Gravitational-wave dispersion over inhomogeneous space-times: General relativity, screened theories of gravity and non-minimal dark energy

Abstract

Gravitational waves (GWs) are direct probes of cosmological gravity, sensitive to space-time inhomogeneities along their propagation. The presence of massive objects breaks homogeneity and isotropy, allowing for new interactions between different GW polarizations, and opening up the intriguing opportunity to test modified gravity theories. This setup generalizes the notion of gravitational deflection and lensing, revealing novel phenomena in modified theories. Any non-minimal theory introduces effective mass terms for GWs, causing \textit{lens-induced dispersion} (LID), a frequency-dependent phase shift on the waveform. We compute GW dispersion in Einstein's general relativity (GR) for a spherical matter distribution, finding a small but non-zero phasing that is potentially accessible to next-generation detectors. We then extend our analysis to scalar-tensor theories, focusing on symmetron gravity as an example of screened theory, combining cosmological deviations and consistency with local gravity tests. We find enhanced GW dispersion in a large region of the symmetron parameter space, compared to both GR and Brans-Dicke theory. We argue that dispersion, associated to an effective mass for the metric fluctuations, can in some cases prevent the propagation of GWs through some astrophysical bodies, turning them into reflectors. Our analysis shows that the Earth becomes an efficient GW shield for a hitherto unconstrained region of the symmetron parameter space, leading to a $\sim 50\%$ fraction of events becoming unobservable or at least displaying a dramatic modification of the detector antenna response. The richness and universality of dispersive phenomena in non-minimal theories open a new avenue to test theories of dynamical dark-energy, relevant in light of recent observational results challenging the $Λ$CDM paradigm.

Figures (8)

• Figure 1: Schematic representation of the scalar field profile $\phi$ in a partially screened spherical object. The screened interior is represented with a blue gradient. A thin shell (cyan) surrounds the core and marks the transition region, beyond which the field grows rapidly. For more details on this figure and a comparison with objects that are fully screened or unscreened, see Sakstein:2014jrq.
• Figure 2: Normalized scalar field profile $\phi/\phi_0$ (left panel) and its radial derivatives (central and right panel) for a spherical Earth-like object. The red dashed circle indicates the boundary $r = R$. The values $\lambda = 10^{-80}$ and $M_s = 10^{12}~\mathrm{GeV}$ are adopted as illustrative benchmarks to show the spatial structure of the field and its gradients. As $M_s$ increases and $\lambda$ pushed towards its limit value bound (see Eq. \ref{['lambda max']}), the smooth thin-shell region visible in the left panel becomes increasingly narrow. This implies a more abrupt variation of the scalar field profile and its derivatives at the boundary.
• Figure 3: Schematic representation of the wave propagation setup. Left panel. The wave is emitted at $z_s$, propagates along the $\hat{z}$-direction with an impact parameter $b$, and crosses a spherically symmetric lens located at $z_{\text{lens}}$. The propagation path is split into regions inside and outside the lens (with $r$ the radial coordinate from the lens center). Right panel. The entire propagation path is divided into three regions: I, before encountering the lens; II, inside the lens (i.e., $|z - z_{\text{lens}}| < R$); and III, after encountering the lens. Each region is treated independently to evaluate the corresponding contribution to the total signal.
• Figure 4: Evolution of the dispersion parameter $\beta_{mm}$, in GR, as a function of the normalized impact parameter $b/R$ for various astrophysical lenses. Dashed lines correspond to near-field objects, specifically the Earth and the Moon, which violate the short-wave condition $GMf \gtrsim 1$ (see Sec. \ref{['numerical results']} for a detailed discussion on this). Solid lines, represent the results of more more massive objects for which $GMf \ll 1$ holds at $f = 100~\mathrm{Hz}$, when approximated as homogeneous spheres (see text).
• Figure 5: This plot shows the dispersion parameter $|\beta_{mm}/GMf|$ as a function of the impact parameter, normalized to the object's radius $b/R$, for two different lenses: Earth (top row, with reference frequency $f_{\rm GW}\sim 100$ Hz) and Moon (bottom row, with $f_{\rm GW}\sim 1$ Hz). Each column corresponds to a different value of the scalar field self-coupling $\lambda$. Different colors in the plot represent different values of the symmetron coupling to matter $M_s$, with the GR result shown as a red dash-dot line for reference. An additional black dotted line at $\beta_{mm}/GMf\sim1$ and a vertical line at $b/R\sim1$ are shown in all panels: the first indicates when dispersive effects become $\mathcal{O}(1)$ and the second indicates the boundary of the lens. On the top of each subplot, we show the impact parameter rescaled by the Schwarzschild radius of the corresponding lens, $b/R_s$.