Modelling type 1 quasar colours in the era of Rubin and Euclid

We construct a parametric SED model which is able to reproduce the average observed SDSS-UKIDSS-WISE quasar colours to within one tenth of a magnitude across a wide range of redshift $(0<z<5)$ and luminosity $(-22>M_i>-29)$. This model is shown to provide accurate predictions for the colours of known quasars which are less luminous than those used to calibrate the model parameters, and also those at higher redshifts $z>5$. Using a single parameter, the model encapsulates an up-to-date understanding of the intra-population variance in the rest-frame ultraviolet and optical emission lines of luminous quasars. At fixed redshift, there are systematic changes in the average quasar colours with apparent i-band magnitude, which we find to be well explained by the contribution from the host galaxy and our parametrization of the emission-line properties. By including redshift as an additional free parameter, the model could be used to provide photometric redshifts for individual objects. For the population as a whole we find that the average emission line and host galaxy contributions can be well described by simple functions of luminosity which account for the observed changes in the average quasar colours across $18.1<i_\textrm{AB}<21.5$. We use these trends to provide predictions for quasar colours at the luminosities and redshifts which will be probed by the Rubin Observatory LSST and ESA-Euclid wide survey. The model code is applicable to a wide range of upcoming photometric and spectroscopic surveys, and is made publicly available.