EDP - Equacions en Derivades Parcials i Aplicacions
http://hdl.handle.net/2117/79726
2021-04-23T04:26:46Z
2021-04-23T04:26:46Z
Quintessential inflation for exponential type potentials: scaling and tracker behavior
Aresté Saló, Llibert
Haro Cases, Jaume
http://hdl.handle.net/2117/344071
2021-04-21T09:20:29Z
2021-04-21T09:18:39Z
Quintessential inflation for exponential type potentials: scaling and tracker behavior
Aresté Saló, Llibert; Haro Cases, Jaume
We will show that for exponential type potentials of the form V(f)~e-¿fn/Mnpl, which are used to depict quintessential inflation, the solutions whose initial conditions take place during the slow roll phase in order to describe correctly the inflationary period, do not belong for large values of the parameter n to the basin of attraction of the scaling solution – a solution of the scalar field equation whose energy density scale as the one of the fluid component of the universe during radiation or the matter domination period –, meaning that a late time mechanism to exit this behavior and depict correctly the current cosmic acceleration is not needed. However, in such cases, namely n large enough, these potentials cannot correctly depict the current cosmic acceleration. This is the reason why the potential must be improved introducing another parameter -as the one in the well-known Peebles–Vilenkin quintessential inflation model, which depends on two parameters, one to describe inflation and the other one to correctly depict the present accelerated evolution – able to deal with the late time acceleration of our universe.
2021-04-21T09:18:39Z
Aresté Saló, Llibert
Haro Cases, Jaume
We will show that for exponential type potentials of the form V(f)~e-¿fn/Mnpl, which are used to depict quintessential inflation, the solutions whose initial conditions take place during the slow roll phase in order to describe correctly the inflationary period, do not belong for large values of the parameter n to the basin of attraction of the scaling solution – a solution of the scalar field equation whose energy density scale as the one of the fluid component of the universe during radiation or the matter domination period –, meaning that a late time mechanism to exit this behavior and depict correctly the current cosmic acceleration is not needed. However, in such cases, namely n large enough, these potentials cannot correctly depict the current cosmic acceleration. This is the reason why the potential must be improved introducing another parameter -as the one in the well-known Peebles–Vilenkin quintessential inflation model, which depends on two parameters, one to describe inflation and the other one to correctly depict the present accelerated evolution – able to deal with the late time acceleration of our universe.
Calibrations and null-Lagrangians for nonlocal perimeters and an application to the viscosity theory
Cabré Vilagut, Xavier
http://hdl.handle.net/2117/343976
2021-04-20T09:40:27Z
2021-04-20T09:35:29Z
Calibrations and null-Lagrangians for nonlocal perimeters and an application to the viscosity theory
Cabré Vilagut, Xavier
For nonnegative even kernels K, we consider the K-nonlocal perimeter functional acting on sets. Assuming the existence of a foliation of space made of solutions of the associated K-nonlocal mean curvature equation in an open set O¿Rn, we built a calibration for the nonlocal perimeter inside O¿Rn. The calibrating functional is a nonlocal null-Lagrangian. As a consequence, we conclude the minimality in O of each leaf of the foliation. As an application, we prove the minimality of K-nonlocal minimal graphs and that they are the unique minimizers subject to their own exterior data. As a second application of the calibration, we give a simple proof of an important result from the seminal paper of Caffarelli, Roquejoffre, and Savin, stating that minimizers of the fractional perimeter are viscosity solutions.
The final publication is available at link.springer.com
2021-04-20T09:35:29Z
Cabré Vilagut, Xavier
For nonnegative even kernels K, we consider the K-nonlocal perimeter functional acting on sets. Assuming the existence of a foliation of space made of solutions of the associated K-nonlocal mean curvature equation in an open set O¿Rn, we built a calibration for the nonlocal perimeter inside O¿Rn. The calibrating functional is a nonlocal null-Lagrangian. As a consequence, we conclude the minimality in O of each leaf of the foliation. As an application, we prove the minimality of K-nonlocal minimal graphs and that they are the unique minimizers subject to their own exterior data. As a second application of the calibration, we give a simple proof of an important result from the seminal paper of Caffarelli, Roquejoffre, and Savin, stating that minimizers of the fractional perimeter are viscosity solutions.
