP and N ∕fb0, the quantities actually measured in practice, should vary according to tP − N ∕fb0 = − ˙fb0N 2∕2f3b0 ≈ − (f ˙b0∕2fb0)t2. Figure 7* shows the results: The dots are the data points, while the curve is the predicted difference using the measured masses and the quadrupole formula for f˙ b0 [409].

The consistency among the constraints provides a test of the assumption that the two bodies behave as “point” masses, without complicated tidal effects, obeying the general relativistic equations of motion including gravitational radiation. It is also a test of strong gravity, in that the highly relativistic internal structure of the neutron stars does not influence their orbital motion, as predicted by the SEP of GR.

Observations [231, 410] indicate that the pulse profile is varying with time, which suggests that the pulsar is undergoing geodetic precession on a 300-year timescale as it moves through the curved spacetime generated by its companion (see Section 4.4.2). The amount is consistent with GR, assuming that the pulsar’s spin is suitably misaligned with the orbital angular momentum. Unfortunately, the evidence suggests that the pulsar beam may precess out of our line of sight by 2025.

6.2 A zoo of binary pulsars

More than 70 binary neutron star systems with orbital periods less than a day are now known. While some are less interesting for testing relativity, some have yielded interesting tests, and others, notably the recently discovered “double pulsar” are likely to continue to produce significant results well into the future. Here we describe some of the more interesting or best studied cases;

The “double” pulsar: J0737–3039A, B.
This binary pulsar system, discovered in 2003 [72], was already remarkable for its extraordinarily short orbital period (0.1 days) and large periastron advance (∘ −1 16.8995 yr), but then the companion was also discovered to be a pulsar [265]. Because two projected semi-major axes could be measured, the mass ratio was obtained directly from the ratio of the two values of ap sin i, and thereby the two masses could be obtained by combining that ratio with the periastron advance, assuming GR. The results are mA = 1.3381 ± 0.0007M ⊙ and mB = 1.2489 ± 0.0007M ⊙, where A denotes the primary (first) pulsar. From these values, one finds that the orbit is nearly edge-on, with sini = 0.9997, a value which is completely consistent with that inferred from the Shapiro delay parameter. In fact, the five measured post-Keplerian parameters plus the ratio of the projected semi-major axes give six constraints on the masses (assuming GR): as seen in Figure 8*, all six overlap within their measurement errors [232]. (Note that Figure 8* is based on more recent data than that quoted in [232], in this discussion and in Table 8.) Because of the location of the system, galactic proper-motion effects play a significantly smaller role in the interpretation of P˙b measurements than they did in B1913+16; this and the reduced effect of interstellar dispersion means that the accuracy of measuring the gravitational-wave damping may soon beat that from the Hulse–Taylor system. It may ultimately be necessary for the data analysis to include second post-Newtonian (2PN) corrections, for example in the pericenter advance. The geodetic precession of pulsar B’s spin axis has also been measured by monitoring changes in the patterns of eclipses of the signal from pulsar A, with a result in agreement with GR to about 13 percent [68]; the constraint on the masses from that effect (assuming GR to be correct) is also shown in Figure 8*. In fact, pulsar B has precessed so much that its signal no longer sweeps by the Earth, so it has gone “silent”. For a recent overview of the double pulsar, see [71].

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Figure 8: Constraints on masses of the pulsar and its companion from data on J0737–3039A,B, assuming GR to be valid. The inset shows the intersection region magnified by a factor of 80. Image courtesy of M. Kramer.

J1738+0333: A white-dwarf companion.
This is a low-eccentricity, 8.5-hour period system in which the white-dwarf companion is bright enough to permit detailed spectroscopy, allowing the companion mass to be determined directly to be 0.181M ⊙. The mass ratio is determined from Doppler shifts of the spectral lines of the companion and of the pulsar period, giving the pulsar mass 1.46M ⊙. Ten years of observation of the system yielded both a measurement of the apparent orbital period decay, and enough information about its parallax and proper motion to account for the substantial kinematic effect to give a value of the intrinsic period decay of ˙ −15 −1 Pb = (− 25.9 ± 3.2) × 10 s s in agreement with the predicted effect [164]. But because of the asymmetry of the system, the result also places a significant bound on the existence of dipole radiation, predicted by many alternative theories of gravity (see Section 6.3 below for discussion). Data from this system were also used to place the tight bound on the PPN parameter α1 shown in Table 4.

