6 Gravitational Wave Tests of 5 Stellar System Tests of 5.3 Binary pulsars and alternative

5.4 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 [147Jump To The Next Citation Point In The Article], one can obtain formulas for the periastron shift, the gravitational redshift/second-order Doppler shift parameter, and the rate of change of orbital period, analogous to Eqs. (60Popup Equation). These formulas depend on the masses of the two neutron stars, on their self-gravitational binding energy, represented by ``sensitivities'' s and tex2html_wrap_inline5427, and on the Brans-Dicke coupling constant tex2html_wrap_inline3785 . First, there is a modification of Kepler's third law, given by


Then, the predictions for tex2html_wrap_inline5279, tex2html_wrap_inline5281 and tex2html_wrap_inline5283 are




where tex2html_wrap_inline5437, and, to first order in tex2html_wrap_inline5439, we have


The quantities tex2html_wrap_inline5441 and tex2html_wrap_inline5443 are defined by


and measure the ``sensitivity'' of the mass tex2html_wrap_inline5445 and moment of inertia tex2html_wrap_inline5447 of each body to changes in the scalar field (reflected in changes in G) for a fixed baryon number N (see TEGP 11, 12 and 14.6 (c) [147Jump To The Next Citation Point In The Article] for further details). The quantity tex2html_wrap_inline5441 is related to the gravitational binding energy. Notice how the violation of SEP in Brans-Dicke 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 tex2html_wrap_inline5455, we recover GR, and all structure dependence disappears. The first term in tex2html_wrap_inline5283 (Eq. (68Popup Equation)) is the effect of quadrupole and monopole gravitational radiation, while the second term is the effect of dipole radiation.

In order to estimate the sensitivities tex2html_wrap_inline5441 and tex2html_wrap_inline5461, one must adopt an equation of state for the neutron stars. It is sufficient to restrict attention to relatively stiff neutron star equations of state in order to guarantee neutron stars of sufficient mass, approximately tex2html_wrap_inline5463 . The lower limit on tex2html_wrap_inline3785 required to give consistency among the constraints on tex2html_wrap_inline5279, tex2html_wrap_inline4351 and tex2html_wrap_inline5283 as in Figure  6 is several hundred [153]. The combination of tex2html_wrap_inline5279 and tex2html_wrap_inline4351 give a constraint on the masses that is relatively weakly dependent on tex2html_wrap_inline4355, thus the constraint on tex2html_wrap_inline4355 is dominated by tex2html_wrap_inline5283 and is directly proportional to the measurement error in tex2html_wrap_inline5283 ; in order to achieve a constraint comparable to the solar system value of tex2html_wrap_inline5485, the error in tex2html_wrap_inline5487 would have to be reduced by more than a factor of ten.

Alternatively, a binary pulsar system with dissimilar objects, such as a white dwarf or black hole companion, would provide potentially more promising tests of dipole radiation. Unfortunately, none has been discovered to date; the dissimilar system B0655+64, with a white dwarf companion is in a highly circular orbit, making measurement of the periastron shift meaningless, and is not as relativistic as 1913+16. From the upper limit on tex2html_wrap_inline5283 (Table  7), one can infer at best the weak bound tex2html_wrap_inline5491 .

Damour and Esposito-Farèse [42] have generalized these results to a broad class of scalar-tensor theories. These theories are characterized by a single function tex2html_wrap_inline5493 of the scalar field tex2html_wrap_inline4205, which mediates the coupling strength of the scalar field. For application to the solar system or to binary systems, one expands this function about a cosmological background field value tex2html_wrap_inline5497 :


A purely linear coupling function produces Brans-Dicke theory, with tex2html_wrap_inline5499 . The function tex2html_wrap_inline5493 acts as a potential function for the scalar field tex2html_wrap_inline4205, and, if tex2html_wrap_inline5505, during cosmological evolution, the scalar field naturally evolves toward the minimum of the potential, i.e. toward tex2html_wrap_inline5507, tex2html_wrap_inline5509, or toward a theory close to, though not precisely GR [47, 48]. Bounds on the parameters tex2html_wrap_inline3777 and tex2html_wrap_inline3779 from solar-system, binary-pulsar and gravitational wave observations (see Sec.  6.3) are shown in Figure  8  [44Jump To The Next Citation Point In The Article]. Negative values of tex2html_wrap_inline3779 correspond to an unstable scalar potential; in this case, objects such as neutron stars can experience a ``spontaneous scalarization'', whereby the interior values of tex2html_wrap_inline4205 can take on values very different from the exterior values, through non-linear interactions between strong gravity and the scalar field, dramatically affecting the stars' internal structure and the consequent violations of SEP. On the other hand, tex2html_wrap_inline5519 is of little practical interest, because, with an unstable tex2html_wrap_inline4205 potential, cosmological evolution would presumably drive the system away from the peak where tex2html_wrap_inline5507, toward parameter values that could easily be excluded by solar system experiments. On the tex2html_wrap_inline5525 plane shown in Figure  8, the tex2html_wrap_inline3777 axis corresponds to pure Brans-Dicke theory, while the origin corresponds to pure GR. As discussed above, solar system bounds (labelled ``1PN'' in Figure  8) still beat the binary pulsars. The bounds labelled ``LIGO-VIRGO'' are discussed in Sec.  6.3 .


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Figure 8: The region of the scalar-tensor theory tex2html_wrap_inline3777 - tex2html_wrap_inline3779 plane allowed by solar-system, binary-pulsar, and future gravitational wave observations. A polytropic equation of state for the neutron stars was assumed. The shaded region is that allowed by all tests. For positive values of tex2html_wrap_inline3779, solar-system bounds (labelled 1PN) still are the best. (From [44Jump To The Next Citation Point In The Article], © 1998 by the American Physical Society, reproduced by permission.)

6 Gravitational Wave Tests of 5 Stellar System Tests of 5.3 Binary pulsars and alternative

image The Confrontation between General Relativity and Experiment
Clifford M. Will
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de