Self-adjoint Dirac operators on domains in R-3
Holzmann, Markus
Mas Blesa, Albert
http://hdl.handle.net/2117/343540
2021-04-18T18:29:20Z
2021-04-12T12:22:50Z
Self-adjoint Dirac operators on domains in R-3
Holzmann, Markus; Mas Blesa, Albert
In this paper, the spectral and scattering properties of a family of self-adjoint Dirac operators in L2(O;C4), where O¿R3 is either a bounded or an unbounded domain with a compact C2-smooth boundary, are studied in a systematic way. These operators can be viewed as the natural relativistic counterpart of Laplacians with boundary conditions as of Robin type. Our approach is based on abstract boundary triple techniques from extension theory of symmetric operators and a thorough study of certain classes of (boundary) integral operators, that appear in a Krein-type resolvent formula. The analysis of the perturbation term in this formula leads to a description of the spectrum and a Birman–Schwinger principle, a qualitative understanding of the scattering properties in the case that O is an exterior domain, and corresponding trace formulas.
2021-04-12T12:22:50Z
Holzmann, Markus
Mas Blesa, Albert
In this paper, the spectral and scattering properties of a family of self-adjoint Dirac operators in L2(O;C4), where O¿R3 is either a bounded or an unbounded domain with a compact C2-smooth boundary, are studied in a systematic way. These operators can be viewed as the natural relativistic counterpart of Laplacians with boundary conditions as of Robin type. Our approach is based on abstract boundary triple techniques from extension theory of symmetric operators and a thorough study of certain classes of (boundary) integral operators, that appear in a Krein-type resolvent formula. The analysis of the perturbation term in this formula leads to a description of the spectrum and a Birman–Schwinger principle, a qualitative understanding of the scattering properties in the case that O is an exterior domain, and corresponding trace formulas.
Stable solutions to semilinear elliptic equations are smooth up to dimension 9
Cabré Vilagut, Xavier
Figalli, Alessio
Ros Oton, Xavier
Serra, Joaquim
http://hdl.handle.net/2117/343155
2021-04-11T19:47:19Z
2021-04-06T10:42:34Z
Stable solutions to semilinear elliptic equations are smooth up to dimension 9
Cabré Vilagut, Xavier; Figalli, Alessio; Ros Oton, Xavier; Serra, Joaquim
In this paper we prove the following long-standing conjecture: stable solutions to semi-linear elliptic equations are bounded (and thus smooth) in dimension n¿9. This result, that was only known to be true for n¿4 , is optimal: log(1/|x|2) is a W1,2 singular stable solution for n¿10. The proof of this conjecture is a consequence of a new universal estimate: we prove that, in dimension n¿9, stable solutions are bounded in terms only of their L1 norm, independently of the non-linearity. In addition, in every dimension we establish a higher integrability result for the gradient and optimal integrability results for the solution in Morrey spaces.
As one can see by a series of classical examples, all our results are sharp. Furthermore, as a corollary, we obtain that extremal solutions of Gelfand problems are W1,2 in every dimension and they are smooth in dimension n¿9. This answers to two famous open problems posed by Brezis and Brezis–Vázquez.
2021-04-06T10:42:34Z
Cabré Vilagut, Xavier
Figalli, Alessio
Ros Oton, Xavier
Serra, Joaquim
In this paper we prove the following long-standing conjecture: stable solutions to semi-linear elliptic equations are bounded (and thus smooth) in dimension n¿9. This result, that was only known to be true for n¿4 , is optimal: log(1/|x|2) is a W1,2 singular stable solution for n¿10. The proof of this conjecture is a consequence of a new universal estimate: we prove that, in dimension n¿9, stable solutions are bounded in terms only of their L1 norm, independently of the non-linearity. In addition, in every dimension we establish a higher integrability result for the gradient and optimal integrability results for the solution in Morrey spaces.
