About the propagation of the Gravitational Waves in an asymptotically de-Sitter space: Comparing two points of view

We analyze the propagation of gravitational waves (GWs) in an asymptotically de-Sitter space by expanding the perturbation around Minkowski and introducing the effects of the Cosmological Constant ($\Lambda$), first as an additional source (de-Donder gauge) and after as a gauge effect ($\Lambda$-gauge). In both cases the inclusion of the Cosmological Constant $\Lambda$ impedes the detection of a gravitational wave at a distance larger than $L_{crit}=(6\sqrt{2}\pi f \hat{h}/\sqrt{5})r_\Lambda^2$, where $r_\Lambda=\frac{1}{\sqrt{\Lambda}}$ and f and $\hat{h}$ are the frequency and strain of the wave respectively. We demonstrate that $L_{crit}$ is just a confirmation of the Cosmic No hair Conjecture (CNC) already explained in the literature.Comment: Accepted for publication in MPL

Newton's laws of motion in form of Riccati equation

We discuss two applications of Riccati equation to Newton's laws of motion. The first one is the motion of a particle under the influence of a power law central potential V(r)=k r^{\epsilon}. For zero total energy we show that the equation of motion can be cast in the Riccati form. We briefly show here an analogy to barotropic Friedmann-Robertson-Lemaitre cosmology where the expansion of the universe can be also shown to obey a Riccati equation. A second application in classical mechanics, where again the Riccati equation appears naturally, are problems involving quadratic friction. We use methods reminiscent to nonrelativistic supersymmetry to generalize and solve such problem

An approach to anomalous diffusion in the n-dimensional space generated by a self-similar Laplacian

We analyze a quasi-continuous linear chain with self-similar distribution of harmonic interparticle springs as recently introduced for one dimension (Michelitsch et al., Phys. Rev. E 80, 011135 (2009)). We define a continuum limit for one dimension and generalize it to $n=1,2,3,..$ dimensions of the physical space. Application of Hamilton's (variational) principle defines then a self-similar and as consequence non-local Laplacian operator for the $n$-dimensional space where we proof its ellipticity and its accordance (up to a strictly positive prefactor) with the fractional Laplacian $-(-\Delta)^\frac{\alpha}{2}$. By employing this Laplacian we establish a Fokker Planck diffusion equation: We show that this Laplacian generates spatially isotropic L\'evi stable distributions which correspond to L\'evi flights in $n$-dimensions. In the limit of large scaled times $\sim t/r^{\alpha} >>1$ the obtained distributions exhibit an algebraic decay $\sim t^{-\frac{n}{\alpha}} \rightarrow 0$ independent from the initial distribution and spacepoint. This universal scaling depends only on the ratio $n/\alpha$ of the dimension $n$ of the physical space and the L\'evi parameter $\alpha$.Comment: Submitted manuscrip