Second order differentiation formula on RCD∗(K;N) spaces

The aim of this paper is to prove a second order differentiation formula for H2;2 functions along geodesics in RCD∗(K;N) spaces with K ∈R and N < ∞. This formula is new even in the context of Alexandrov spaces, where second order differentiation is typically related to semiconvexity. We establish this result by showing that W2-geodesics can be approximated up to second order, in a sense which we shall make precise, by entropic interpolations. In turn this is achieved by proving new, even in the smooth setting, estimates concerning entropic interpolations which we believe are interesting on their own. In particular we obtain: • equiboundedness of densities along entropic interpolations, • local equi-Lipschitz continuity of Schrödinger potentials, • uniform weighted L2 control of the Hessian of such potentials. Finally, the techniques adopted in this paper can be used to show that in the RCD setting the viscous solution of the Hamilton-Jacobi equation can be obtained via a vanishing viscosity method, as in the smooth case. With respect to a previous version, where the space was assumed to be compact, in this paper the second order differentiation formula is proved in full generality

Einstein static universe in scalar-fluid theories

A new Lagrangian framework has recently been proposed to describe interactions between relativistic perfect fluids and scalar fields. In this paper we investigate the Einstein static universe in this new class of theories, which have been named scalar-fluid theories. The stability of the static solutions to both homogeneous and inhomogeneous perturbations is analyzed deriving the relevant cosmological perturbation equations at the linear order. We can find several configurations corresponding to an Einstein static universes which are stable against inhomogeneous perturbations, but unstable against homogeneous perturbations. This shows the possible applications of scalar-fluid theories to the inflationary emergent universe scenario

Interacting quintessence from a variational approach. I. Algebraic couplings

We present a new approach to build models of quintessence interacting with dark or baryonic matter. We use a variational approach for relativistic fluids to realize an effective description of matter fields at the Lagrangian level. The coupling is introduced directly in the action by considering a single function mixing the dynamical degrees of freedom of the theory. The resulting gravitational field equations are derived by variations with respect to the independent variables. New interesting phenomenology can be obtained at both small scales, where new screening mechanisms for scalar fields can be realized, and large scales, where one finds an original and rich class of interacting quintessence models. The background cosmology of two of these models is studied in detail using dynamical system techniques. We find a variety of interesting results: for instance, these models contain dark energy dominated late-time attractors and scaling solutions, both with early-time matter dominated epochs and a possible inflationary origin. In general this new approach provides the starting point for future in-depth studies on new interacting quintessence models

Interacting quintessence from a variational approach. II. Derivative couplings

We consider an original variational approach for building new models of quintessence interacting with dark or baryonic matter. The coupling is introduced at the Lagrangian level using a variational formulation for relativistic fluids, where the interacting term generally depends on both the dynamical degrees of freedom of the theory and their spacetime derivatives. After deriving the field equations from the action, we consider applications in the context of cosmology. Two simple models are studied using dynamical system techniques showing the interesting phenomenology arising in this framework. We find that these models contain dark energy dominated late-time attractors with early-time matter dominated epochs and also obtain a possible dynamical crossing of the phantom barrier. The formulation and results presented here complete and expand the analysis exposed in the first part of this work, where only algebraic couplings, without spacetime derivatives, were considered

Dynamical systems in dark energy models

This PhD thesis is devoted to the study of dynamical systems appearing in theoretical models of dark energy. The quest for understanding the origin of the observed cosmic acceleration has led physicists to advance a large number of phenomenological explanations based on different fundamental theories. The best approach to analyse the background cosmological impli- cations of all these models consists in employing dynamical systems tech- niques. In this thesis, after reviewing elements of dynamical systems theory and basic cosmology, several dynamical systems, which arise in dark energy models ranging from scalar fields to modified gravity, will be studied using both analytical and numerical methods. The work is organised in order to present as many details as possible for the simpler and well known models, while outlining major results and referring to the literature for the less stud- ied ones. This choice aims at providing the reader with a complete overview and summary of dynamical systems in dark energy applications

Rotational elasticity and couplings to linear elasticity

It is the aim of the paper to present a new point of view on rotational elasticity in a nonlinear setting using orthogonal matrices. The proposed model, in the linear approximation, can be compared to the well known equilibrium equations of static linear elasticity. An appropriate kinetic energy is identified and we present a dynamical model of rotational elasticity. The propagation of elastic waves in such a medium is studied and we find two classes of waves, transversal rotational waves and longitudinal rotational waves, both of which are solutions of the nonlinear partial differential equations. For certain parameter choices, the transversal wave velocity can be greater than the longitudinal wave velocity. Moreover, parameter ranges are identified where the model describes an auxetic material. However, in all cases the potential energy functional is positive definite. Finally, we couple the rotational waves to linear elastic waves to study the behaviour of the coupled system. We find wave like solutions to the coupled equations and can visualise our results with the help of suitable figures

Massive black hole binaries in LISA: multimessenger prospects and electromagnetic counterparts

