Category: Cosmology

Scalar Gravitational Waves in a Dark Energy Dominated Universe

In the recent years, I have been curious if the notion of “gravitational waves” ought to be enlarged once the background spacetime in question is not Minkowski spacetime, but the large scale expanding universe

g_{\mu\nu}[\eta,\vec{x}] = a[\eta]^2 \eta_{\mu\nu}.

In particular, the massless helicity-2 character of the gravitational wave traveling in a flat background is no longer the only allowed polarization pattern. In a previous post, I discussed the existence of spin-0 gravitational polarizations in a radiation dominated universe.

Dark Energy versus Cosmological Constant

In my recent work with Li-Ying Chou (my former undergrad, now Master’s student) and Yen-Wei Liu (my former postdoc), I turned my attention to a nearly de Sitter spacetime, engendered not by a cosmological constant in Einstein’s equations, but by a slowly rolling canonical scalar field with Lagrangian density

\mathcal{L}_\varphi \equiv \frac{1}{2} (\nabla\varphi)^2 - V[\varphi].

On a cosmological constant driven de Sitter geometry, the linearized Einstein’s equations have been studied quite a bit over the past decade and a half or so; and the result is that gravitational radiation does only have massless spin-2 components. However, it has been clear to me for quite some time now, that when this de Sitter-like behavior is instead due to a Dark Energy scalar–or, more generally, some extra field degrees of freedom–there will be mixing between the first order perturbations of this Dark Energy with those of the scalar metric ones. In other words, there will be a discontinuous change in the number of allowed gravitational polarization modes when comparing the two scenarios, even though the background geometry is nearly indistinguishable. For the specific case of a Dark Energy scalar driven universe, I expected massless helicity zero gravitational waves.

Traceless Tidal Forces

The key object we computed was \delta_1 C^i_{\phantom{i}0j0}, the electric portion of the linearized Weyl tensor, as sourced by some isolated astrophysical system. By recalling the geodesic deviation equation, we see that these components describe the traceless tidal forces acting on a pair of infinitesimally nearby free-falling test masses. Importantly, the linearized Weyl tensor in an expanding universe is invariant under infinitesimal coordinate transformations because of conformal/Weyl invariance. This, in turn, allows us to un-ambiguously identify the “scalar” and “tensor” contributions to these traceless tidal forces.

We found that these canonical scalar Dark Energy cosmological spin-0 gravitational waves, unlike their radiation dominated cousins, propagate at the speed of light; and, in particular, the null cone spin-0 contribution to the traceless tidal force reads

(1) \delta_1 C^i_{\phantom{i}0j0}[\text{spin}-0,\text{ null cone}] \\ \approx -\frac{G_N}{2 a[\eta] r} \left( \delta_{ij} - 3 \widehat{r}_i \widehat{r}_j \right) \mathcal{H}[\eta] \mathcal{H}[\eta_r] \sqrt{ \delta w[\eta] \delta w[\eta_r] } \left( M[\eta_r] + \frac{ \ddot{Q}_{\ell\ell}[\eta_r] }{2 a^2[\eta_r] } + \dots \right),

where \eta_r \equiv \eta - r is retarded conformal time; a[\eta] \cdot r is the source-observer proper spatial distance; and M is the system’s total mass. We have assumed the source is non-relativistic; and here, \ddot{Q}_{\ell\ell} is the (conformal time) acceleration of the spatial trace of its mass quadrupole moment. Moreover, we have parametrized the Dark Energy equation-of-state (its pressure-to-energy density) as

w = -1 + \delta w ,

where the deviation \delta w is presumably small; and the retarded time is \eta_r \equiv \eta - r.

Of course, even though the number of polarization modes are different — as one might expect — the scalar gravitational ones are suppressed relative to their tensor cousins due to the presence of \mathcal{H}[\eta] \equiv \dot{a}/a in eq. (1). On the other hand, note that this spin-0 signal is directly proportional to \sqrt{\delta w[\eta] \delta w[\eta_r]}; i.e., it is directly sensitive to the Dark Energy equation of state.

