Scalar, vector and tensor fields on $dS_3$ with arbitrary sources: harmonic analysis and antipodal maps

The scalar, vector and tensor spherical harmonics on three-dimensional de Sitter spacetime are defined and analyzed. Each harmonic defines two sets of asymptotic data on the two sphere in the asymptotic expansion close to both the past and the future of de Sitter spacetime. For each case, we explicit the antipodal relationship of both sets of asymptotic data between past and future infinity, which can be non-local. A procedure is defined to extract these asymptotic data in the presence of sources. This provides for each class of propagating field on de Sitter the relationship between two independent sets of data defined on the sphere in the asymptotic future with the corresponding data defined in the asymptotic past. We also provide several theorems on the decomposition of vector and tensors on de Sitter such as one proving that a large class of tensors obeying an inhomogeneous wave equation can be expressed locally in terms of a symmetric transverse traceless tensor. These results are instrumental in the description of interacting four-dimensional asymptotically flat fields at spatial infinity.

💡 Research Summary

The paper presents a comprehensive harmonic analysis of scalar, vector, and tensor fields on three‑dimensional de Sitter space (dS₃) with arbitrary smooth sources, and it elucidates the antipodal relationships between data defined at past and future infinity. The authors begin by fixing global coordinates ((\tau,\theta,\phi)) with metric (q_{ab}=-d\tau^{2}+\cosh^{2}\tau,\gamma_{AB}). They solve the eigenvalue problem (\Box\psi_{n}=-(n+1)^{2}+1)(\psi_{n}) for integer (n\ge-1), obtaining two families of scalar harmonics (\psi^{(p)}{n\ell m}) and (\psi^{(q)}{n\ell m}). The “p‑type” solutions are built from associated Legendre functions of the first kind (P), while the “q‑type” solutions involve the second‑kind functions (Q). For (\ell<n+1) the time dependence is expressed through (P_{\ell+1/2}^{,n+1/2}(i\sinh\tau)) and (Q_{\ell+1/2}^{,n+1/2}(i\sinh\tau)); for (\ell\ge n+1) the dependence is through (P_{n+1}^{\ell}(\tanh\tau)) and (Q_{n+1}^{\ell}(\tanh\tau)).

A combined parity‑flip and time‑reversal map (\Upsilon_{H}) (the antipodal map) sends ((\tau,\theta,\phi)) to ((-,\tau,\pi-\theta,\phi+\pi)). Under this map the p‑type harmonics acquire eigenvalue ((-1)^{n+1}) and the q‑type harmonics ((-1)^{n}). This dichotomy is central to the later matching of data at (\mathcal{I}^{\pm}).

The authors introduce the Klein‑Gordon inner product ((\psi_{1},\psi_{2}){KG}) on a constant‑(\tau) slice, normalize the harmonics so that ((\psi^{(c)}{n\ell m},\psi^{(d)}{n\ell’ m’}){KG}=\delta_{\ell\ell’}\delta_{mm’}\epsilon_{cd}) with (\epsilon_{pq}=-\epsilon_{qp}=1). By forming (\psi^{\pm}_{n\ell m}=\frac12(\psi^{(p)}\pm i\psi^{(q)})) they obtain positive‑energy and negative‑energy modes suitable for canonical quantization.

When a smooth source (s(\tau,x^{A})) is present, the inhomogeneous equation ((\Box-n(n+2))\hat\psi_{n}=s) is solved via a Green‑function expansion. The solution reads
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