DefiningkinG(k)

AbstractWe show how the field of definitionkof ak-isotropic absolutely almost simplek-groupG“lives” in the groupG(k) ofk-rational points. The construction which is inspired by the fundamental work of Borel-Tits is as follows: We choose an element inside the center of the unipotent radical of a minimal parabolick-subgroupP; the orbit under the action of the centerZof a Levik-subgroup ofPgenerates a one-dimensional vector space which then carries the additive field structure in a natural way. The multiplicative structure is induced by the action ofZ. IfGisk-simple, our construction yields a finite extensionlofk.As an immediate consequence we obtain an answer to a question of Borovik–Nesin under the additional assumption thatGisk-isotropic:Theorem. IfGis ak-simplek-isotropic group such thatG(k) has finite Morley rank, thenkis either algebraically closed or real closed. IfGis absolutely simplek-isotropic, thenkis algebraically closed

Tree indiscernibilities, revisited

We give definitions that distinguish between two notions of indiscernibility for a set \{a_\eta \mid \eta \in \W\} that saw original use in \cite{sh90}, which we name \textit{\s-} and \textit{\n-indiscernibility}. Using these definitions and detailed proofs, we prove \s- and \n-modeling theorems and give applications of these theorems. In particular, we verify a step in the argument that TP is equivalent to TP$_1$ or TP$_2$ that has not seen explication in the literature. In the Appendix, we exposit the proofs of \citep[{App. 2.6, 2.7}]{sh90}, expanding on the details.Comment: submitte

Non-linear inflationary perturbations

We present a method by which cosmological perturbations can be quantitatively studied in single and multi-field inflationary models beyond linear perturbation theory. A non-linear generalization of the gauge-invariant Sasaki-Mukhanov variables is used in a long-wavelength approximation. These generalized variables remain invariant under time slicing changes on long wavelengths. The equations they obey are relatively simple and can be formulated for a number of time slicing choices. Initial conditions are set after horizon crossing and the subsequent evolution is fully non-linear. We briefly discuss how these methods can be implemented numerically in the study of non-Gaussian signatures from specific inflationary models.Comment: 10 pages, replaced to match JCAP versio

Non-Gaussian perturbations from multi-field inflation

We show how the primordial bispectrum of density perturbations from inflation may be characterised in terms of manifestly gauge-invariant cosmological perturbations at second order. The primordial metric perturbation, zeta, describing the perturbed expansion of uniform-density hypersurfaces on large scales is related to scalar field perturbations on unperturbed (spatially-flat) hypersurfaces at first- and second-order. The bispectrum of the metric perturbation is thus composed of (i) a local contribution due to the second-order gauge-transformation, and (ii) the instrinsic bispectrum of the field perturbations on spatially flat hypersurfaces. We generalise previous results to allow for scale-dependence of the scalar field power spectra and correlations that can develop between fields on super-Hubble scales.Comment: 11 pages, RevTex; minor changes to text; conclusions unchanged; version to appear in JCA

Non-Gaussianity in braneworld and tachyon inflation

We calculate the bispectrum of single-field braneworld inflation, triggered by either an ordinary scalar field or a cosmological tachyon, by means of a gradient expansion of large-scale non-linear perturbations coupled to stochastic dynamics. The resulting effect is identical to that for single-field 4D standard inflation, the non-linearity parameter being proportional to the scalar spectral index in the limit of collapsing momentum. If the slow-roll approximation is assumed, braneworld and tachyon non-Gaussianities are subdominant with respect to the post-inflationary contribution. However, bulk physics may considerably strengthen the non-linear signatures. These features do not change significantly when considered in a non-commutative framework.Comment: 17 pages; v2: added references and previously skipped details in the derivation of the result; v3: improved discussio

Non-Gaussianities in two-field inflation

We study the bispectrum of the curvature perturbation on uniform energy density hypersurfaces in models of inflation with two scalar fields evolving simultaneously. In the case of a separable potential, it is possible to compute the curvature perturbation up to second order in the perturbations, generated on large scales due to the presence of non-adiabatic perturbations, by employing the $\delta N$-formalism, in the slow-roll approximation. In this case, we provide an analytic formula for the nonlinear parameter $f_{NL}$. We apply this formula to double inflation with two massive fields, showing that it does not generate significant non-Gaussianity; the nonlinear parameter at the end of inflation is slow-roll suppressed. Finally, we develop a numerical method for generic two-field models of inflation, which allows us to go beyond the slow-roll approximation and confirms our analytic results for double inflation.Comment: 29 pages, 6 figures. v2, comparison with previous estimates. v3, JCAP version; Revisions based on Referee's comment, corrected typos, added few eqs and refs, conclusions unchange

Multiple-field inflation and the CMB

In this paper, we investigate some consequences of multiple-field inflation for the cosmic microwave background radiation (CMB). We derive expressions for the amplitudes, the spectral indices and the derivatives of the indices of the CMB power spectrum in the context of a very general multiple-field theory of slow-roll inflation, where the field metric can be non-trivial. Both scalar (adiabatic, isocurvature and mixing) and tensor perturbations are treated and the differences with single-field inflation are discussed. From these expressions, several relations are derived that can be used to determine the importance of multiple-field effects observationally from the CMB. We also study the evolution of the total entropy perturbation during radiation and matter domination and the influence of this on the isocurvature spectral quantities.Comment: 24 pages. References added, some very minor textual changes, matches version to be published in CQ

Diagrammatic approach to non-Gaussianity from inflation

We present Feynman type diagrams for calculating the n-point function of the primordial curvature perturbation in terms of scalar field perturbations during inflation. The diagrams can be used to evaluate the corresponding terms in the n-point function at tree level or any required loop level. Rules are presented for drawing the diagrams and writing down the corresponding terms in real space and Fourier space. We show that vertices can be renormalised to automatically account for diagrams with dressed vertices. We apply these rules to calculate the primordial power spectrum up to two loops, the bispectrum including loop corrections, and the trispectrum.Comment: 17 pages, 13 figures. v2: Comments and references added, v3: Introduction expanded, subsection on evaluating loop diagrams added, minor errors corrected, references adde