Preferred-location effects: Variation of Newton’s constant

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

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 [7

] 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 $|G˙∕G | ≤ (1.4– 3.2) × 10 −11 yr− 1$ [7

]. These values are similar to those obtained by solar-system experiments such as laser ranging to the Viking Lander on Mars (see, e.g., [113 , 62 ]). Several other millisecond pulsars, once “Shklovskii” and Galactic-acceleration corrections are taken into account, have similarly large characteristic ages (see, e.g., [29 , 137 ]).

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

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

(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 (29

), this leads to a bound of $G˙∕G = (− 9 ± 18) × 10−12 yr−1$ , which Arzoumanian [7

] corrects using Equation (30

) 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 [135 ] 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 5 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 $G$ .

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 [90 ], 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.