Improving Fisher matrix forecasts for galaxy surveys: window function, bin cross-correlation, and bin redshift uncertainty

The Fisher matrix is a widely used tool to forecast the performance of future experiments and approximate the likelihood of large data sets. Most of the forecasts for cosmological parameters in galaxy clustering studies rely on the Fisher matrix approach for large-scale experiments like DES, Euclid, or SKA. Here we improve upon the standard method by taking into account three effects: the finite window function, the correlation between redshift bins, and the uncertainty on the bin redshift. The first two effects are negligible only in the limit of infinite surveys. The third effect, on the contrary, is negligible for infinitely small bins. Here we show how to take into account these effects and what the impact on forecasts of a Euclid-type experiment will be. The main result of this article is that the windowing and the bin cross-correlation induce a considerable change in the forecasted errors, of the order of 10-30% for most cosmological parameters, while the redshift bin uncertainty can be neglected for bins smaller than $\Delta z = 0.1$ roughly.