Research Lines 2015

Research Lines:

  • Loop Quantum Gravity and Cosmology.
  • Classical and Quantum General Relativity.
  • Quantum cosmology.
  • Black holes.
  • Combinatorics.
  • Quantum Field Theory.
  • Models of quantum gravity and multi-fractal geometry.
  • Mathematical Physics.

Research Activity

Quantum Cosmology, Black Holes, and Fractals

We have further developed a characteristic research line of the department focused on the quantization of cosmological systems with inhomogeneities. More specifically, we have analyzed the application of the hybrid quantization formalism proposed by us to the study of cosmological perturbations. This formalism establishes a hierarchy in the quantum geometry phenomena, therefore allowing the combination of quantum cosmology techniques with more conventional methods typical of field theories in curved spacetimes. We have discussed in detail the consequences of the hybrid quantization for inflationary scenarios. Our quantization scheme extends the cosmological dynamics to a pre-inflationary era when one would expect that quantum gravity effects were important. This allows us to compare theoretical predictions of quantum models with recent observations of the cosmic microwave background radiation. It is worth emphasizing that our study of a homogeneous and isotropic universe with a scalar field in the presence of perturbation is based on a formulation that is specially designed to preserve the covariance of General Relativity, because we describe the perturbations in terms of gauge invariants and never fix the gauge at the classical level. We have also carried on further investigations on the hybrid quantization of the Gowdy model, which is inhomogeneous but with symmetries in certain directions, so that the local degrees of freedom propagate on an anisotropic background. We have generalized the construction of approximate solutions to the Hamiltonian constraint with a matter content consisting of a scalar field. The solutions have a very interesting behavior as they represent states that are indeed inhomogeneous, but turn out to be approximate solutions of a homogeneous and isotropic cosmological model, with matter terms that may be given by any kind of perfect fluid (e.g. a cosmological constant) and with possible changes to the geometry, similar to those studied in modified theories of gravity, like f (R).

Besides, with the aim of extending to fermionic fields previous results of our group about the uniqueness of the Fock quantization of fields propagating in curved spacetimes, we have investigated the case of a Dirac field on a homogeneous and isotropic cosmological background with space sections whose topology is the three- sphere. We have built an equivalence class of Fock representations that is selected by the criteria of possessing a vacuum that is invariant under the action of the isometry group of the cosmological background, and admitting a unitary implementation of the quantum dynamics.

There exist several proposals for regularizing the classical singularity of black holes so that their formation and evaporation do not lead to information-loss problems. One characteristic is shared by most of these proposals: These regularly evaporating black holes present long-lived trapping horizons. We have elaborated on an alternative regularization of the classical singularity, which leads to a completely different scenario. In our scheme the collapse of a stellar object would result in a genuine time-symmetric bounce, which in geometrical terms amounts to the connection of a black-hole geometry with a white-hole geometry. The duration of the bounce as seen by external observers is very brief. This scenario motivates the search for new forms of stellar equilibrium.

We have analyzed also the implications of the violation of the strong Huygens principle in the transmission of information from the early universe to the current era via massless fields. We show that much more information reaches us through timelike channels, and not mediated by real photons.

Moving particles outside a star will generally experience quantum friction caused by Unruh radiation reaction. There exist however radial trajectories that lack this effect. Along these trajectories, observers perceive just stellar emission, without further contribution from the Unruh effect. They turn out to have the property that the variations of the Doppler and the gravitational shifts compensate each other. They are not geodesics, and their proper acceleration obeys an inverse square law, which means that could in principle be generated by outgoing stellar radiation. In the case of a black hole emitting Hawking radiation, this may lead to a buoyancy scenario.

We have also investigated the cosmological constant problem, which can be understood as the failure of the decoupling principle behind effective field theory, so that some quantities in the low-energy theory are extremely sensitive to the high-energy properties. Following this intuition, we have considered a minimal modification of the structure of General Relativity which, as an effective theory, permits to work consistently at low energies, i.e., below the quantum gravity scale. This effective description preserves the classical phenomenology of General Relativity and the particle spectrum of the standard model, at the price of changing our conceptual and mathematical picture of spacetime.

An independent line of research has been the theoretical and experimental development of theories on multi-scale spacetimes, where geometry changes with the scale in a manner similar to that of multi-fractals. We studied for the first time the Standard Model of elementary particles both in the classical case and in its quantum aspects. After formulating the fundamental action in the electroweak and strong sectors of the the theories with weighted and “q” derivatives, we have obtained for the first time observational constraints on the characteristic scales of the geometric measure, stemming from the mean lifetime of the muon and from the Lamb-shift effect in hydrogenic atoms.

Moreover, we have studied the spectral dimension (which indicates the number of effective dimensions felt by a test particle) in quantum gravities with discrete geometries, including spin foams, group field theory and loop quantum gravity. We have shown that the geometry of these theories is affected by the discrete and combinatorial structure so that the effective spacetime dimension changes with the scale and, in some cases, can be declared as fractal. This result allows one to pinpoint in a precise manner the scales starting from which one can define the classical limit.

Quantum field theories in bounded spatial domains are important in several branches in theoretical physics reaching from the study of black holes to condensed matter systems. One of the outstanding features of general relativity is its diffeomorphism invariance, a well-known source of conceptual difficulties and a standing roadblock in its consistent and successful quantization. Among the most popular ideas used to understand the issue of diff-invariance in field theories and probe possible approaches towards its resolution is the introduction of the so called parametrized models where embeddings play an important role as additional configuration variables. These can be used to avoid the introduction of particular foliations and somehow sidestep some of the issues related with the so called problem of time. The interactions between parametrization, the presence of spatial boundaries and gauge symmetries are quite intricate from the mathematical point of view and must be duly disentangled in order to get the Hamiltonian formulation for these models and approach their quantization. Two particular kinds of systems, that have somehow eluded a satisfactory treatment, have been considered by our group: the parametrized electromagnetic field (for which a partial, but not fully satisfactory treatment was provided by Kuchař and collaborators) and the parametrized scalar field in bounded spatial regions subject to Dirichlet, Neumann or Robin boundary conditions (considered by Marolf and collaborators but, again, only partially understood). By resorting to geometric Hamiltonian methods we have been able to obtain a complete and detailed Hamiltonian formulation for these models. The most striking and unexpected feature of this description is the appearance of “sectors” in the primary constraint hypersurface. These are associated with the fact that the (generalized) rank of the pullback of the symplectic form to it is not uniform. This can be seen to be related with a bifurcation of the Dirac algorithm that is traditionally used to deal with constrained/singular mechanical systems. The fact that systems for which this bifurcation takes place are usually discarded as unphysical is one of the reasons why they have not been considered in detailed and are deemed as pathological. The fact that, such natural models as the ones that we have looked at, display this type of behavior at their very core shows the need to approach them with new tools such as the ones that we have employed. The availability of these tools, honed with the study of the models mentioned above, opens up the possibility of studying more complicated systems (general relativity) and use their parametrized versions to approach some of the difficulties currently encountered by more popular approaches. Our hope is that we will be able to tackle some of them in the next months.


Research Lines 2014

Research Lines:

  • Loop Quantum Gravity and Cosmology.
  • Classical and Quantum General Relativity.
  • Computational methods in gravitational physics.
  • Black hole analogues in condensed matter.
  • Combinatorics.
  • Quantum Field Theory.
  • Models of quantum gravity and fractal geometry.

