arXiv subjects

Marek Szydlowski

Publications and source records attributed to Marek Szydlowski.

At least 37 records · Page 2Linked to original sources

arXiv2015

Cosmological dynamics with non-minimally coupled scalar field and a constant potential function

Dynamical systems methods are used to investigate global behavior of the spatially flat Friedmann-Robertson-Walker cosmological model in gravitational theory with a non-minimally coupled scalar field and a constant potential function. We show that the system can be reduced to an autonomous three-dimensional dynamical system and additionally is equipped with an invariant manifold corresponding to an accelerated expansion of the universe. Using this invariant manifold we find an exact solution of the reduced dynamics. We investigate all solutions for all admissible initial conditions using theory of dynamical systems to obtain a classification of all evolutional paths. The right-hand sides of the dynamical system depend crucially on the value of the non-minimal coupling constant therefore we study bifurcation values of this parameter under which the structure of the phase space changes qualitatively. We found a special bifurcation value of the non-minimal coupling constant which is distinguished by dynamics of the model and may suggest some additional symmetry in matter sector of the theory.

gr-qc↗

arXiv2015

Dynamical Emergence of FRW Cosmological Models

Recent astronomical observations strongly indicate that the current Universe is undergoing an accelerated phase of expansion. The discovery of this fact was unexpected and resulted in the comeback of cosmological constant. The conception of standard cosmological model has its roots in this context. The paper relates to the methodological status of effective theories in the context of cosmological investigations. We argue that the standard cosmological model (LCDM model) as well as the CDM have a status of effective theories only, similarly to the standard model of particle physics. The LCDM model is studied from the point of view of the methodological debate on reductionism and epistemological emergence in the science. It is shown in the paper that bifurcation as well as structural instability notion can be useful in the detection of emergence the LCDM model from the CDM model. We demonstrate that the structural stability of the LCDM model can explain the flexibility of the model to accommodation of the observational data. We show that LCDM model can be derived from CDM as the bifurcation. It is an example of acausal derivation of Lambda term. The case study of emergence of LCDM model suggests that it can be understood in terms of bifurcation and structural stability issue. The reduction from the upper models represented in terms of dynamical system to low-level ones can be realized in any case by application of a mathematical limit (boundary crossing) with respect to the model parameter. It is a simple consequence of mathematical theorem about smooth dependence solutions with respect to time, initial condition and the parameters.

gr-qc↗

arXiv2015

Cosmology with decaying cosmological constant -- exact solutions and model testing

We study dynamics of $Λ(t)$ cosmological models which are a natural generalization of the standard cosmological model (the $Λ$CDM model). We consider a class of models: the ones with a prescribed form of $Λ(t)=Λ_{\text{bare}}+\frac{α^2}{t^2}$. This type of a $Λ(t)$ parametrization is motivated by different cosmological approaches. We interpret the model with running Lambda ($Λ(t)$) as a special model of an interacting cosmology with the interaction term $-dΛ(t)/dt$ in which energy transfer is between dark matter and dark energy sectors. For the $Λ(t)$ cosmology with a prescribed form of $Λ(t)$ we have found the exact solution in the form of Bessel functions. Our model shows that fractional density of dark energy $Ω_e$ is constant and close to zero during the early evolution of the universe. We have also constrained the model parameters for this class of models using the astronomical data such as SNIa data, BAO, CMB, measurements of $H(z)$ and the Alcock-Paczy{ń}ski test. In this context we formulate a simple criterion of variability of $Λ$ with respect to $t$ in terms of variability of the jerk or sign of estimator $(1-Ω_{\text{m},0}-Ω_{Λ,0})$. The case study of our model enable us to find an upper limit $α^2 < 0.012$ ($2σ$ C.L.) describing the variation from the cosmological constant while the LCDM model seems to be consistent with various data.

