The concept of attractors, well-known in classical mechanics, proved to be very productive in supergravity, in the theory of black holes and inflationary cosmology. We start with attractors in supersymmetric black holes and discuss also non-BPS black hole attractors. Recently the non-BPS case helped to explain, via enhanced dualitiy symmetry, mysterious cancellation of ultraviolet divergences in 82 Feynman diagrams in 4-loop superamplitude in $N=5$ supergravity. We discuss the implications of these results for the possibility of the all-loop finiteness of $N > 4$ 4D supergravities. We continue with the description of inflationary $\alpha$-attractors. This large class of inflationary models gives predictions that are stable with respect to even very significant modifications of inflationary potentials. These predictions match all presently available CMB-related cosmological data. These models provide targets for the future satellite mission LiteBIRD, which will attempt to detect primordial gravitational waves. We describe some of the recent advanced versions of cosmological attractors which have a beautiful fractal landscape structure. Invited contribution to "Half a century of Supergravity", eds.~A. Ceresole and G.~Dall'Agata (Cambridge Univ. Press, to appear)

We introduce a new family of cosmological attractors with non-minimal coupling of gravity and non-canonical kinetic terms. In the Einstein frame, these models transform into a class of exponential and polynomial attractors with the spectral index n_s spanning a broad range 1-2/N ≤ n_s < 1-1/N, and r can decrease to zero in the limit ξ→∞. This is sufficient to match any combination of Planck, BICEP/Keck, ACT, SPT, and DESI data. We present a supergravity implementation of these models.

We study the impact of spatial curvature (Ω_k) and dynamical dark energy (parametrized by w_0 and w_a) on the spectral index n_s using a combination of cosmic microwave background datasets (Planck, SPT, and ACT), and spectroscopic galaxy samples from DESI, including both BAO and full-shape clustering measurements. We show that a small negative curvature, Ω_k≃ 3× 10^-3, lowers the value of n_s, bringing it closer to predictions of the Starobinsky, Higgs, and simplest α-attractor inflationary models. In particular, we find n_s= 0.9667±0.0041 (using Planck and DESI data) or n_s= 0.9692±0.0035 (adding ACT and SPT). Allowing for time-evolving dark energy also reduces the spectral index, leading to n_s=0.9716±0.0032 (from the combined dataset), or n_s=0.9694±0.0035 in combination with a small negative curvature. Our results demonstrate that the tension between current observational data and the Starobinsky, Higgs, and simplest α-attractor models holds only for ΛCDM, and can be mitigated in extended cosmological models. We discuss implications of these findings for inflationary models in an open universe and/or with dynamical dark energy, including scenarios with quantum tunneling and non-standard topology. Furthermore, we briefly describe a special class of α-attractor models, where one can make n_s arbitrarily large, and we describe the α-attractor quintessence model. Such models may be of particular relevance when future data from DESI, as well as DESI-II, SPHEREx, Euclid, Rubin, and Roman, becomes available.

The scalar spectral index ns is a powerful test of inflationary models. The tightest constraint on ns to date derives from the combination of cosmic microwave background (CMB) data with baryon acoustic oscillation (BAO) data. The resulting ns constraint is shifted significantly upward relative to the constraint from CMB alone, with the consequence that previously preferred inflationary models are seemingly disfavored by >= 26. Here we show that this shift in ns is the combined effect of a degeneracy between ns and BAO parameters exhibited by CMB data and the tension between CMB datasets and the Dark Energy Spectroscopic Instrument BAO data under the assumption of the standard cosmological model. Given the crucial role of ns in discriminating between inflationary models, we urge caution in interpreting CMB + BAO constraints on ns until the BAO-CMB tension is resolved.

The superconformal action can be gauge-fixed in a gauge where it leads to the Einstein frame supergravity defined by a Kähler potential 𝒦(z,z) , or in a gauge where it leads to a Jordan frame supergravity defined by the frame function Ω(z,z) , in addition to 𝒦(z,z) . We present new supergravity ξ-attractor models with non-minimal gravity coupling. They describe potentials with an exponential and a polynomial approach to the plateau. The previously known ξ-attractors in the large ξ limit predicted the spectral index ns = 1 − 2/N and the tensor-to-scalar ratio r→12/N^2 . The new ξ-attractors in the large ξ limit predict ns ≥ 1 − 2/N and r → 0, which provides a better match to the recent observational data and makes these models a target for the future B-mode experiments with small r. New ξ-attractors have some features similar to those of Palatini attractors. However, we show that the Palatini gravity with nonminimal scalar coupling and an independent affine connection has no supergravity embedding.