Research Activity

Gravitation and Cosmology

The Gravitation and Cosmology Group has continued the work on the quantization of inhomogeneous cosmologies during 2014, focusing on the consequences of the quantization of the geometry on cosmological perturbations and the imprints on the Cosmic Microwave Background. We studied the evolution of a homogeneous and isotropic spacetime (a Friedmann-Robertson-Walker spacetime) coupled to a massless scalar field with small scalar perturbations within Loop Quantum Cosmology, using a proposal for the effective dynamics based on a previous hybrid quantization completed by us. We introduced a convenient gauge fixing and adopted reduced canonical variables adapted to that hybrid quantum description. Besides, we kept backreaction contributions on the background coming from terms quadratic in the perturbations in the action of the system. We carried out a numerical analysis assuming that the inhomogeneities were initially (at distant past) in a massless vacuum state. At distant future, we observed a statistical amplification of the modes amplitude in the infrared region, as well as a phase synchronization arising from quantum gravity phenomena. Finally, we analyzed some consequences of the backreaction in our effective description.

We also studied cosmological perturbations in the framework of Loop Quantum Cosmology using Mukhanov- Sasaki variables in the hybrid quantization approach. The formulation in terms of these gauge invariants allows one to clarify the independence of the results on choices of gauge and facilitates the comparison with other approaches proposed to deal with cosmological perturbations in the context of Loop Quantum Theory. We employed a kind of Born-Oppenheimer ansatz to extract the dynamics of the inhomogeneous perturbations, separating them from the degrees of freedom of the Friedmann-Robertson-Walker geometry. With this ansatz, we derived an approximate Schrödinger equation for the cosmological perturbations and studied its range of validity. We also proved that, with an alternate factor ordering, the dynamics deduced for the perturbations is similar to the one found in the the so-called dressed metric approach, apart from a possible scaling of the matter field in order to preserve its unitary evolution in the regime of Quantum Field Theory in a curved background and some quantization prescription issues. Finally, we obtained the effective equations that are naturally associated with the Mukhanov-Sasaki variables, both with and without introducing the Born-Oppenheimer ansatz.

In addition to all this, and in order to contemplate the possibility of signature changes in the quantum geometry, we studied the quantization of Klein-Gordon type scalar fields in nonstationary spacetimes, such as those found in cosmology, assuming that all the relevant spatial dependence in the field equations is contained in the Laplacian. We showed that the field description and the Fock representation for the quantization of the field are fixed indeed in a unique way (except for unitary transformations that do not affect the physical predictions) if we adopt the combined criterion of (a) imposing the invariance of the vacuum under the group of spatial symmetries of the field equations and (b) requiring a unitary implementation of the dynamics in the quantum theory. Besides, we provided a spacetime interpretation of the field equations as those corresponding to a scalar field in a cosmological spacetime that is conformally ultrastatic. In the privileged Fock quantization, we investigated the generalization of the evolution of physical states from the hyperbolic dynamical regime to an elliptic regime, owing to processes with signature change in the spacetime where the field propagates. We discussed the behavior of the background geometry when the change happens, proving that the spacetime metric degenerates. Finally, we argued that this kind of signature change leads naturally to a phenomenon of particle creation, with exponential production.

In our study of inhomogeneities in Loop Quantum Cosmology, on the other hand, we also developped approximation methods in the hybrid quantization of the Gowdy model with linear polarization and a massless scalar field, for the case of three-torus spatial topology. The Gowdy models are cosmological inhomogeneous spacetimes with symmetries that simplify the analysis. The Loop Quantization of the homogeneous gravitational sector of the Gowdy model and the presence of inhomogeneities lead to a very complicated Hamiltonian constraint. Therefore, the extraction of physical results calls for the introduction of well justified approximations. We first showed how to approximate the homogeneous part of the Hamiltonian constraint, corresponding to anisotropic Bianchi I geometries, as if it described a isotropic Friedmann-Robertson-Walker model corrected with anisotropies. This approximation is valid in the sector of high energies of the Friedmann-Robertson-Walker geometry and for anisotropy profiles that are sufficiently smooth. In addition, for certain families of states related to regimes of physical interest, with negligible quantum effects of the anisotropies and small inhomogeneities, one can approximate the Hamiltonian constraint of the inhomogeneous system by that of a Friedmann-Robertson- Walker geometry with simple matter content, and then obtain its solutions.

Besides, we have considered the problem of quantum correlations in the geometry across gravitational horizons. Classically, different spacetime regions separated by horizons are not related to each other. We carried out a canonical quantization of a Kantowski-Sachs minisuperspace model whose classical solutions exhibit both an event horizon and a cosmological horizon in order to check whether the above statement also holds from the quantum gravitational point of view. Our analysis showed that, in fact, this is not the case: Quantum gravitational states with support in spacetime configurations that exclusively describe either the region between horizons or outside them are not consistent in the sense that there exist unitary operators describing a natural notion of evolution that connect them.

We also have given a detailed description of electrodynamics as an emergent theory from condensed-matter-like structures, not only per se but also as a warm-up for the study of the much more complex case of gravity. We started with Maxwell's mechanical model for electrodynamics. We next took a superfluid 3He-like system as representative of a broad class of fermionic quantum systems whose low-energy physics reproduces classical electrodynamics.

The white-hole sector of Kruskal's solution is almost never used in physical applications. However, it might contain the solution to many of the problems associated with gravitational collapse and evaporation. There exist some bouncing geometries that make a democratic use of the black- and white-hole sectors. These types of behavior could be perfectly natural in some approaches to the next physical level beyond classical general relativity.

Concerning other issues, it is commonly accepted that general relativity is the only solution to the consistency problem that appears when trying to build a theory of interacting gravitons (massless spin-2 particles). We have presented the self-coupling problem in detail and explicitly solved the infinite-iterations scheme associated with it for the simplest theory of a graviton field. We have made explicit the non-uniqueness problem by finding an entire family of solutions to the self-coupling problem. Then we have shown that the only resulting theory which implements a deformation of the original gauge symmetry happens to have essentially the structure of unimodular gravity. This makes plausible the possibility of a natural solution to the first cosmological constant problem in theories of emergent gravity. Later on we have changed to Fierz-Pauli theory, an equivalent theory but with a larger gauge symmetry. As one requires the (deformed) preservation of internal gauge invariance, one naturally recovers the structure of unimodular gravity or general relativity but in a version that explicitly shows the underlying Minkowski spacetime, in the spirit of Rosen's flat-background bimetric theory.

One of the most peculiar features of the quantization used in Loop Quantum Gravity is the non-separability of the kinematical Hibert spaces used in this setting. The expectation is that this will not be an issue in the final formulation of quantum gravity as the physical Hilbert spaces determined by the solutions to the quantum constraints will, very likely, be separable. In any case there are a number of questions that have been not answered in a completely satisfactory way which, hence, deserve some attention: in particular the statistical mechanics of these models, superselection, and the characterization of the ambiguities introduced by the polymer scale. A natural setting to study them is provided by the so called polymer quantum mechanics, which incorporates some of the most unusual features of the quantization of general relativity used in LQG and is especially relevant for Loop Quantum Cosmology. In recent years we have paid attention to the study of simple systems —such as the polymerized harmonic oscillator— in one dimension with the hope of gaining a good grasp on the less-familiar features associated with non-separability. In the last year we have considered, in collaboration with Dr. Tomasz Pawlowski —a former member of the group—, the extension of some of our results to the study of some particular models in quantum cosmology. The key idea has been to replace non-separable spaces with separable ones while introducing appropriate measures in the spectrum of the relevant operators defining quantum observables. Although this idea may finally turn out to be too simple it is quite useful to understand non-separability and issues related to superselection.

The work carried out by the group on black hole entropy in LQG in recent years led to the development of several mathematical tools which were needed to solve the combinatorial problems involved in the entropy computations. A mathematical spin-off of this work has led to the solution of a long standing problem in combinatorics stated in the renowned book Concrete Mathematics by Graham, Knuth and Patashnik. The problem in question asked to solve and characterize the solutions to a very general six-parametric linear recurrence whose solutions encompass broad families of well-known combinatorial numbers including binomial coefficients, Stirling numbers, Eulerian numbers and many others. The resolution was based on the use of bivariate generating functions similar to the ones used by the group to study black holes. A surprising discovery was the identification of a phenomenon that just begs to be called gauge invariance, i.e. the existence of different sets of parameters leading to the same families of combinatorial numbers. The work has been published in the prestigious Journal of Combinatorial Theory, Series A and has been followed by a paper on applications of the formalism developed therein (under consideration at the present moment in the Electronic Journal of Combinatorics).

