General solution for scalar perturbations in bouncing cosmologies

Bouncing cosmologies, suggested by String/M-theory, may provide an alternative to standard inflation to account for the origin of inhomogeneities in our universe. The fundamental question regards the correct way to evolve the scalar perturbations through the bounce. In this work, we determine the evolution of perturbations and the final spectrum for an arbitrary (spatially flat) bouncing cosmology, with the only assumption that the bounce is governed by a single physical scale. In particular, we find that the spectrum of the pre-bounce growing mode of the Bardeen potential (which is scale-invariant in some limit, and thus compatible with observations) survives unaltered in the post-bounce only if the comoving pressure perturbation is directly proportional to the Bardeen potential rather than its Laplacian, as for any known form of ordinary matter. If some new physics acting at the bounce justifies such relation, then bouncing cosmologies are entitled to become a real viable alternative for the generation of the observed inhomogeneities. Our treatment also includes some class of models with extra-dimensions, whereas we show that bounces induced by positive spatial curvature are structurally different from all bounces in spatially flat universes, requiring a distinct analysis.Comment: 24 pages, 2 figure

Caustics in special multiple lenses

Despite its mathematical complexity, the multiple gravitational lens can be studied in detail in every situation where a perturbative approach is possible. In this paper, we examine the caustics of a system with a lens very far from the others with respect to their Einstein radii, and a system where mutual distances between lenses are small compared to the Einstein radius of the total mass. Finally we review the case of a planetary system adding some new information (area of caustics, duality and higher order terms).Comment: 11 pages, 7 figures, submitted to A &

Regular two-component bouncing cosmologies and perturbations therein

We present a full investigation of scalar perturbations in a rather generic model for a regular bouncing universe, where the bounce is triggered by an effective perfect fluid with negative energy density. Long before and after the bounce the universe is dominated by a source with positive energy density, which may be a perfect fluid, a scalar field, or any other source with an intrinsic isocurvature perturbation. Within this framework, we present an analytical method to accurately estimate the spectrum of large-scale scalar perturbations until their reentry, long after the bounce. We also propose a simple way to identify non-singular gauge-invariant variables through the bounce and present the results of extensive numerical tests in several possible realizations of the scenario. In no case do we find that the spectrum of the pre-bounce growing mode of the Bardeen potential can be transferred to a post-bounce constant mode.Comment: 19 pages, 9 figure

Scalar perturbations in regular two-component bouncing cosmologies

We consider a two-component regular cosmology bouncing from contraction to expansion, where, in order to include both scalar fields and perfect fluids as particular cases, the dominant component is allowed to have an intrinsic isocurvature mode. We show that the spectrum of the growing mode of the Bardeen potential in the pre-bounce is never transferred to the dominant mode of the post-bounce. The latter acquires at most a dominant isocurvature component, depending on the relative properties of the two fluids. Our results imply that several claims in the literature need substantial revision.Comment: 10 pages, 1 figur

A solution to the anisotropy problem in bouncing cosmologies

Bouncing cosmologies are often proposed as alternatives to standard inflation for the explanation of the homogeneity and flatness of the universe. In such scenarios, the present cosmological expansion is preceded by a contraction phase. However, during the contraction, in general the anisotropy of the universe grows and eventually leads to a chaotic mixmaster behavior. This would either be hard to reconcile with observations or even lead to a singularity instead of the bounce. In order to preserve a smooth and isotropic bounce, the source for the contraction must have a super-stiff equation of state with $P/\rho=w>1$. In this letter we propose a new mechanism to solve the anisotropy problem for any low-energy value of $w$ by arguing that high energy physics leads to a modification of the equation of state, with the introduction of non-linear terms. In such a scenario, the anisotropy is strongly suppressed during the high energy phase, allowing for a graceful isotropic bounce, even when the low-energy value of $w$ is smaller than unity.Comment: 9 pages, accepted for publication in JCA

Caustics of 1/rn1/r^n binary gravitational lenses: from galactic haloes to exotic matter