The textbook N=1 supergravity has an F-term potential depending on a superpotential W (zi) and a Kähler potential K(z^i,z^i) , with the scalar potential V(z^i,z^i) = eK(|DW|2 − 3|W|2). In this approach, it is not always easy to find the potential V(z^i,z^i) with the required properties. We show that in supergravity with a nilpotent superfield and with any Kähler potential K(z^i,z^i) one can obtain any desired potential V(z^i,z^i) by a proper choice of the Kähler metric of the nilpotent superfield. This construction is particularly suitable for cosmological and particle physics applications, which may require maximal freedom in the choice of kinetic terms and scalar potentials.

Hybrid α-attractor models can have significantly greater values of n_s and smaller r, while preserving the relation r≅ 3α(1-n_s)^2, which is valid for exponential T- and E-models at large values of the inflaton field. Here we study single-field α-attractors with features inspired by hybrid models: one can uplift the potential, and one can also have a waterfall regime that leads to a premature termination of inflation near the critical point φ_c. This allows one to increase the effective number of e-foldings N_c in formulas like n_s≃ 1-2 N_c, r≃12 α N_c^2. By changing the waterfall's steepness and location, one can continuously move the predictions along the curves with r≅ 3α(1-n_s)^2 as n_s increases and r decreases. We also study the effect of waterfall insertions and uplift on n_s in quintessential α-attractors that describe inflation and dynamical dark energy.

Inflationary cx-attractor models naturally appear in supergravity with hyperbolic geometry. The simplest versions of cx-attractors, T-and E-models, originate from theories with non-singular potentials. In canonical variables, these potentials have a plateau that is approached exponentially fast at large values of the inflaton field y. In a closely related class of polynomial cx-attractors, or P-models, the potential is not singular, but its derivative is singular at the boundary. The resulting inflaton potential also has a plateau, but it is approached polynomially. In this paper, we will consider a more general class of potentials, which can be singular at the boundary of the moduli space, S-models. These potentials may have a short plateau, after which the potential may grow polynomially or exponentially at large values of the inflaton field. We will show that this class of models may provide a simple solution to the initial conditions problem for cx-attractors and may account for a very broad range of possible values of ns matching the recent ACT, SPT, and DESI data.

We suggest that the surprising one-loop finiteness of 6D half-maximal supergravities recently discovered in [1] might be related to anomaly-free SO(5, 21) G-duality symmetry in (2, 0) supergravity with 21 tensor multiplets and to anomaly-free SO(4, 20) G-duality symmetry of (1, 1) supergravity with 20 vector multiplets. Our argument is reminiscent of the analogous role of the anomaly-free E7(7) G-duality symmetry in the UV finiteness of a maximal 4D supergravity at 3 loops.

We introduce a family of simple α-attractor models that can interpolate between exponential and polynomial cosmological attractors. By varying the interpolation parameter μ in these models, one can scan a wide range of values of the spectral index n_s matching any combination of CMB and DESI data.

Quintessential α-attractor models of single-field inflation and evolving dark energy were constructed about a decade ago. Recently, it was pointed out that some of them might be disfavored due to dark-radiation constraints on gravitational waves and the higher values of n_s favored by ACT. Here we present a class of updated quintessential α-attractor models in which a single field describes both inflation and evolving dark energy, yields higher values of n_s, and admits reheating scenarios consistent with the dark-radiation bound on ΔN_ eff. Depending on the value of the cosmological constant Λ, these models interpolate between ΛCDM with Λ> 0 (future dS universe), dynamical dark energy with Λ= 0 (future Minkowski universe), and dynamical dark energy with Λ< 0 (future cosmological collapse). We also study quintessential α-attractor models based on an axion-inflaton complex scalar field with hyperbolic geometry, which describe inflation and dark energy of a “phantom illusion” type compatible with DESI DR2.