An important problem also suggested by the activity of the group on black hole physics is the rigorous study of the Hamiltonian formulation for field theories in the presence of boundaries. Remember that the model presently used to describe black holes in the LQG framework hinges upon the use of the so called isolated horizons and the formulation of general relativity in space-time manifolds with inner boundaries. The origin of the quantum degrees of freedom which account for the entropy is somewhat mysterious and, in particular, the connection between the classical and quantum degrees of freedom is not completely clear. With the aim of clarifying this issue and developing a rigorous understanding of the sector of general relativity containing space-time boundaries we have started to study the rigorous Hamiltonian formulation for field theories in bounded spatial regions. One of the goals of this work is to characterize those field theories for which boundary degrees of freedom appear in a natural sense. The first paper on this subject, published in Classical and Quantum gravity and distinguished as a Highlight of the year 2014 by this prestigious journal, was devoted to the study of scalar and electromagnetic fields in bounded and smooth regions subject to several types of boundary conditions (Dirichlet, Neumann, Robin and others specific of the EM field). In those examples we have been able to rigorously prove that no boundary degrees of freedom arise. In the present moment we are extending the results of this work to parametrized fields and also to models where fields are coupled to point particles. These works —in progress or under consideration for publication— will shed further light on the issue of the existence of genuine boundary degrees of freedom and black hole entropy.

Within loop quantum cosmology, we have studied the possibility to obtain the homogeneous LQC equations from group field theory, a background-independent theory supposedly more fundamental than loop gravity. In this case, we have used a condensate quantum state to reproduce a cosmological background in the continuum limit. We have also explored other theories of quantum gravity, where we stressed the change of properties of the effective geometry with the change in the resolution. (i) In the nonlocal theory proposed by G. Calcagni and L. Modesto, we have constructed a gravitational Lagrangian in 11 dimensions with nonlocal operators, reproducing various characteristics of M theory and of string field theory. Concerning cosmology, we have shown that it is possible to replace the big-bang singularity with a classical bounce due to the presence of nonlocal operators. We have also discussed the same nonlocality in string theory. (ii) We have developed the electrodynamics (classical and quantum) and the cosmology of multiscale spacetimes, recently proposed theories where geometry changes with the scale in a manner similar to that of multifractals. The geometric measure naturally produces an era of cyclic contractions and expansions that can leave a unique imprint in the primordial inflationary spectra. The cosmological constant dominating the late universe is reinterpreted as an effective “potential” generated by the geometric measure and necessary for the self-consistency of the solutions. (iii) We placed observational constraints on noncommutative, brane, and quantum-tunneling cosmological models, thus ruling out experimentally the first two classes of models thanks to Planck data. (iv) We have started the study of the spectral dimension (which indicates the number of effective dimensions felt by a test particle) in quantum gravities with discrete geometries, including spin foams and loop gravity.

Last, we have installed and tested successfully the FORTRAN program CosmoMC (Monte Carlo cosmological simulations) on the Trueno cluster. CosmoMC is a fundamental instrument to place observational constraints on theoretical models of the early universe and it will allow us to verify experimentally and directly the inflationary predictions of many of the models discussed here. IEM-CSIC thus becomes the first national node equipped with this important numerical-cosmology code.


Research Lines 2013

Research Lines:

  • Loop Quantum Gravity and Cosmology.
  • Classical and Quantum General Relativity.
  • Black hole analogs in condensed matter.

Research Activity

Gravitation and Cosmology

During 2013 we have continued our study of inhomogeneous cosmologies within the framework of Loop Quantum Cosmology, paying special attention to the treatment of cosmological perturbations around homogenous and isotropic universes like the one in which we live, at a first approximation. More specifically, we have presented a complete quantization of an approximately homogeneous and isotropic universe with small scalar perturbations in the case in which the matter content is a minimally coupled scalar field and the spatial sections are flat and compact. The quantization has been carried out along the lines that were put forward by our group in previous works for spherical topology. We have truncated the action of the system at second order in perturbations, and fixed the local gauge freedom at the classical level, although we have discussed different gauges and shown that they lead to equivalent conclusions. Moreover, we have considered also descriptions in terms of gauge-invariant quantities. The reduced system obtained in this way has been proven to admit a symplectic structure, and its dynamical evolution is dictated by a Hamiltonian constraint. Then, we have proceeded to quantize the background geometry with loop techniques, while a Fock representation has been adopted for the inhomogeneities. The latter has been selected by uniqueness criteria proposed by our group in the context of quantum field theory in curved spacetimes. These criteria determine also a specific scaling of the perturbations. In our hybrid quantization, combining loop and Fock methods, we have promoted the Hamiltonian constraint to an operator on the kinematical Hilbert space. Then, if the zero mode of the scalar field is interpreted as a relational time, a suitable ansatz for the dependence of the physical states on the loop degrees of freedom leads to a quantum wave equation for the evolution of the perturbations. Alternatively, the solutions to the quantum constraint can be characterized by their initial data on a minimumvolume section. The physical implications of this model are being addressed at present, in order to check whether they are compatible with cosmological observations.

An important point in our hybrid quantization is the possibility of selecting a unique Fock representation for the perturbation fields by adhering to certain criteria that our group introduced in recent years. These criteria consist in the requirements of (i) invariance of the vacuum under the spatial symmetries of the field equations and (ii) unitary implementability of the dynamics in the quantum theory. In the case of cosmological perturbations, the fields that describe the inhomogeneities go through an inflationary period that can be understood as the propagation in a de Sitter background. With this motivation, we have proven that, under the standard conformal scaling used in cosmology, a massless field in de Sitter spacetime admits an O(4)-invariant Fock quantization such that time evolution is indeed unitarily implemented, and that this quantization is essentially unique. This result disproves previous claims in the literature. We have also discussed the relationship between this privileged quantization with unitary dynamics and the family of O(4)-invariant Hadamard states given by Allen and Folacci, as well as with the Bunch-Davies vacuum, standard in inflationary cosmology.

Moreover, we have demonstrated the robustness of our criteria for the selection of a unique Fock quantization of scalar fields of Klein-Gordon type in nonstationary scenarios with compact spatial sections by allowing also for different field descriptions that are related by means of certain nonlocal linear canonical transformations that depend on time. More specifically, we have considered transformations that do not mix eigenmodes of the Laplace-Beltrami operator, which are supposed to be dynamically decoupled. Canonical transformations of this kind are found in the study of scalar perturbations in inflationary cosmologies, relating for instance the physical degrees of freedom of these perturbations after gauge fixing with gauge invariants. Hence, our results have direct implications for our hybrid quantization of scalar perturbations in Loop Quantum Cosmology. We have characterized all possible transformations of the considered type and shown that, independently of the initial field description, our criteria of spatial symmetry invariance and unitary dynamics lead indeed to a unique Fock quantization, modulo unitary transformations which do not affect the physical predictions.

Another issue that we have analyzed in detail in the context of Loop Quantum Cosmology is the resolution of singularities. Let us recall that one of the most remarkable phenomena in Loop Quantum Cosmology is that, at least for homogeneous cosmological models, the Big Bang is replaced with a Big Bounce that connects our universe with a previous branch without passing through a cosmological singularity. For the first time in the literature, we have studied the existence of singularities in Loop Quantum Cosmology including inhomogeneities and checked that the behavior obtained in the purely homogeneous setting continues to be valid. We have focused our attention on Gowdy cosmologies with linearly polarized gravitational waves and used effective dynamics to carry out the analysis. For the considered cosmological model, we have proven that all potential cosmological singularities are avoided, generalizing the results about resolution of singularities to this scenario with inhomogeneities. We have also demonstrated that, if a bounce in the (homogenous background) volume occurs, the inhomogeneities increase the value of this volume at the bounce.

