Comment on "Note on the Analytical Solution of the Rabi Model"
We make several comments on "Note on the Analytical Solution of the Rabi Model" (arXiv:1210.4946).
quant-ph↗
arXiv subjects
Tomasz Stachowiak
Publications and source records attributed to Tomasz Stachowiak.
How to calculate spectra of Rabi and related models
We show how to use properly the Bargmann space of entire functions in the analysis of the Rabi model. We are able to correct a serious error in recent papers on the topic and develop a corrected method of finding the spectrum applicable also to other, more general systems.
math-ph↗
A differential algorithm for the Lyapunov spectrum
We present a new algorithm for computing the Lyapunov exponents spectrum based on a matrix differential equation. The approach belongs to the so called continuous type, where the rate of expansion of perturbations is obtained for all times, and the exponents are reached as the limit at infinity. It does not involve exponentially divergent quantities so there is no need of rescaling or realigning of the solution. We show the algorithm's advantages and drawbacks using mainly the example of a particle moving between two contracting walls.
math.DS↗
On solvable Dirac equation with polynomial potentials
One dimensional Dirac equation is analysed with regard to the existence of exact (or closed-form) solutions for polynomial potentials. The notion of Liouvillian functions is used to define solvability, and it is shown that except for the linear potentials the equation in question is not solvable.
math-ph↗
Vortex in axion condensate as a dark matter halo
We study the possibility of the vortices formation in axion condensate on the galactic scale. Such vortices can occur as a result of global rotation of the early universe. We study analytical models of vortices and calculate exemplary galaxy rotation curves. Depending on the setup it is possible to obtain a variety of shapes which give a good qualitative agreement with observational results. However, as we show, the extremely low velocity dispersions of the axions velocity are required to form the single vortex on the galactic scales. We find that the required velocity dispersion is of the order of $σ\approx 10^{-12} \frac{m}{s}$. This is much smaller that predicted within the present understanding of the axion physics. The vortices in axion condensate can be however formed on the much smaller scales and give seeds to the galaxy formation and to their angular momenta. On the other hand, the vortices can be formed on the galactic scales, but only if the mass of the axion-like particles is of the order of $10^{-30}$eV. In this case, the particle de Broglie wavelength is comparable with the galactic diameter. This condition must be fulfilled in order to keep the coherence of the quantum condensate on the galactic scales.
astro-ph↗
Integrability and Chaos - algebraic and geometric approach
This thesis presents two descriptions of complexity in dynamical systems. The algebraic approach deals with the differential Galois group theory and its restrictions on integrability. The geometric part is a formulation of dynamics in the language of differential geometry with particular application to Lyapunov exponents and variational equations. The algorithm for calculating the Lyapunov spectrum is illustrated with three examples.
math-ph↗
Kepler versus Akaike
I use the example of the Earth's orbit to illustrate the principle behind the Akaike Information Criterion, and refute the misconception that the criterion, by definition, discards more complex models in favour of simpler ones.
physics.data-an↗
Global integrability of cosmological scalar fields
We investigate the Liouvillian integrability of Hamiltonian systems describing a universe filled with a scalar field (possibly complex). The tool used is the differential Galois group approach, as introduced by Morales-Ruiz and Ramis. The main result is that the generic systems with minimal coupling are non-integrable, although there still exist some values of parameters for which integrability remains undecided; the conformally coupled systems are only integrable in four known cases. We also draw a connection with chaos present in such cosmological models, and the issues of integrability restricted to the real domain.
math-ph↗
Testing and selection of cosmological models with $(1+z)^6$ corrections
In the paper we check whether the contribution of $(-)(1+z)^6$ type in the Friedmann equation can be tested. We consider some astronomical tests to constrain the density parameters in such models. We describe different interpretations of such an additional term: geometric effects of Loop Quantum Cosmology, effects of braneworld cosmological models, non-standard cosmological models in metric-affine gravity, and models with spinning fluid. Kinematical (or geometrical) tests based on null geodesics are insufficient to separate individual matter components when they behave like perfect fluid and scale in the same way. Still, it is possible to measure their overall effect. We use recent measurements of the coordinate distances from the Fanaroff-Riley type IIb (FRIIb) radio galaxy (RG) data, supernovae type Ia (SNIa) data, baryon oscillation peak and cosmic microwave background radiation (CMBR) observations to obtain stronger bounds for the contribution of the type considered. We demonstrate that, while $ρ^2$ corrections are very small, they can be tested by astronomical observations -- at least in principle. Bayesian criteria of model selection (the Bayesian factor, AIC, and BIC) are used to check if additional parameters are detectable in the present epoch. As it turns out, the $Λ$CDM model is favoured over the bouncing model driven by loop quantum effects. Or, in other words, the bounds obtained from cosmography are very weak, and from the point of view of the present data this model is indistinguishable from the $Λ$CDM one.
gr-qc↗
Exact solutions for Big Bounce in loop quantum cosmology
In this paper we study the flat (k=0) cosmological FRW model with holonomy corrections of Loop Quantum Gravity. The considered universe contains a massless scalar field and the cosmological constant Lambda. We find analytical solutions for this model in different configurations and investigate its dynamical behaviour in the whole phase space. We show the explicit influence of Lambda on the qualitative and quantitative character of solutions. Even in the case of positive Lambda the oscillating solutions without the initial and final singularity appear as a generic case for some quantisation schemes.
