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We investigate, in dark matter and galaxy mocks, the effects of approximatingthe galaxy power spectrum-bispectrum estimated covariance as a diagonal matrix,for an analysis that aligns with the specifications of recent and upcominggalaxy surveys. We find that, for a joint power spectrum and bispectrumdata-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 covarianceapproximation recovers $\sim 10\%$ larger error-bars on the parameters$\{\sigma_8,f,\alpha_\parallel,\alpha_\bot\}$ with respect to the fullcovariance case, while still underestimating the corresponding true errors onthe recovered parameters by $\sim 10\%$. This is caused by the diagonalapproximations weighting the elements of the data-vector in a sub-optimal way,resulting in a less efficient estimator, with poor coverage properties, thanthe maximum likelihood estimator featuring the full covariance matrix. Wefurther investigate intermediate approximations to the full covariance matrix,with up to $\sim 80\%$ of the matrix elements being zero, which could beadvantageous for theoretical and hybrid approaches. We expect these results tobe qualitatively insensitive to variations of the total cosmological volume,depending primarily on the bin size and shot-noise, thus making themparticularly significant for present and future galaxy surveys.

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