We investigate the caustic topologies for binary gravitational lenses made up of two objects whose gravitational potential declines as $1/r^n$. With $n<1$ this corresponds to power-law dust distributions like the singular isothermal sphere. The $n>1$ regime can be obtained with some violations of the energy conditions, one famous example being the Ellis wormhole. Gravitational lensing provides a natural arena to distinguish and identify such exotic objects in our Universe. We find that there are still three topologies for caustics as in the standard Schwarzschild binary lens, with the main novelty coming from the secondary caustics of the close topology, which become huge at higher $n$. After drawing caustics by numerical methods, we derive a large amount of analytical formulae in all limits that are useful to provide deeper insight in the mathematics of the problem. Our study is useful to better understand the phenomenology of galaxy lensing in clusters as well as the distinct signatures of exotic matter in complex systems.Comment: 28 pages, 19 figures, focus expanded to galactic haloe

Observing gravitational lensing effects by Sgr A* with GRAVITY

The massive black hole at the Galactic center Sgr A* is surrounded by a cluster of stars orbiting around it. Light from these stars is bent by the gravitational field of the black hole, giving rise to several phenomena: astrometric displacement of the primary image, the creation of a secondary image that may shift the centroid of Sgr A*, magnification effects on both images. The near-to-come second generation VLTI instrument GRAVITY will perform observations in the Near Infrared of the Galactic Center at unprecedented resolution, opening the possibility of observing such effects. Here we investigate the observability limits for GRAVITY of gravitational lensing effects on the S-stars in the parameter space [DLS,gamma,K], where DLS is the distance between the lens and the source, gamma is the alignment angle of the source, and K is the source apparent magnitude in the K-band. The easiest effect to be observed in the next years is the astrometric displacement of primary images. In particular the shift of the star S17 from its Keplerian orbit will be detected as soon as GRAVITY becomes operative. For exceptional configurations it will be possible to detect effects related to the spin of the black hole or Post-Newtonian orders in the deflection.Comment: 29 pages, 9 figures, in press on Ap

Gravitational Lensing of stars in the central arcsecond of our Galaxy

In the neighborhood of Sgr A*, several stars (S2, S12, S14, S1, S8, S13) enjoy an accurate determination of their orbital parameters. General Relativity predicts that the central black hole acts as a gravitational lens on these stars, generating a secondary image and two infinite series of relativistic images. For each of these six stars, we calculate the light curves for the secondary and the first two relativistic images, in the Schwarzschild black hole hypothesis, throughout their orbital periods. The curves are peaked around the periapse epoch, but two subpeaks may arise in nearly edge-on orbits, when the source is behind or in front of Sgr A*. We show that for most of these stars the secondary image should be observable during its brightness peak. In particular, S14 is the best candidate, since its secondary image reaches K=23.3 with an angular separation of 0.125 mas from the apparent horizon of the central black hole. The detection of such images by future instruments could represent the first observation of gravitational lensing beyond the weak field approximation.Comment: 28 pages, 9 figures, in press on Ap

Time Delay in Black Hole Gravitational Lensing as a Distance Estimator

We calculate the time delay between different relativistic images formed by black hole gravitational lensing in the strong field limit. For spherically symmetric black holes, it turns out that the time delay between the first two images is proportional to the minimum impact angle. Their ratio gives a very interesting and precise measure of the distance of the black hole. Moreover, using also the separation between the images and their luminosity ratio, it is possible to extract the mass of the black hole. The time delay for the black hole at the center of our Galaxy is just few minutes, but for supermassive black holes with M=10^8 - 10^9 solar masses in the neighbourhood of the Local Group the time delay amounts to few days, thus being measurable with a good accuracy.Comment: 8 pages, 3 figure

Relativistic iron lines in accretion disks: the contribution of higher order images in the strong deflection limit

The shapes of relativistic iron lines observed in spectra of candidate black holes carry the signatures of the strong gravitational fields in which the accretion disks lie. These lines result from the sum of the contributions of all images of the disk created by gravitational lensing, with the direct and first-order images largely dominating the overall shapes. Higher order images created by photons tightly winding around the black holes are often neglected in the modeling of these lines, since they require a substantially higher computational effort. With the help of the strong deflection limit, we present the most accurate semi-analytical calculation of these higher order contributions to the iron lines for Schwarzschild black holes. We show that two regimes exist depending on the inclination of the disk with respect to the line of sight. Many useful analytical formulae can be also derived in this framework.Comment: 23 pages, 13 figure