Abstract
The existing transformation from a relativistic real scalar field to a complex non-relativistic scalar field by Namjoo, Guth, and Kaiser is generalized from Minkowski space to a more general background metric. In that case the transformation is not purely algebraic any more but determined by a differential equation. We apply the generalized transformation to a real scalar with $\phi^4$ interaction on an Friedmann-Lema\^itre-Robertson-Walker cosmologically expanding background and calculate the resulting non-relativistic action up to second order in small parameters. We also show that the transformation can be interpreted as a Bogoliubov transformation between relativistic and non-relativistic creation and annihilation operators and comment on emerging symmetries in the non-relativistic theory.
Highlights
The exact nature of dark matter is an open problem that reaches across multiple areas of physics from cosmology to particle physics
Going back to Minkowski space-time gμν 1⁄4 ημν, we can choose uμ 1⁄4 ð1; 0; 0; 0Þ and argðαÞ 1⁄4 ðm − V0Þt which leads to the same differential equation for γ as in the previous section and the action reduces to (38)
The Lagrangian of the free real scalar field has for a general space-time metric a Z2 symmetry
Summary
The exact nature of dark matter is an open problem that reaches across multiple areas of physics from cosmology to particle physics. The precise relation between the relativistic theory and an effective nonrelativistic description is important for interesting phenomenological questions, such as whether scalar field dark matter resembles a Bose-Einstein condensate or whether it could show superfluid behavior [6,7,8,9,10,11,12,13,14,15,16,17,18,19]. II A in which the factors of c are explicitly written out and eventually taken to the limit c → ∞
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