We study the covariance in the angular power spectrum estimates of CMB fluctuations when the primordial fluctuations are non-Gaussian. The non-Gaussian covariance comes from a nonzero connected four-point correlation function -- or the trispectrum in Fourier space -- and can be large when long-wavelength (super-CMB) modes are strongly coupled to short-wavelength modes. The effect of such non-Gaussian covariance can be modeled through additional freedom in the theoretical CMB angular power spectrum and can lead to different inferred values of the standard cosmological parameters relative to those in $Lambda$CDM. Taking the collapsed limit of the primordial trispectrum in the quasi-single field inflation model as an example, we study how the six standard $Lambda$CDM parameters shift when two additional parameters describing the trispectrum are allowed. The reduced statistical significance of the Hubble tension in the extended model allows us to combine the {it Planck} temperature data and the type Ia supernovae data from Panstarrs with the distance-ladder measurement of the Hubble constant. This combination of data shows strong evidence for a primordial trispectrum-induced non-Gaussian covariance, with a likelihood improvement of $Delta chi^2 approx -15$ (with two additional parameters) relative to $Lambda$CDM.
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