J1141–6545: A white-dwarf companion.
This system is similar in some ways to the Hulse–Taylor binary: short orbital period (0.20 days), significant orbital eccentricity (0.172), rapid periastron advance (5.3 degrees per year) and massive components (Mp = 1.27 ± 0.01M ⊙, Mc = 1.02 ± 0.01M ⊙). The key difference is that the companion is again a white dwarf. The intrinsic orbit period decay has been measured in agreement with GR to about six percent, again placing limits on dipole gravitational radiation [46].

J0348+0432: The most massive neutron star.
Discovered in 2011 [264, 19], this is another neutron-star white-dwarf system, in a very short period (0.1 day), low eccentricity (−6 2 × 10) orbit. Timing of the neutron star and spectroscopy of the white dwarf have led to mass values of 0.172M ⊙ for the white dwarf and 2.01 ± 0.04M ⊙ for the pulsar, making it the most massive accurately measured neutron star yet. This supported an earlier discovery of a 2M ⊙ pulsar [127]; such large masses rule out a number of heretofore viable soft equations of state for nuclear matter. The orbit period decay agrees with the GR prediction within 20 percent and is expected to improve steadily with time.

J0337+1715: Two white-dwarf companions.
This remarkable system was reported in 2014 [332]. It consists of a 2.73 millisecond pulsar (M = 1.44M ⊙) with extremely good timing precision, accompanied by two white dwarfs in coplanar circular orbits. The inner white dwarf (M = 0.1975M ⊙) has an orbital period of 1.629 days, with e = 6.918 × 10 −4, and the outer white dwarf (M = 0.41M ⊙) has a period of 327.26 days, with e = 3.536 × 10 −2. This is an ideal system for testing the Nordtvedt effect in the strong-field regime. Here the inner system is the analogue of the Earth-Moon system, and the outer white dwarf plays the role of the Sun. Because the outer semi-major axis is about 1/3 of an astronomical unit, the basic driving perturbation is comparable to that provided by the Sun. However, the self-gravitational binding energy per unit mass of the neutron star is almost a billion times larger than that of the Earth, greatly amplifying the size of the Nordtvedt effect. Depending on the details, this system could exceed lunar laser ranging in testing the Nordtvedt effect by several orders of magnitude.

Other binary pulsars.
Two of the earliest binary pulsars, B1534+12 and B2127+11C, discovered in 1990, failed to live up to their early promise despite being similar to the Hulse–Taylor system in most respects (both were believed to be double neutron-star systems). The main reason was the significant uncertainty in the kinematic effect on ˙ Pb of local accelerations, galactic in the case of B1534+12, and those arising from the globular cluster that was home to B2127+11C.

Table 8: Parameters of other binary pulsars. References may be found in the text. Values for orbit period derivatives include corrections for galactic kinematic effects
J0737–3039(A, B)
(i) Keplerian:
apsini (s) 1.415032(1)∕1.516(2) 0.34342913 (2) 1.858922(6)
e 0.0877775(9) (3.4 ± 1.1) × 10−7 0.171884(2)
Pb (day) 0.10225156248(5) 0.354790739872 (1) 0.1976509593 (1)
(ii) Post-Keplerian:
⟨˙ω⟩ (∘ yr−1) 16.8995(7) 5.3096(4)
′ γ (ms) 0.386(3) 0.77(1)
P˙b (10−12) − 1.25(2) − 0.026 (3) − 0.401(25)
r (μs) 6.2(3)
s = sin i 0.9997(4 )

6.3 Binary pulsars and alternative theories

Soon after the discovery of the binary pulsar it was widely hailed as a new testing ground for relativistic gravitational effects. As we have seen in the case of GR, in most respects, the system has lived up to, indeed exceeded, the early expectations.

In another respect, however, the system has only partially lived up to its promise, namely as a direct testing ground for alternative theories of gravity. The origin of this promise was the discovery [139, 415] that alternative theories of gravity generically predict the emission of dipole gravitational radiation from binary star systems. In GR, there is no dipole radiation because the “dipole moment” (center of mass) of isolated systems is uniform in time (conservation of momentum), and because the “inertial mass” that determines the dipole moment is the same as the mass that generates gravitational waves (SEP). In other theories, while the inertial dipole moment may remain uniform, the “gravity wave” dipole moment need not, because the mass that generates gravitational waves depends differently on the internal gravitational binding energy of each body than does the inertial mass (violation of SEP). Schematically, in a coordinate system in which the center of inertial mass is at the origin, so that mI,1x1 + mI,2x2 = 0, the dipole part of the retarded gravitational field would be given by

