EDP  Equacions en Derivades Parcials i Aplicacions
http://hdl.handle.net/2117/79726
Fri, 23 Apr 2021 06:24:13 GMT
20210423T06:24:13Z

Quintessential inflation for exponential type potentials: scaling and tracker behavior
http://hdl.handle.net/2117/344071
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 wellknown 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.
Wed, 21 Apr 2021 09:18:39 GMT
http://hdl.handle.net/2117/344071
20210421T09:18:39Z
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 wellknown 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 nullLagrangians for nonlocal perimeters and an application to the viscosity theory
http://hdl.handle.net/2117/343976
Calibrations and nullLagrangians for nonlocal perimeters and an application to the viscosity theory
Cabré Vilagut, Xavier
For nonnegative even kernels K, we consider the Knonlocal perimeter functional acting on sets. Assuming the existence of a foliation of space made of solutions of the associated Knonlocal 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 nullLagrangian. As a consequence, we conclude the minimality in O of each leaf of the foliation. As an application, we prove the minimality of Knonlocal 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
Tue, 20 Apr 2021 09:35:29 GMT
http://hdl.handle.net/2117/343976
20210420T09:35:29Z
Cabré Vilagut, Xavier
For nonnegative even kernels K, we consider the Knonlocal perimeter functional acting on sets. Assuming the existence of a foliation of space made of solutions of the associated Knonlocal 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 nullLagrangian. As a consequence, we conclude the minimality in O of each leaf of the foliation. As an application, we prove the minimality of Knonlocal 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.

Selfadjoint Dirac operators on domains in R3
http://hdl.handle.net/2117/343540
Selfadjoint Dirac operators on domains in R3
Holzmann, Markus; Mas Blesa, Albert
In this paper, the spectral and scattering properties of a family of selfadjoint Dirac operators in L2(O;C4), where O¿R3 is either a bounded or an unbounded domain with a compact C2smooth 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 Kreintype 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.
Mon, 12 Apr 2021 12:22:50 GMT
http://hdl.handle.net/2117/343540
20210412T12:22:50Z
Holzmann, Markus
Mas Blesa, Albert
In this paper, the spectral and scattering properties of a family of selfadjoint Dirac operators in L2(O;C4), where O¿R3 is either a bounded or an unbounded domain with a compact C2smooth 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 Kreintype 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
http://hdl.handle.net/2117/343155
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 longstanding conjecture: stable solutions to semilinear 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/x2) 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 nonlinearity. 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.
Tue, 06 Apr 2021 10:42:34 GMT
http://hdl.handle.net/2117/343155
20210406T10:42:34Z
Cabré Vilagut, Xavier
Figalli, Alessio
Ros Oton, Xavier
Serra, Joaquim
In this paper we prove the following longstanding conjecture: stable solutions to semilinear 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/x2) 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 nonlinearity. 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
http://hdl.handle.net/2117/341948
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 nongaussianities 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 nongaussianities, the calculation of the 3point 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
Thu, 18 Mar 2021 10:46:41 GMT
http://hdl.handle.net/2117/341948
20210318T10:46:41Z
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 nongaussianities 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 nongaussianities, the calculation of the 3point 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
http://hdl.handle.net/2117/341942
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
Thu, 18 Mar 2021 10:26:03 GMT
http://hdl.handle.net/2117/341942
20210318T10:26:03Z
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 Starobinskytype potentials
http://hdl.handle.net/2117/341480
Note on the reheating temperature in Starobinskytype potentials
Haro Cases, Jaume; Aresté Saló, Llibert
The relation between the reheating temperature, the number of efolds 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
Thu, 11 Mar 2021 11:31:19 GMT
http://hdl.handle.net/2117/341480
20210311T11:31:19Z
Haro Cases, Jaume
Aresté Saló, Llibert
The relation between the reheating temperature, the number of efolds 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
http://hdl.handle.net/2117/335822
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 socalled 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
Fri, 22 Jan 2021 11:28:39 GMT
http://hdl.handle.net/2117/335822
20210122T11:28:39Z
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 socalled 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
http://hdl.handle.net/2117/332252
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 FriedmannLemaîtreRobertsonWalker (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 wellknown 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
Mon, 16 Nov 2020 12:50:33 GMT
http://hdl.handle.net/2117/332252
20201116T12:50:33Z
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 FriedmannLemaîtreRobertsonWalker (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 wellknown 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
http://hdl.handle.net/2117/328290
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
Wed, 02 Sep 2020 11:04:01 GMT
http://hdl.handle.net/2117/328290
20200902T11:04:01Z
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