Characteristic Evolution and Matching

I review the development of numerical evolution codes for general relativity based upon the characteristic initial value problem. Progress in characteristic evolution is traced from the early stage of 1D feasibility studies to 2D axisymmetric codes that accurately simulate the oscillations and gravitational collapse of relativistic stars and to current 3D codes that provide pieces of a binary black hole spacetime. Cauchy codes have now been successful at simulating all aspects of the binary black hole problem inside an artificially constructed outer boundary. A prime application of characteristic evolution is to extend such simulations to null infinity where the waveform from the binary inspiral and merger can be unambiguously computed. This has now been accomplished by Cauchy-characteristic extraction, where data for the characteristic evolution is supplied by Cauchy data on an extraction worldtube inside the artificial outer boundary. The ultimate application of characteristic evolution is to eliminate the role of this outer boundary by constructing a global solution via Cauchy-characteristic matching. Progress in this direction is discussed.Comment: New version to appear in Living Reviews 2012. arXiv admin note: updated version of arXiv:gr-qc/050809

Small-xx resummation from HELL

Small-$x$ logarithmic enhancements arising from high-energy gluon emissions affect both the evolution of collinearly-factorized parton densities and partonic coefficient functions. With the higher collider energy reached by the LHC, the prospect of a future high-energy collider, and the recent deep-inelastic scattering (DIS) results at small-$x$ from HERA, providing phenomenological tools for performing small-$x$ resummation has become of great relevance. In this paper we discuss a framework to perform small-$x$ resummation for both parton evolution and partonic coefficient functions and we describe its implementation in a computer code named High-Energy Large Logarithms (HELL). We present resummed and matched results for the DGLAP splitting functions and, as a proof of principle, for the massless structure functions in DIS.Comment: Version accepted by EPJ C. 26 pages, 7 figures. Section 2.4 largely re-written. Added estimate of theoretical uncertainty and comparison to CCS

Correlation functions in a cascading N=1 gauge theory from supergravity

We study fluctuations around the warped conifold supergravity solution of Klebanov and Tseytlin [hep-th/0002159], known to be dual to a cascading N=1 gauge theory. Although this supergravity background is not asymptotically AdS, corresponding to a non-conformal field theory, it is possible to apply the usual methods of AdS/CFT duality to extract the high energy behavior of field theory correlators by solving linearized equations of motion for fluctuations around the background. We consider the Goldstone vector dual to the anomalous R-symmetry current and compute its mass, which exactly matches the general prediction of [hep-th/0009156]. We find the high energy 2-point functions for the R-current and two other vectors. As expected, the R-current 2-point function has a longitudinal part because R-symmetry is broken. We also calculate the high energy 2-point function of the energy-momentum tensor from fluctuations of modes in the graviton sector. This 2-point function has a trace part corresponding to broken conformal symmetry.Comment: JHEP, 29 page

Scaling and superscaling solutions from the functional renormalization group

We study the renormalization group flow of $\mathbb{Z}_2$-invariant supersymmetric and non-supersymmetric scalar models in the local potential approximation using functional renormalization group methods. We focus our attention to the fixed points of the renormalization group flow of these models, which emerge as scaling solutions. In two dimensions these solutions are interpreted as the minimal (supersymmetric) models of conformal field theory, while in three dimension they are manifestations of the Wilson-Fisher universality class and its supersymmetric counterpart. We also study the analytically continued flow in fractal dimensions between 2 and 4 and determine the critical dimensions for which irrelevant operators become relevant and change the universality class of the scaling solution. We also include novel analytic and numerical investigations of the properties that determine the occurrence of the scaling solutions within the method. For each solution we offer new techniques to compute the spectrum of the deformations and obtain the corresponding critical exponents.Comment: 23 pages, 14 figures; v2: several improvements, new references, version to appear in PR

Cosmological Perturbations in a Big Crunch/Big Bang Space-time

A prescription is developed for matching general relativistic perturbations across singularities of the type encountered in the ekpyrotic and cyclic scenarios i.e. a collision between orbifold planes. We show that there exists a gauge in which the evolution of perturbations is locally identical to that in a model space-time (compactified Milne mod Z_2) where the matching of modes across the singularity can be treated using a prescription previously introduced by two of us. Using this approach, we show that long wavelength, scale-invariant, growing-mode perturbations in the incoming state pass through the collision and become scale-invariant growing-mode perturbations in the expanding hot big bang phase.Comment: 47 pages, 4 figure

Holography of AdS vacuum bubbles

We consider the fate of AdS vacua connected by tunneling events. A precise holographic dual of thin-walled Coleman--de Luccia bounces is proposed in terms of Fubini instantons in an unstable CFT. This proposal is backed by several qualitative and quantitative checks, including the precise calculation of the instanton action appearing in evaluating the decay rate. Big crunches manifest themselves as time dependent processes which reach the boundary of field space in a finite time. The infinite energy difference involved is identified on the boundary and highlights the ill-defined nature of the bulk setup. We propose a qualitative scenario in which the crunch is resolved by stabilizing the CFT, so that all attempts at crunching always end up shielded from the boundary by the formation of black hole horizons. In all these well defined bulk processes the configurations have the same asymptotics and are finite energy excitations.Comment: version submitted to journal. Note added referring to previous work on holographic instantons

Numerical Relativity: A review

Computer simulations are enabling researchers to investigate systems which are extremely difficult to handle analytically. In the particular case of General Relativity, numerical models have proved extremely valuable for investigations of strong field scenarios and been crucial to reveal unexpected phenomena. Considerable efforts are being spent to simulate astrophysically relevant simulations, understand different aspects of the theory and even provide insights in the search for a quantum theory of gravity. In the present article I review the present status of the field of Numerical Relativity, describe the techniques most commonly used and discuss open problems and (some) future prospects.Comment: 2 References added; 1 corrected. 67 pages. To appear in Classical and Quantum Gravity. (uses iopart.cls

Summing Divergent Perturbative Series in a Strong Coupling Limit. The Gell-Mann - Low Function of the \phi^4 Theory

An algorithm is proposed for determining asymptotics of the sum of a perturbative series in the strong coupling limit using given values of the expansion coefficients. Operation of the algorithm is illustrated by test examples, method for estimating errors is developed, and an optimization procedure is described. Application of the algorithm to the $\phi^4$ theory gives a behavior $\beta(g)\approx 7.4 g^{0.96}$ at large $g$ for its Gell-Mann -- Low function. The fact that the exponent is close to unity can be interpreted as a manifestation of the logarithmic branching of the type $\beta(g)\sim g (\ln g)^{-\gamma}$ (with $\gamma\approx 0.14$), which is confirmed by independent evidence. In any case, the $\phi^4$ theory is internally consistent. The procedure of summing perturbartive series with arbitrary values of expansion parameter is discussed.Comment: 23 pages, PD