We study the Hybrid Natural Inflation (HNI) model and some of its realisations in the light of recent CMB observations, mainly Planck temperature and WMAP-9 polarization, and compare with the recent release of BICEP2 dataset. The inflationary sector
of HNI is essentially given by the potential $V(phi) = V_0(1+acos (frac{phi}{f} ) )$, where $a$ is a positive constant smaller or equal to one and $f$ is the scale of (pseudo Nambu-Goldstone) symmetry breaking. We show that to describe the HNI model realisations we only need two observables; the spectral index $n_s$, the tensor-to-scalar ratio, and a free parameter in the amplitude of the cosine function $a$. We find that in order to make the HNI model compatible with the BICEP2 observations, we require a large positive running of the spectra. We find that this could over-produce primordial black holes in the most consistent case of the model. This situation could be aleviated if, as recently argued, the BICEP2 data do not correspond to primordial gravitational waves.
The current concordance model of cosmology is dominated by two mysterious ingredients: dark matter and dark energy. In this paper, we explore the possibility that, in fact, there exist two dark-energy components: the cosmological constant $Lambda$, w
ith equation-of-state parameter $w_Lambda=-1$, and a `missing matter component $X$ with $w_X=-2/3$, which we introduce here to allow the evolution of the universal scale factor as a function of conformal time to exhibit a symmetry that relates the big bang to the future conformal singularity, such as in Penroses conformal cyclic cosmology. Using recent cosmological observations, we constrain the present-day energy density of missing matter to be $Omega_{X,0}=-0.034 pm 0.075$. This is consistent with the standard $Lambda$CDM model, but constraints on the energy densities of all the components are considerably broadened by the introduction of missing matter; significant relative probability exists even for $Omega_{X,0} sim 0.1$, and so the presence of a missing matter component cannot be ruled out. As a result, a Bayesian model selection analysis only slightly disfavours its introduction by 1.1 log-units of evidence. Foregoing our symmetry requirement on the conformal time evolution of the universe, we extend our analysis by allowing $w_X$ to be a free parameter. For this more generic `double dark energy model, we find $w_X = -1.01 pm 0.16$ and $Omega_{X,0} = -0.10 pm 0.56$, which is again consistent with the standard $Lambda$CDM model, although once more the posterior distributions are sufficiently broad that the existence of a second dark-energy component cannot be ruled out. The model including the second dark energy component also has an equivalent Bayesian evidence to $Lambda$CDM, within the estimation error, and is indistinguishable according to the Jeffreys guideline.