In the next decade, the Laser Interferometer Space Antenna (LISA) will detect the coalescence of massive black hole binaries (MBHBs) in the range $[10^4, 10^8] \, \rm M_{\odot}$, up to $z\sim10$. Their gravitational wave (GW) signal is expected to be accompanied by an electromagnetic counterpart (EMcp), generated by the gas accreting on the binary or on the remnant BH. In this work, we present the number and characteristics (such as redshift and mass distribution, apparent magnitudes or fluxes) of EMcps detectable jointly by LISA and some representative EM telescopes. We combine state-of-the-art astrophysical models for the galaxies formation and evolution to build the MBHBs catalogues, with Bayesian tools to estimate the binary sky position uncertainty from the GW signal. Exploiting additional information from the astrophysical models, such as the amount of accreted gas and the BH spins, we evaluate the expected EM emission in the soft X-ray, optical and radio bands. Overall, we predict between 7 and 21 EMcps in 4 yrs of joint observations by LISA and the considered EM facilities, depending on the astrophysical model. We also explore the impact of the hydrogen and dust obscuration of the optical and X-ray emissions, as well as of the collimation of the radio emission: these effects reduce the number to EMcps to 2 or 3, depending on the astrophysical model, again in 4 yrs of observations. Most of the EMcps are characterised by faint EM emission, challenging the observational capabilities of future telescopes. Finally, we also find that systems with multi-modal sky position posterior distributions represent only a minority of cases and do not affect significantly the number of EMcps.Comment: 28 pages, 18 figures. Submitted to PR

Constraining the evolution of Newton's constant with slow inspirals observed from spaceborne gravitational-wave detectors

Spaceborne gravitational-wave (GW) detectors observing at milli-Hz and deci-Hz frequencies are expected to detect large numbers of quasi-monochromatic signals. The first and second time-derivative of the GW frequency ($\dot f_0$ and $\ddot f_0$) can be measured for the most favourable sources and used to look for negative post-Newtonian corrections, which can be induced by the source's environment or modifications of general relativity. We present an analytical, Fisher-matrix-based approach to estimate how precisely such corrections can be constrained. We use this method to estimate the bounds attainable on the time evolution of the gravitational constant $G(t)$ with different classes of quasi-monochromatic sources observable with LISA and DECIGO, two representative spaceborne detectors for milli-Hz and deci-Hz GW frequencies. We find that the most constraining source among a simulated population of LISA galactic binaries could yield $\dot G/G_0 \lesssim 10^{-6}\text{yr}^{-1}$, while the best currently known verification binary will reach $\dot G/G_0 \lesssim 10^{-4}\text{yr}^{-1}$. We also perform Monte-Carlo simulations using quasi-monochromatic waveforms to check the validity of our Fisher-matrix approach, as well as inspiralling waveforms to analyse binaries that do not satisfy the quasi-monochromatic assumption. We find that our analytical Fisher matrix produces good order-of-magnitude constraints even for sources well beyond its regime of validity. Monte-Carlo investigations also show that chirping stellar-mass compact binaries detected by DECIGO-like detectors at cosmological distances of tens of Mpc can yield constraints as tight as $\dot G/G_0 \lesssim 10^{-11}\text{yr}^{-1}$.Comment: 10 pages, 3 figure

Constraining the evolution of Newton's constant with slow inspirals observed from spaceborne gravitational-wave detectors

Spaceborne gravitational-wave (GW) detectors observing at milli-Hz and deci-Hz frequencies are expected to detect large numbers of quasi-monochromatic signals. The first and second time-derivative of the GW frequency ($\dot f_0$ and $\ddot f_0$) can be measured for the most favourable sources and used to look for negative post-Newtonian corrections, which can be induced by the source's environment or modifications of general relativity. We present an analytical, Fisher-matrix-based approach to estimate how precisely such corrections can be constrained. We use this method to estimate the bounds attainable on the time evolution of the gravitational constant $G(t)$ with different classes of quasi-monochromatic sources observable with LISA and DECIGO, two representative spaceborne detectors for milli-Hz and deci-Hz GW frequencies. We find that the most constraining source among a simulated population of LISA galactic binaries could yield $\dot G/G_0 \lesssim 10^{-6}\text{yr}^{-1}$, while the best currently known verification binary will reach $\dot G/G_0 \lesssim 10^{-4}\text{yr}^{-1}$. We also perform Monte-Carlo simulations using quasi-monochromatic waveforms to check the validity of our Fisher-matrix approach, as well as inspiralling waveforms to analyse binaries that do not satisfy the quasi-monochromatic assumption. We find that our analytical Fisher matrix produces good order-of-magnitude constraints even for sources well beyond its regime of validity. Monte-Carlo investigations also show that chirping stellar-mass compact binaries detected by DECIGO-like detectors at cosmological distances of tens of Mpc can yield constraints as tight as $\dot G/G_0 \lesssim 10^{-11}\text{yr}^{-1}$