3.1 Strong Equivalence Principle: Nordtvedt effect

The possibility of direct tests of the SEP through Lunar Laser Ranging (LLR) experiments was first pointed out by Nordtvedt [103]. As the masses of Earth and the Moon contain different fractional contributions from self-gravitation, a violation of the SEP would cause them to fall differently in the Sun’s gravitational field. This would result in a “polarization” of the orbit in the direction of the Sun. LLR tests have set a limit of |η| < 0.001 (see, e.g., [47150]), where η is a combination of PPN parameters:
10 2 2 1 η = 4β − γ − 3 − --ξ − α1 + --α2 − -ζ1 − --ζ2. (12 ) 3 3 3 3

The strong-field formalism instead uses the parameter Δi [43Jump To The Next Citation Point], which for object “i” may be written as

( mgrav ) -------- = 1 + Δi minertial i ( ) ( ) Egrav ′ Egrav 2 = 1 + η mc2 i + η mc2 i + .... (13 )
Pulsar–white-dwarf systems then constrain Δnet = Δpulsar − Δcompanion [43Jump To The Next Citation Point]. If the SEP is violated, the equations of motion for such a system will contain an extra acceleration Δnetg, where g is the gravitational field of the Galaxy. As the pulsar and the white dwarf fall differently in this field, this Δnetg term will influence the evolution of the orbit of the system. For low-eccentricity orbits, by far the largest effect will be a long-term forcing of the eccentricity toward alignment with the projection of g onto the orbital plane of the system. Thus, the time evolution of the eccentricity vector will not only depend on the usual GR-predicted relativistic advance of periastron (˙ω), but will also include a constant term. Damour and Schäfer [43Jump To The Next Citation Point] write the time-dependent eccentricity vector as
e(t) = eF + eR(t), (14 )
where e (t) R is the ˙ω-induced rotating eccentricity vector, and e F is the forced component. In terms of Δnet, the magnitude of eF may be written as [43Jump To The Next Citation Point145Jump To The Next Citation Point]
3--Δnetg-⊥-- |eF| = 2ω˙a (2π∕P ), (15 ) b
where g⊥ is the projection of the gravitational field onto the orbital plane, and a = x∕sin i is the semi-major axis of the orbit. For small-eccentricity systems, this reduces to
1---Δnetg-⊥c2--- |eF| = 2F GM (2π ∕Pb)2, (16 )
where M is the total mass of the system, and, in GR, F = 1 and G is Newton’s constant.

Clearly, the primary criterion for selecting pulsars to test the SEP is for the orbital system to have a large value of Pb2∕e, greater than or equal to 107 days2 [145Jump To The Next Citation Point]. However, as pointed out by Damour and Schäfer [43Jump To The Next Citation Point] and Wex [145Jump To The Next Citation Point], two age-related restrictions are also needed. First of all, the pulsar must be sufficiently old that the ˙ω-induced rotation of e has completed many turns and eR(t) can be assumed to be randomly oriented. This requires that the characteristic age τc be ≫ 2 π∕˙ω, and thus young pulsars cannot be used. Secondly, ω˙ itself must be larger than the rate of Galactic rotation, so that the projection of g onto the orbit can be assumed to be constant. According to Wex [145Jump To The Next Citation Point], this holds true for pulsars with orbital periods of less than about 1000 days.

View Image

Figure 4: “Polarization” of a nearly circular binary orbit under the influence of a forcing vector g, showing the relation between the forced eccentricity eF, the eccentricity evolving under the general-relativistic advance of periastron eR(t), and the angle 𝜃. (After [145Jump To The Next Citation Point].)

Converting Equation (16View Equation) to a limit on Δnet requires some statistical arguments to deal with the unknowns in the problem. First is the actual component of the observed eccentricity vector (or upper limit) along a given direction. Damour and Schäfer [43Jump To The Next Citation Point] assume the worst case of possible cancellation between the two components of e, namely that |e | ≃ |e | F R. With an angle 𝜃 between g ⊥ and e R (see Figure 4View Image), they write |eF | ≤ e∕ (2 sin (𝜃∕2)). Wex [145Jump To The Next Citation Point146Jump To The Next Citation Point] corrects this slightly and uses the inequality

( |{ 1∕ sin 𝜃 for 𝜃 ∈ [0,π∕2 ), |eF | ≤ e ξ1(𝜃), ξ1(𝜃) = 1 for 𝜃 ∈ [π∕2, 3π∕2], (17 ) |( − 1∕ sin 𝜃 for 𝜃 ∈ (3π ∕2,2π),
where e = |e|. In both cases, 𝜃 is assumed to have a uniform probability distribution between 0 and 2π.

Next comes the task of estimating the projection of g onto the orbital plane. The projection can be written as

|g⊥| = |g|[1 − (cosi cosλ + sin i sinλ sinΩ )2]1∕2, (18 )
where i is the inclination angle of the orbital plane relative to the line of sight, Ω is the line of nodes, and λ is the angle between the line of sight to the pulsar and g [43Jump To The Next Citation Point]. The values of λ and |g | can be determined from models of the Galactic potential (see, e.g., [84Jump To The Next Citation Point2Jump To The Next Citation Point]). The inclination angle i can be estimated if even crude estimates of the neutron star and companion masses are available, from statistics of NS masses (see, e.g., [136Jump To The Next Citation Point]) and/or a relation between the size of the orbit and the WD companion mass (see, e.g., [112]). However, the angle Ω is also usually unknown and also must be assumed to be uniformly distributed between 0 and 2π.

Damour and Schäfer [43Jump To The Next Citation Point] use the PSR B1953+29 system and integrate over the angles 𝜃 and Ω to determine a 90% confidence upper limit of Δnet < 1.1 × 10−2. Wex [145] uses an ensemble of pulsars, calculating for each system the probability (fractional area in 𝜃Ω space) that Δnet is less than a given value, and then deriving a cumulative probability for each value of Δnet. In this way he derives −3 Δnet < 5 × 10 at 95% confidence. However, this method may be vulnerable to selection effects; perhaps the observed systems are not representative of the true population. Wex [146Jump To The Next Citation Point] later overcomes this problem by inverting the question. Given a value of Δ net, an upper limit on |𝜃| is obtained from Equation (17View Equation). A Monte Carlo simulation of the expected pulsar population (assuming a range of masses based on evolutionary models and a random orientation of Ω) then yields a certain fraction of the population that agree with this limit on |𝜃 |. The collection of pulsars ultimately gives a limit of Δnet < 9 × 10− 3 at 95% confidence. This is slightly weaker than Wex’s previous limit but derived in a more rigorous manner.

Prospects for improving the limits come from the discovery of new suitable pulsars, and from better limits on eccentricity from long-term timing of the current set of pulsars. In principle, measurement of the full orbital orientation (i.e., Ω and i) for certain systems could reduce the dependence on statistical arguments. However, the possibility of cancellation between |eF| and |e | R will always remain. Thus, even though the required angles have in fact been measured for the millisecond pulsar J0437–4715 [139Jump To The Next Citation Point], its comparatively large observed eccentricity of ∼ 2 × 10–5 and short orbital period mean it will not significantly affect the current limits.

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