Fractional Lindstedt series

The parametric equations of the surfaces on which highly resonant quasi-periodic motions develop (lower-dimensional tori) cannot be analytically continued, in general, in the perturbation parameter, i.e. they are not analytic functions of the perturbation parameter. However rather generally quasi-periodic motions whose frequencies satisfy only one rational relation ("resonances of order 1") admit formal perturbation expansions in terms of a fractional power of the perturbation parameter, depending on the degeneration of the resonance. We find conditions for this to happen, and in such a case we prove that the formal expansion is convergent after suitable resummation.Comment: 40 pages, 6 figure

Melnikov's approximation dominance. Some examples

We continue a previous paper to show that Mel'nikov's first order formula for part of the separatrix splitting of a pendulum under fast quasi periodic forcing holds, in special examples, as an asymptotic formula in the forcing rapidity.Comment: 46 Kb; 9 pages, plain Te

Resummation of perturbation series and reducibility for Bryuno skew-product flows

We consider skew-product systems on T^d x SL(2,R) for Bryuno base flows close to constant coefficients, depending on a parameter, in any dimension d, and we prove reducibility for a large measure set of values of the parameter. The proof is based on a resummation procedure of the formal power series for the conjugation, and uses techniques of renormalisation group in quantum field theory.Comment: 30 pages, 12 figure

Summation of divergent series and Borel summability for strongly dissipative equations with periodic or quasi-periodic forcing terms

We consider a class of second order ordinary differential equations describing one-dimensional systems with a quasi-periodic analytic forcing term and in the presence of damping. As a physical application one can think of a resistor-inductor-varactor circuit with a periodic (or quasi-periodic) forcing function, even if the range of applicability of the theory is much wider. In the limit of large damping we look for quasi-periodic solutions which have the same frequency vector of the forcing term, and we study their analyticity properties in the inverse of the damping coefficient. We find that already the case of periodic forcing terms is non-trivial, as the solution is not analytic in a neighbourhood of the origin: it turns out to be Borel-summable. In the case of quasi-periodic forcing terms we need Renormalization Group techniques in order to control the small divisors arising in the perturbation series. We show the existence of a summation criterion of the series in this case also, but, however, this can not be interpreted as Borel summability.Comment: 24 pages, 16 figure

Stability for quasi-periodically perturbed Hill's equations

We consider a perturbed Hill's equation of the form $\ddot \phi + (p_{0}(t) + \epsilon p_{1}(t)) \phi = 0$, where $p_{0}$ is real analytic and periodic, $p_{1}$ is real analytic and quasi-periodic and \eps is a ``small'' real parameter. Assuming Diophantine conditions on the frequencies of the decoupled system, i.e. the frequencies of the external potentials $p_{0}$ and $p_{1}$ and the proper frequency of the unperturbed ($\epsilon=0$) Hill's equation, but without making non-degeneracy assumptions on the perturbing potential $p_{1}$, we prove that quasi-periodic solutions of the unperturbed equation can be continued into quasi-periodic solutions if $\epsilon$ lies in a Cantor set of relatively large measure in $[-\epsilon_0,\epsilon_0]$, where $\epsilon_0$ is small enough. Our method is based on a resummation procedure of a formal Lindstedt series obtained as a solution of a generalized Riccati equation associated to Hill's problem.Comment: 40 pages, 4 figure

Interpolating point spread function anisotropy

Planned wide-field weak lensing surveys are expected to reduce the statistical errors on the shear field to unprecedented levels. In contrast, systematic errors like those induced by the convolution with the point spread function (PSF) will not benefit from that scaling effect and will require very accurate modeling and correction. While numerous methods have been devised to carry out the PSF correction itself, modeling of the PSF shape and its spatial variations across the instrument field of view has, so far, attracted much less attention. This step is nevertheless crucial because the PSF is only known at star positions while the correction has to be performed at any position on the sky. A reliable interpolation scheme is therefore mandatory and a popular approach has been to use low-order bivariate polynomials. In the present paper, we evaluate four other classical spatial interpolation methods based on splines (B-splines), inverse distance weighting (IDW), radial basis functions (RBF) and ordinary Kriging (OK). These methods are tested on the Star-challenge part of the GRavitational lEnsing Accuracy Testing 2010 (GREAT10) simulated data and are compared with the classical polynomial fitting (Polyfit). We also test all our interpolation methods independently of the way the PSF is modeled, by interpolating the GREAT10 star fields themselves (i.e., the PSF parameters are known exactly at star positions). We find in that case RBF to be the clear winner, closely followed by the other local methods, IDW and OK. The global methods, Polyfit and B-splines, are largely behind, especially in fields with (ground-based) turbulent PSFs. In fields with non-turbulent PSFs, all interpolators reach a variance on PSF systematics $\sigma_{sys}^2$ better than the $1\times10^{-7}$ upper bound expected by future space-based surveys, with the local interpolators performing better than the global ones

Testing baryon-induced core formation in Λ\LambdaCDM: A comparison of the DC14 and coreNFW dark matter halo models on galaxy rotation curves

Recent cosmological hydrodynamical simulations suggest that baryonic processes, and in particular supernova feedback after bursts of star formation, can alter the structure of dark matter haloes and transform primordial cusps into shallower cores. To assess whether this mechanism offers a solution to the cusp-core controversy, simulated haloes must be compared to real dark matter haloes inferred from galaxy rotation curves. For this purpose, two new dark matter density profiles were recently derived from simulations of galaxies in complementary mass ranges: the DC14 halo ($10^{10} < M_{\text{halo}}/M_{\odot} < 8 \times 10^{11}$) and the coreNFW halo ($10^{7} < M_{\text{halo}}/M_{\odot} < 10^{9}$). Both models have individually been found to give good fits to observed rotation curves. For the DC14 model, however, the agreement of the predicted halo properties with cosmological scaling relations was confirmed by one study, but strongly refuted by another. A next question is whether the two models converge to the same solution in the mass range where both should be appropriate. To investigate this, we tested the DC14 and cNFW halo models on the rotation curves of a selection of galaxies with halo masses in the range $4 \times 10^{9}$ - $7 \times 10^{10}$ $M_{\odot}$. We further applied the DC14 model to a set of rotation curves at higher halo masses, up to $9 \times 10^{11}$ $M_{\odot}$, to verify the agreement with the cosmological scaling relations. We find that both models are generally able to reproduce the observed rotation curves, in line with earlier results, and the predicted dark matter haloes are consistent with the cosmological $c-M_{\text{halo}}$ and $M_{*}-M_{\text{halo}}$ relations. The DC14 and cNFW models are also in fairly good agreement with each other, even though DC14 tends to predict slightly less extended cores and somewhat more concentrated haloes than cNFW.Comment: 19 pages, 12 figures, accepted for publication in A&

Combined Solar System and rotation curve constraints on MOND

The Modified Newtonian Dynamics (MOND) paradigm generically predicts that the external gravitational field in which a system is embedded can produce effects on its internal dynamics. In this communication, we first show that this External Field Effect can significantly improve some galactic rotation curves fits by decreasing the predicted velocities of the external part of the rotation curves. In modified gravity versions of MOND, this External Field Effect also appears in the Solar System and leads to a very good way to constrain the transition function of the theory. A combined analysis of the galactic rotation curves and Solar System constraints (provided by the Cassini spacecraft) rules out several classes of popular MOND transition functions, but leaves others viable. Moreover, we show that LISA Pathfinder will not be able to improve the current constraints on these still viable transition functions.Comment: 13 pages, 7 figures, accepted for publication in MNRA