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Superconformal Inflationary $\alpha$-Attractors
Renata Kallosh, Andrei Linde, Diederik Roest
TL;DR
The paper asks how to generalize superconformal inflationary models with universal predictions. It introduces α-attractors whose parameter is tied to Kähler curvature, finding interpolation from chaotic inflation at large α to the universal attractor regime at small α. The resulting family targets the region favored by Planck data, subject to stability restrictions in one superconformal realization.
Problem
Superconformal inflationary models exhibit universal predictions, but a broader class is needed to describe how those predictions connect to generic chaotic inflation.
Source: §1 · pp. 2–3Method
The paper constructs α-attractors in supergravity and interprets α geometrically as the inverse curvature of the inflaton’s Kähler manifold.
Source: §1 · p. 3, §3 · p. 6, §6 · p. 11Results
For large α the models reproduce chaotic-inflation predictions, while decreasing α continuously moves them toward n_s = 1 − 2/N and r = 12α/N^2.
Source: §1 · p. 4, §6 · p. 11Takeaways & Limitations
The generalized attractors provide a continuous family connecting chaotic-inflation values with a universal attractor region in the (n_s, r) plane.
Source: §6 · pp. 11–12In the chiral-superfield realization, α < 1/3 makes the scalar superpartner tachyonic and the model unstable, although the bosonic and α−β models remain stable at small α.
Source: §6 · p. 12
Abstract
from arXiv · showhide
Recently a broad class of superconformal inflationary models was found leading to a universal observational prediction $n_s=1-2/N$ and $r=12/N^2$. Here we generalize this class of models by introducing a parameter $\alpha$ inversely proportional to the curvature of the inflaton Kahler manifold. In the small curvature (large $\alpha$) limit, the observational predictions of this class of models coincide with the predictions of generic chaotic inflation models. However, for sufficiently large curvature (small $\alpha$), the predictions converge to the universal attractor regime with $n_s=1-2/N$ and $r=12\alpha/N^2$, which corresponds to the part of the $n_s-r$ plane favored by the Planck data.
1 Introduction
The paper generalizes superconformal inflationary attractors by introducing α, tied to the inflaton manifold’s curvature, so predictions interpolate between chaotic inflation and a universal attractor regime.
Background
- Earlier superconformal and nonminimal-coupling models already produced the attractor prediction and connected it continuously to chaotic-inflation scenarios.These results motivate identifying a broader, geometrically organized family of models.Source: §1 · p. 2
Contribution
- α is inversely related to the inflaton’s Kähler curvature through R_K = −2/(3α), making geometry the parameter controlling the generalized attractors.The same geometric interpretation applies to both model classes considered.Source: §1 · p. 3
Roadmap
- The paper analyzes an α−β model, reviews α = 1 superconformal models, generalizes them to arbitrary α, and derives their observational predictions.The construction uses chiral multiplets and relates the geometric field to the canonical inflaton through a tanh mapping.Source: §1 · pp. 3–4
Contribution
- The paper introduces α-attractors, a broad class of supergravity models designed to realize the universal attractor regime at large N.Their inflaton potential is an arbitrary function of tanh(ϕ/√(6α)).Source: §1 · p. 3
Phenomenology
- As α increases, predictions approach those of chaotic inflation, while decreasing α drives r below 12/N^2 toward the Planck-favored region.The change reflects the growing difference between the bounded geometric field and the unbounded canonical field.Source: §1 · pp. 3–4
- The α-attractor predictions continuously interpolate between standard chaotic-inflation predictions and the universal attractor values n_s = 1 − 2/N and r = 12α/N^2.This interpolation spans a broad range of tensor-to-scalar ratios because α may be large or arbitrarily small.Source: §1 · pp. 3–4
2 Phenomenology of the α −β model
The α−β model’s predictions vary continuously with α, connecting quadratic chaotic inflation, the Starobinsky model, and the attractor limit.
Benchmark limits
- At α=1, the potential coincides with the Starobinsky model, while large α reproduces the simplest quadratic chaotic model.The mass parameter is m2 = 4V0/(3α).Source: §2 · p. 4
Phenomenology
- The predictions interpolate continuously from quadratic chaotic inflation at α→∞ through the Starobinsky model at α=1 to ns = 1 − 2/N, r = 0 at α→0.The numerical trajectory spans the full range of r from 8/N to 0.Source: §2 · p. 5
Large-α limit
- For α≫160 at N=60, the potential is consistently approximated by quadratic chaotic inflation.The quadratic approximation requires α≫8N/3.Source: §2 · p. 5
Tensor predictions
- The model covers a broad range of tensor-to-scalar ratios as α decreases continuously from infinity to zero.The trajectory’s final segment near α of order one follows the attractor regime.Source: §2 · p. 5
3 Superconformal attractors for α = 1
The α=1 superconformal construction uses three chiral multiplets and broad superpotential choices to realize universal large-N attractor predictions.