In addition, we have identified a signature of quantum gravitational effects that survives from the early universe to the current era: Fluctuations of quantum fields as seen by comoving observers are significantly influenced by the history of the early universe. The existence (or not) of a quantum bounce leaves a trace in the background quantum noise that is not damped and would be non-negligible even nowadays. We have estimated an upper bound to the typical energy and length scales where quantum effects are relevant. We have also presented the graviton selfcoupling problem in detail and explicitly solved the infinite-iterations scheme associated with it, thus complementing previous results in the literature concerning the recovery of General Relativity within this setup. We have concluded that, as long as one requires the (deformed) preservation of gauge invariance, one naturally recovers the structure of General Relativity, but in a version explicitly showing an underlying Minkowski spacetime, in the spirit of Rosen’s flat-background bimetric theory.

During this last year, on the other hand, we have developed also lines of investigation in quantum and anomalous geometry. Gianluca Calcagni has dedicated the whole 2013 in consolidating a class of models of multiscale spacetime where the geometry of the universe shows the typical properties of multifractals. The goals have been 1) to develop a self-consistent theory which could explain analytically certain universal features of spacetime which appear in many independent models of quantum gravity, 2) to develop its phenomenology, especially in classical and quantum field theory and in cosmology, and 3) to test its experimental viability. Goals 1) and 2) have been fully met and we have greatly clarified the fundamental structure of these geometries, their properties in the presence of metric curvature and their cosmological consequences for the very early universe and the cosmological constant, while 3) is still in progress. In parallel, Calcagni has worked on the cosmology of braneworld and noncommutative scenarios as well as on anomalous geometries and the calculation of their spectral dimension in various approaches including string theory, asymptotic safety, black holes, loop quantum gravity, and a super-renormalizable theory of quantum gravity proposed by L. Modesto and others, where cosmological bouncing solutions were found.

Regarding the study of general aspects of quantum field theory, we have devoted special attention to the Hamiltonian formalism for theories defined in bounded spatial regions. The main importance of these models in the context of quantum gravity lies in the fact that the usual treatment of the black holes in loop quantum gravity is carried out by introducing isolated horizons, a type of internal space-time boundary where natural boundary conditions --that give rise to a quasi-local horizon-- are imposed. The main problem encountered when dealing with these systems is the interpretation of these boundary conditions, in particular finding out whether they are constraints in the usual sense and their type (first or second class). This is crucial in order to determine how they must be dealt with when quantizing.

From a technical point of view the presence of boundaries forces us to use mathematical schemes more elaborate than the ones that are formally obtained by generalizing the traditional Dirac method employed in the study of singular Lagrangian systems. The appropriate framework to study such systems in the presence of boundaries is provided by the geometric methods developed for this purpose by Gotay, Nester and Hinds. These authors developed a precise geometric scheme (which, in the particular case of field theories, is based on the use of differential manifolds modeled upon infinite dimensional Banach spaces) that is perfectly suited to the kind of problem that interests us. Notice that, from a technical point of view, working with infinite dimensional manifolds requires the use of functional analytic methods that introduce significant mathematical difficulties in the formalism.

During the year 2013 we have started an ambitious program to work on this issue that has resulted in the complete and systematic study of scalar field theories subject to different sets of boundary conditions Dirichlet and Robin type) and Maxwell fields (with perfect conductor and Neumann boundary conditions) . Our work has given rise to a publication in Classical and Quantum Gravity (in collaboration with Jorge Prieto and Eduardo Sánchez Villaseñor) that has appeared in January 2014. At the present moment we are extending our results to interacting theories (scalar fields and Yang -Mills) and other models that are expected to have physical degrees of freedom associated with boundaries (Maxwell - Chern -Simons on 2+1 dimensions) .

Within the context of Loop Quantum Gravity and Loop Quantum Cosmology it is important to understand the physical consequences of the introduction of non-separable Hilbert spaces. Several models have been considered in the literature in this respect. Among them one of the most popular ones is the polymeric harmonic oscillator, for which the Hamiltonian has been studied by several authors. During last year we have studied the spectrum of this type of Hamiltonians in a precise mathematical sense. The most important result that we have obtained was to determine that the generic polymeric spectrum Hamiltonian consists of bands (as in solids). However, at variance with these, the spectrum is a pure point one (and therefore consists of actual eigenvalues) . Our results have led to a publi-cation in Classical and Quantum Gravity (in collaboration with Jorge Prieto and Eduardo Sánchez Villaseñor). At the present moment we are exploring the implications of these results, in particular in the context of Loop Quantum Cosmology. We are currently proposing alternatives to the use of non-separable Hilbert spaces.

Finally, we have devoted considerable effort to the study of combinatorial problems using the generating function methods that we have developed to study black hole entropy in loop quantum gravity. Our work has focused on the comprehensive study of an important class of generalized linear recurrences defining important families of combinatorial numbers (which include Euler, Stirling, Ward, Lah numbers among many others). We have written two works on this subject (in partnership with Jesus Salas and Eduardo Sanchez Villaseñor of the Carlos III University of Madrid).


Research Lines 2012

Research Lines:

  • Loop Quantum Gravity and Cosmology.
  • Classical and Quantum General Relativity.
  • Computational methods in gravitational physics.
  • Black hole analogs in condensed matter.

Research Activity

Gravitation

The Gravitation and Cosmology Group has kept on during this year carrying on its research on the effects of inhomogeneities in Quantum Cosmology. This research has focused on two fronts. On the one hand, a detailed study has been performed on the quantization of scalar fields in nonstationary curved spacetimes, a generic situation in cosmology. This study has led to powerful theorems about the uniqueness of the quantization when two rather natural requirements are imposed: first, the invariance of the vacuum under the spatial symmetries of the field equations; and second, the unitarity of the quantum dynamics, in order to preserve the standard probabilistic interpretation of Quantum Mechanics. The case of spacetimes with compact spatial sections of flat topology has been studied with special care, owing to its physical interest, since present observations of the Universe support its spatial flatness. In this way, it has been possible to prove the uniqueness both of the Fock representation employed in the quantization and of the choice of canonical pair for the field when time dependent canonical transformations are allowed in the system. It is natural to consider this type of transformations in nonstationary backgrounds. Their inclusion modifies the dynamics, and would imply an ambiguity in the description selected for the field if it we could not appeal to the uniqueness results mentioned above. The conclusions reached in the case of flat topolology have been extended later to any compact spatial topology, attaining conclusions of a remarkable generality. These uniqueness theorems provide robustness to the physical predictions of the theory, which would be otherwise plagued with an infinite ambiguity.

In the same topic of fields in nonstationary curved spacetimes, besides, a theoretical formalism has been developed for the description of cosmological perturbations in shear-free Bianchi III type models. This formalism allows one to face the study of primordial perturbations in scenarios with anisotropy, and hence to discuss the possible observational consequences, a study of great interest owing to the recent indications suggesting that privileged axis might exist in cosmology.

In addition, the theory of cosmological perturbations has been revisited in the case of isotropic and homogeneous spacetimes, conventional in cosmology, applying the results of the Group about the uniqueness of the Fock quantization of such perturbations. Furthermore, this privileged Fock quantization has been combined with the quantization of the homogeneous and isotropic background by means of loop techniques, in the framework of Loop Quantum Cosmology. This hybrid quantization approach, proposed by the IEM Group, has permitted the construction of a complete and consistent quantum model for the description of primordial perturbations in a Universe filled with a massive scalar field, case in which it is known that there exist regimes in which enough inflation is generated, therefore providing a physically realistic scenario.

