Higgs-Dilaton Cosmology: From the Early to the Late Universe

TL;DR

This work develops a minimal, scale-invariant extension of the Standard Model coupled to gravity and implemented within Unimodular Gravity to generate all scales via spontaneous symmetry breaking. Inflation is driven by the Higgs-dilaton sector, while a massless dilaton becomes a thawing quintessence field responsible for late-time dark energy, yielding a run-away potential controlled by an integration constant $\Lambda_0$. The model makes sharp, testable predictions: a scalar spectral tilt $n_s<0.97$, a small running $-0.0006<d\ln n_s/d\ln k\lesssim-0.00015$, and a scalar-to-tensor ratio $0.0009\lesssim r<0.0033$, with $ω_{DE}^0>-1$; in the special case $\beta=0$, $n_s$ and the present DE equation of state $w_{DE}^0$ become tightly linked, yielding $0<1+w_{DE}^0<0.02$ and a relation between the running and DE dynamics. The findings connect early and late Universe observables, offering Planck-era tests and future dark-energy surveys as consistency checks of the Higgs-Dilaton framework.

Abstract

We consider a minimal scale-invariant extension of the Standard Model of particle physics combined with Unimodular Gravity formulated in \cite{Shaposhnikov:2008xb}. This theory is able to describe not only an inflationary stage, related to the Standard Model Higgs field, but also a late period of Dark Energy domination, associated with an almost massless dilaton. A number of parameters can be fixed by inflationary physics, allowing to make specific predictions for any subsequent period. In particular, we derive a relation between the tilt of the primordial spectrum of scalar fluctuations, $n_s$, and the present value of the equation of state parameter of dark energy, $ω_{DE}^0$. We find bounds for the scalar tilt, $n_s<0.97$, the associated running, $-0.0006<d\ln n_s/d\ln k\lesssim-0.00015$, and for the scalar-to-tensor ratio, $0.0009\lesssim r<0.0033$, which will be critically tested by the results of the Planck mission. For the equation of state of dark energy, the model predicts $ω_{DE}^0>-1$. The relation between $n_s$ and $ω_{DE}^0$ allows us to use the current observational bounds on $n_s$ to further constrain the dark energy equation of state to $0< 1+ω_{DE}^0< 0.02$, which is to be confronted with future dark energy surveys.

Figures (11)

• Figure 1: These plots show the shape of the E-frame potential $\tilde{U} (h,\chi)$ (equation \ref{['potE0']}) for $\Lambda_0=0$, $\Lambda_0>0$ and $\Lambda_0<0$ respectively.
• Figure 2: The blue region is the slow-roll region for $\xi_\chi\ll 1$, $\xi_h\gg 1$ and $\alpha=0$ given by $\epsilon^{(\mathrm{SR})}<1$. The inclusion of the second slow-roll condition $\eta^{(\mathrm{SR})}<1$ does not change the essential properties of this region. The general features of the slow-roll region are the same whenever $\xi_\chi<\frac{1}{2}$ and $\xi_h>\frac{1}{2}$. For $\xi_\chi<\frac{1}{2}$ and $\xi_h<\frac{1}{2}$ the central fast-rolling region vanishes. For $\xi_\chi>\frac{1}{2}$ the slow-roll region does not extend to infinity along the $\chi$-axis in which case the scalar fields can not act as dark energy in the late stage of evolution. The shaded region corresponds to the scale-invariant region delimited by $\upsilon_1<1$ and $\upsilon_2<1$. This is the region where the influence of $\Lambda_0\neq 0$ is small. The presence of the slow roll-region along the $\chi$-axis such as the central fast-roll region are effects of $\Lambda_0>0$. For $\Lambda_0=0$ the slow-roll region is simply given by the triangles delimited by the two diagonal lines. Note that in this case the units of the axis have to be chosen differently. The red line represents a trajectory of type a, never leaving the SR-region. The blue line is a trajectory of the type b, which leaves the SR-region and oscillates strongly before rolling down the valley. These trajectories were found by numerically solving the exact equations \ref{['sfe']}.
• Figure 3: This plot shows the parameter regions for which the amplitude $P_\zeta(k_0)$ and the tilt $n_s(k_0)$ of the scalar spectrum lie in the observationally allowed region (WMAP7 + BAO + $H_0$ at $99\%$ confidence level), for $\lambda=1$. (The variation of the result induced by variation of $\lambda$ in the interval $0.1<\lambda<1$ is negligible.) The red region is obtained for $\varrho_{rh}=\varrho_{rh}^{max}$ (instantaneous reheating), while the blue region corresponds to $\varrho_{rh}=\varrho_{rh}^{min}$ (long reheating). The fact that the bands are cut on the right comes from the constraint on the scalar tilt $n_s(k_0)$, cf. \ref{['tiltcal']}, while the band-shape is due to the constraint on the amplitude $P_\zeta(k_0)$, cf. \ref{['ampcal']}.
• Figure 4: The spectral tilt as a function of the non-minimal coupling parameter $\xi_\chi$. The other parameters are set to $\xi_h=65000$ and $\lambda=1$. Note, however, that changing the ratio $\xi_h/\sqrt{\lambda}$ in the observationally allowed range affects the result only by a negligible amount. The solid curves correspond to the numerical results, the blue one is obtained for $\varrho_{rh}=\varrho_{rh}^{min}$ (long reheating) and the red one for $\varrho_{rh}=\varrho_{rh}^{max}$ (instantaneous reheating). The blue and red dashed curves are obtained from the analytical approximation \ref{['tiltcal']} for $\bar{N}^*_{min}$ and $\bar{N}^*_{max}$. The black dashed curve represents the asymptotic solution \ref{['nsassol']}, which is a good approximation if $\frac{1}{4N^*}<\xi_\chi\ll1$ . The horizontal line and the shaded regions correspond to the observational mean value and the $1\sigma$ and $3\sigma$ confidence intervals, cf. \ref{['tiltobs']}.
• Figure 5: The running of the scalar spectral tilt and the tensor-to-scalar ratio (right) as a function of the coupling $\xi_\chi$. The other parameters are set to $\xi_h=65000$ and $\lambda=1$. Note, however, that changing the ratio $\xi_h/\sqrt{\lambda}$ in the observationally allowed range affects the result only by a negligible amount. Blue solid curves show the numerical results for $\varrho_{rh}=\varrho_{rh}^{min}$ (long reheating), while the red solid curves are the numerical results for the case $\varrho_{rh}=\varrho_{rh}^{max}$ (instantaneous reheating). The dashed curves are obtained from the approximate expressions \ref{['rcal']} and \ref{['runcal']}, inserting $\bar{N}^*_{min}$ and $\bar{N}^*_{max}$