Supersymmetric Hybrid Inflation in light of Atacama Cosmology Telescope Data Release 6, Planck 2018 and LB-BK18

Supersymmetry-based hybrid inflation models (referred to as `spontaneously broken supersymmetry’ by the Planck collaboration) are attractive for several reasons, including the appealing feature that inflation is associated with local gauge symmetry breaking in the early universe. Following the Planck collaboration’s notation, the inflationary potential with sub-Planckian inflaton field values is given by: $V={\mathrm{\Lambda }}^4[1+{\alpha }_h\log (\phi \mathrm{/}{M}_{Pl})]-m_{3\mathrm{/}2}{\mathrm{\Lambda }}^2\phi +{\mathrm{\Lambda }}^{4}O((\phi \mathrm{/}{M}_{Pl}{)}^4)$. Here, $\mathrm{\Lambda }=\sqrt{\kappa }M$ denotes the energy scale of inflation, $M$ is the gauge symmetry breaking scale, $\kappa$ is a dimensionless parameter that sets the inflaton mass ($\sqrt{2}\kappa M$), and ${\alpha }_h$ is determined from quantum corrections in terms of $\kappa$ and the underlying gauge group. A soft supersymmetry-breaking term proportional to the gravitino mass $m_{3\mathrm{/}2}$ (~10 TeV) and linear in the inflaton field $\phi$ is also present during inflation. The final term in $V$ represents the leading-order supergravity correction. (Note that the last two terms were not taken into account in the Planck analysis.) We provide estimates for the parameters $\kappa$ (and ${\alpha }_h$) that yield a scalar spectral index $n_s$ in the range 0.96 to 0.98, which is fully consistent with recent P-ACT-LB measurements presented by the Atacama Cosmology Telescope, as well as earlier measurements by Planck. We recall that in the absence of the soft SUSY-breaking term proportional to $m_{3\mathrm{/}2}$ in $V$, the spectral index $n_s=1-1\mathrm{/}N=0.98$, where $N=50$ denotes the number of e-foldings. The tensor-to-scalar ratio $r$ in this minimal model is tiny, but it can reach potentially observable values ($r\lesssim 0.01$) in non-minimal models.