3.4 Preferred-location effects: Variation of Newton’s constant

Theories that violate the SEP by allowing for preferred locations (in time as well as space) may permit Newton’s constant G to vary. In general, variations in G are expected to occur on the timescale of the age of the Universe, such that G˙∕G ∼ H ∼ 0.7 × 10− 10 yr− 1 0, where H 0 is the Hubble constant. Three different pulsar-derived tests can be applied to these predictions, as a SEP-violating time-variable G would be expected to alter the properties of neutron stars and white dwarfs, and to affect binary orbits.

3.4.1 Spin tests

By affecting the gravitational binding of neutron stars, a non-zero ˙ G would reasonably be expected to alter the moment of inertia of the star and hence change its spin on the same timescale [34]. Goldman [56] writes

( ) ( ) P˙ -∂ ln-I G˙ P ˙= ∂ lnG G , (28 ) G N
where I is the moment of inertia of the neutron star, about 1045 g cm2, and N is the (conserved) total number of baryons in the star. By assuming that this represents the only contribution to the observed ˙P of PSR B0655+64, in a manner reminiscent of the test of αˆ3 described above, Goldman then derives an upper limit of ˙ −11 − 1 |G ∕G | ≤ (2.2– 5.5) × 10 yr, depending on the stiffness of the neutron star equation of state. Arzoumanian [7Jump To The Next Citation Point] applies similar reasoning to PSR J2019+2425 [101], which has a characteristic age of 27 Gyr once the “Shklovskii” correction is applied [100]. Again, depending on the equation of state, the upper limits from this pulsar are |G˙∕G | ≤ (1.4– 3.2) × 10 −11 yr− 1 [7Jump To The Next Citation Point]. These values are similar to those obtained by solar-system experiments such as laser ranging to the Viking Lander on Mars (see, e.g., [11362]). Several other millisecond pulsars, once “Shklovskii” and Galactic-acceleration corrections are taken into account, have similarly large characteristic ages (see, e.g., [29137]).

3.4.2 Orbital decay tests

The effects on the orbital period of a binary system of a varying G were first considered by Damour, Gibbons, and Taylor [41], who expected

( ) P˙b G˙ --- = − 2--. (29 ) Pb ˙G G
Applying this equation to the limit on the deviation from GR of the P˙b for PSR 1913+16, they found a value of ˙G∕G = (1.0 ± 2.3) × 10 −11 yr− 1. Nordtvedt [102] took into account the effects of ˙ G on neutron-star structure, realizing that the total mass and angular momentum of the binary system would also change. The corrected expression for ˙Pb incorporates the compactness parameter ci and is
( ) P˙b [ (m1c1 + m2c2 ) 3 ( m1c2 + m2c1 )] G˙ P-- = − 2 − --m---+-m---- − 2- -m---+-m----- G-. (30 ) b G˙ 1 2 1 2
(Note that there is a difference of a factor of − 2 in Nordtvedt’s definition of ci versus the Damour definition used throughout this article.) Nordtvedt’s corrected limit for PSR B1913+16 is therefore slightly weaker. A better limit actually comes from the neutron-star–white-dwarf system PSR B1855+09, with a measured limit on P˙b of (0.6 ± 1.2 ) × 10 −12 [74]. Using Equation (29View Equation), this leads to a bound of G˙∕G = (− 9 ± 18) × 10−12 yr−1, which Arzoumanian [7Jump To The Next Citation Point] corrects using Equation (30View Equation) and an estimate of NS compactness to ˙ −11 −1 G ∕G = (− 1.3 ± 2.7) × 10 yr. Prospects for improvement come directly from improvements to the limit on P˙b. Even though PSR J1012+5307 has a tighter limit on P˙b [85], its shorter orbital period means that the G˙ limit it sets is a factor of 2 weaker than obtained with PSR B1855+09.

3.4.3 Changes in the Chandrasekhar mass

The Chandrasekhar mass, MCh, is the maximum mass possible for a white dwarf supported against gravitational collapse by electron degeneracy pressure [30]. Its value – about 1.4M ⊙ – comes directly from Newton’s constant: MCh ∼ (¯h c∕G )3∕2∕m2n, where ¯h is Planck’s constant and mn is the neutron mass. All measured and constrained pulsar masses are consistent with a narrow distribution centred very close to MCh: 1.35 ± 0.04M ⊙ [136]. Thus, it is reasonable to assume that MCh sets the typical neutron star mass, and to check for any changes in the average neutron star mass over the lifetime of the Universe. Thorsett [135Jump To The Next Citation Point] compiles a list of measured and average masses from 5 double-neutron-star binaries with ages ranging from 0.1 Gyr to 12 or 13 Gyr in the case of the globular-cluster binary B2127+11C. Using a Bayesian analysis, he finds a limit of ˙ −12 −1 G ∕G = (− 0.6 ± 4.2) × 10 yr at the 95% confidence level, the strongest limit on record. Figure 5View Image illustrates the logic applied.

While some cancellation of “observed” mass changes might be expected from the changes in neutron-star binding energy (cf. Section 3.4.2 above), these will be smaller than the MCh changes by a factor of order the compactness and can be neglected. Also, the claimed variations of the fine structure constant of order −5 Δ α∕α ≃ − 0.72 ± 0.18 × 10 [140] over the redshift range 0.5 < z < 3.5 could introduce a maximum derivative of 1∕(¯hc) ⋅ d (¯hc )∕dt of about 5 × 10–16 yr–1 and hence cannot influence the Chandrasekhar mass at the same level as the hypothesized changes in G.

View Image

Figure 5: Measured neutron star masses as a function of age. The solid lines show predicted changes in the average neutron star mass corresponding to hypothetical variations in G, where ζ− 12 = 10 implies G˙∕G = 10 × 10−12 yr −1. (From [135], used by permission.)

One of the five systems used by Thorsett has since been shown to have a white-dwarf companion [138], but as this is one of the youngest systems, this will not change the results appreciably. The recently discovered PSR J1811–1736 [90Jump To The Next Citation Point], a double-neutron-star binary, has a characteristic age of only τc ∼ 1 Gyr and, therefore, will also not significantly strengthen the limit. Ongoing searches for pulsars in globular clusters stand the best chance of discovering old double-neutron-star binaries for which the component masses can eventually be measured.

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