As one can see by a series of classical examples, all our results are sharp. Furthermore, as a corollary, we obtain that extremal solutions of Gelfand problems are W1,2 in every dimension and they are smooth in dimension n¿9. This answers to two famous open problems posed by Brezis and Brezis–Vázquez.
Matter Bounce Scenario in F(T) gravity
Haro Cases, Jaume
Amorós Torrent, Jaume
http://hdl.handle.net/2117/341948
2021-03-21T19:26:55Z
2021-03-18T10:46:41Z
Matter Bounce Scenario in F(T) gravity
Haro Cases, Jaume; Amorós Torrent, Jaume
It is shown that teleparallel F(T) theories of gravity combined with holonomy corrected Loop Quantum Cosmology (LQC) support a Matter Bounce Scenario (MBS) which is a potential alternative to the inflationary paradigm. The Matter Bounce Scenario is reviewed and, according to the current observational data provided by PLANCK’s team, we have summarized all the conditions that it has to satisfy in order to be a viable alternative to inflation, such as to provide a theoretical value of the spectral index and its running compatible with the latest PLANCK data, to have a reheating process via gravitational particle production, or to predict some signatures in the non-gaussianities of the power spectrum. The calculation of the power spectrum for scalar perturbations and the ratio of tensor to scalar perturbations has been done, in the simplest case of an exact matter dominated background, for both holonomy corrected LQC and teleparallel F(T) gravity. Finally, we have discussed the challenges (essentially, dealing with non-gaussianities, the calculation of the 3-point function in flat spatial geometries for theories beyond General Relativity) and problems (Jeans instabilities in the case of holonomy corrected LQC or local Lorentz dependence in teleparallelism) that arise in either bouncing scenario
2021-03-18T10:46:41Z
Haro Cases, Jaume
Amorós Torrent, Jaume
It is shown that teleparallel F(T) theories of gravity combined with holonomy corrected Loop Quantum Cosmology (LQC) support a Matter Bounce Scenario (MBS) which is a potential alternative to the inflationary paradigm. The Matter Bounce Scenario is reviewed and, according to the current observational data provided by PLANCK’s team, we have summarized all the conditions that it has to satisfy in order to be a viable alternative to inflation, such as to provide a theoretical value of the spectral index and its running compatible with the latest PLANCK data, to have a reheating process via gravitational particle production, or to predict some signatures in the non-gaussianities of the power spectrum. The calculation of the power spectrum for scalar perturbations and the ratio of tensor to scalar perturbations has been done, in the simplest case of an exact matter dominated background, for both holonomy corrected LQC and teleparallel F(T) gravity. Finally, we have discussed the challenges (essentially, dealing with non-gaussianities, the calculation of the 3-point function in flat spatial geometries for theories beyond General Relativity) and problems (Jeans instabilities in the case of holonomy corrected LQC or local Lorentz dependence in teleparallelism) that arise in either bouncing scenario
Viability of the Matter Bounce Scenario
Haro Cases, Jaume
Amorós Torrent, Jaume
http://hdl.handle.net/2117/341942
2021-03-21T19:29:07Z
2021-03-18T10:26:03Z
Viability of the Matter Bounce Scenario
Haro Cases, Jaume; Amorós Torrent, Jaume
It is shown that teleparallel F(T) theories of gravity combined with Loop Quantum Cosmology support a Matter Bounce Scenario which is an alternative to the inflation scenario in the Big Bang paradigm. It is checked that these bouncing models provide theoretical data that fits well with the current observational data, allowing the viability of the Matter Bounce Scenario
2021-03-18T10:26:03Z
Haro Cases, Jaume
Amorós Torrent, Jaume
It is shown that teleparallel F(T) theories of gravity combined with Loop Quantum Cosmology support a Matter Bounce Scenario which is an alternative to the inflation scenario in the Big Bang paradigm. It is checked that these bouncing models provide theoretical data that fits well with the current observational data, allowing the viability of the Matter Bounce Scenario
Note on the reheating temperature in Starobinsky-type potentials
Haro Cases, Jaume
Aresté Saló, Llibert
http://hdl.handle.net/2117/341480
