Poincare gauge theory of gravity: Friedman cosmology with even and odd parity modes. Analytic part

We propose a cosmological model in the framework of the Poincar\'e gauge theory of gravity (PG). The gravitational Lagrangian is quadratic in curvature and torsion. In our specific model, the Lagrangian contains (i) the curvature scalar $R$ and the curvature pseudo-scalar $X$ linearly and quadratically (including an $RX$ term) and (ii) pieces quadratic in the torsion {\it vector} $\cal V$ and the torsion {\it axial} vector $\cal A$ (including a ${\cal V}{\cal A}$ term). We show generally that in quadratic PG models we have nearly the same number of parity conserving terms (`world') and of parity violating terms (`shadow world'). This offers new perspectives in cosmology for the coupling of gravity to matter and antimatter. Our specific model generalizes the fairly realistic `torsion cosmologies' of Shie-Nester-Yo (2008) and Chen et al.\ (2009). With a Friedman type ansatz for an orthonormal coframe and a Lorentz connection, we derive the two field equations of PG in an explicit form and discuss their general structure in detail. In particular, the second field equation can be reduced to first order ordinary differential equations for the curvature pieces $R(t)$ and $X(t)$. Including these along with certain relations obtained from the first field equation and curvature definitions, we present a first order system of equations suitable for numerical evaluation. This is deferred to the second, numerical part of this paper.