Higgs-dilaton cosmology: An inflation–dark-energy connection and forecasts for future galaxy surveys

The Higgs-dilaton model is a scale-invariant extension of the Standard Model nonminimally coupled to gravity and containing just one additional degree of freedom on top of the Standard Model particle content. This minimalistic scenario predicts a set of measurable consistency relations between the inflationary observables and the dark-energy equation-of-state parameter. We present an alternative derivation of these consistency relations that highlights the connections and differences with the $\alpha$-attractor scenario. We study how far these constraints allow one to distinguish the Higgs-dilaton model from $\mathrm{\Lambda }\mathrm{CDM}$ and $w\mathrm{CDM}$ cosmologies. To this end we first analyze existing data sets using a Markov chain Monte Carlo approach. Second, we perform forecasts for future galaxy surveys using a Fisher matrix approach, both for galaxy clustering and weak lensing probes. Assuming that the best fit values in the different models remain comparable to the present ones, we show that both Euclid- and SKA2-like missions will be able to discriminate a Higgs-dilaton cosmology from $\mathrm{\Lambda }\mathrm{CDM}$ and $w\mathrm{CDM}$.

Figure 1

The HD Einstein-frame potential $U_E=U+U_{{\mathrm{\Lambda }}_0}$ in the ${\mathrm{\Lambda }}_0>0$ case. In order to better visualize the almost degenerate valleys at $h_0=\pm \sqrt{\alpha }{\chi }_0$ we used a rather large and unphysical value of $\alpha$. For illustration purposes we also included a typical Higgs-dilaton trajectory. Inflation takes place in the asymptotically flat region, where the effective potential can be well approximated by the $U$ term and scale invariance is approximately realized. The conservation of the Noether current associated to this continuous symmetry restricts the motion of the fields to ellipsoidal trajectories in the $\{h,\chi \}$ plane. After inflation, heating and entropy production take place and the fields eventually relax to the valleys of the potential. Due to the $U_{{\mathrm{\Lambda }}_0}$ term in Eq. (11), these valleys are slightly tilted towards $\chi \to \infty$. At early times, the Hubble friction is so large that the fields stay essentially “frozen.” When the decreasing SM energy density equals the approximately constant term $U_{{\mathrm{\Lambda }}_0}$, the fields roll down the valley and asymptotically approach the ground state of the system at $\chi \to \infty$.

Figure 2

Results of the MCMC run for different scenarios. The plots in the bottom left corner compare $\mathrm{\Lambda }\mathrm{CDM}$ (red, dashed) with the HD model (blue, solid). The comparison of the latest scenario with $w\mathrm{CDM}$ (green, dotted) is shown in the upper right corner. Note that the tensor-to-scalar ratio $r$ is plotted on a logarithmic scale (see Fig. 2 for a detailed view of the $r-n_s$ contours in the HD model).

Figure 3

(Left) Detailed view of the $1\sigma$ and $2\sigma$ contours for the spectral tilt and tensor-to-scalar ratio in a HD cosmology. These values are obtained by means of the consistency relations (43)–(45). This $r-n_s$ consistency relation strongly restricts the possible values for the spectral tilt $n_s$ and the tensor-to-scalar ratio $r$ as compared to the $\mathrm{\Lambda }\mathrm{CDM}$ and $w\mathrm{CDM}$ scenarios (cf. Fig. 2). (Right) Detailed view of $1\sigma$ and $2\sigma$ contours for the spectral tilt and the present dark-energy equation of state in the HD model (blue, solid) and the $w\mathrm{CDM}$ scenario (green, dotted). The blue solid line stands for the expected HD consistency relation (43) evaluated at the mean values of the MCMC run ($N_{*}=60.7$, ${\mathrm{\Omega }}_{\mathrm{DE},0}=0.696$).

Figure 4

$1\sigma$ and $2\sigma$ constraints on the model parameters ${\xi }_{\mathrm{eff}}$ and $c$. The contours are obtained by combining Eqs. (28) and (29) with the observationally allowed regions for the curvature power spectrum and the spectral tilt, cf. Fig. 2. The blue solid line stands for the expected HD relation evaluated at the mean values of the MCMC run [$N_{*}=60.7$, $\ln ({10}^{10}{A}_s)=3.07$]. Note that for not too small values of the Higgs self-coupling, e.g. $\lambda \sim \mathcal{O}({10}^{-3})$, the condition ${\xi }_{\mathrm{eff}}/\sqrt{\lambda }\gg 1$ translates into a large value of the Higgs nonminimal coupling, ${\xi }_{\mathrm{eff}}\simeq {\xi }_h\gg 1$.

Figure 5

Comparison of a $\mathrm{\Lambda }\mathrm{CDM}$ (red, dashed) and a HD cosmology (blue, solid). The leftmost column shows the fully marginalized 1D Gaussian probability distribution for the $n_s$ parameter. The second and third columns show the $1\sigma$ and $2\sigma$ 2D Gaussian error contours. Each model is centered on the fiducial values obtained from its own MCMC run. We plot (i) the constraints from the MCMC run in a Fisher approximation, and add to this (ii) the constraints for a DESI-like mission considering GC on linear scales only, (iii) the constraints for a Euclid-like mission, combining GC on linear scales and WL nonlinear, and (iv) the combination of constraints for an SKA2-like survey, with both GC and WL nonlinear. All remaining parameters are marginalized over. A HD cosmology can be ruled out by a future measurement of large spectral tilt.

Figure 6

Comparison of a HD cosmology (blue, solid) and a $w\mathrm{CDM}$ model (green, dotted) for the same observables and data as in Fig. 5. Note that the $w\mathrm{CDM}$ case is centered on the Higgs-dilaton fiducial values (for example the MCMC mean value for $w_0$ in $w\mathrm{CDM}$ is $w_0=-0.939\pm 0.061$) to facilitate the comparison with the HD cosmology and to illustrate the impact of the consistency relations on parameter uncertainties, especially on the estimation of the present dark-energy equation-of-state parameter $w_0$. A HD cosmology can be ruled out by a measurement of a large $w_0$ or if phantom dark energy is preferred by future data. For a futuristic SKA2-like survey, the cosmological constant scenario could be ruled out at the $2\sigma$ level.

Figure 7

Correlation matrix for the HD model forecast for a Euclid-like probe. The $+1$ and $-1$ limits stand respectively for totally correlated and totally anticorrelated.