Constraints on the varying electron mass and early dark energy in light of ACT DR6 and DESI DR2 and the implications for inflation

Primarily motivated by the Hubble tension, we analyze the varying electron mass model and axionlike early dark energy model (EDE) using baryon acoustic oscillation data from DESI DR2 data and including the recent results from ACT DR6. Our analysis indicates that $m_e\mathrm{/}{m}_{e0}=1.0081\pm 0.0046$ and the energy fraction of EDE is constrained as $f_{\mathrm{E}\mathrm{D}\mathrm{E}}<0.016$. Since those cosmological models fit with different spectral index $n_s$, we show the posterior of those models on the ($n_s-r$) plane and point out that, for example, Starobinsky inflation works for varying electron mass model while the standard supersymmetric hybrid inflation is preferred in the EDE model.