Still in the context of quantum field theory in curved spacetimes, the structure of the renormalized stress-energy tensor of a massless scalar field in a 1-dimensional curved spacetime, as obtained by two different strategies, has been explicitly compared. The two strategies are: normal-mode construction of the field operator and one-loop effective action. These two potentially different renormalized stress-energy tensors have been shown to be actually equal, when using vacuum-state choices appropriately related. Some of the hybrid classical-quantum models proposed in the literature have been reappraised, with the goal of retrieving some of their common characteristics. The formalisms used in those models have been shown that generally entail the necessity of dealing with additional degrees of freedom beyond those in the straight complete quantization of the system.

On the other hand, the evolution of the entanglement of a non-trivial initial quantum field state has been analyzed when it undergoes a gravitational collapse, discussing what kind of problems can be tackled using the formalism spelled out here, as well as singling out future avenues of research. From another viewpoint, a vacuum state in a black hole spacetime can be analyzed in terms of how it is perceived (in terms of particle content) by different observers by means of an effective-temperature function. A general analytic expression for the effective-temperature function has been found which, apart from the vacuum state choice, depends on the position, the local velocity, and the acceleration of the specific observer. A clear physical interpretation of the quantities appearing in the expression has been given, and its potentiality illustrated with a few examples.

Another trend of investigation is the one focused on effective quantum spacetimes with multifractal geometry, whose properties change with the probed scale. The main objectives are to construct a self-consistent quantum field theory dynamics and explore its renormalization properties, and with that proceed to develop formalism for classical gravity in multifractal spacetimes, cosmological scenarios and their phenomenology.

As far as the study of black hole entropy in Loop Quantum Gravity is concerned, we have started a program to rigorously study quantum field theories in the presence of boundaries, concentrated, especially in their Hamiltonian formulation. One of the most important conceptual problems in this area is related to the interpretation of the degrees of freedom responsible for the black hole entropy. Black holes in LQG are modeled by introducing the so-called isolated horizons. At present the standard interpretation of the resulting model suggests that there are no classical degrees of freedom that can be exclusively associated to the boundary that represents the horizon and, therefore, the degrees of freedom that account for the entropy must have a purely quantum origin. The problem with this point of view view is that the identification of the classical degrees of freedom demands a rigorous treatment incorporating, in particular, important aspects related to the functional spaces to which the classical fields belong. In order to achieve this goal the methods based on the traditional Dirac algorithm are not good enough and it is necessary to use geometrical techniques adapted to the infinitedimensional character of the configuration spaces of field theories. Among these methods the geometric algorithm of Gotay-Nester-Hinds is especially useful. It has been used already to rigorously derive the Hamiltonian formalism for linear theories in the presence of boundaries and determine if the boundary conditions admit an interpretation as constraints in the traditional sense. The models considered so far have been the scalar and electromagnetic fields with Dirichlet and Robin boundary conditions. They have been fully characterized and the constraint differential submanifolds where the dynamics takes place have been found. In addition we have obtained a detailed description of the Hamiltonian vector field whose integral curves define the dynamics. In order to achieve this goal it has been necessary to introduce the appropriate functional spaces (Sobolev spaces associated to some of the differential operators that play a role in these models). At present, these results are being extended to parameterized theories to understand in detail several aspects associated with their polymer quantizations.


Research Lines 2011

Research Lines:

  • Classical and Quantum General Relativity.
  • Quantum Cosmology.
  • Loop Quantum Gravity.
  • Black hole physics.
  • Computational methods in grvitational physics.

Research Activity

Quantum Gravity & Quantum Cosmology

During the year 2011 we have carried to completion the work that we had been developing in the last years on the study of black hole entropy in loop quantum gravity (LQG). Specifically we have been able to finish the study of the asymptotic behavior of the entropy as a function of the horizon area. The most important open problem in this regard was to determine if the intriguing structure observed for low areas was also present in the asymptotic regime. To accomplish this goal, during 2010 we developed a number of statistical methods that led to an efficient approximation procedure for the statistical entropy that clearly explained why the low area substructure had to disappear at large scales. The result that we have just described can be understood, from a completely different perspective, if we rely on the formalism of statistical mechanics and, in particular, on some aspects related to its mathematical foundations. In this context it is of basic importance to understand the mathematical properties of the entropy as a function of the energy (the relevant variable in standard statistical mechanics). It is very important, for example, to determine under which conditions the entropy is a sufficiently smooth function of the energy and its convexity properties. The first point is relevant because thermodynamical properties (such as the temperature of a given system) are defined as derivatives of the entropy, whereas the second is central to the understanding of the stability of physical systems. The classical theorems on these issues prove that in the so called thermodynamic limit the entropy satisfies reasonable smoothness and convexity properties.

During the last year we have devoted a considerable effort to understand the thermodynamic limit for black holes by using the combinatorial methods that we have developed in the preceding years. As in the case of interest for us it is possible to effectively build both the microcanonical and the canonical (area) ensembles, we have been able to study the behavior of black holes in the thermodynamic limit (not to be confused with the large area limit). The most important conclusion of our work is that, in this limit, the entropy is indeed a smooth and concave function of the area and it also satisfies the Bekenstein-Hawking law. It is important to highlight, nonetheless, the fact that subdominant corrections (proportional to the logarithm of the area) are different for the statistical (counting) entropy and the true thermodynamical entropy. This fact is relevant not only within the context of LQG but also for other approaches to quantum gravity such as string theories.

The methods of LQG have also been employed to analyze the quantum realm in cosmological systems, in the field of specialization that is nowadays known as loop quantum cosmology (LQC). In particular, during 2011 we have implemented and compared the so-called improved dynamics prescriptions that exist for LQC in the literature, studying homogeneous and isotropic spacetimes containing a massless scalar field. We have checked that all these prescriptions lead to the same qualitative results for semiclassical states in such cosmological spacetimes, and that the physical behavior is in fact similar even for states which are not so semiclassical or in regimes where the quantum effects start to be not totally negligible, although there exist discrepancies. What is more important, not all of these prescriptions have the same properties from the viewpoint of numerical computations. In particular, a prescription introduced by us seems especially simple to implement and reduces considerably the time of computation. We have optimized the codes of our numerical library for simulations in LQC in order to take full advantage of the features of this specific prescription.

During this year we have also successfully applied loop techniques to inhomogeneous cosmologies of the Gowdy type including a massless scalar field, reaching for the first time a complete and consistent quantization in the framework of LQC for a model with local degrees of freedom in the matter content and in the geometry. We have developed a proposal for a hybrid quantization of this model, and implemented it to completion, proving the consistency of the approach. The aim of this work has been manifold: a) to obtain an exact quantum description of inhomogeneous cosmologies with matter fields that includes effects of the loop quantum geometry, b) to determine the space of physical states and a complete set of physical observables, c) to prove that the quantum dynamics is well posed, d) to demonstrate that the cosmological singularities are resolved in this framework, and e) to show that one recovers the standard Fock quantization for the inhomogeneities in appropriate regimes for physical states. Moreover, restricting the analysis to the vacuum case, we have also discussed the effect of the inhomogeneities on the Big Bounce that replaces the Big Bang singularity, using the effective theory associated with our exact quantum model. The analytical study has confirmed the qualitative robustness of the bounce. Numerical simulations have shown that this robustness is also quantitative. Besides, this numerical analysis has demonstrated that the amplitudes of the inhomogeneities do not change statistically in the bouncing process, except when they are small, case in which they seem to be enlarged by a kind of enhancing mechanism. This mechanism might explain the relatively large amplitude of primordial fluctuations if it is confirmed in more realistic cosmological models.