2021-03-15T14:40:11Z
2021-03-11T11:31:19Z
Note on the reheating temperature in Starobinsky-type potentials
Haro Cases, Jaume; Aresté Saló, Llibert
The relation between the reheating temperature, the number of e-folds and the spectral index is shown for the Starobinsky model and some of its descendants through a very detailed calculation of these three quantities. The conclusion is that for viable temperatures between 1 MeV and 109 GeV the corresponding values of the spectral index enter perfectly in its 2s C.L., which shows the viability of this kind of models
2021-03-11T11:31:19Z
Haro Cases, Jaume
Aresté Saló, Llibert
The relation between the reheating temperature, the number of e-folds and the spectral index is shown for the Starobinsky model and some of its descendants through a very detailed calculation of these three quantities. The conclusion is that for viable temperatures between 1 MeV and 109 GeV the corresponding values of the spectral index enter perfectly in its 2s C.L., which shows the viability of this kind of models
The spectrum of gravitational waves, their overproduction in quintessential inflation and its influence in the reheating temperature
Haro Cases, Jaume
Aresté Saló, Llibert
http://hdl.handle.net/2117/335822
2021-01-25T07:25:45Z
2021-01-22T11:28:39Z
The spectrum of gravitational waves, their overproduction in quintessential inflation and its influence in the reheating temperature
Haro Cases, Jaume; Aresté Saló, Llibert
One of the most important issues in an inflationary theory as standard or quintessential inflation is the mechanism to reheat the universe after the end of the inflationary period in order to match with the Hot Big Bang universe. In quintessential inflation two mechanisms are frequently used, namely the reheating via gravitational particle production which is, as we will see, very efficient when the phase transition from the end of inflation to a kinetic regime (all the energy of the inflaton field is kinetic) is very abrupt, and the so-called instant preheating which is used for a very smooth phase transition because in that case the gravitational particle production is very inefficient. In the present work, a detailed study of these mechanisms is done, obtaining bounds for the reheating temperature and the range of the parameters involved in each reheating mechanism in order that the Gravitational Waves (GWs) produced at the beginning of kination do not disturb the Big Bang Nucleosynthesis (BBN) success
2021-01-22T11:28:39Z
Haro Cases, Jaume
Aresté Saló, Llibert
One of the most important issues in an inflationary theory as standard or quintessential inflation is the mechanism to reheat the universe after the end of the inflationary period in order to match with the Hot Big Bang universe. In quintessential inflation two mechanisms are frequently used, namely the reheating via gravitational particle production which is, as we will see, very efficient when the phase transition from the end of inflation to a kinetic regime (all the energy of the inflaton field is kinetic) is very abrupt, and the so-called instant preheating which is used for a very smooth phase transition because in that case the gravitational particle production is very inefficient. In the present work, a detailed study of these mechanisms is done, obtaining bounds for the reheating temperature and the range of the parameters involved in each reheating mechanism in order that the Gravitational Waves (GWs) produced at the beginning of kination do not disturb the Big Bang Nucleosynthesis (BBN) success
Understanding the phenomenology of interacting dark energy scenarios and their theoretical bounds
Pan, Supriya
Haro Cases, Jaume
Yang, Weiqiang
Amorós Torrent, Jaume
http://hdl.handle.net/2117/332252
2020-11-22T18:03:11Z
2020-11-16T12:50:33Z
Understanding the phenomenology of interacting dark energy scenarios and their theoretical bounds
Pan, Supriya; Haro Cases, Jaume; Yang, Weiqiang; Amorós Torrent, Jaume
