Preferred-frame effects and non-conservation of momentum

3.2 Preferred-frame effects and non-conservation of momentum

3.2.1 Limits on $αˆ1$

A non-zero $ˆα1$ implies that the velocity $w$ of a binary pulsar system (relative to a “universal” background reference frame given by the Cosmic Microwave Background, or CMB) will affect its orbital evolution. In a manner similar to the effects of a non-zero $Δnet$ , the time evolution of the eccentricity will depend on both $ω˙$ and a term that tries to force the semi-major axis of the orbit to align with the projection of the system velocity onto the orbital plane.

The analysis proceeds in a similar fashion to that for $Δ net$ , except that the magnitude of $e F$ is now written as [37 , 19 ]

where $w ⊥$ is the projection of the system velocity onto the orbital plane. The angle $λ$ , used in determining this projection in a manner similar to that of Equation (18

), is now the angle between the line of sight to the pulsar and the absolute velocity of the binary system.

The figure of merit for systems used to test $ˆα1$ is $P 1b∕3∕e$ . As for the $Δnet$ test, the systems must be old, so that $τc ≫ 2π∕ω˙$ , and $˙ω$ must be larger than the rate of Galactic rotation. Examples of suitable systems are PSR J2317+1439 [28 , 19 ] with a last published value of $−6 e < 1.2 × 10$ in 1996 [29 ], and PSR J1012+5307, with $− 7 e < 8 × 10$ [85 ]. This latter system is especially valuable because observations of its white-dwarf component yield a radial velocity measurement [25 ], eliminating the need to find a lower limit on an unknown quantity. The analysis of Wex [146 ] yields a limit of $αˆ1 < 1.4 × 10 −4$ . This is comparable in magnitude to the weak-field results from lunar laser ranging, but incorporates strong field effects as well.

3.2.2 Limits on $ˆα3$

Tests of $ˆα3$ can be derived from both binary and single pulsars, using slightly different techniques. A non-zero $αˆ3$ , which implies both a violation of local Lorentz invariance and non-conservation of momentum, will cause a rotating body to experience a self-acceleration $aself$ in a direction orthogonal to both its spin $Ω S$ and its absolute velocity $w$ [105 ]:

Thus, the self-acceleration depends strongly on the compactness of the object, as discussed in Section 3 above.

An ensemble of single (isolated) pulsars can be used to set a limit on $αˆ 3$ in the following manner. For any given pulsar, it is likely that some fraction of the self-acceleration will be directed along the line of sight to the Earth. Such an acceleration will contribute to the observed period derivative $P˙$ via the Doppler effect, by an amount

where $ˆn$ is a unit vector in the direction from the pulsar to the Earth. The analysis of Will [149

] assumes random orientations of both the pulsar spin axes and velocities, and finds that, on average, $|P˙ˆα | ≃ 5 × 10−5|ˆα3| 3$ , independent of the pulse period. The sign of the $ˆα3$ contribution to $P˙$ , however, may be positive or negative for any individual pulsar; thus, if there were a large contribution on average, one would expect to observe pulsars with both positive and negative total period derivatives. Young pulsars in the field of the Galaxy (pulsars in globular clusters suffer from unknown accelerations from the cluster gravitational potential and do not count toward this analysis) all show positive period derivatives, typically around 10^–14 s/s. Thus, the maximum possible contribution from $αˆ3$ must also be considered to be of this size, and the limit is given by $−10 |ˆα3| < 2 × 10$ [149

Bell [17 ] applies this test to a set of millisecond pulsars; these have much smaller period derivatives, on the order of 10^–20 s/s. Here, it is also necessary to account for the “Shklovskii effect” [117 ] in which a similar Doppler-shift addition to the period derivative results from the transverse motion of the pulsar on the sky:

where $μ$ is the proper motion of the pulsar and $d$ is the distance between the Earth and the pulsar. The distance is usually poorly determined, with uncertainties of typically 30% resulting from models of the dispersive free electron density in the Galaxy [131

, 32

]. Nevertheless, once this correction (which is always positive) is applied to the observed period derivatives for isolated millisecond pulsars, a limit on $|αˆ3 |$ on the order of 10^–15 results [17 , 20

In the case of a binary-pulsar–white-dwarf system, both bodies experience a self-acceleration. The combined accelerations affect both the velocity of the centre of mass of the system (an effect which may not be readily observable) and the relative motion of the two bodies [20 ]. The relative-motion effects break down into a term involving the coupling of the spins to the absolute motion of the centre of mass, and a second term which couples the spins to the orbital velocities of the stars. The second term induces only a very small, unobservable correction to $Pb$ and $ω˙$ [20 ]. The first term, however, can lead to a significant test of $ˆα3$ . Both the compactness and the spin of the pulsar will completely dominate those of the white dwarf, making the net acceleration of the two bodies effectively

where $cp$ and $ΩSp$ denote the compactness and spin angular frequency of the pulsar, respectively, and $w$ is the velocity of the system. For evolutionary reasons (see, e.g., [22

]), the spin axis of the pulsar may be assumed to be aligned with the orbital angular momentum of the system, hence the net effect of the acceleration will be to induce a polarization of the eccentricity vector within the orbital plane. The forced eccentricity term may be written as

where $β$ is the (unknown) angle between $w$ and $ΩSp$ , and $P$ is, as usual, the spin period of the pulsar: $P = 2π ∕ΩSp$ .

The figure of merit for systems used to test $ˆα3$ is $P2∕(eP ) b$ . The additional requirements of $τc ≫ 2π∕ω˙$ and $˙ω$ being larger than the rate of Galactic rotation also hold. The 95% confidence limit derived by Wex [146 ] for an ensemble of binary pulsars is $− 19 ˆα3 < 1.5 × 10$ , much more stringent than for the single-pulsar case.

3.2.3 Limits on $ζ2$

Another PPN parameter that predicts the non-conservation of momentum is $ζ2$ . It will contribute, along with $α3$ , to an acceleration of the centre of mass of a binary system [148 , 149 ]

where $n p$ is a unit vector from the centre of mass to the periastron of $m 1$ . This acceleration produces the same type of Doppler-effect contribution to a binary pulsar’s $˙ P$ as described in Section 3.2.2. In a small-eccentricity system, this contribution would not be separable from the $P˙$ intrinsic to the pulsar. However, in a highly eccentric binary such as PSR B1913+16, the longitude of periastron advances significantly – for PSR B1913+16, it has advanced nearly 120° since the pulsar’s discovery. In this case, the projection of $a cm$ along the line of sight to the Earth will change considerably over the long term, producing an effective second derivative of the pulse period. This $¨ P$ is given by [148

, 149

]

where $X = m1 ∕m2$ is the mass ratio of the two stars and an average value of $cosω$ is chosen. As of 1992, the 95% confidence upper limit on $¨ P$ was 4 × 10^–30 s^–1 [132

, 148

]. This leads to an upper limit on $(α3 + ζ2)$ of 4 × 10^–5 [148 ]. As $α3$ is orders of magnitude smaller than this (see Section 3.2.2), this can be interpreted as a limit on $ζ2$ alone. Although PSR B1913+16 is of course still observed, the infrequent campaign nature of the observations makes it difficult to set a much better limit on $¨ P$ (J. Taylor, private communication, as cited in [76

]). The other well-studied double-neutron-star binary, PSR B1534+12, yields a weaker test due to its orbital parameters and very similar component masses. A complication for this test is that an observed $P¨$ could also be interpreted as timing noise (sometimes seen in recycled pulsars [74

]) or else a manifestation of profile changes due to geodetic precession [80 , 76 ].