In fieldlike systems like this, which possess an infinite number of degrees of freedom, a severe problem are the ambiguities found in the quantization process, which affect the final outcome in the quantum theory. In the case of a Fock quantization, where one can reach a concept of particle for the field (at least to a certain extent), an important part of these ambiguities are those arising in the choice of a quantum representation. This problem is encountered in standard quantum field theory on curved backgrounds, but also in the case of the quantization of the inhomogeneities within the hybrid scheme put forward by us in the framework of LQC. In stationary settings, symmetry or energy criteria are known to select a unique Fock quantization. Quite remarkably, we have been able to prove recently that, even in non-stationary settings, one can pick up a unique Fock quantization by introducing the criteria of a) invariance of the vacuum under the symmetries of the field equations, and b) unitarity in the dynamical evolution (with respect to an emergent time related to area or volume elements). This uniqueness provides a considerable robustness to the results of the quantization and its physical predictions. Strictly speaking, the uniqueness theorems reached so far apply to the case of scalar fields on any compact spatial topology and in any spatial dimension equal or smaller than three. Applications for an unambiguous quantization of perturbations in cosmology are almost direct.

Another research line that we have followed has been the analysis of Hawking radiation in a Schwarzschild black hole as perceived by different observers. The method is based on the introduction of an effective temperature function that varies with time. First we introduce a non-stationary vacuum state which mimics the process of switching on Hawking radiation in a stationary spacetime. Then, we analyse this vacuum state from the perspective of static observers at different radial positions, observers undergoing a free-fall trajectory from infinity, and observers standing at rest at a radial distance and then released to fall freely towards the horizon. We find that generic freely-falling observes do not perceive vacuum when crossing the horizon, but an effective temperature a few times larger than the one that they perceived when started to free-fall. We explain this phenomenon as due to a diverging Doppler effect at horizon crossing. From a different perspective, we use the trans-Planckian problem of Hawking radiation as a guiding principle in searching for a compelling scenario for the evaporation of black holes or black-hole-like objects. We argue that there exist only three possible scenarios, depending on whether the classical notion of long-lived horizon is preserved by high-energy physics and on whether the dark and compact astrophysical objects that we observe have long-lived horizons in the first place. Along the way, we find that a) a theory with high-energy superluminar signalling and a long-lived trapping horizon would be extremely unstable in astrophysical terms and that b) stellar pulsations of objects hovering right outside but extremely close to their gravitational radius can result in a mechanism for Hawking-like emission.


Research Lines 2010

Research Lines:

  • Classical and Quantum General Relativity.
  • Loop Quantum Gravity and Cosmology.
  • Black hole analogues in condensed matter.
  • Computational methods in gravitational physics.

Research Activity

Quantum Gravity

During the year 2010 we have continued with our work on black hole entropy in loop quantum gravity with the goal of understanding its behavior, as customarily defined in this framework, for small horizon areas. In particular we have developed several statistical methods necessary to obtain a smoothing of the very complicated black hole degeneracy spectrum (and its integrated version giving the entropy). The smoothed approximations can be successfully used to understand the behavior of the entropy for large areas. The main idea makes use of modified generating functions for the black hole entropy that are designed to allow the extraction of relevant statistical information (the mean and the variance of the distribution defining the peaks in the degeneracy spectrum). At the present moment there is an essentially complete understanding of the structure of the entropy as a function of the area.

Regarding our work on the quantization of symmetry reductions of general relativity we have completed a paper for the series of Living Reviews on Relativity on the quantization of midisuperspace models. These are symmetry reductions of general relativity with an infinite number of physical degrees of freedom. In contrast with the more familiar minisuperspace models these are genuine field theories and, in many instances, keep a residual diffeomorphism invariance. For these reasons this kind of models constitutes a very good testbed for quantization techniques and plays a central role in the current work on loop quantum gravity and cosmology. Being a review, the paper does not give new results but, rather, gives a comprehensive view of the field that takes into account all the relevant approaches to the subject including both the more traditional geometrodynamical methods and the ones inspired in loop quantum gravity.

Concerning specifically the application of the loop quantization techniques to such midisuperspaces, we have continued the development of the quantization of Gowdy cosmologies which contain linearly polarized gravitational waves. Actually, we have completed their quantization using a hybrid approach which combines a loop quantization of the zero modes of the geometry with a Fock quantization of the modes which describe inhomogeneities of the gravitational field. We have successfully implemented the new prescriptions for an improved dynamics in loop quantum cosmology, determined the structure of the superselection sectors of the model, proven that the initial value problem is well posed in the quantum theory, and found the Hilbert space of physical states. Besides, we have carried out a numerical analysis of the corresponding effective dynamics, showing that the (initial) cosmological singularity is eluded and replaced with a quantum bounce. Furthermore, this big bounce scenario is not qualitatively affected by the presence of inhomogeneities. We have studied also the behavior of the inhomogeneous modes through the bounce, showing that their amplitudes are preserved statistically, except when they are small, case in which they are amplified by the bounce. This is the first study of the behavior of inhomogeneities in a fully quantized model within the framework of loop quantum cosmology.


Bounce

Quantum bounce of the wave function of the universe ψ for a Friedmann-Robertson-Walker spacetime with a scalar field ϕ. v corresponds to the volume of the universe in Planck units.


The choice of the Fock quantization in the proposed hybrid approach to inhomogeneous quantum cosmology is based on some uniqueness theorems about quantum scalar fields in non-stationary backgrounds recently proven by us. We have demonstrated that the selection of both a field description (among the family of fields related by time dependent canonical transformations), and of a Fock representation for it, is fixed up to unitary equivalence if one demands invariance of the vacuum under the symmetries of the field equations and unitarity of the dynamical evolution (in the deparametrized theory). These theorems remove the field theory ambiguities in the quantum predictions, e.g., for inhomogeneous fields in cosmological backgrounds and for cosmological perturbations about Friedmann-Robertson-Walker spacetimes.

Another line of research that we have developed is the study of gravitational analogs in condensed matter theory. Fluctuations around a Bose-Einstein condensate can be described by means of Bogoliubov theory or by means of nontrivially modified Klein-Gordon equation. The concepts of quasiparticle derived from both approaches are actually the same. In a stationary configuration containing an acoustic horizon, there are several possible choices of a regular vacuum state, including a regular generalization of the Boulware vacuum. Issues such as Hawking radiation crucially depend on this vacuum choice. On a different front, we have determined the degree of entanglement of a bipartite system Alice-Rob (when Rob is in the proximities of a Schwarzschild black hole while Alice is free falling into it) as a function of the distance of Rob to the event horizon, the mass of the black hole, and the frequency. All the interesting phenomena occur in the vicinity of the event horizon and the presence of event horizons do not effectively degrade the entanglement when Rob is far off the black hole. On the other hand, the vacuum state can evolve to an entangled state in a dynamical gravitational collapse. We have shown that this entanglement could even reach the maximal entanglement limit for low frequencies or very small black holes. It provides quantum information resources between the modes in the asymptotic future (thermal Hawking radiation) and those which fall to the event horizon. We have also shown that fermions are more sensitive than bosons to this quantum entanglement generation.


Research Lines 2009

Research Lines:
  • Gravitation and Cosmology.
  • Condensed Matter Theory.
Research Sublines:
  • Loop Quantum Gravity and Cosmology.
  • Classical and Quantum General Relativity.
  • Computational methods in gravitational physics.
  • Black hole analogs in condensed matter.
  • Strongly correlated and mesoscopic systems.
Employed Techniques:
  • Theoretical and mathematical physics.
  • Computational methods.

Research Activity

Loop Quantum Cosmology

During 2009, the Gravitational Physics Group has continued his research in the field of Loop Quantum Cosmology. The investigation has been developed in three main directions: the consolidation of the foundations of Loop Quantum Cosmology, the extension of the results of homogeneous and isotropic models to anisotropic cases, and the inclusion of inhomogeneities to allow the study of cosmological perturbations.

