In LQG, the dynamics can be expressed using two different languages : the canonical one, given using a Hamiltonian, and the covariant one, where dynamics is coded by a transition amplitude obtained summing over histories. The Spinfoam theory is this covariant version of LQG. In this formalism the amplitudes give the probability to pass from a given configuration of space-time to another. The main objective of my thesis has been the application of this covariant formalism to cosmology for the first time. This objective has been reached using two different strategies: the first starts from full LQG, the second from LQC.
The first approach is to look for a covariant version of quantum cosmology by starting from the complete LQG quantum theory . This is what I consider my most original contribution. It is a new approach to quantum cosmology. Quantum cosmology is traditionally conceived as follows: one starts from General Relativity, then the assumption of homogeneity and isotropy reduces the gravitational field to a system with a single degree of freedom. Working with one or a few degrees of freedom, instead of the infinite degrees of freedom of the complete theory, renders the quantization problem much easier. In the 60', when the study of the quantum theory of gravity was still in its infancy, this was the only feasible strategy for studying the quantum properties of the early universe. In this way the Wheeler-deWitt equation for cosmology and all the quantum mini-superspace models were born. LQC can be considered a fruitful realization of this research program. However, today we have a well-defined theory of quantum gravity in the framework of loop quantization, and we can try to construct a theory of quantum cosmology starting directly from the full quantum theory, trying to understand which degrees of freedom are relevant for cosmology. This is what I have attempted to do in my research.
The advantage of this approach is two-fold. First, it will allow us to better understand the relation between usual quantum cosmology and the full theory. But what in my opinion is more interesting would be to concretely exploit the resulting model in order to study the early the universe, in the regime that classically corresponds to the big bang (here replaced by a bounce). The richness of this model allows to describe the quantum geometry with some of its inhomogeneities beyond usual perturbation theory, and hopefully will provide an appropriate framework to explain the seeds of structure formation.
This research line has started with a model  that I introduced with Carlo Rovelli in 2008. This is sometimes known as Dipole Cosmology and the dynamics was defined canonically by a Hamiltonian constraint. In a collaboration with Eugenio Bianchi and Carlo Rovelli, we have been able to compute the cosmological transition amplitude from the full covariant spinfoam expansion , using a dipole graph. Finally, in collaboration with Thomas Krajewski, we also studied the amplitude in de Sitter space . I have recently shown that this result does not depends to the graph used, but holds for every regular discretization .
This approach has been explored so far on the base of Spinfoam dynamics, but it could be applied to other theory. Group Field Theory  would be a perfect candidate.
The main motivation to study spinfoam cosmology is that it could provide a useful setting in order to describe the quantum geometry at the bounce. In particular, I want to understand how the quantum fluctuations of geometry affect structure formation beyond the classical perturbative approach. To do this, one has to relax the assumptions of homogeneity and isotropy. In spinfoam cosmology it is possible to choose how many degrees of freedom one wants to describe by computing the dynamics on an appropriate graph .
The recent development of a spinor framework in LQG  could provide a new tool to explore this issue. My research plan is to study this framework in order to address two concrete questions. The first concerns the definition of homogeneity and isotropy: the U(N) symmetry in the spinor framework make it possible to impose homogeneity and isotropy in a very elegant way. This has already been studied only for the "dipole graph"  and it would be interesting to generalize this result to more complex situations. The second concerns gravitational waves in LQG. I would like to understand how the change of spins propagate in the spinnetwork. My idea is to describe such a propagation in terms of the spinor creation and annihilation operators.
