r-mode instability: Analytical solution with gravitational radiation reaction

Abstract

Analytical r-mode solutions are investigated within the linearized theory in the case of a slowly rotating, Newtonian, barotropic, non-magnetized, perfect-fluid star in which the gravitational radiation (GR) reaction force is present. For the GR reaction term we use the 3.5 post-Newtonian order expansion of the GR force, in order to include the contribution of the current quadrupole moment. We find the explicit expression for the r-mode velocity perturbations and we conclude that they are sinusoidal with the same frequency as the well-known GR force-free linear r-mode solution, and that the GR force drives the r-modes unstable with a growth timescale that agrees with the expression first found by Lindblom, Owen and Morsink. We also show that the amplitude of these velocity perturbations is corrected, relatively to the GR force-free case, by a term of order Ω 6 , where Ω is the angular velocity of the star.

Neutrino-Less Double Beta Decay and Particle Physics

“…Those cases have been studied in Refs. 272,253,273 , and constraints on the masses and mixings with the SM fermions have been obtained. Majoron emission is also possible in those models, because typically neutral scalars with lepton number exists, whose vevs induce spontaneous violation of lepton number, see Section 5.5.…”

ANALYTICAL r-MODE SOLUTION WITH GRAVITATIONAL RADIATION REACTION FORCE

“…The velocity perturbations δ (1) v contain also a piece proportional to αγ(A − 1)Ω 6 , where A is an arbitrary constant fixed by the choice of initial data. If we choose this constant A to be of order unity, then this part of the solution could be neglected [4].…”

“…The linearized hydrodynamic equations ( 8)- (10), with the first-order Eulerian change in the GR force, δ (1) F GR , given by Eq. ( 17), admit the solution [4] δ (1) v r = 0, (18a)…”

“…In this work, in which the main results of Ref. [4] are reported, we present an explicit expression for the r-mode velocity perturbations that solves the linearized hydrodynamics equations with the GR force. Our conclusions are that (i) these velocity perturbations are sinusoidal with the same frequency as the well-known GR force-free linear r-mode solution, (ii) the GR force drives the r-modes unstable with a growth timescale that agrees with the expression found in Refs.…”

Gravitational waves from r-modes

“…When the gravitational radiation reaction force is taken into account, the hydrodynamic equations (1-3) yield velocity perturbations which have the same sinusoidal behavior and the same frequency as the gravitational radiation forcefree velocity perturbations, but with an amplitude which grows exponentially (Dias and Sá 2005).…”