Scalar two-point functions at the late-time boundary of de Sitter

We calculate two-point functions of scalar fields of mass m and their conjugate momenta at the late-time boundary of de Sitter with Bunch-Davies boundary conditions, in general d + 1 spacetime dimensions. We perform the calculation using the wavefunction picture and using canonical quantization. With the latter one clearly sees how the late-time field and conjugate momentum operators are linear combinations of the normalized late-time operators α_N and β_N that correspond to unitary irreducible representations of the de Sitter group with well-defined inner products. The two-point functions resulting from these two different methods are equal and we find that both the autocorrelations of α_N and β_N and their cross correlations contribute to the late-time field and conjugate momentum two-point functions. This happens both for light scalars $\left(m<\frac{d}{2}H\right)$ \( \left(m<\frac{d}{2}H\right) \) , corresponding to complementary series representations, and heavy scalars $\left(m>\frac{d}{2}H\right)$ \( \left(m>\frac{d}{2}H\right) \) , corresponding to principal series representations of the de Sitter group, where H is the Hubble scale of de Sitter. In the special case m = 0, only the β_N autocorrelation contributes to the conjugate momentum two-point function in any dimensions and we gather hints that suggest α_N to correspond to discrete series representations for this case at d = 3.