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

Abstract

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 $\alphaN$ and $\betaN$ 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 $\alphaN$ and $\betaN$ and their cross correlations contribute to the late-time field and conjugate momentum two-point functions. This happens both for light scalars ($m<\frac{d}{2}H$), corresponding to complementary series representations, and heavy scalars ($m>\frac{d}{2}H$), 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 $\betaN$ autocorrelation contributes to the conjugate momentum two-point function in any dimensions and we gather hints that suggest $\alphaN$ to correspond to discrete series representations for this case at $d=3$.