The Continuous Spontaneous Localisation (CSL) theory in the cosmological context is subject to uncertainties related to the choice of the collapse operator. In this paper, we constrain its form based on generic arguments. We show that, if the collapse operator is even in the field variables, it is unable to induce the collapse of the wavefunction. Instead, if it is odd, we find that only linear operators are such that the outcomes are distributed according to Gaussian statistics, as required by measurements of the cosmic microwave background. We discuss implications of these results for previously proposed collapse operators. We conclude that the cosmological CSL collapse operator should be linear in the field variables.