Compact Binary System

Since mass is conserved on astrophysical timescales, the high frequency part of the spin-0 traceless tidal forces in eq. (1) is the \ddot{Q}_{\ell\ell} term. For a compact binary system with total mass m, reduced mass \mu, orbital angular frequency \omega_a, and eccentricity e, the high frequency portion of eq. (1) becomes

(2) \delta_1 C^i_{\phantom{i}0j0}[\text{High freq. spin}-0,\text{ null cone}] \\ \approx -\frac{G_N^{5/3} (\omega_a \cdot m)^{2/3} \mu}{2 a[\eta] r} \frac{e}{1-e^2} \left( \delta_{ij} - 3 \widehat{r}_i \widehat{r}_j \right) \mathcal{H}[\eta] \mathcal{H}[\eta_r] \sqrt{ \delta w[\eta] \delta w[\eta_r]} \cos \psi ,

where \psi is the (retarded) orientation angle of the binary on the 2D plane they lie on. On the other hand, the spin-2 contribution reads

(3) \delta_1 C^i_{\phantom{i}0j0}[\text{spin}-2, \text{ null cone}] \\ \approx -\frac{8G_N^{5/3} (\omega_a \cdot m)^{2/3} \mu}{2 a[\eta] r} \omega_a^2 \left( \frac{1+\cos^2\theta}{2} \cos[2(\psi-\phi)] e^+_{ij} + \cos\theta \sin[2(\psi-\phi)] e^\times_{ij} \right) ,

where the (\theta,\phi) are spherical coordinates defined by treating the orbital plane of the binary system as the (1,2)-plane; and the helicity-2 polarization tensors e^{\times,+}_{ij} are perpendicular to the propagation direction — i.e., the unit radial vector \widehat{r} — so that e^{\times,+}_{ij} \delta^{ij} = 0 = e^{\times,+}_{ij} \widehat{r}^i. By comparing equations (2) and (3), one readily recognizes the isotropic character of the former spin-0 signal. Moreover, eq. (2) is directly sensitive to the eccentricity of the orbital motion engendering the gravitational radiation.

Summary

Because we deliberately did not couple ordinary matter to the Dark Energy scalar, I expect the existence of scalar gravitational waves to be rather generic — albeit highly suppressed — in Dark Energy models of accelerated cosmic expansion, provided of course the theory itself is valid down to astrophysical length scales. I believe eq. (2) is the first concrete illustration of such spin-0 gravitational radiation emitted from the sort of compact binary systems LIGO, Virgo, etc. have been hearing from to date.

References

  • L.Y.Chou, Y.Z.Chu and Y.W.Liu, “Scalar Gravitational Waves Can Be Generated Even Without Direct Coupling Between Dark Energy and Ordinary Matter,” [arXiv:2310.14547 [gr-qc]].
  • A.Ashtekar, B.Bonga and A.Kesavan, “Asymptotics with a positive cosmological constant: I. Basic framework,” Class. Quant. Grav. 32, no.2, 025004 (2015) [arXiv:1409.3816 [gr-qc]].
  • A.Ashtekar, B.Bonga and A.Kesavan, “Asymptotics with a positive cosmological constant. II. Linear fields on de Sitter spacetime,” Phys. Rev. D 92, no.4, 044011 (2015) [arXiv:1506.06152 [gr-qc]].
  • A.Ashtekar, B.Bonga and A.Kesavan, “Asymptotics with a positive cosmological constant: III. The quadrupole formula,” Phys. Rev. D 92, no.10, 104032 (2015)[arXiv:1510.05593 [gr-qc]].
  • H.J.de Vega, J.Ramirez and N.G.Sanchez, “Generation of gravitational waves by generic sources in de Sitter space-time,” Phys. Rev. D 60, 044007 (1999) [arXiv:astro-ph/9812465 [astro-ph]].
  • Y.Z.Chu, “Gravitational Wave Memory In dS\,_{4+2n} and 4D Cosmology,” Class. Quant. Grav. 34, no.3, 035009 (2017) [arXiv:1603.00151 [gr-qc]].
  • B.Bonga and J.S.Hazboun, “Power radiated by a binary system in a de Sitter Universe,” Phys. Rev. D 96, no.6, 064018 (2017) [arXiv:1708.05621 [gr-qc]].