A Minimal Single-Parameter Exponential Inflation Model: Dynamical and Observational Validation via Numerical Optimization

We present a refined single-field inflationary model based on a deformed exponential potential of the form \( V(φ) = V_0 exp(-λ φ^n) \), where the parameter \(n\) introduces a controlled deviation from the standard exponential behavior. This deformation provides additional flexibility in shaping the inflationary dynamics, improving its compatibility with current observational constraints. To assess the viability of the model, we construct a loss function that incorporates the latest Planck 2018 bounds on the scalar spectral index \(n_s\), the tensor-to-scalar ratio \(r\), and the scalar amplitude \(A_s\). We then apply a Differential Evolution (DE) optimization algorithm to identify the optimal parameter configuration. The best-fit values are found to be \(n = 1.5001\), \(λ = 0.007820\), and \(N ≈ 53.64\), yielding \(n_s ≈ 0.9736\), \(r ≈ 0.0551\), and \(A_s = 2.1 × 10⁻⁹\), all of which are in excellent agreement with current observations. The model’s behavior and predictions are further analyzed through a comprehensive set of visualizations, including heatmaps and 3D surface plots illustrating the sensitivity of inflationary observables to the parameters \((n, λ)\); the shape of the potential \(V(φ)\); the evolution of slow-roll parameters \(ε(φ)\) and \(η(φ)\); the scalar field trajectory \(φ(N)\); inflationary predictions in the \((n_s, r)\) plane compared to Planck confidence contours; and the landscape of the loss function over the parameter space. What distinguishes this model is its combination of mathematical simplicity and physical flexibility: the deformation parameter \(n\) allows for a continuous interpolation between different inflationary regimes, providing a natural fit to modern cosmological data.