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We study the system of double Rashba wires brought into the proximity to an$s$-wave superconductor. The time reversal invariant topologicalsuperconductivity is realized if the interwire pairing corresponding to crossedAndreev reflection dominates over the standard intrawire pairing. We derive thetopological criterion and show that the system hosts zero energy Andreev boundstates such as a Kramers pair of Majorana fermions. We classify symmetry of theCooper pairs focusing on the four degrees of freedom, $i.e.$, frequency, spin,spatial parity inside wires, and spatial parity between wires. The magnitude ofthe odd-frequency pairing is strongly enhanced in the topological state. Wealso explore properties of junctions occurring in such double wire systems. Ifone section of the junction is in the topological state and the other is in thetrivial state, the energy dispersion of Andreev bound states is proportional to$\sim\pm\sin\varphi$, where $\varphi$ denotes the macroscopic phase differencebetween two sections. This behavior can be intuitively explained by thecouplings of a Kramers pair of Majorana fermions and spin-singlet $s$-waveCooper pair and can also be understood by analyzing an effective continuummodel of the $s+p$/$s$-wave superconductor hybrid system.

Theory of time reversal topological superconductivity in double Rashba wires -- symmetries of Cooper pair and Andreev bound states