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Recently, gravitational gauge theories with torsion have been discussed by an increasing number of authors from a classical as well as from a quantum field theoretical point of view. The Einstein-Cartan(-Sciama-Kibble) Lagrangian has been enriched by the parity odd pseudoscalar curvature (Hojman, Mukku, and Sayed) and by torsion square and curvature square pieces, likewise of even and odd parity. (i) We show that the inverse of the so-called Barbero-Immirzi parameter multiplying the pseudoscalar curvature, because of the topological Nieh-Yan form, can only be appropriately discussed if torsion square pieces are included. (ii) The quadratic gauge Lagrangian with both parities, proposed by Obukhov et al. and Baekler et al., emerges also in the framework of Diakonov et al. (2011). We establish the exact relations between both approaches by applying the topological Euler and Pontryagin forms in a Riemann-Cartan space expressed for the first time in terms of irreducible pieces of the curvature tensor. (iii) Only in a Riemann-Cartan spacetime, that is, in a spacetime with torsion, parity violating terms can be brought into the gravitational Lagrangian in a straightforward and natural way. Accordingly, Riemann-Cartan spacetime is a natural habitat for chiral fermionic matter fields.PACS numbers: 04.50. Kd, 11.30.Er, 98.80.Jk ‡ We follow the conventions of [6]. We have the coframe 1-form ϑ α = e i α dx i and the frame vectors e β = e j β ∂ j , with e β ⌋ϑ α = δ α β . Greek indices are raised and lowered by means of the Minkowski metric g αβ = diag (−1, 1, 1, 1). The volume 4-form is denoted by η, and η α = ⋆ ϑ α , η αβ = ⋆ ϑ αβ , η αβγ = ⋆ ϑ αβγ , η αβγδ = ⋆ ϑ αβγδ , where ⋆ is the Hodge star operator and ϑ αβ := ϑ α ∧ ϑ β , etc. § The second minus sign on the right-hand-side of this equation is corrected. In [6], Eq.(5.9.18) was a sign error. * By using some simple algebra, Eq.(16) can alternatively be derived in a straightforward way from Eqs.(10.17) to (10.22) of Obukhov [4]. and curvature invariants with even and odd parity including all boundary terms

We propose a cosmological model in the framework of the Poincaré gauge theory of gravity (PG). The gravitational Lagrangian is quadratic in both curvature and torsion. In our specific model, the Lagrangian contains (i) the curvature scalar R and the curvature pseudoscalar X linearly and quadratically (including an RX term) and (ii) pieces quadratic in the torsion vector V and the torsion axial vector A (including a V 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 RðtÞ and XðtÞ. 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.

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We show that the Einstein-aether theory of Jacobson and Mattingly (J&M) can be understood in the framework of the metric-affine (gauge theory of) gravity (MAG).We achieve this by relating the aether vector field of J&M to certain post-Riemannian nonmetricity pieces contained in an independent linear connection of spacetime.Then, for the aether, a corresponding geometrical curvature-square Lagrangian with a massive piece can be formulated straightforwardly. We find an exact spherically symmetric solution of our model.