It is possible to include matter and the cosmological constant in spinfoam theory. In my future research I would like to study how to include matter as an effective scalar field. On the other hand, a formulation of the dynamics of fermions and Yang-Mills fields in spinfoam theory has recently been developed  and it deserves further investigation (in particular to evaluate the asymptotic behavior). It would be very interesting to specialize this result to cosmology. The role of fermions in cosmology should be further understood, and I think that it could be particularly interesting in LQC. I would like to investigate it in connection with inflation: in fact, a fermion condensate can play the role of inflaton field , and we know that in LQC, assuming the presence of an inflaton field, the conditions for inflation are naturally achieved . As for the cosmological constant, we have recently add it in the spinfoam amplitude as a "face term" . Part of my future research would be devoted to a further study of the properties of this term, in particular in order to relate it with the observed value of the cosmological constant. A more fundamental way to include the cosmological constant is to use quantum groups, where the parameter deforming the algebra can be related to the value of Λ . I would like to study how to connect the two approaches.
A second approach for applying the covariant formalism to cosmology, is to study how a path integral can be defined from the Hamiltonian constraint of LQC, trying to mimic the spinfoam expansion that we have in the full theory. Starting form a proposal by Ashtekar and collaborators , we have been able to obtain an expression that can be put in a one-to-one correspondence with the spinfoam amplitude . The procedure to perform the expansion is generic, and can be applied to different Hamiltonian based on loop quantization. As in the spinfoam theory, the resulting path integral is fully covariant, marking an improvement with respect to the perturbative evaluation of the "wave function of the universe" à la Halliwell.
In a paper with Carlo Rovelli  we have proposed a regulator to correctly implement the proposal in , and in  we have shown, in collaboration also with Adam Henderson and Edward Wilson- Ewing of the PennState University, that the regulated expression is local and therefore it matches the form of the amplitude in spinfoam theory.
The resolution of classical singularity in the context of quantum gravity seems to be robust for every model based on the kinematical Hilbert space of LQG. More precisely, what has been shown in LQC is that the model leads to the resolution of the (strong) cosmological singularities. The most important singularity is of course the big bang. In LQC the big bang is replaced by a bounce of the universe from a contracting to an expanding phase . This happens because of the quantum nature of the theory, without the introduction of matter violating the energy conditions. On the other hand, if one assumes exotic matter, other types of singularity can appear in classical cosmology, for instance situations where the curvature or its derivatives diverge. In collaboration with Param Singh, I have studied how LQC cures also these singularities (big rip, sudden singularity, big brake...). We have considered a model where various types of singularity can be produced classically by varying a few parameters. I have extended a previous result in the k=0 case  to the curved case with k=±1: I have shown that all strong singularities are cured by quantum effects in LQC. An additional reason of interest of this result is that it partially fills the void of the literature on the classic analysis of future singularities for curved space-time. The original part of my work has regarded a re-derivation of the equations for curved LQC, and a numerical study of the singularities in the classic and in the quantum case.
Cosmology provides a privileged framework to study the relation between the canonical and the covariant LQG. There are several issues that can be explored. As an example I would like to mention in particular the presence of ambiguities in the regularization of the Hamiltonian constraint. In LQC an "improved dynamics"  has been introduced, where the regularization is fixed by requiring a good semiclassical behavior. I would like to understand how to obtain the same prescription in Spinfoam Cosmology, making contact between the two frameworks.
Ambiguities in the Hamiltonian of LQC
The regularization issue is not the only ambiguity suffered by the LQG Hamiltonian. I'm interested in exploring what is the effect of using a generalized cellular decomposition (instead of a triangulation or a cubic lattice), in particular for the symmetry-reduced Hamiltonian. Furthermore, LQC is always defined by using the Thiemann's trick, namely the use of a classical Poison-bracket identity, in order to avoid using the inverse-volume operator: this is nonetheless a particular choice of the Hamiltonian, which leads to quantum corrections different from those without the trick . I would like to understand the effect of these differences.
An interesting aspect of this framework is the possibility of a cyclic universe. This yields a problem, originally noticed by Tolman, regarding the behavior of the entropy. For this reason, I got interested with the definition of entropy for the gravitational field. Inspired by a result in graph theory , I proposed with Carlo Rovelli a definition of gravitational entropy  obtained considering a test particle on a gravitational field described by a spinnetwork. The interest of this work is that it is the first attempt to make contact between LQG and thermodynamics, outside of the context of the computation of black hole entropy.