Gravitational waves in the spectral action of noncommutative geometry

The spectral triple approach to noncommutative geometry allows one to develop the entire standard model (and supersymmetric extensions) of particle physics from a purely geometry standpoint and thus treats both gravity and particle physics on the same footing. The bosonic sector of the theory contains a modification to Einstein-Hilbert gravity, involving a nonconformal coupling of curvature to the Higgs field and conformal Weyl term (in addition to a nondynamical topological term). In this paper we derive the weak-field limit of this gravitational theory and show that the production and dynamics of gravitational waves are significantly altered. In particular, we show that the graviton contains a massive mode that alters the energy lost to gravitational radiation, in systems with evolving quadrupole moment. We explicitly calculate the general solution and apply it to systems with periodically varying quadrupole moments, focusing, in particular, on the well-known energy loss formula for circular binaries.

Figure 1

The points are the numerical evaluation of $f_{\mathrm{c}}^2(x,z)+f_{\mathrm{s}}^2(x,z)$ (for three different values of $z<1$) and the lines are plots of the corresponding fitted function given in Eq. (63). Notice that the fitting function breaks down as we approach $z\to 1$, which corresponds to $2\omega \to \beta c$.

Figure 2

The points are the numerical evaluation of $f_{\mathrm{c}}^2(x,z)+f_{\mathrm{s}}^2(x,z)$ (for three different values of $z>1$) and the lines are plots of the corresponding fitted function given in Eq. (64). Notice that the fitting function remain a good approximation even as $z\to 1$, which corresponds to $2\omega \to \beta c$.

Figure 3

The retarded Green’s function is defined by extending the contour of the integral into the positive half of the imaginary plane around each of the four poles, $\omega =\pm |\mathbf{k}|$ and $\omega =\pm \sqrt{|\mathbf{k}{|}^2+{\beta }^2}$.