Inflation, the Hubble Tension and Early Dark Energy: an alternative overview

I review and discuss the possible implications for inflation resulting from considering new physics in light of the Hubble tension. My study is motivated by a simple argument that the constraints on inflationary parameters, most typically the spectral index $n_s$, depend to some extent on the cosmological framework. To avoid broadening the uncertainties resulting from marginalizing over additional parameters (typical in many alternative models), I first adopt the same alternative viewpoint of previous studies and analyze what happens if a physical theory can fix extra parameters to non-standard values. Focusing on the dark energy equation of state $w$ and the effective number of relativistic species $N_{\rm{eff}}$, I confirm that physical theories able to fix $w \approx -1.2$ or $N_{\rm{eff}} \approx 3.9$ produce values of $H_0$ from CMB and BAO in line with the local distance ladder estimate. While in the former case I do not find any relevant implications for inflation, in the latter scenarios, I observe a shift towards $n_s \approx 1$. From a model-selection perspective, both cases are strongly disfavored compared to $\Lambda$CDM. However, models with $N_{\rm{eff}} \approx 3.3 - 3.4$ could bring the $H_0$ tension down to $\sim 3\sigma$ while being moderately disfavored. Yet, this is enough to change the constraints on inflation so that the most accredited models (e.g., Starobinsky inflation) would no longer be favored by data. I then focus on Early Dark Energy (EDE), arguing that an EDE fraction $f_{\rm{EDE}}\sim 0.04 - 0.06$ (only able to mildly reduce the $H_0$-tension down to $\sim 3\sigma$) could already require a similar change in perspective on inflation. In fact, performing a full joint analysis of EDE and Starobinsky inflation, I find that the two models can hardly coexist for $f_{\rm{EDE}}\gtrsim 0.06$.