A volume-weighted measure for eternal inflation

I propose a new volume-weighted probability measure for cosmological "multiverse" scenarios involving eternal inflation. The "reheating-volume (RV) cutoff" calculates the distribution of observable quantities on a portion of the reheating hypersurface that is conditioned to be finite. The RV measure is gauge-invariant, does not suffer from the "youngness paradox," and is independent of initial conditions at the beginning of inflation. In slow-roll inflationary models with a scalar inflaton, the RV-regulated probability distributions can be obtained by solving nonlinear diffusion equations. I discuss possible applications of the new measure to "landscape" scenarios with bubble nucleation. As an illustration, I compute the predictions of the RV measure in a simple toy landscape.Comment: Version accepted for publication in Phys.Re

Growth of preferential attachment random graphs via continuous-time branching processes

A version of ``preferential attachment'' random graphs, corresponding to linear ``weights'' with random ``edge additions,'' which generalizes some previously considered models, is studied. This graph model is embedded in a continuous-time branching scheme and, using the branching process apparatus, several results on the graph model asymptotics are obtained, some extending previous results, such as growth rates for a typical degree and the maximal degree, behavior of the vertex where the maximal degree is attained, and a law of large numbers for the empirical distribution of degrees which shows certain ``scale-free'' or ``power-law'' behaviors.Comment: 20 page

Infinite dimensional stochastic differential equations of Ornstein-Uhlenbeck type

We consider the operator \sL f(x)=\tfrac12 \sum_{i,j=1}^\infty a_{ij}(x)\frac{\del^2 f}{\del x_i \del x_j}(x)-\sum_{i=1}^\infty \lam_i x_i b_i(x) \frac{\del f}{\del x_i}(x). We prove existence and uniqueness of solutions to the martingale problem for this operator under appropriate conditions on the $a_{ij}, b_i$, and \lam_i. The process corresponding to \sL solves an infinite dimensional stochastic differential equation similar to that for the infinite dimensional Ornstein-Uhlenbeck process

Atomic and Molecular Absorption in Redshifted Radio Sources

We report on a survey for associated HI 21-cm and OH 18-cm absorption with the Giant Metrewave Radio Telescope at redshifts z = 0.2-0.4. Although the low redshift selection ensures that our targets are below the critical ultra-violet luminosity, which is hypothesised to ionise all of the neutral gas in the host galaxy, we do not obtain any detections in the six sources searched. Analysing these in context of the previous surveys, in addition to the anti-correlation with the ultra-violet luminosity (ionising photon rate), we find a correlation between the strength of the absorption and the blue -- near-infrared colour, as well as the radio-band turnover frequency. We believe that these are due to the photo-ionisation of the neutral gas, an obscured sight-line being more conducive to the presence of cold gas and the compact radio emission being better intercepted by the absorbing gas, maximising the flux coverage, respectively. Regarding the photo-ionisation, the compilation of the previous surveys increases the significance of the critical ionising photon rate, above which all of the gas in the host galaxy is hypothesised to be ionised, to >5 sigma. This reaffirms that this is an ubiquitous effect, which has profound implications for the detection of neutral gas in these objects with the Square Kilometre Array.Comment: Accepted by MNRA

Spiraling of approximations and spherical averages of Siegel transforms

This is the author accepted manuscript. The final version is available from the publisher via the DOI in this record.We consider the question of how approximations satisfying Dirichlet’s theorem spiral around vectors in Rd. We give pointwise almost everywhere results (using only the Birkhoff ergodic theorem on the space of lattices). In addition, we show that for every unimodular lattice, on average, the directions of approximates spiral in a uniformly distributed fashion on the d − 1 dimensional unit sphere. For this second result, we adapt a very recent proof of Marklof and Strombergsson [19] to show a spherical average result for Siegel transforms on SLd+1(R)/ SLd+1(Z). Our techniques are elementary. Results like this date back to the work of Eskin-Margulis-Mozes [9] and KleinbockMargulis [14] and have wide-ranging applications. We also explicitly construct examples in which the directions are not uniformly distributedJ.S.A. partially supported by NSF grant DMS 1069153, and NSF grants DMS 1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties” (the GEAR Network). A.G. partially supported by the Royal Society. J.T. acknowledges the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 291147

Ergodic theory and Diophantine approximation for translation surfaces and linear forms

This is the author accepted manuscript. The final version is available from IOP Publishing via the DOI in this record.We derive results on the distribution of directions of saddle connections on translation surfaces using only the Birkhoff ergodic theorem applied to the geodesic flow on the moduli space of translation surfaces. Our techniques, together with an approximation argument, also give an alternative proof of a weak version of a classical theorem in multi-dimensional Diophantine approximation due to Schmidt (1960 Can. J. Math. 12 619–31, 1964 Trans. Am. Math. Soc. 110 493–518). The approximation argument allows us to deduce the Birkhoff genericity of almost all lattices in a certain submanifold of the space of unimodular lattices from the Birkhoff genericity of almost all lattices in the whole space and similarly for the space of affine unimodular lattices.JSA partially supported by NSF grant DMS 1069153, and NSF grants DMS 1107452, 1107263, 1107367 ‘RNMS: GEometric structures And Representation varieties’ (the GEAR Network), and NSF CAREER grant DMS 1351853. JT acknowledges the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 291147 and acknowledges support by the Heilbronn Institute for Mathematical Research