Forecast constraints on null tests of the $\Lambda$CDM model with SPHEREx

In this work we quantify the ability of the upcoming SPHEREx survey to constrain cosmological observables and test the internal consistency of the cosmological constant and cold dark matter ($\Lambda$CDM) model. Using Fisher matrix forecasting, we assess the expected precision on Baryon Acoustic Oscillations (BAO) observables, such as the angular diameter distance $D_\mathrm{A}(z)$ and the Hubble parameter $H(z)$. We further explore SPHEREx's potential to probe some of the fundamental assumptions of large-scale spatial homogeneity and isotropy, through model-independent reconstructions of several consistency tests of the $\Lambda$CDM model. In addition, we also examine the effect of the model dependence of the resulting Fisher and covariance matrices, using a neural network (NN) classification approach. We find that, while it is commonly assumed the covariance matrix depends weakly on the model, in fact the NN can very accurately ($\sim 98\%$) detect the underlying fiducial cosmological model based solely on the covariance matrix of the data, thus challenging this assumption. This model dependence, often neglected in standard analyses, can be naturally incorporated within simulation-based inference frameworks, which offer a flexible alternative for capturing such effects.