Loop Quantum Gravity has emerged in recent years as a solid candidate for a nonperturbative quantum theory of General Relativity. It is a background independent theory based on a description of the gravitational field in terms of holonomies and fluxes. In order to discuss its physical implications, a lot of attention has been paid to the application of the quantization techniques of Loop Quantum Gravity to symmetry reduced models with cosmological solutions. This line of research is what has been called Loop Quantum Cosmology.

Most of the activity in this field has been focused on the analysis of simple homogeneous and isotropic models. In particular, the flat Friedmann-Robertson-Walker (homogeneous and isotropic) universe with a massless scalar field is a paradigmatic model in Loop Quantum Cosmology. In spite of the prominent role that the model has played in the development of this branch of physics, there still remained some aspects of its quantization which deserved a more detailed discussion. These aspects included the kinematical resolution of the cosmological big-bang singularity, the possibility of identifying superselection sectors as simple as possible, and a clear comprehension of the Wheeler-DeWitt limit associated with the theory in those sectors. We have proposed an alternative operator to represent the Hamiltonian constraint which is especially suitable to deal with these issues in a satisfactory way. In particular, our constraint operator superselects simple sectors for the universe volume, with a support contained in a single semiaxis of the real line and for which the basic functions that encode the information about the geometry possess optimal physical properties. Namely, they provide a no-boundary description around the cosmological singularity and admit a well-defined Wheeler-DeWitt limit in terms of standing waves. Both properties explain the presence of a generic quantum bounce replacing the big-bang singularity at a fundamental level, in contrast with previous studies where the bounce was proved in concrete regimes and focusing on states with a marked semiclassical behavior.

We have also shown that the global dynamics of a homogeneous universe in Loop Quantum Cosmology can be viewed as a scattering process. This picture can be employed to build a flexible method of verifying the preservation of the semiclassicality of the states through the bounce. The method has been applied in detail to two simple examples: an isotropic Friedmann-Robertson-Walker universe and the isotropic sector of a Bianchi I universe. In both cases, the dispersions in the logarithm of the volume and in the logarithm of the scalar field momentum are related in the distant future and past. This leads to a strict preservation of the semiclassicality.

In addition, we have studied the self-adjointness of the evolution operator corresponding to a flat Friedmann- Robertson-Walker universe with a massless scalar field and a positive cosmological constant, described in the framework of Loop Quantum Cosmology. It has been shown that, if the cosmological constant is smaller than a certain value, the operator admits many self-adjoint extensions, each of them with a purely discrete spectrum. On the other hand, when the cosmological constant is larger than the mentioned value, the operator is essentially self- adjoint, although the physical Hilbert space does not contain any physically interesting state in this case.

Bounce

We have also studied thoroughly the anisotropic cosmological model consisting of a vacuum Bianchi I universe, as an example to investigate the concept of physical evolution in Loop Quantum Cosmology in the absence of the massless scalar field which is frequently used as an internal time. In order to retrieve the system dynamics when no such a suitable clock field is present, we have explored different constructions of families of unitarily related partial observables. These observables are parameterized, respectively, by one of the components of the densitized triad, or by its conjugate momentum; each of them playing the role of an evolution parameter. Exploiting the properties of the considered example, we have investigated in detail the domains of applicability of the introduced constructions. In both cases the observables possess a neat physical interpretation only in an approximate sense. However, whereas in the first case such interpretation is reasonably accurate only for a portion of the evolution of the universe, in the second case it remains so during all the evolution. We have used these families of observables to describe the evolution of the Bianchi I universe. Our analysis confirms the robustness of the bounces, also in absence of matter fields, as well as the preservation of the semiclassicality through them.

Besides, as a necessary step towards the extraction of realistic results from Loop Quantum Cosmology, we have analyzed the physical consequences of including inhomogeneities. We have considered the quantization of a gravitational model in vacuo which possesses local degrees of freedom, namely, the linearly polarized Gowdy cosmologies with the spatial topology of a three-torus. We have carried out a hybrid quantization which combines loop and Fock techniques. This quantization resolves the cosmological big-bang singularity, and permits a rigorous definition of the quantum constraints, as well as the construction of their solutions. We have determined the Hilbert structure of the space of physical states and, furthermore, recovered from it a conventional Fock quantization for the inhomogeneities.

The theoretical research of the group has been complemented with work on the numerical front of classical and quantum general relativity. For instance, we have created a tensor computer algebra package called xPert, for fast construction and manipulation of the equations of metric perturbation theory, around arbitrary backgrounds. This package is based on the combination of explicit combinatorial formulas for the n-th order perturbation of curvature tensors and their gauge changes, and the use of highly efficient techniques of index canonicalization, provided by the underlying tensor system xAct (also made by one of us) for Mathematica.

In addition, using efficient symbolic manipulations tools developed by us, we have presented a general gauge- invariant formalism to study arbitrary radiative second-order perturbations of a Schwarzschild black hole. In particular, we have constructed the second order Zerilli and Regge-Wheeler equations under the presence of any two first-order modes, reconstructed the perturbed metric in terms of the master scalars, and computed the radiated energy at null infinity. Our results enable systematic studies of generic second order perturbations of the Schwarzschild spacetime. In particular, studies of mode-mode coupling and non-linear effects in gravitational radiation, the second-order stability of the Schwarzschild spacetime, or the geometry of the black hole horizon.

On a different front, we have analyzed the Hawking radiation process due to collapsing configurations in the presence of superluminal modifications of the dispersion relation. With such superluminal dispersion relations, the horizon effectively becomes a frequency-dependent concept. In particular, at every moment of the collapse, there is a critical frequency above which no horizon is experienced. We have shown that, as a consequence, the late-time radiation suffers strong modifications, both quantitatively and qualitatively, compared to the standard Hawking picture. Concretely, the radiation spectrum becomes dependent on the measuring time, on the surface gravities associated with different frequencies, and on the critical frequency. Even if the critical frequency is well above the Planck scale, important modifications still show up.

Quantum Gravity

One of the most important problems in quantum gravity is the identification of the microscopic degrees of freedom that give rise to the entropy of black holes. If these can be unambiguously described and counted, the entropy of a black hole can be computed. The relationship between entropy and area is given by the famous Bekenstein- Hawking formula that states that the entropy is one fourth of the horizon area (in the appropriate units). The two leading candidates for quantum theories of gravity -string theory and loop quantum gravity- both account for this law. In the case of loop quantum gravity the result has been known for more than a decade now, however, it has been recently discovered that the entropy of microscopic black holes, as described within loop quantum gravity, displays an unexpected behavior consisting in a periodic modulation of the otherwise linear growth with the area. The understanding of this phenomenon has been one of the research problems that our group has worked on during the last year. To this end we have successfully developed, together with our collaborators in Valencia and Warsaw, methods based on number theory and combinatorics that have allowed us to gain a deeper understanding on this issue. Among them the most useful ones have been the identification of a set of diophantine equations that help in the exact solution of the counting problem and the successful introduction of generating functions for the entropy. We have also found a way to connect these results with the more standard methods using Laplace transforms.

The results that we have obtained are manifold. First we have been able to show that the observed modulation in the entropy can be present for macroscopic areas (though the exact asymptotic behavior has yet to be obtained). This result is a consequence of the subtle way in which the asymptotic behavior of the entropy is given by the integral expressions (inverse Laplace transforms) that express the entropy as a function of the area. Specifically the accumulation of the real parts of the poles in the integrand to the value that fixes the linear growth of the entropy is compatible with the linear behavior given by the Bekenstein law but leaves room to a subdominant periodic correction that could explain the observed behavior of the microscopic black hole entropy.