( ) 1--d ηm-- mGW,1-- mGW,2-- h ∼ R dt(mGW,1x1 + mGW,2x2 ) ⋅ N ∼ R v ⋅ N mI,1 − mI,2 , (110 )
where v = v1 − v2, N is a unit vector directed toward the observer, and η and m are defined using inertial masses. In theories that violate SEP, the difference between gravitational-wave mass and inertial mass is a function of the internal gravitational binding energy of the bodies. This additional form of gravitational radiation damping could, at least in principle, be significantly stronger than the usual quadrupole damping, because it depends on fewer powers of the orbital velocity v, and it depends on the gravitational binding energy per unit mass of the bodies, which, for neutron stars, could be as large as 20 percent (see TEGP 10 [420*] for further details). As one fulfillment of this promise, Will and Eardley worked out in detail the effects of dipole gravitational radiation in the bimetric theory of Rosen, and, when the first observation of the decrease of the orbital period was announced in 1979, the Rosen theory suffered a terminal blow. A wide class of alternative theories also fails the binary pulsar test because of dipole gravitational radiation (TEGP 12.3 [420*]).

On the other hand, the early observations of PSR 1913+16 already indicated that, in GR, the masses of the two bodies were nearly equal, so that, in theories of gravity that are in some sense “close” to GR, dipole gravitational radiation would not be a strong effect, because of the apparent symmetry of the system. The Rosen theory, and others like it, are not “close” to GR, except in their predictions for the weak-field, slow-motion regime of the solar system. When relativistic neutron stars are present, theories like these can predict strong effects on the motion of the bodies resulting from their internal highly relativistic gravitational structure (violations of SEP). As a consequence, the masses inferred from observations of the periastron shift and γ′ may be significantly different from those inferred using GR, and may be different from each other, leading to strong dipole gravitational radiation damping. By contrast, the Brans–Dicke theory is “close” to GR, roughly speaking within 1∕ωBD of the predictions of the latter, for large values of the coupling constant ωBD. Thus, despite the presence of dipole gravitational radiation, the Hulse–Taylor binary pulsar provides at present only a weak test of pure Brans–Dicke theory, not competitive with solar-system tests.

However, the discovery of binary pulsar systems with a white dwarf companion, such as J1738+0333, J1141–6545 and J0348+0432 has made it possible to perform strong tests of the existence of dipole radiation. This is because such systems are necessarily asymmetrical, since the gravitational binding energy per unit mass of white dwarfs is of order −4 10, much less than that of the neutron star. Already, significant bounds have been placed on dipole radiation using J1738+0333 and J1141–6545 [164, 46].

Because the gravitational-radiation and strong-field properties of alternative theories of gravity can be dramatically different from those of GR and each other, it is difficult to parametrize these aspects of the theories in the manner of the PPN framework. In addition, because of the generic violation of the strong equivalence principle in these theories, the results can be very sensitive to the equation of state and mass of the neutron star(s) in the system. In the end, there is no way around having to analyze every theory in turn. On the other hand, because of their relative simplicity, scalar–tensor theories provide an illustration of the essential effects, and so we shall discuss binary pulsars within this class of theories.

6.4 Binary pulsars and scalar–tensor gravity

Making the usual assumption that both members of the system are neutron stars, and using the methods summarized in TEGP 10 – 12 [420*] (see also [286]) one can obtain formulas for the periastron shift, the gravitational redshift/second-order Doppler shift parameter, the Shapiro delay coefficients, and the rate of change of orbital period, analogous to Eqs. (108*). These formulas depend on the masses of the two neutron stars, on their sensitivities sA, and on the scalar–tensor parameters, as defined in Table 6 (and on a new sensitivity κ∗, defined below). First, there is a modification of Kepler’s third law, given by