astro-ph.CO↗

arXiv2015

Cosmological models with running cosmological term and decaying dark matter

We investigate the dynamics of the generalized $Λ$CDM model, which the $Λ$ term is running with the cosmological time. The $Λ(t)$ term emerges from the covariant theory of the scalar field $ϕ$ with the self-interacting potential $V(ϕ)$. On the example of the model $Λ(t)=Λ_{\text{bare}}+\frac{α^2}{t^2}$ we show the existence of a mechanism of the modification of the scaling law for energy density of dark matter: $ρ_{\text{dm}}\propto a^{-3+λ(t)}$. We discuss the evolution of $Λ(t)$ term and pointed out that during the cosmic evolution there is a long phase in which this term is approximately constant. This effect justifies Alcaniz and Lima's approach to $Λ(H)$ cosmologies. We also present the statistical analysis of both the $Λ(t)$CDM model with dark energy and decaying dark matter and the $Λ$CDM standard cosmological model. We divide the observational data into two groups: low $z$ data (SNIa, BAO, $H(z)$ and AP test) and high $z$ data (Planck, WP and lensing). While for the former we find the best fit value of the parameter $λ$ is positive ($λ=0.0338$, energy transfer is from the dark energy to dark matter sector), for the latter we find that $λ$ is $-0.0199$ which is an evidence that the energy transfer is from decaying dark matter. This disagreement correlates with estimated values of $H_0$ ($67.77$ km/(s Mpc) and $65.62$ km/(s Mpc) respectively). The decaying dark matter causes to lowering a mass of dark matter particles which are lighter than CDM particles and remain relativistic. The rate of the process of decaying matter is estimated. We show that in the models of decaying dark matter, the cosmological constant problem disappears naturally. The model with decaying dark matter possesses one parameter more but in light of the AIC it is better than the $Λ$CDM standard cosmological model.

astro-ph.CO↗

arXiv2015

Cosmology with a Decaying Vacuum Energy Parametrization Derived from Quantum Mechanics

Within the quantum mechanical treatment of the decay problem one finds that at late times $t$ the survival probability of an unstable state cannot have the form of an exponentially decreasing function of time $t$ but it has an inverse power-like form. This is a general property of unstable states following from basic principles of quantum theory. The consequence of this property is that in the case of false vacuum states the cosmological constant becomes dependent on time: $Λ- Λ_{\text{bare}}\equiv Λ(t) -Λ_{\text{bare}} \sim 1/t^{2}$. We construct the cosmological model with decaying vacuum energy density and matter for solving the cosmological constant problem and the coincidence problem. We show the equivalence of the proposed decaying false vacuum cosmology with the $Λ(t)$ cosmologies (the $Λ(t)$CDM models). The cosmological implications of the model of decaying vacuum energy (dark energy) are discussed. We constrain the parameters of the model with decaying vacuum using astronomical data. For this aim we use the observation of distant supernovae of type Ia, measurements of $H(z)$, BAO, CMB and others. The model analyzed is in good agreement with observation data and explain a small value of the cosmological constant today.

astro-ph.CO↗

arXiv2015

Cosmological model with decaying vacuum energy law from principles of quantum mechanics

We construct the cosmological model to explain the cosmological constant problem. We built the extension of the standard cosmological model $Λ$CDM by consideration of decaying vacuum energy represented by the running cosmological term. From the principles of quantum mechanics one can find that in the long term behavior survival probability of unstable states is a decreasing function of the cosmological time and has the inverse power-like form. This implies that cosmological constant $ρ_{\text{vac}} = Λ(t) = Λ_{\text{bare}} + \fracα{t^2}$ where $Λ_{\text{bare}}$ and $α$ are constants. We investigate the dynamics of this model using dynamical system methods due to a link to the $Λ(H)$ cosmologies. We have found the exact solution for the scale factor as well as the indicators of its variability like the deceleration parameter and the jerk. From the calculation of the jerk we obtain a simple test of the decaying vacuum in the FRW universe. Using astronomical data (SNIa, $H(z)$, CMB, BAO) we have estimated the model parameters and compared this model with the $Λ$CDM model. Our statistical results indicate that the decaying vacuum model is a little worse than the $Λ$CDM model. But the decaying vacuum cosmological model explains the small value of the cosmological constant today.

astro-ph.CO↗

arXiv2014

AIC, BIC, Bayesian evidence against the interacting dark energy model

Recent astronomical observations have indicated that the Universe is in the phase of accelerated expansion. While there are many cosmological models which try to explain this phenomenon, we focus on the interacting $Λ$CDM model where the interaction between the dark energy and dark matter sectors takes place. This model is compared to its simpler alternative---the $Λ$CDM model. To choose between these models the likelihood ratio test was applied as well as the model comparison methods (employing Occam's principle): the Akaike information criterion (AIC), the Bayesian information criterion (BIC) and the Bayesian evidence. Using the current astronomical data: SNIa (Union2.1), $h(z)$, BAO, Alcock--Paczynski test and CMB we evaluated both models. The analyses based on the AIC indicated that there is less support for the interacting $Λ$CDM model when compared to the $Λ$CDM model, while those based on the BIC indicated that there is the strong evidence against it in favor the $Λ$CDM model. Given the weak or almost none support for the interacting $Λ$CDM model and bearing in mind Occam's razor we are inclined to reject this model.

