Constraining $$\beta $$-exponential inflation with the latest ACT observations
Abstract
Abstract Recent observations from the Atacama Cosmology Telescope (ACT), especially when combined with DESI baryon acoustic oscillation data, indicate a scalar spectral index $$ n_s $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:math> higher than the value reported by Planck 2018, placing tension on universal inflationary attractor models. Motivated by this discrepancy, we investigate the inflationary predictions of the $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> -exponential potential, $$ V(\phi )=V_0\left( 1-\lambda \beta \phi /M_p\right) ^{1/\beta }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>V</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ϕ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>V</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:msup> <mml:mfenced> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>λ</mml:mi> <mml:mi>β</mml:mi> <mml:mi>ϕ</mml:mi> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:mfenced> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mi>β</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> considering both minimally and non-minimally coupled realizations. This potential generalizes standard exponential inflation and naturally arises in braneworld scenarios. We derive analytical expressions for the slow-roll parameters and inflationary observables using a perturbative expansion in the non-minimal coupling $$\xi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ξ</mml:mi> </mml:math> , and validate these results through numerical calculations. In the minimally coupled case, the model predicts $$ n_s \simeq 0.976 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mo>≃</mml:mo> <mml:mn>0.976</mml:mn> </mml:mrow> </mml:math> and $$ r \simeq 0.035 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>≃</mml:mo> <mml:mn>0.035</mml:mn> </mml:mrow> </mml:math> for $$N=50$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>50</mml:mn> </mml:mrow> </mml:math> and moderate values of $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> , remaining compatible with ACT+DESI (P-ACT-LB) constraints at the $$1\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mi>σ</mml:mi> </mml:mrow> </mml:math> level while yielding a spectral tilt larger than the universal attractor prediction. Introducing a small non-minimal coupling significantly improves agreement with observations by suppressing the tensor-to-scalar ratio while preserving the enhanced scalar tilt. For $$N=60 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>60</mml:mn> </mml:mrow> </mml:math> , $$ \lambda \sim 0.3\!-\!0.5 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>∼</mml:mo> <mml:mn>0.3</mml:mn> <mml:mspace/> <mml:mo>-</mml:mo> <mml:mspace/> <mml:mn>0.5</mml:mn> </mml:mrow> </mml:math> , and $$ \beta \sim \mathcal {O}(1\!-\!5) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>∼</mml:mo> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mspace/> <mml:mo>-</mml:mo> <mml:mspace/> <mml:mn>5</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , the non-minimally coupled model yields $$ n_s \