A bstract We review the structure of superconformal anomalies in 4d $$ \mathcal{N} $$ N = 4 conformal supergravity (CSG) coupled to a number N v of $$ \mathcal{N} $$ N = 4 vector multiplets and 6d (2,0) CSG coupled to N T of (2,0) tensor multiplets. Anomalies cancel if N v = 4 and N T = 26 respectively. If the CSG part of the action is dropped and N v = 6 + n v , the first theory is classically equivalent to the 4d $$ \mathcal{N} $$ N = 4 Poincaré supergravity (PSG) coupled to n v vector multiplets, while the second one with N T = 5 + n T is classically equivalent to the 6d (2,0) PSG coupled to n T tensor multiplets. We suggest that these facts imply that divergences in the 4d PSG with n v vectors should be proportional to n v + 2 and similarly in the 6d PSG with n T tensors to n T – 21. This conjecture appears to be consistent with most of the known results of explicit scattering amplitude computations in these 4d and 6d PSG theories, apart from one coefficient in the 3-loop mixed vector scattering amplitude computed in arXiv:1305.4876.

SL(2,ℤ) invariant α -attractor models proposed in [1] have plateau potentials with respect to the inflaton and axion fields. The slope of the potential in the inflaton direction is exponentially suppressed at large values of the inflaton field, but the slope of the potential in the axion direction is double-exponentially suppressed . Therefore, the axion field remains nearly massless and practically does not change during inflation. The inflationary trajectory in such models is stable with respect to quantum fluctuations of the axion field. We show that isocurvature perturbations do not feed into the curvature perturbations during inflation, and discuss the possibility of such transfer at the post-inflationary stage.

We show that the simplest generalization of the chaotic inflation model 1/2m^{2}ϕ^{2} with nonminimal coupling to gravity (1+ϕ)R provides a good match to the results of the latest data release of the Atacama Cosmology Telescope and South Pole Telescope, with r≈10^{-2}.

We give a brief review of the basic principles of inflationary theory and discuss the present status of the simplest inflationary models which can describe Planck/BICEP/Keck observational data by choice of a single model parameter. In particular, we discuss the Starobinsky model, Higgs inflation, and α -attractors, including the recently developed α -attractor models with SL(2,ℤ) invariant potentials. We also describe inflationary models providing a good fit to the recent ACT data, as well as the polynomial chaotic inflation models with three parameters, which can account for any values of the three main CMB-related inflationary parameters A_s , n_s and r.

We show that multi-field inflationary models with negligible turning in field space during inflation can lead to an effective sourcing of adiabatic from entropic perturbations after the end of inflation. We illustrate this general phenomenon with a detailed analysis of an inflationary model whose scalar potential is determined by modular invariance. Its entropic perturbations are frozen during inflation, but instead, they are converted into adiabatic perturbations in the first post-inflationary e-folds. The curvature power spectrum, giving rise to CMB fluctuations, reaches a novel and enhanced plateau in this process; we address the implications for the inflationary observables A_s, n_s and r.

We explain why the surprising one-loop finiteness of 6D half-maximal supergravities recently discovered by Huang et al [1] is the result of the cancellation of the six-dimensional gravitational and gauge anomalies in (2,0) supergravity with 21 tensor multiplets and (1,1) supergravity with 20 vector multiplets.

We show that the simplest generalization of the chaotic inflation model 12 m^2ϕ^2 with nonminimal coupling to gravity (1+ϕ) R provides a good match to the results of the latest data release of the Atacama Cosmology Telescope, with r ≈10^-2.

We study cosmological theory where the kinetic term and potential have SL(2,ℤ) symmetry. Potentials have a plateau at large values of the inflaton field, where the axion forms a flat direction. Due to the underlying hyperbolic geometry and special features of SL(2,ℤ) potentials, the theory exhibits an α-attractor behavior: its cosmological predictions are stable with respect to significant modifications of the SL(2,ℤ) invariant potentials. We present a supersymmetric version of this theory in the framework of D 3 induced geometric inflation. The choice of α is determined by underlying string compactification. For example, in a CY compactification with T 2 , one has 3 α = 1, the lowest discrete Poincaré disk target for LiteBIRD.

We investigate the global structure of the recently discovered family of SL(2,ℤ)-invariant potentials describing inflationary α-attractors. These potentials have an inflationary plateau consisting of the fundamental domain and its images fully covering the upper part of the Poincaré half-plane. Meanwhile, the lower part of the half-plane is covered by an infinitely large number of ridges, which, at first glance, are too sharp to support inflation. However, we show that this apparent sharpness is just an illusion created by hyperbolic geometry, and each of these ridges is physically equivalent to the inflationary plateau in the upper part of the Poincaré half-plane.