$α$-Attractors: Planck, LHC and Dark Energy

TL;DR

The paper develops four-parameter supergravity frameworks based on cosmological $\boldsymbol{\alpha}$-attractors to model inflation and dark energy while keeping $\frac{\delta\rho}{\rho}$, $n_s$, and $\Lambda$ fixed by observations, and allowing the gravitino mass $m_{3/2}$ and tensor-to-scalar ratio $r$ to vary. It employs Killing-adapted variables and a nilpotent goldstino to realize spontaneous SUSY breaking at a de Sitter minimum, with the attractor geometry controlled by $\alpha$ and a tunable SUSY-breaking scale $M$ linked to collider constraints. The work presents reconstruction methods to embed arbitrary inflationary potentials into these supergravity models and introduces generalizations that interpolate between polynomial inflaton potentials and attractor predictions, preserving Planck compatibility. By connecting inflation, dark energy, and SUSY breaking in a string-inspired setting, the framework provides testable links between $B$-mode observations and LHC-scale physics, and clarifies how a tiny cosmological constant can arise in a landscape-inspired context.

Abstract

We develop four-parameter supergravity models of inflation and dark energy, constrained so that ${δρ\over ρ}$, $n_s$ and the cosmological constant $Λ$ take their known observable values, but where the mass of gravitino $m_{3/2}$ and the tensor-to-scalar ratio $r$ are free parameters. We focus on generalized cosmological $α$-attractor models, with logarithmic Kahler potentials, a nilpotent goldstino and spontaneously broken supersymmetry at the de Sitter minimum. The future data on B-modes will specify the parameter $α$, measuring the geometry of the Kahler, manifold. The string landscape idea for dark energy is supported in these models via an incomplete cancellation of the universal positive goldstino and negative gravitino contribution. The scale of SUSY breaking M related to the mass of gravitino in our models is a controllable parameter, independent on the scale of inflation, it will be constrained by LHC data and future collider Energy-frontier experiments.

Figures (5)

• Figure 1: Examples of supergravity T- models with $r$-dependence in logarithmic scale in $r$. For potentials $V=\tanh^{2n} {\varphi\over \sqrt{6\alpha}}$, the predictions of these models interpolate between the predictions of various polynomial models $\varphi^{2n}$ at very large $\alpha$ and the vertical attractor line for $\alpha\leq O(10)$. When $\alpha \rightarrow \infty$ the models approach the ones with $\varphi^{2n}$ potentials. This attractor line beginning with the red star corresponds to the predictions of the simplest models $V=\tanh^{2n} {\varphi\over \sqrt{6\alpha}}$ with $n=1$.
• Figure 2: The cosmological observables $(n_s,r)$, in a logarithmic scale in $r$, for simple examples of E-models, with $V= (1- e^{-{ \sqrt {2\over 3 \alpha} \varphi}})^{2n}$ with $n = (1/2, 3/4, 7/8, 1, 3/2, 2, 3)$ starting from the right, increasing to the left, with the vertical line for $n=1$ in the middle. When $\alpha \rightarrow \infty$ the models approach the ones with $\varphi^{2n}$ potentials. The attractor line, common for all $n$, starts below $r\approx 10^{-3}$ and goes down, unlimited.
• Figure 3: Cosmological predictions of the simplest T-model (\ref{['Tpot']}) with SUSY breaking and a non-vanishing cosmological constant $\Lambda \sim 10^{{-120}}$.
• Figure 4: The potential for the supergravity model in eq. (\ref{['newRexample']}) as a function of $\varphi$ and $\vartheta$. It has a de Sitter minimum at $\varphi=\vartheta=0$ where $V_{\rm min} =\Lambda$. Supersymmetry is broken at this minimum with $D_S W=M$, the mass of gravitino is $m^2_{3/2}= {M^2\over 3} (1-{\Lambda\over M^2})$. The inflationary de Sitter valleys have a nice feature known for models with Minkowski minimum with unbroken SUSY, studied in Carrasco:2015rva. These valleys provide nice initial conditions for the inflation to start in these models.
• Figure 5: Predictions of the model \ref{['KdiskExample']} for $20> \alpha > 1/3$ are shown by the thin green line. The top of the line indicated by the dark red star corresponds to $\alpha = 20$. The line ends at the pink star corresponding to $\alpha = 1/3$.