Higher-Order Analytical Expansion of Thawing Dark Energy with an Exponential Potential

Motivated by recent DESI results suggesting dynamical dark energy, we investigate the thawing scenario in quintessence with an exponential potential, $V=V_0e^{-λφ/m_{\mathrm{pl}}}$, by analytically expanding the deviation of the equation of state parameter $w_φ$ from $-1$ in powers of $λ$. In addition to the previously known leading-order result at $O(λ^2)$, we derive the $O(λ^4)$ correction as a function of the density parameter $Ω_φ$. We show that a consistent determination of the redshift dependence of $w_φ$ through $O(λ^4)$ requires corrections to the background expansion. We obtain the required correction by expanding $Ω_φ$ in powers of $λ$ around its $Λ\mathrm{CDM}$ value. Comparison with numerical solutions demonstrates that the $O(λ^4)$ expansion provides a more accurate approximation than the leading-order result. Our analytical approximation, which consistently incorporates the $O(λ^4)$ correction, will provide a potentially useful tool for distinguishing the exponential quintessence model from other dark energy models in future observations.