Model construction
- Three chiral multiplets—a conformon X0, inflaton X1 = Φ, and sGoldstino X2 = S—define the superconformal models.The construction is specified by a real Kähler function and a holomorphic superpotential.Source: §3 · p. 6
Symmetry
- A constant f preserves the SO(1,1) subgroup, whereas deviations from a constant break that remaining symmetry.After stabilization of extra moduli, the construction produces an inflationary model.Source: §3 · p. 6
Universal attractor
- For a broad class of nonsingular functions f(z), the models yield the universal prediction (1.1) in the large-N limit.The functions have a zero and increase monotonically toward z = 1.Source: §3 · p. 6
Corrections
- For f(z) = λz^n, the expressions for ns and r include higher-order corrections in 1/N.The leading large-N terms are shared across the models, with differences appearing only in subleading corrections.Source: §3 · p. 6
4 A general family of α-attractors
The paper generalizes the α=1 construction to α-attractors whose inflaton geometry has constant negative curvature controlled by α.
Generalization
- The generalized superconformal models lead to a family of α-attractors beyond the previously considered models.For constant f and α = 1, the superpotential preserves SO(1,1), a subgroup of SU(1,1); if f is not constant or α ≠ 1, SO(1,1) is deformed.Source: §4 · p. 7
Single-field reduction
- For real Φ, the action reduces to a single canonically normalized field ϕ through a rapidity-like field redefinition.The truncation sets S = Φ−¯Φ = 0 and yields the effective Lagrangian used for inflation.Source: §4 · p. 7
Special case
- For constant f, the potential is independent of ϕ and α and describes de Sitter vacuum.Source: §4 · p. 7
Geometry
- During inflation at S = 0, all α-models are defined by the SU(1,1)/U(1) Kähler potential, whose metric corresponds to a symmetric space with negative constant curvature.Source: §4 · pp. 7–8
Stability
- Stability of the inflationary trajectory requires α > 1/3 and g > 1/6 up to slow-roll-suppressed corrections.These conditions follow from the masses of the extra scalar fields.Source: §4 · p. 8
5 Phenomenology of the α-attractors
The α-attractor phenomenology connects monomial chaotic inflation at large α with a curvature-controlled universal attractor at small α.
Scope
- The phenomenological analysis uses the single-field Lagrangian when f(Φ/3) is holomorphic and the trajectory S = Φ−¯Φ = 0 is stable.The stability restriction α > 1/3 is observationally relevant in the analyzed regime.Source: §5 · pp. 8–9
Large-α behavior
- At large α, tanh2n(ϕ/√(6α)) models recover the corresponding chaotic monomial models ϕ2n.This follows because rapidity approaches velocity when α is large.Source: §5 · p. 9
Interpolation
- For n2≪α≪nN, r grows linearly with α toward r≈8n/N, while ns depends linearly on r for α≫n2.These relations produce trajectories interpolating between chaotic models and the universal attractor.Source: §5 · p. 9
Attractor limit
- For α→0, the trajectories approach ns = 1 − 2/N and r = 0, while different monomial powers produce distinct large-α limits.For N=60, the trajectories form a fan at large α and merge for α = O(1).Source: §5 · p. 10
Small-α phenomenology
- Near α≈1, 1/3 < α < 3 gives 10^-3 < r < 10^-2 with very weak dependence on n.In this regime, the tensor amplitude is determined primarily by the Kähler-manifold curvature.Source: §5 · p. 10
Departure from attractor
- For α>3, nonuniversal terms separate the trajectories into a fan covering a significant part of the dark-blue region in the ns−r plane.The near-attractor behavior occupies only a small region around the red-star point.Source: §5 · pp. 10–11
Model dependence
- Different choices of f generate additional trajectories interpolating between chaotic-inflation models and the attractor regime.For n=1, the construction recovers the α−β potential analyzed earlier.Source: §5 · p. 11
6 Discussion
The paper generalizes inflationary attractors by labeling them with α, which has a geometric interpretation as the inverse curvature of the inflaton’s scalar manifold. These models interpolate between chaotic-inflation predictions at large α and universal attractor behavior at small α, with stability and phenomenological regimes depending on the realization.
6 Discussion
- The α-attractor family generalizes earlier attractor values to a one-parameter family, with α corresponding to the inverse curvature of the inflaton’s scalar manifold.The paper identifies the curvature through the underlying Kähler geometry and distinguishes geometric fields from canonically normalized fields.Source: §6 · p. 11
- For potentials V(Φ) ∼ Φ^n, the models continuously interpolate between chaotic-inflation and universal-attractor values of (n_s, r).At large α, these models agree with conventional chaotic inflation using V(ϕ) ∼ ϕ^n.Source: §6 · p. 11
- For large α, the models reproduce chaotic-inflation predictions, while decreasing α toward order one drives convergence toward the universal attractor point.This transition corresponds to moving from a nearly flat Kähler manifold toward one with order-one curvature.Source: §6 · p. 11
- For moderate 1/3 < α < 3, r varies universally between 10^-3 and 10^-2 within the attractor regime.The corresponding curvature interval is identified as subject to attractor behavior.Source: §6 · p. 12
- The chiral-superfield realization becomes unstable for α < 1/3 because the scalar superpartner of the inflaton has negative mass squared.The bosonic model and α−β model avoid this stability issue and approach r = 0 as α → 0.Source: §6 · p. 12
- The paper highlights the convergence of many inflationary models in the same (n_s, r) region as a target for further theoretical and observational investigation.The authors caution that this convergence does not establish that the evolution of our universe is described by one of these theories.Source: §6 · p. 12