On the Rabinowitz Floer homology of twisted cotangent bundles

Consider the cotangent bundle of a Riemannian manifold $(M,g)$ of dimension 2 or more, endowed with a twisted symplectic structure defined by a closed weakly exact 2-form $\sigma$ on $M$ whose lift to the universal cover of $M$ admits a bounded primitive. We compute the Rabinowitz Floer homology of energy hypersurfaces $\Sigma_{k}=H^{-1}(k)$ of mechanical (kinetic energy + potential) Hamiltonians $H$ for the case when the energy value k is greater than the Mane critical value c. Under the stronger condition that k>c_{0}, where c_{0} denotes the strict Mane critical value, Abbondandolo and Schwarz recently computed the Rabinowitz Floer homology of such hypersurfaces, by means of a short exact sequence of chain complexes involving the Rabinowitz Floer chain complex and the Morse (co)chain complex associated to the free time action functional. We extend their results to the weaker case k>c, thus covering cases where $\sigma$ is not exact. As a consequence, we deduce that the hypersurface corresponding to the energy level k is never displaceable for any k>c. Moreover, we prove that if dim M > 1, the homology of the free loop space of $M$ is infinite dimensional, and if the metric is chosen generically, a generic Hamiltonian diffeomorphism has infinitely many leaf-wise intersection points in $\Sigma_{k}$.Comment: V4 - final version, accepted for publication in CVPD

Large Scale Structure Observations

Galaxy Surveys are enjoying a renaissance thanks to the advent of multi-object spectrographs on ground-based telescopes. The last 15 years have seen the fruits of this experimental advance, including the 2-degree Field Galaxy Redshift Survey (2dFGRS; Colless et al. 2003) and the Sloan Digital Sky Survey (SDSS; York et al. 2000). Most recently, the Baryon Oscillation Spectroscopic Survey (BOSS; Dawson et al. 2013), part of the SDSS-III project (Eisenstein et al. 2011), has provided the largest volume of the low-redshift Universe ever surveyed with a galaxy density useful for high-precision cosmology. This set of lecture notes looks at some of the physical processes that underpin these measurements, the evolution of measurements themselves, and looks ahead to the next 15 years and the advent of surveys such as the enhanced Baryon Oscillation Spectroscopic Survey (eBOSS), the Dark Energy Spectroscopic Instrument (DESI) and the ESA Euclid satellite mission.Comment: Lectures given at Post-Planck Cosmology, Ecole de Physique des Houches, Les Houches, July 8-Aug 2, 2013, eds. B. Wandelt, C. Deffayet, P. Peter, to be published by Oxford University Press, and New Horizons for Observational Cosmology, International School of Physics Enrico Fermi, Varenna, July 1-6, 2013, eds. A. Melchiorri, A. Cooray, E. Komatsu, to be published by the Italian Society of Physic

Galaxy 2-Point Covariance Matrix Estimation for Next Generation Surveys

We perform a detailed analysis of the covariance matrix of the spherically averaged galaxy power spectrum and present a new, practical method for estimating this within an arbitrary survey without the need for running mock galaxy simulations that cover the full survey volume. The method uses theoretical arguments to modify the covariance matrix measured from a set of small-volume cubic galaxy simulations, which are computationally cheap to produce compared to larger simulations and match the measured small-scale galaxy clustering more accurately than is possible using theoretical modelling. We include prescriptions to analytically account for the window function of the survey, which convolves the measured covariance matrix in a non-trivial way. We also present a new method to include the effects of supersample covariance and modes outside the small simulation volume which requires no additional simulations and still allows us to scale the covariance matrix. As validation, we compare the covariance matrix estimated using our new method to that from a brute force calculation using 500 simulations originally created for analysis of the Sloan Digital Sky Survey Main Galaxy Sample (SDSS-MGS). We find excellent agreement on all scales of interest for large scale structure analysis, including those dominated by the effects of the survey window, and on scales where theoretical models of the clustering normally break-down, but the new method produces a covariance matrix with significantly better signal-to-noise. Although only formally correct in real-space, we also discuss how our method can be extended to incorporate the effects of Redshift Space Distortions.Comment: 18 pages, 9 figures. Accepted for publication in MNRAS. Added new references to introduction and slightly updated text accordingl

Quantum revivals and carpets in some exactly solvable systems

We consider the revival properties of quantum systems with an eigenspectrum E_{n} proportional to n^{2}, and compare them with the simplest member of this class - the infinite square well. In addition to having perfect revivals at integer multiples of the revival time t_{R}, these systems all enjoy perfect fractional revivals at quarterly intervals of t_{R}. A closer examination of the quantum evolution is performed for the Poeschel-Teller and Rosen-Morse potentials, and comparison is made with the infinite square well using quantum carpets.Comment: 5 pages, 5 figures (1 new), minor additions, to appear in J. Phys.

