On approximations of the redshift-space bispectrum and power spectrum multipoles covariance matrix

We investigate, in dark matter and galaxy mocks, the effects of approximating the galaxy power spectrum-bispectrum estimated covariance as a diagonal matrix, for an analysis that aligns with the specifications of recent and upcoming galaxy surveys. We find that, for a joint power spectrum and bispectrum data-vector, with corresponding $k$-ranges of $0.02<k\,[h{\rm Mpc}^{-1}]<0.15$ and $0.02<k\,[h{\rm Mpc}^{-1}]<0.12$ each, the diagonal covariance approximation recovers $\sim 10\%$ larger error-bars on the parameters $\{\sigma_8,f,\alpha_\parallel,\alpha_\bot\}$ with respect to the full covariance case, while still underestimating the corresponding true errors on the recovered parameters by $\sim 10\%$. This is caused by the diagonal approximations weighting the elements of the data-vector in a sub-optimal way, resulting in a less efficient estimator, with poor coverage properties, than the maximum likelihood estimator featuring the full covariance matrix. We further investigate intermediate approximations to the full covariance matrix, with up to $\sim 80\%$ of the matrix elements being zero, which could be advantageous for theoretical and hybrid approaches. We expect these results to be qualitatively insensitive to variations of the total cosmological volume, depending primarily on the bin size and shot-noise, thus making them particularly significant for present and future galaxy surveys.