( ) αm-- 1∕2 2πfb = a3 . (111 )
Then the predictions for ⟨˙ω⟩, ′ γ, ˙ Pb, r and s are
⟨˙ω⟩ = 6πfb (2παmfb )2∕3(1 − e2)−1𝒫, (112 ) m [ m ] γ ′ = e(2πfb )−1(2παmfb )2∕3--2α −1 1 − 2ζs2 + α --2 + ζκ∗1(1 − 2s2) , (113 ) m m P˙ = − 192-π(2π αℳf )5∕3F ′(e) − 8π ζ(2πμf )𝒮2G (e), (114 ) b 5 b b r = m2 (1 − ζ), (115 ) s = sini, (116 )
1-( ¯ ¯ ) 𝒫 = 1 + 3 2¯γ − β+ + ψ β− , (117 ) 1 [ ( 7 1 ) 1 ( 1 ) ] F ′(e) = ---(1 − e2)−7∕2 κ1 1 + -e2 + --e4 − --κ2e2 1 + -e2 , (118 ) 12 ( ) 2 2 2 2 2 −5∕2 1 2 G (e) = (1 − e ) 1 + --e , (119 ) 2
where κ1 and κ2 are defined in Eq. (104*). The quantity κ∗A is defined by
( ) κ∗ = ∂-(ln-IA) , (120 ) A ∂(lnϕ )
and measures the “sensitivity” of the moment of inertia IA of each body to changes in the scalar field for a fixed baryon number N (see TEGP 11, 12 and 14.6 (c) [420*] for further details). The sensitivities sA and κ∗A will depend on the neutron-star equation of state. Notice how the violation of SEP in scalar–tensor theory introduces complex structure-dependent effects in everything from the Newtonian limit (modification of the effective coupling constant in Kepler’s third law) to gravitational radiation. In the limit ζ → 0, we recover GR, and all structure dependence disappears. The first term in ˙ Pb (see Eq. (114*)) is the combined effect of quadrupole and monopole gravitational radiation, post-Newtonian corrections to dipole radiation, and a dipole-octupole coupling term, all contributing at 0PN order, while the second term is the effect of dipole radiation, contributing at the dominant –1PN order.

Unfortunately, because of the near equality of neutron star masses in typical double neutron star binary pulsars, dipole radiation is somewhat suppressed, and the bounds obtained are typically not competitive with the Cassini bound on γ, except for those generalized scalar–tensor theories, with β0 < 0 where the strong gravity of the neutron stars induces spontaneous scalarization effects [106*]. Figure 9* illustrates this: the bounds on α0 and β0 from the three binary neutron star systems B1913+16, J0737–3039, and B1534+12 are not close to being competitive with the Cassini bound on α0, except for very negative values of β0 (recall that α0 = (3 + 2 ω0)−1∕2).

On the other hand, a binary pulsar system with dissimilar objects, such as a white dwarf or black hole companion, provides potentially more promising tests of dipole radiation. As a result, the neutron-star–white-dwarf systems J1141–6545 and J1738+0333 yield much more stringent bounds. Indeed, the latter system surpasses the Cassini bound for β > 1 0 and β < − 2 0, and is close to that bound for the pure Brans–Dicke case β0 = 0 [164].

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Figure 9: Bounds on the scalar–tensor parameters α 0 and β 0 from solar-system and binary pulsar measurements. Bounds from tests of the Nordtvedt effect using lunar laser ranging and circular pulsar–white-dwarf binary systems are denoted LLR and SEP, respectively. Image reproduced with permission from [164], copyright by the authors.

It is worth pointing out that the bounds displayed in Figure 9* have been calculated using a specific choice of scalar–tensor theory, in which the function A (φ) is given by

[ ] A (φ) = exp α (φ − φ ) + 1β (φ − φ )2 , (121 ) 0 0 2 0 0
where α0, and β0, are arbitrary parameters, and φ0 is the asymptotic value of the scalar field. In the notation for scalar–tensor theories used here, this theory corresponds to the choice
( ) 1- -----1------ ω (ϕ) = − 2 3 − α2 − β0 lnϕ , (122 ) 0
where −2 ϕ0 = A (φ0) = 1. The parameters ζ and λ are given by
α20 ζ = -----2, 1 + α 0 λ = 1--β0---. (123 ) 21 + α20
The bounds were also calculated using a polytropic equation of state, which tends to give lower maximum masses for neutron stars than do more realistic equations of state.

Bounds on various versions of TeVeS theories have also been established, with the tightest constraints again coming from neutron-star–white-dwarf binaries [164]; in the case of TeVeS, the theory naturally predicts γ = 1 in the post-Newtonian limit, so the Cassini measurements are irrelevant here. Strong constraints on the Einstein-Æther and Khronometric theories have also been placed using binary pulsar measurements, exploiting both gravitational-wave damping data, and data related to preferred-frame effects [443, 442].

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