The challenge to make cosmological predictions ...

... based on a spectral Pati-Salam like extension of the standard model

Classical cosmology, tested by a variety of precise astrophysical measurements, is built upon Einstein’s theory of General Relativity and the Cosmological Principle. General Relativity is however a classical theory, hence its validity breaks down at very high energy scales. The Cosmological Principle, namely the assumption of a continuous space-time characterised by homogeneity and isotropy on large scales, is valid only once we consider late-eras of our universe, characterised by energies far below the Planck scale. At very early times, very close to the Big Bang and the so-called Planck era, quantum corrections can no longer be neglected and geometry may altogether lose the meaning we are familiar with. To describe the physics near the Big Bang, a Quantum Gravity theory and the associated appropriate space-time geometry is hence required. 
The available Quantum Gravity proposals can be divided into two classes. On the one hand, there is String Theory/M-theory, according which matter consists of one-dimensional objects, strings, which can be either closed or open (without ends). Different string vibrations would represent different particles; splitting and joining of strings would then correspond to different particle interactions. On the other hand, there are non-perturbative approaches to quantum gravity; some examples are Loop Quantum Gravity, a Euclidean approach to quantum gravity like Causal Dynamical Triangulations, and Group Field Theory. The latter class of models adopts the hypothesis that space is not infinitely divisible, instead it has a granular structure, hence it is made out of quanta of space. In the former class of models, matter is the important ingredient; in the latter one, matter is, so far, (rather artificially) added. These two classes of models can be considered as following a top-down approach, whilst they both inspire cosmological models leading to several observational consequences. 
In the following, I will consider a bottom-up approach, in the sense that I will focus on a proposal attempting to guess the small-scale structure of space-time near the Planck era, using our knowledge of well-tested particle physics at the electroweak scale. More precisely, I will focus on Noncommutative Spectral Geometry (NCSG). One may argue that at the Planck energy scale, quantum gravity implies that space-time is a wildly noncommutative manifold. However, at an intermediate scale, one may assume that the algebra of coordinates is only a mildly noncommutative algebra of matrix valued functions, which if appropriately chosen, may lead to the Standard Model of particles physics coupled to gravity. It is important to note that according to the NCSG proposal, to construct a quantum theory of gravity coupled to matter, the gravity-matter interaction is the most important ingredient to determine the dynamics; this consideration is not the case for either of the {String Theory/M-theory or Loop Quantum Gravity}... mentioned previously ... 
{in the noncommutative spectral standard model} one may wonder whether the Higgs field, through its non-minimal coupling to the background geometry, could play the rôle of the inflaton [17, 18]. To address this question one looks for a flat region of the Higgs potential. Considering the renormalisation of the Higgs self-coupling up to two-loops, one finds that for each value of the top quark mass, there is a value of the Higgs mass where the Higgs potential is locally flattened [18]. However, the flat region is very narrow and to achieve a sufficiently long inflationary era, the slow-roll must be very slow, leading to an amplitude of density perturbations incompatible with Cosmic Microwave Background data (CMB) [18].
It remains an open question of whether inflation, if at all needed within a wildly noncommutative manifold, can be naturally incorporated. The known scalar fields, appearing in the NCSG action, could provide through their non-minimal coupling to the background geometry an era of accelerated expansion but fail to match the cosmic microwave background temperature anisotropies data. Unfortunately, the successful R2 -type inflation [25], favoured by the Planck CMB [26] data, cannot be applied in the higher derivative gravitational theory obtained by noncommutative spectral geometry. It is not clear yet whether one can accommodate a dilaton-type inflation [27] or use a scalar field, in a beyond the Standard Model scenario like the Pati-Salam model [28], as a successful inflaton candidate.
(Submitted on 12 May 2016)

 The general case for a heavy singlet scalar unitarizing Higgs inflaton  
The idea that the Higgs boson could play the role of the inflaton is very intriguing. A scalar theory with quartic interaction in the potential and large non-minimal coupling ξ to the curvature can support inflation. However, the inflationary dynamics occurs at such large values of the scalar field that the identification of the inflaton with the Higgs boson remains suspicious, in view of the existence of the intermediate scale MP/ξ at which the theory around its true vacuum violates unitarity. If we insist that the theory can be extended up to the Planck mass MP, it is quite plausible that the necessary new physics occurring at the scale MP/ξ will modify the Higgs potential in the regime relevant for inflation. Any conclusion about the viability of Higgs inflation will then require knowledge of the new dynamics that unitarizes the theory.  
We have considered a simple model, with one additional scalar field σ, which cures the unitarity violation at the intermediate scale and allows for an extrapolation of the theory up to MP. The procedure we followed to construct the model is reminiscent of the unitarization of the non-linear sigma model into its linear version.  
In our model, the σ field has a mass of order MP/ξ and the effective theory below this scale essentially corresponds to the original model of Higgs inflation [1], namely the SM with a large non-minimal coupling between the Higgs and the curvature. The analysis of our model in the regime above MP/ξ shows that the theory can support inflation in a way completely analogous to the case of the original Higgs inflation. The predictions for the slow-roll parameters and the spectral index are identical in both theories. It is interesting that, in our model, the mass of the heavy mode increases with the field background. Being equal to MP/ξ around the true vacuum, the mass of the new state is about MP/√ξ during inflation, giving an explicit realization of the mechanism advocated in ref. [9], for which the scale of unitarity violation is raised at large background field. In spite of the similarities with the original Higgs inflation, the σ field plays a crucial role. Besides unitarizing the theory, σ directly participates in the inflationary dynamics. Its role is also reflected in the fact that the relation between ξ and the inflationary scale does not only depend on the measurable Higgs quartic coupling, but also on unknown coupling constants determining the σ interactions. The reheating temperature in our model can be smaller than in the original Higgs inflation.
(Submitted on 7 Oct 2010 (v1), last revised 27 Oct 2010 (this version, v2))

The COBE constraint precisely fixes the sigma mass in the vacuum to be Mσ ≈ 1013 GeV ... the sigma field drives a slow-roll inflation while the Higgs field is stabilized at a large VEV during inflation. The difference from a single-field inflation with non-minimal coupling is that the Higgs field contributes a large vacuum energy during inflation and participates in the reheating process...
In the unitarized Higgs inflation, the reheating temperature is sensitive to the Higgs component of the inflaton, which is determined by the mixing coupling λHσ between the Higgs and sigma fields. ... for N=62(59), the tensor-to-scalar ratio is given by r=0.0030(0.0033).
(Submitted on 9 Jan 2013 (v1), last revised 18 Apr 2013 (this version, v3))