gr-qc↗
Global dynamics of cosmological scalar fields -- Part II
This is the second part of integrability analysis of cosmological models with scalar fields. Here, we study systems with conformal coupling, and show that apart from four cases, where explicit first integrals are known, the generic system is not integrable. We also comment on some chaotic properties of the system, and the issues of integrability restricted to the real domain.
gr-qc↗
Exact solutions in bouncing cosmology
We discuss the effects of a (possibly) negative $(1+z)^6$ type contribution to the Friedmann equation. No definite answer can be given as to the presence and magnitude of a particular mechanism, because any test using the general relation $H(z)$ is able to estimate only the total of all sources of such a term. That is why we describe four possibilities: 1) geometric effects of loop quantum cosmology, 2) braneworld cosmology, 3) metric-affine gravity, and 4) cosmology with spinning fluid. We find the exact solutions for the models with $ρ^2$ correction in terms of elementary functions, and show all evolutional paths on their phase plane. Instead of the initial singularity, the generic feature is now a bounce.
gr-qc↗
Global dynamics of cosmological scalar fields -- Part I
We investigate the Liouvillian integrability of Hamiltonian systems describing a universe filled with a scalar field (possibly complex). The tool used is the differential Galois group approach, as introduced by Morales-Ruiz and Ramis. The main result is that the generic systems are non-integrable, although there still exist some values of parameters for which integrability remains undecided. We also draw a connection with chaos present in such cosmological models. The first part of the article deals with minimally coupled fields, and the second treats the conformal couping.
gr-qc↗
Non-integrability of density perturbations in the FRW universe
We investigate the evolution equation of linear density perturbations in the Friedmann-Robertson-Walker universe with matter, radiation and the cosmological constant. The concept of solvability by quadratures is defined and used to prove that there are no "closed form" solutions except for the known Chernin, Heath, Meszaros and simple degenerate ones. The analysis is performed applying Kovacic's algorithm. The possibility of the existence of other, more general solutions involving special functions is also investigated.
gr-qc↗
Phantom Friedmann Cosmologies and Higher-Order Characteristics of Expansion
We discuss a more general class of phantom ($p < -\varrho$) cosmologies with various forms of both phantom ($w < -1$), and standard ($w > -1$) matter. We show that many types of evolution which include both Big-Bang and Big-Rip singularities are admitted and give explicit examples. Among some interesting models, there exist non-singular oscillating (or "bounce") cosmologies, which appear due to a competition between positive and negative pressure of variety of matter content. From the point of view of the current observations the most interesting cosmologies are the ones which start with a Big-Bang and terminate at a Big-Rip. A related consequence of having a possibility of two types of singularities is that there exists an unstable static universe approached by the two asymptotic models - one of them reaches Big-Bang, and another reaches Big-Rip. We also give explicit relations between density parameters $Ω$ and the dynamical characteristics for these generalized phantom models, including higher-order observational characteristics such as jerk and "kerk". Finally, we discuss the observational quantities such as luminosity distance, angular diameter, and source counts, both in series expansion and explicitly, for phantom models. Our series expansion formulas for the luminosity distance and the apparent magnitude go as far as to the fourth-order in redshift $z$ term, which includes explicitly not only the jerk, but also the "kerk" (or "snap") which may serve as an indicator of the curvature of the universe.
hep-th↗
Towards testing interacting cosmology by distant type Ia supernovae
We investigate the possibility of testing cosmological models with interaction between matter and energy sector. We assume the standard FRW model while the so called energy conservation condition is interpreted locally in terms of energy transfer. We analyze two forms of dark energy sectors: the cosmological constant and phantom field. We find a simple exact solution of the models in which energy transfer is described by a Cardassian like term in the relation of $H^{2}(z)$, where $H$ is Hubble's function and $z$ is redshift. The considered models have two additional parameters $(Ω_{\text{int}},n)$ (apart the parameters of the $Λ$CDM model) which can be tested using SNIa data. In the estimation of the model parameters Riess et al.'s sample is used. We also confront the quality of statistical fits for both the $Λ$CDM model and the interacting models with the help of the Akaike and Bayesian informative criteria. Our conclusion from standard best fit method is that the interacting models explains the acceleration of the Universe better but they give rise to a universe with high matter density. However, using the tools of information criteria we find that the two new parameters play an insufficient role in improving the fit to SNIa data and the standard $Λ$CDM model is still preferred. We conclude that high precision detection of high redshift supernovae could supply data capable of justifying adoption of new parameters.
astro-ph↗
Phantom cosmologies
We discuss a class of phantom ($p < - \varrho$) cosmological models. Except for phantom we admit various forms of standard types of matter and discuss the problem of singularities for these cosmologies. The singularities are different from those of standard matter cosmology since they appear for infinite values of the scale factor. We also find an interesting relation between the phantom models and standard matter models which is like the duality symmetry of string cosmology.
hep-th↗