Semi-analytic stellar structure in scalar-tensor gravity

Abstract

Precision tests of gravity can be used to constrain theproperties of hypothetical very light scalar fields, but thesetests depend crucially on how macroscopic astrophysical objectscouple to the new scalar field. We study the equations of stellarstructure using scalar-tensor gravity, with the goal of seeing howstellar properties depend on assumptions made about the scalarcoupling at a microscopic level. In order to make the studyrelatively easy for different assumptions about microscopiccouplings, we develop quasi-analytic approximate methods forsolving the stellar-structure equations rather than simplyintegrating them numerically. (The approximation involved assumesthe dimensionless scalar coupling at the stellar center is weak,and we compare our results with numerical integration in order toestablish its domain of validity.) We illustrate these methods byapplying them to Brans-Dicke scalars, and their generalization inwhich the scalar-matter coupling slowly runs — or `walks' — as afunction of the scalar field: a(ϕ) ≃ as+bsϕ.(Such couplings can arise in extra-dimensional applications, forinstance.) The four observable parameters that characterize thefields external to a spherically symmetric star are thestellar radius, R, mass,M, scalar `charge', Q, and the scalar's asymptotic value,ϕ∞. These are subject to two relations because of thematching to the interior solution, generalizing the usualmass-radius, M(R), relation of General Relativity. Sinceϕ∞ is common to different stars in a given region (suchas a binary pulsar), all quantities can be computed locally interms of the stellar masses. We identify how these relationsdepend on the microscopic scalar couplings, agreeing with earlierworkers when comparisons are possible. Explicit analyticalsolutions are obtained for the instructive toy model ofconstant-density stars, whose properties we compare to morerealistic equations of state for neutron star models.