This article is meant as a summary and introduction to the ideas of effective field theory as applied to gravitational systems, ideas which provide the theoretical foundations for the modern use of general relativity as a theory from which precise predictions are possible.

We investigate the possibility of self-tuning of the effective 4D cosmological constant in 6D supergravity, to see whether it could naturally be of order 1/r 4 when compactified on two dimensions having Kaluza-Klein masses of order 1/r. In the models we examine supersymmetry is broken by the presence of nonsupersymmetric 3-branes (on one of which we live). If r were sub-millimeter in size, such a cosmological constant could describe the recently-discovered dark energy. A successful self-tuning mechanism would therefore predict a connection between the observed size of the cosmological constant, and potentially observable effects in submillimeter tests of gravity and at the Large Hadron Collider. We do find self tuning inasmuch as 3-branes can quite generically remain classically flat regardless of the size of their tensions, due to an automatic cancellation with the curvature and dilaton of the transverse two dimensions. We argue that in some circumstances six-dimensional supersymmetry might help suppress quantum corrections to this cancellation down to the bulk supersymmetry-breaking scale, which is of order 1/r. We finally examine an explicit realization of the mechanism, in which 3-branes are inserted into an anomaly-free version of Salam-Sezgin gauged 6D supergravity compactified on a 2sphere with nonzero magnetic flux. This realization is only partially successful due to a topological constraint which relates bulk couplings to the brane tension, although we give arguments why these relations may be stable against quantum corrections.

We use the power-counting formalism of effective field theory to study the size of loop corrections in theories of slow-roll inflation, with the aim of more precisely identifying the limits of validity of the usual classical inflationary treatments. We keep our analysis as general as possible in order to systematically identify the most important corrections to the classical inflaton dynamics. Although most slow-roll models lie within the semiclassical domain, we find the consistency of the Higgs-Inflaton scenario to be more delicate due to the proximity between the Hubble scale during inflation and the upper bound allowed by unitarity on the new-physics scale associated with the breakdown of the semiclassical approximation within the effective theory. Similar remarks apply to curvature-squared inflationary models.

We introduce a simple string model of inflation, in which the inflaton field can take trans-Planckian values while driving a period of slow-roll inflation. This leads naturally to a realisation of large field inflation, inasmuch as the inflationary epoch is well described by the single-field scalar potential V = V 0 3 − 4e −φ/ √ 3 . Remarkably, for a broad class of vacua all adjustable parameters enter only through the overall coefficient V 0 , and in particular do not enter into the slow-roll parameters. Consequently these are determined purely by the number of e -foldings, N e , and so are not independent: ε ≃ 3 2 η 2 . This implies similar relations among observables like the primordial scalar-to-tensor amplitude, r, and the scalar spectral tilt, n s : r ≃ 6(n s − 1) 2 . N e is itself more model-dependent since it depends partly on the post-inflationary reheat history. In a simple reheating scenario a reheating temperature of T rh ≃ 10 9 GeV gives N e ≃ 58, corresponding to n s ≃ 0.970 and r ≃ 0.005, within reach of future observations. The model is an example of a class that arises naturally in the context of type IIB string compactifications with large-volume moduli stabilisation, and takes advantage of the generic existence there of Kähler moduli whose dominant appearance in the scalar potential arises from string loop corrections to the Kähler potential. The inflaton field is a combination of Kähler moduli of a K3-fibered Calabi-Yau manifold. We believe there are likely to be a great number of models in this class -'high-fibre models' -in which the inflaton starts off far enough up the fibre to produce observably large primordial gravity waves.

We present warped compactification solutions of six-dimensional supergravity, which are generalizations of the Randall-Sundrum (RS) warped brane world to codimension two and to a supersymmetric context. In these solutions the dilaton varies over the extra dimensions, and this makes the electroweak hierarchy only power-law sensitive to the proper radius of the extra dimensions (as opposed to being exponentially sensitive as in the RS model). Warping changes the phenomenology of these models because the Kaluza-Klein gap can be much larger than the internal space's inverse proper radius. We provide examples both for Romans' nonchiral supergravity and Salam-Sezgin chiral supergravity, and in both cases the solutions break all of the supersymmetries of the models. We interpret the solution as describing the fields sourced by a 3-brane and a boundary 4-brane (Romans' supergravity) or by one or two 3-branes (Salam-Sezgin supergravity), and we identify the topological constraints which are required by this interpretation. For both types of solutions the 3-branes are flat for all topologically-allowed values of the brane tensions. We identify the general mechanism for and limitations of the self-tuning of the effective 4D cosmological constant in higher-dimensional supergravity which these models illustrate.Keywords: brane-world, supergravity, warped geometries. 42 7. Appendix: Explicit Solution with λ 4 = −ζ 4 = 1 2 44 -1 - 4 We are outlining the calculation for the case where the vacuum expectation value of the gauge field lies in the U (1) subgroup, but the result is the same in the case that it lies in SU (2). 5 If b 2 = 0, such as when A = 0, then instead one finds r 2 3 = B/b 1 .

This review summarizes Effective Field Theory techniques, which are the modern theoretical tools for exploiting the existence of hierarchies of scale in a physical problem. The general theoretical framework is described, and explicitly evaluated for a simple model. Powercounting results are illustrated for a few cases of practical interest, and several applications to Quantum Electrodynamics are described.

We reexamine recent claims that Einstein-frame scattering in the Higgs inflation model is unitary above the cut-off energy Λ ≃ M p /ξ. We show explicitly how unitarity problems arise in both the Einstein and Jordan frames of the theory. In a covariant gauge they arise from non-minimal Higgs self-couplings, which cannot be removed by field redefinitions because the target space is not flat. In unitary gauge, where there is only a single scalar which can be redefined to achieve canonical kinetic terms, the unitarity problems arise through non-minimal Higgs-gauge couplings.

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