KiDS-1000 Methodology: Modelling and inference for joint weak gravitational lensing and spectroscopic galaxy clustering analysis

We present the methodology for a joint cosmological analysis of weak gravitational lensing from the fourth data release of the ESOKilo-Degree Survey (KiDS-1000) and galaxy clustering from the partially overlapping Baryon Oscillation Spectroscopic Survey(BOSS) and the 2-degree Field Lensing Survey (2dFLenS). Cross-correlations between BOSS and 2dFLenS galaxy positions andsource galaxy ellipticities have been incorporated into the analysis, necessitating the development of a hybrid model of non-linearscales that blends perturbative and non-perturbative approaches, and an assessment of signal contributions by astrophysical effects.All weak lensing signals were measured consistently via Fourier-space statistics that are insensitive to the survey mask and displaylow levels of mode mixing. The calibration of photometric redshift distributions and multiplicative gravitational shear bias has beenupdated, and a more complete tally of residual calibration uncertainties was propagated into the likelihood. A dedicated suite of morethan 20 000 mocks was used to assess the performance of covariance models and to quantify the impact of survey geometry andspatial variations of survey depth on signals and their errors. The sampling distributions for the likelihood and theχ2goodness-of-fitstatistic have been validated, with proposed changes for calculating the effective number of degrees of freedom. The prior volumewas explicitly mapped, and a more conservative, wide top-hat prior on the key structure growth parameter S8=σ8(Ωm/0.3)1/2wasintroduced. The prevalent custom of reportingS8weak lensing constraints via point estimates derived from its marginal posterioris highlighted to be easily misinterpreted as yielding systematically low values ofS8, and an alternative estimator and associatedcredible interval are proposed. Known systematic effects pertaining to weak lensing modelling and inference are shown to biasS8byno more than 0.1 standard deviations, with the caveat that no conclusive validation data exist for models of intrinsic galaxy alignments.Compared to the previous KiDS analyses, S8constraints are expected to improve by 20 % for weak lensing alone and by 29 % for thejoint analysis.