astro-ph↗

arXiv2014

Dynamics and cosmological constraints on Brans-Dicke cosmology

We investigate observational constraints on the Brans-Dicke cosmological model using observational data coming from distant supernovae type Ia, the Hubble function $H(z)$ measurements, information coming from the Alcock-Paczy{ń}ski test, and baryon acoustic oscillations. Our analysis is based on the modified Friedmann function resulting form dynamical investigations of Brans-Dicke cosmology in the vicinity of a de Sitter state. The qualitative theory of dynamical systems enables us to obtain three different behaviors in the vicinity of this state. We find for a linear approach to the de Sitter state $ω_{\textrm{BD}}=-0.8606^{+0.8281}_{-0.1341}$, for an oscillatory approach to the de Sitter state $ω_{\textrm{BD}}=-1.1103^{+0.1872}_{-0.1729}$, and for the transient de Sitter state represented by a saddle-type critical point $ω_{\textrm{BD}}=-2.3837^{+0.4588}_{-4.5459}$. We obtain the mass of the Brans-Dicke scalar field at the present epoch as $m_ϕ\sim H_{0}$. The Bayesian methods of model comparison are used to discriminate between obtained models. We show that observational data point toward vales of the $ω_{\textrm{BD}}$ parameter close to the value suggested by the low-energy limit of the bosonic string theory.

astro-ph.CO↗

arXiv2013

Dynamical complexity of the Brans-Dicke cosmology

The dynamics of the Brans-Dicke theory with a quadratic scalar field potential function and barotropic matter is investigated. The dynamical system methods are used to reveal complexity of dynamical evolution in homogeneous and isotropic cosmological models. The structure of phase space crucially depends on the parameter of the theory $ω_{\textrm{BD}}$ as well as barotropic matter index $w_{m}$. In our analysis these parameters are treated as bifurcation parameters. We found sets of values of these parameters which lead to generic evolutional scenarios. We show that in isotropic and homogeneous models in the Brans-Dicke theory with a quadratic potential function the de Sitter state appears naturally. Stability conditions of this state are fully investigated. It is shown that these models can explain accelerated expansion of the Universe without the assumption of the substantial form of dark matter and dark energy. The Poincare construction of compactified phase space with a circle at infinity is used to show that phase space trajectories in a physical region can be equipped with a structure of a vector field on nontrivial topological closed space. For $ω_{\textrm{BD}}<-3/2$ we show new types of early and late time evolution leading from the anti-de Sitter to the de Sitter state through an asymmetric bounce. In the theory without a ghost we find bouncing solutions and the coexistence of the bounces and the singularity. Following the Peixoto theorem some conclusions about structural stability are drawn.

gr-qc↗

arXiv2013

Brans-Dicke theory and the emergence of ΛCDM model

The dynamics of the Brans-Dicke theory with a scalar field potential function is investigated. We show that the system with a barotropic matter content can be reduced to an autonomous three-dimensional dynamical system. For an arbitrary potential function we found the values of the Brans-Dicke parameter for which a global attractor in the phase space representing de Sitter state exists. Using linearized solutions in the vicinity of this critical point we show that the evolution of the Universe mimics the $Λ$CDM model. From the recent Planck satellite data, we obtain constraints on the variability of the effective gravitational coupling constant as well as the lower limit of the mass of the Brans-Dicke scalar field at the de Sitter state.

gr-qc↗

arXiv2013

Scalar Field Cosmology -- Geometry of Dynamics

We study the Scalar Field Cosmology (SFC) using the geometric language of the phase space. We define and study an ensemble of dynamical systems as a Banach space with a Sobolev metric. The metric in the ensemble is used to measure a distance between different models. We point out the advantages of visualisation of dynamics in the phase space. It is investigated the genericity of some class of models in the context of fine tuning of the form of the potential function in the ensemble of SFC. We also study the symmetries of dynamical systems of SFC by searching for their exact solutions. In this context we stressed the importance of scaling solutions. It is demonstrated that scaling solutions in the phase space are represented by unstable separatrices of the saddle points. Only critical point itself located on two dimensional stable submanifold can be identified as scaling solution. We have also found a class of potentials of the scalar fields forced by the symmetry of differential equation describing the evolution of the universe. A class of potentials forced by scaling (homology) symmetries was given. We point out the role of the notion of a structural stability in the context of the problem of indetermination of the potential form of the Scalar Field Cosmology. We characterise also the class of potentials which reproduces the ΛCDM model, which is known to be structurally stable. We show that the structural stability issue can be effectively used is selection of the scalar field potential function. This enables us to characterise a structurally stable and therefore a generic class of SFC models. We have found a nonempty and dense subset of structurally stable models. We show that these models possess symmetry of homology.

