We study the infrared scale expansion of single field cosmological models using the Hamilton-Jacobi formalism, showing that its specialization at unit scale parameter recovers the slow roll expansion. In particular, we show that the latter coincides with a Laurent expansion of the Hamilton-Jacobi function in powers of the Planck mass, whose terms are controlled by certain recursively-defined polynomials. This allows us to give an explicit recursion procedure for constructing all higher order terms of the slow roll expansion. We also discuss the corresponding effective potential and the action of the universal similarity group.

Romania

Calin Iuliu Lazaroiu

On the slow roll expansion of one-field cosmological models

We study the scaling behavior of classical multifield cosmological models with complete scalar manifold <math><mo>(</mo><mi>M</mi><mo>,</mo><mi>G</mi><mo>)</mo></math> and positive smooth scalar potential Φ, introducing a dynamical renormalization group action which relates their UV and IR limits. We show that the RG flow of such models interpolates between a modification of the geodesic flow of <math><mo>(</mo><mi>M</mi><mo>,</mo><mi>G</mi><mo>)</mo></math> (obtained in the UV limit) and the gradient flow of <math><mo>(</mo><mi>M</mi><mo>,</mo><mi>G</mi><mo>,</mo><mi>V</mi><mo>)</mo></math> (obtained in the IR limit), where the classical effective potential V is proportional to <math><msqrt><mrow><mn>2</mn><mi>Φ</mi></mrow></msqrt></math>. Using this fact, we show that two-field models whose scalar manifold has constant Gaussian curvature equal to −1, 0 or 1 are infrared universal in the sense that they suffice to describe the first order IR approximants of cosmological orbits for all two-field models with positive smooth scalar potential.

Dynamical renormalization and universality in classical multifield cosmological models

We study the first order infared behavior of tame hyperbolizable two-field cosmological models, defined as those classical two-field models whose scalar manifold is a connected, oriented and topologically finite hyperbolizable Riemann surface <math><mo>(</mo><mi>Σ</mi><mo>,</mo><mi>G</mi><mo>)</mo></math> and whose scalar potential Φ admits a positive and Morse extension to the end compactification of Σ. We achieve this by determining the universal forms of the asymptotic gradient flow of the classical effective potential V with respect to the uniformizing metric G near all interior critical points and ends of Σ, finding that some of the latter act like fictitious but exotic stationary points of the gradient flow. We also compare these results with numerical studies of cosmological orbits. For critical cusp ends, we find that cosmological curves have transient quasiperiodic behavior but are eventually attracted or repelled by the cusp along principal geodesic orbits determined by the extended effective potential. This behavior is approximated in the infrared by that of gradient flow curves near the cusp.

Elena Mirela Babalic

The infrared behavior of tame two-field cosmological models

We study generalized two-field α-attractor models whose rescaled scalar manifold is the triply-punctured sphere endowed with its complete hyperbolic metric, whose underlying complex manifold is the modular curve <math><mi>Y</mi><mo>(</mo><mn>2</mn><mo>)</mo></math>. Using an explicit embedding into the end compactification, we compute solutions of the cosmological evolution equations for a few globally well-behaved scalar potentials, displaying particular trajectories with inflationary behavior as well as more general cosmological trajectories of surprising complexity. In such models, the orientation-preserving isometry group of the scalar manifold is isomorphic with the permutation group on three elements, acting on <math><mi>Y</mi><mo>(</mo><mn>2</mn><mo>)</mo></math> as the group of anharmonic transformations. When the scalar potential is preserved by this action, α-attractor models of this type provide a geometric description of two-field “modular invariant j-models” in terms of gravity coupled to a non-linear sigma model with topologically non-trivial target and with a finite (as opposed to discrete but infinite) group of symmetries. The precise relation between the two perspectives is provided by the elliptic modular function λ, which can be viewed as a field redefinition that eliminates almost all of the countably infinite unphysical ambiguity present in the Poincaré half-plane description of such models.

South Korea

Generalized two-field -attractor Models From the Hyperbolic Triply-punctured Sphere

We study “constrained generalized Killing (s)pinors,” which characterize supersymmetric flux compactifications of supergravity theories. Using geometric algebra techniques, we give conceptually clear and computationally effective methods for translating supersymmetry conditions into differential and algebraic constraints on collections of differential forms. In particular, we give a synthetic description of Fierz identities, which are an important ingredient of such problems. As an application, we show how our approach can be used to efficiently treat <math id="M1"><mi>N</mi><mo>=</mo><mn>1</mn></math> compactification of M-theory on eight manifolds and prove that we recover results previously obtained in the literature.