Breaking a Dark Degeneracy with Gravitational Waves

TL;DR

The paper identifies a Horndeski scalar-tensor model whose background expansion and linear scalar perturbations are indistinguishable from ΛCDM, yet it yields self-acceleration via a conformal mapping between Einstein-Friedmann and Jordan frames. It shows that although the scalar sector remains degenerate, the tensor sector experiences modified propagation, with gravitational waves propagating at up to 95% of light speed and experiencing reduced damping relative to GR. By leveraging gravitational-wave observations, notably GW150914 and possible EM counterparts, the authors argue that one can break the dark degeneracy and constrain the underlying modified gravity, linking α_M and α_T to observable GW signatures. They also discuss standard sirens and arrival-time measurements as complementary probes, concluding that a combined GW-EM approach can robustly test or rule out broad classes of self-accelerated Horndeski models, while nonlinear effects remain an open area for future work.

Abstract

We identify a scalar-tensor model embedded in the Horndeski action whose cosmological background and linear scalar fluctuations are degenerate with the concordance cosmology. The model admits a self-accelerated background expansion at late times that is stable against perturbations with a sound speed attributed to the new field that is equal to the speed of light. While degenerate in scalar fluctuations, self-acceleration of the model implies a present cosmological tensor mode propagation at < 95% of the speed of light with a damping of the wave amplitude that is > 5% less efficient than in general relativity. We show that these discrepancies are endemic to self-accelerated Horndeski theories with degenerate large-scale structure and are tested with measurements of gravitational waves emitted by events at cosmological distances. Hence, gravitational-wave cosmology breaks the dark degeneracy in observations of the large-scale structure between two fundamentally different explanations of cosmic acceleration - a cosmological constant and a scalar-tensor modification of gravity. The gravitational wave event GW150914 recently detected with the aLIGO instruments and its potential association with a weak short gamma-ray burst observed with the Fermi GBM experiment may have provided this crucial measurement.

Figures (4)

• Figure 1: Cosmic acceleration from a modification of gravity. In a genuinely self-accelerated modified gravity model, the conformal factor $\Omega$ maps a non-accelerated scale factor $\tilde{a}(\tilde{t})$ in the Einstein--Friedmann frame to the observed accelerated scale factor $a(t)$ in the Jordan frame, which matches here the expansion history $H(t)$ of the concordance cosmology $\Lambda$CDM. For ${\rm d}^2\tilde{a}/{\rm d}\tilde{t}^2\lesssim0$ in the late-time universe, we require $\Omega_+\lesssim-0.1$.
• Figure 2: The EFT coefficients (left panel) and modified gravity parameters $\alpha_i$ (right panel) characterising the cosmological background evolution and linear perturbations of the linearly shielded Horndeski scalar-tensor model. The model recovers a GR cold dark matter universe at early times with deviations in the EFT coefficients from their $\Lambda$CDM values of order $\Omega_+$ at late times.
• Figure 3: Left panel: The linearly shielded scalar-tensor model satisfies stability conditions, whereby the sound speed of the scalar field fluctuation equals the speed of light $c_{\rm s}^2=1$ and the tensor modes propagate at subluminal speed. Right panel: Comparison of scalar fluctuations in the linearly shielded model against $\Lambda$CDM. For explicitness, we have evaluated the modified perturbation equations at the Hubble scale $k=H_0$, where one would typically expect signatures from a self-acceleration in the modified gravity model. Deviations between the perturbative quantities $\epsilon$ in the two different models, however, remain within $|\epsilon/\epsilon_{\Lambda{\rm CDM}}-1|\lesssim2\times10^{-5}$ at all $a$. This degeneracy extends to all scales with maximal deviations confined to the sub-percent level. Hence, the model cannot be tested with ongoing and planned surveys of the large-scale structure.
• Figure 4: Prospective constraints on a running Planck mass $\alpha_{\rm M}\leq0$ and a modified speed of tensor mode propagation $-1<\alpha_{\rm T}\leq0$ with gravitational wave detectors measuring the decay of the wave amplitude with standard sirens and the arrival time with respect to the electromagnetic emission from a cosmological event. Shaded regions indicate allowed regions. As described in Eq. (\ref{['eq:nonzeroaMaT']}) scalar-tensor theories require at least either $\alpha_{\rm M}$ or $\alpha_{\rm T}'$ to be $\mathcal{O}(0.1-1)$ for a self-accelerated cosmic expansion due to modified gravity, here schematically set by $|\Omega'/\Omega|_{z=0}\gtrsim0.5$ and assuming $\alpha_{\rm T}'/\alpha_{\rm T}\approx4$. The dark degeneracy model introduced in Sec. \ref{['sec:model']} cannot be discriminated from $\Lambda$CDM with measurements of the cosmological background expansion and large-scale structure but its self-accelerating regime can be tested with gravitational wave cosmology. More generally, the simultaneous measurement of the gravitational wave and electromagnetic emissions of a cosmological event implies $\alpha_{\rm T}\simeq0$ and limits self-acceleration to an effect due to $\mathcal{O}(\alpha_{\rm M})\gtrsim0.5$, which could either be ruled out by future constraints from large-scale structure (LSS) or from standard sirens, challenging the concept of self-acceleration from a scalar-tensor modification of gravity. Note that such a simultaneous measurement may potentially already have been achieved with the recent gravitational wave observation GW150914 with aLIGO GW150914 and a weak short gamma-ray burst measured by the Fermi GBM experiment GRB150914.