gr-qc↗

arXiv2012

Dynamics of the Bianchi I model with non-minimally coupled scalar field near the singularity

Dynamical systems methods are used to study the evolution of the Bianchi I model with a scalar field. We show that inclusion of the non-minimal coupling term between the scalar field and the curvature changes evolution of the model compared with the minimally coupled case. In the model with the non-minimally coupled scalar field there is a new type of singularity dominated by the non-minimal coupling term. We examine the impact of the non-minimal coupling on the anisotropy evolution and demonstrate the existence of its minimal value in the generic case.

gr-qc↗

arXiv2012

Scalar Field Cosmology -- Improving the Cosmological Evolutional Scenario

We study evolution of cosmological models filled with the scalar field and barotropic matter. We consider the scalar field minimally and non-minimally coupled to gravity. We demonstrated the growth of degree of complexity of evolutional scenario through the description of matter content in terms of the scalar field. In study of all evolutional paths for all initial conditions methods of dynamical systems are used. Using linearized solutions we present simple method of derivation corresponding form of the Hubble function of the scale factor $H(a)$.

astro-ph.CO↗

arXiv2011

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↗

arXiv2010

Uniting cosmological epochs through the twister solution in cosmology with non-minimal coupling

We investigate dynamics of a flat FRW cosmological model with a barotropic matter and a non-minimally coupled scalar field (both canonical and phantom). In our approach we do not assume any specific form of a potential function for the scalar field and we are looking for generic scenarios of evolution. We show that dynamics of universe can be reduced to a 3-dimensional dynamical system. We have found the set of fixed points and established their character. These critical points represent all important epochs in evolution of the universe : (a) a finite scale factor singularity, (b) an inflation (rapid-roll and slow-roll), (c) a radiation domination, (d) a matter domination and (e) a quintessence era. We have shown that the inflation, the radiation and matter domination epochs are transient ones and last for a finite amount of time. The existence of the radiation domination epoch is purely the effect of a non-minimal coupling constant. We show the existence of a twister type solution wandering between all these critical points.

astro-ph.CO↗

arXiv2010

Scalar field cosmology -- toward description of dynamic complexity of cosmological evolution

We study the dynamical evolution of cosmological models with the Robertson-Walker symmetry with a scalar field non-minimally coupled to gravity and barotropic matter. For this aim we use dynamical system methods. We have found a type of evolutional path which links between all important events during the evolution, the cosmological singularity of finite time, inflation, radiation and matter dominating epoch and the accelerated phase expansion of the universe. We point out importance of finding the new generic solution called a twister solution for a deeper description of the evolution of the Universe. We demonstrate that including the non-minimal coupling leads to a new, richer evolutional cosmological scenario in comparison to the case of minimal coupling.

astro-ph.CO↗

arXiv2010

Twister quintessence scenario

We study generic solutions in a non-minimally coupled to gravity scalar field cosmology. It is shown that dynamics for both canonical and phantoms scalar fields with the potential can be reduced to the dynamical system from which the exact forms for an equation of the state parameter can be derived. We have found the stationary solutions of the system and discussed their stability. Within the large class of admissible solutions we have found a non-degenerate critical points and we pointed out multiple attractor type of trajectory travelling in neighborhood of three critical points at which we have the radiation dominating universe, the barotropic matter dominating state and finally the de Sitter attractor. We have demonstrated the stability of this trajectory which we call the twister solution. Discovered evolutional path is only realized if there exist the non-minimal coupling constant. We have found simple duality relations between twister solutions in phantom and canonical scalar fields in the radiation domination phase. For the twister trajectory we have found an oscillating regime of approaching the de Sitter attractor.

astro-ph.CO↗

arXiv2010

Bayesian reasoning in cosmology

We discuss epistemological and methodological aspects of the Bayesian approach in astrophysics and cosmology. The introduction to the Bayesian framework is given for a further discussion concerning the Bayesian inference in physics. The interplay between the modern cosmology, Bayesian statistics, and philosophy of science is presented. We consider paradoxes of confirmation, like Goodman's paradox, appearing in the Bayesian theory of confirmation. As in Goodman's paradox the Bayesian inference is susceptible to some epistemic limitations in the logic of induction. However Goodman's paradox applied to cosmological hypotheses seems to be resolved due to the evolutionary character of cosmology and accumulation new empirical evidences. We argue that the Bayesian framework is useful in the context of falsificability of quantum cosmological models, as well as contemporary dark energy and dark matter problem.

physics.data-an↗