Consistency tests of $\Lambda$CDM from the early integrated Sachs-Wolfe effect: Implications for early-time new physics and the Hubble tension

New physics increasing the expansion rate just prior to recombination is among the least unlikely solutions to the Hubble tension, and would be expected to leave an important signature in the early Integrated Sachs-Wolfe (eISW) effect, a source of Cosmic Microwave Background (CMB) anisotropies arising from the time-variation of gravitational potentials when the Universe was not completely matter dominated. Why, then, is there no clear evidence for new physics from the CMB alone, and why does the $\Lambda$CDM model fit CMB data so well? These questions and the vastness of the Hubble tension theory model space motivate general consistency tests of $\Lambda$CDM. I perform an eISW-based consistency test of $\Lambda$CDM introducing the parameter $A_{\rm eISW}$, which rescales the eISW contribution to the CMB power spectra. A fit to Planck CMB data yields $A_{\rm eISW}=0.988 \pm 0.027$, in perfect agreement with the $\Lambda$CDM expectation $A_{\rm eISW}=1$, and posing an important challenge for early-time new physics, which I illustrate in a case study focused on early dark energy (EDE). I explicitly show that the increase in $\omega_c$ needed for EDE to preserve the fit to the CMB, which has been argued to worsen the fit to weak lensing and galaxy clustering measurements, is specifically required to lower the amplitude of the eISW effect, which would otherwise exceed $\Lambda$CDM's prediction by $\approx 20\%$: this is a generic problem beyond EDE and likely applying to most models enhancing the expansion rate around recombination. Early-time new physics models invoked to address the Hubble tension are therefore faced with the significant challenge of making a similar prediction to $\Lambda$CDM for the eISW effect, while not degrading the fit to other measurements in doing so.