Nongravitational interaction between dark matter and dark energy has been considered in a spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) universe. The interaction rate is assumed to be linear in the energy densities of dark matter and dark energy and it is also proportional to the Hubble rate of the FLRW universe. This kind of interaction model leads to an autonomous linear dynamical system, and depending on the coupling parameters, could be solved analytically by calculating the exponential of the matrix, defining the system. We show here that such interaction rate has a very deep connection with some well-known cosmological theories. We then investigate the theoretical bounds on the coupling parameters of the interaction rate in order that the energy densities of the dark sector remain positive throughout the evolution of the universe and asymptotically converge to zero at very late times. Our analyses also point out that such linear interacting model may encounter with finite time future singularities depending on the coupling parameters as well as the dark energy state parameter. © 2020 American Physical Society
2020-11-16T12:50:33Z
Pan, Supriya
Haro Cases, Jaume
Yang, Weiqiang
Amorós Torrent, Jaume
Nongravitational interaction between dark matter and dark energy has been considered in a spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) universe. The interaction rate is assumed to be linear in the energy densities of dark matter and dark energy and it is also proportional to the Hubble rate of the FLRW universe. This kind of interaction model leads to an autonomous linear dynamical system, and depending on the coupling parameters, could be solved analytically by calculating the exponential of the matrix, defining the system. We show here that such interaction rate has a very deep connection with some well-known cosmological theories. We then investigate the theoretical bounds on the coupling parameters of the interaction rate in order that the energy densities of the dark sector remain positive throughout the evolution of the universe and asymptotically converge to zero at very late times. Our analyses also point out that such linear interacting model may encounter with finite time future singularities depending on the coupling parameters as well as the dark energy state parameter. © 2020 American Physical Society
Scaling solutions in quintessential inflation
Haro Cases, Jaume
Amorós Torrent, Jaume
Pan, Supriya
http://hdl.handle.net/2117/328290
2020-09-06T20:07:16Z
2020-09-02T11:04:01Z
Scaling solutions in quintessential inflation
Haro Cases, Jaume; Amorós Torrent, Jaume; Pan, Supriya
In quintessence scalar field theories, the presence of scaling solutions are important during the radiation and matter epoch due to having their attractor character. Usually, it is assumed that the initial conditions of the quintessence field are in the basin of attraction of the scaling solutions. However, in order to reproduce the current cosmic acceleration, at late times, a mechanism to exit this behavior is needed. In the present work we show that the quintessential inflation models could be an excellent candidate to exhibit the above behavior. However, the crucial point of quintessential inflation is that the initial conditions has to be taken during the inflation, and at the beginning of the radiation era, the scalar field does not belong to the basin of attraction of the scaling solution. This means that, in the case where quintessence is depicted via exponential potentials, only a single exponential in the tail of the quintessential inflation potential is enough to depict the evolution of our univers
2020-09-02T11:04:01Z
Haro Cases, Jaume
Amorós Torrent, Jaume
Pan, Supriya
In quintessence scalar field theories, the presence of scaling solutions are important during the radiation and matter epoch due to having their attractor character. Usually, it is assumed that the initial conditions of the quintessence field are in the basin of attraction of the scaling solutions. However, in order to reproduce the current cosmic acceleration, at late times, a mechanism to exit this behavior is needed. In the present work we show that the quintessential inflation models could be an excellent candidate to exhibit the above behavior. However, the crucial point of quintessential inflation is that the initial conditions has to be taken during the inflation, and at the beginning of the radiation era, the scalar field does not belong to the basin of attraction of the scaling solution. This means that, in the case where quintessence is depicted via exponential potentials, only a single exponential in the tail of the quintessential inflation potential is enough to depict the evolution of our univers