Unbiased clustering estimation in the presence of missing observations

In order to be efficient, spectroscopic galaxy redshift surveys do not obtain redshifts for all galaxies in the population targeted. The missing galaxies are often clustered, commonly leading to a lower proportion of successful observations in dense regions. One example is the close-pair issue for SDSS spectroscopic galaxy surveys, which have a deficit of pairs of observed galaxies with angular separation closer than the hardware limit on placing neighbouring fibers. Spatially clustered missing observations will exist in the next generations of surveys. Various schemes have previously been suggested to mitigate these effects, but none works for all situations. We argue that the solution is to link the missing galaxies to those observed with statistically equivalent clustering properties, and that the best way to do this is to rerun the targeting algorithm, varying the angular position of the observations. Provided that every pair has a non-zero probability of being observed in one realisation of the algorithm, then a pair-upweighting scheme linking targets to successful observations, can correct these issues. We present such a scheme, and demonstrate its validity using realisations of an idealised simple survey strategy.Comment: 14 pages, 8 figures, published in MNRA

An accurate linear model for redshift space distortions in the void-galaxy correlation function

Redshift space distortions within voids provide a unique method to test for environmental dependence of the growth rate of structures in low density regions, where effects of modified gravity theories might be important. We derive a linear theory model for the redshift space void-galaxy correlation that is valid at all pair separations, including deep within the void, and use this to obtain expressions for the monopole $\xi^s_0$ and quadrupole $\xi^s_2$ contributions. Our derivation highlights terms that have previously been neglected but are important within the void interior. As a result our model differs from previous works and predicts new physical effects, including a change in the sign of the quadrupole term within the void radius. We show how the model can be generalised to include a velocity dispersion. We compare our model predictions to measurements of the correlation function using mock void and galaxy catalogues modelled after the BOSS CMASS galaxy sample using the Big MultiDark $N$-body simulation, and show that the linear model with dispersion provides an excellent fit to the data at all scales, $0\leq s\leq120\;h^{-1}$Mpc. While the RSD model matches simulations, the linear bias approximation does not hold within voids, and care is needed in fitting for the growth rate. We show that fits to the redshift space correlation recover the growth rate $f(z=0.52)$ to a precision of $2.7\%$ using the simulation volume of $(2.5\;h^{-1}\mathrm{Gpc})^3$.Comment: 16 pages, 12 figures. v3: updated to match version published in MNRAS. Several minor changes to text for better explanations, with reference to subsequent results (arXiv:1805.09349). No changes to theory, results or conclusion

Using correlations between CMB lensing and large-scale structure to measure primordial non-Gaussianity

We apply a new method to measure primordial non-Gaussianity, using the cross-correlation between galaxy surveys and the CMB lensing signal to measure galaxy bias on very large scales, where local-type primordial non-Gaussianity predicts a $k^2$ divergence. We use the CMB lensing map recently published by the Planck collaboration, and measure its external correlations with a suite of six galaxy catalogues spanning a broad redshift range. We then consistently combine correlation functions to extend the recent analysis by Giannantonio et al. (2013), where the density-density and the density-CMB temperature correlations were used. Due to the intrinsic noise of the Planck lensing map, which affects the largest scales most severely, we find that the constraints on the galaxy bias are similar to the constraints from density-CMB temperature correlations. Including lensing constraints only improves the previous statistical measurement errors marginally, and we obtain $f_{\mathrm{NL}} = 12 \pm 21$ (1$\sigma$) from the combined data set. However, the lensing measurements serve as an excellent test of systematic errors: we now have three methods to measure the large-scale, scale-dependent bias from a galaxy survey: auto-correlation, and cross-correlation with both CMB temperature and lensing. As the publicly available Planck lensing maps have had their largest-scale modes at multipoles $l<10$ removed, which are the most sensitive to the scale-dependent bias, we consider mock CMB lensing data covering all multipoles. We find that, while the effect of $f_{\mathrm{NL}}$ indeed increases significantly on the largest scales, so do the contributions of both cosmic variance and the intrinsic lensing noise, so that the improvement is small.Comment: 5 pages, 3 figures. Additional references added. Submitted to MNRA

Post-Newtonian constraints on f(R) cosmologies in metric formalism

We compute the complete post-Newtonian limit of the metric form of f(R) gravities using a scalar-tensor representation. By comparing the predictions of these theories with laboratory and solar system experiments, we find a set of inequalities that any lagrangian f(R) must satisfy. The constraints imposed by those inequalities allow us to find explicit bounds to the possible nonlinear terms of the lagrangian. We conclude that the lagrangian f(R) must be almost linear in R and that corrections that grow at low curvatures are incompatible with observations. This result shows that modifications of gravity at very low cosmic densities cannot be responsible for the observed cosmic speed-up.Comment: 10 pages, no figures, revtex