We have taken advantage of the methods that we have developed in order to compute the black hole entropy for some alternative non-standard proposals. In particular we have considered a different definition for the area operator that is used in the definition of the entropy. As is well known, in the case of isolated horizons (as opposed to arbitrary closed surfaces) it is possible to consider definitions for the area different from the usual one, [though it must be said that the standard one, with its non-equally spaced spectrum, is the preferred one]. Among these there is a very simple and natural choice that gives rise to an equally spaced spectrum. This has some consequences as far as the definition of the entropy is concerned. In collaboration with Jerzy Lewandowski (Warsaw University) we have explored the physical consequences of adopting the modified definition and have checked that the expected entropy behavior is indeed present. A remarkable feature of this work is the fact that we have been able to obtain exact, closed form expressions for the entropy and reproduced the Bekenstein-Hawking law. Finally another application of our methods has been the computation of the entropy in the new scheme recently put forward by Engle, Noui and Perez to define black hole entropy in loop quantum gravity by using a SU(2) Chern-Simons theory at the horizon to describe the black hole degrees of freedom.

During the last year our group has continued with its traditional line of work on the quantization of midisuperspace models -in particular the Gowdy universes- both within standard quantum field theory and loop quantum gravity. In this respect we have devoted some effort to develop in detail the aspects referring to the quantization of the harmonic oscillator with time-dependent frequency, that could be exported to the Gowdy models. We would like to highlight that the Gowdy models can be thought of as field theories described by an infinite number of such time-dependent oscillators. The methods used in this analysis have been manifold and have significantly extended previous existing work on this area. By relying on them a Schrödinger functional representation has been constructed and several problems have been addressed, in particular the unitary definition of the dynamics and the existence of semiclassical states.

The research activity in gravitational physics has been completed with the following contributions:

  • Modified Gravity: We have deepened into the study of modified theories of gravity in the Palatini formalism to determine the ability of those theories to generate nonsingular cosmologies. We have focused our attention on theories of the f(R) type in Friedmann-Robertson-Walker Universes. We have found that the conditions for the existence of a cosmic bounce are quite generic, being determined solely by the existence of a zero in the derivative of the Lagrangian f(R). Remarkably, we have obtained an f(R) Lagrangian which exactly reproduces the effective dynamics of LQC with a massless scalar, which shows the potential of this kind of theories to capture certain aspects of quantum gravity phenomenology and motivates further research in this direction.

  • Inflationary Cosmology: We have studied the generation of primordial perturbations in a (single-field) slow-roll inflationary universe. In momentum space, these (Gaussian) perturbations are characterized by a zero mean and a non-zero variance. However, according to the standard treatments in the literature, in position space the variance diverges in the ultraviolet. We have reconsidered the calculation of the variance in position space using well- established methods of renormalization of quantum fields in curved space. We have found that this affects the predicted scalar and tensorial power spectra for wavelengths that today are at observable scales. As a consequence, the imprint of slow-roll inflation on the CMB anisotropies is significantly altered as compared to the standard predictions found in the literature. In particular, we have found a non-trivial change in the consistency condition that relates the tensor-to-scalar ratio to the spectral indices. The influence of relic gravitational waves on the CMB may soon come within the range of planned measurements, offering a non-trivial test of the new predictions.

  • Black Holes: A disturbing aspect of Hawking's derivation of black hole radiance is the need to invoke extreme conditions for the quantum field that originates the emitted quanta. It is widely argued that the derivation requires the validity of the conventional relativistic field theory to arbitrarily high, trans-Planckian scales. Using the correlation functions of the matter quantum field, we have shown that this is not necessarily the case if the question is presented in a covariant way. We have found that Hawking radiation is robust against an invariant Planck-scale cutoff.

Electronic Properties of Graphene

During 2009 we have continued the theoretical investigation of the electronic properties of the so-called graphene, that is, a material made of a one-atom-thick sheet of carbon atoms, coordinated with the typical honeycomb structure of the graphite layers. Since its discovery in 2004, the research on graphene has raised mounting interest in the condensed-matter community, as the genuine two-dimensional character of the material has led to expect unconventional properties regarding its electronic, optical, and elastic behavior. Our investigation has focused on the analysis of superconducting instabilities of graphene at large doping levels, as well as on the study of the effects of curvature on this peculiar metallic membrane.

The investigation of the ground state of graphene at large doping levels has been undertaken in collaboration with a prominent experimental group from the Lawrence Berkeley National Laboratory. One of the key observations of this group has been that, under suitable conditions of chemical doping, the Fermi level can be tuned to the Van Hove singularity in the conduction band of the two-dimensional carbon material. It is known that the divergent density of states in the vicinity of that kind of singularity may lead to different types of instabilities in the two- dimensional electron liquid. In our case, we have modeled the specific geometry of the extended saddle-points that arise in the conduction band of graphene, which tend to develop ridges with almost flat dispersion along the boundary of the Brillouin zone. This extended character of the singularity had only precedent in the observation of the band dispersion of high-temperature superconductors. The theoretical analysis has shown indeed that superconductivity is one of the possible electronic instabilities that may appear when graphene is close to the Van Hove singularity in the conduction band. We have reached that conclusion by computing the BCS vertex conveniently dressed within the random-phase approximation, and finding the renormalized e-e couplings in the different representations of the point symmetry group. Near the Van Hove singularity, the strong modulation of the density of states along the Fermi line leads always to the existence of a negative coupling, which triggers the superconducting instability at sufficiently low temperature. We have complemented this analysis by looking also for a possible magnetic instability in the system, which could arise as a consequence of the large density of states near the Van Hove singularity. For that purpose, we have computed the magnetic susceptibility as a function of the temperature to check its divergence against the strength of the pairing instability, finding out that the latter prevails throughout all the range of relevant values of the bare Coulomb repulsion and chemical potential about the Van Hove singularity. Combined with the experimental side of the collaboration, our results have supported the idea that doped graphene can achieve an electronically-mediated superconductivity, provided that the doping is in the vicinity of the Van Hove singularity and the lattice symmetry is preserved, i.e. the chemical dopants do not introduce new states near the singularity to disrupt the bandstructure.

On the other hand, we have studied the effects of strong curvature in graphene continuing with our investigation of the carbon nanotube-graphene junctions. In this regard, it is quite promising that such hybrid structures have been already produced in the Fujitsu laboratories, while it is also conceivable that graphene wormholes may be fabricated starting from the easily available graphene bilayers. Our theoretical analysis has focused on wormhole geometries in which a short nanotube acts as a bridge between two graphene sheets, where the honeycomb carbon lattice is curved from the presence of 12 heptagonal defects. By taking nanotube bridges of very small length compared to the radius, we have developed an effective theory of Dirac fermions to account for the low-energy electronic properties of the wormholes in the continuum limit. In this construction, we have included appropriately the effect that the heptagonal carbon rings induce on the Dirac fields encoding the low-energy electronic excitations of the carbon material. This action has been mimicked by attaching a line of fictitious gauge flux at each topological defect, following the same procedure applied long time ago in the case of the fullerene lattices. The graphene wormholes represent actually an instance which can be considered to some extent dual to the case of the fullerenes, as the 12 pentagonal carbon rings in those closed lattices play a role opposite to that of the heptagonal defects in the wormhole. We have found in particular that, when the effective gauge flux from the topological defects becomes maximal, the zero-energy modes of the Dirac equation can be arranged into two triplets, which can be thought as the counterpart of the two triplets of zero modes that arise in the continuum limit of large spherical fullerenes. We have further investigated the graphene wormhole spectra by performing a numerical diagonalization of tight-binding Hamiltonians for very large lattices realizing the wormhole geometry. In this way, we have shown the correspondence between the number of localized electronic states observed in the numerical approach and the effective gauge flux predicted in the continuum limit. We have then concluded that graphene wormholes can be consistently described by an effective theory of two Dirac fermion fields in the curved geometry of the wormhole, opening the possibility of using real samples of the carbon material as a playground to experiment with the interaction between the background curvature and the Dirac fields. It is therefore plausible that the study of these condensed matter systems may allow the investigation of relevant gravitational effects related to the Dirac character of the electron quasiparticles, which otherwise would be only accessible at the much higher energies typical of the astrophysical phenomena.