The Dark side of the torsion: Dark Energy from kinetic torsion D. Benisty, 1, 2, 3, * E. I. Guendelman, 2, 3, 4, † A. van de Venn, 3, ‡ D. Vasak, 3, § J. Struckmeier, 3, 5, 6, ¶ and H. Stoecker 3, 5, 6, ** 1 DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom 2 Physics Department, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel 3 Frankfurt Institute for Advanced Studies (FIAS), Ruth-Moufang-Strasse 1, 60438 Frankfurt am Main, Germany 4 Bahamas Advanced Study Institute and Conferences, 4A Ocean Heights, Hill View Circle, Stella Maris, Long Island, The Bahamas 5 Fachbereich Physik, Goethe-Universität, Max-von-Laue-Strasse 1, 60438 Frankfurt am Main, Germany 6 GSI Helmholtzzentrum für Schwerionenforschung GmbH, Planckstrasse 1, 64291 Darmstadt, Germany An extension to the Einstein-Cartan (EC) action is discussed in terms of cosmological solutions. The torsion incorporated in the EC Lagrangian is assumed to be totally anti-symmetric, and written by of a vector S μ . Then this torsion model, compliant with the Cosmological Principle, is made dynamical by introducing its quadratic, totally anti-symmetric derivative. The EC Lagrangian then splits up into the Einstein-Hilbert portion and a (mass) term ∼ s 2 0 . While for the quintessence model, dark energy arises from the potential, here the kinetic term, 1 μ 2 ˙ s 2 0 , plays the role of dark energy. The quadratic torsion term, on the other hand, gives rise to a stiff fluid that leads to a bouncing solution. A bound on the bouncing solution is calculated. I. INTRODUCTION The nature of dark energy is a long-standing unre- solved problem in current cosmology. Einstein’s cosmo- logical constant, Λ, in General Relativity (GR) has been invoked to account for the observed accelerating expan- sion of the universe, but failed to be understood yet in terms of field theoretical considerations. An alternative direction to account for these observations has been to to re-formulate GR in terms of torsion (“Teleparallel Grav- ity”) [1–8], to try various extensions of GR, see e.g. [9– 11], or to formulate gravity as a gauge theory in analogy to Yang-Mills [12–15]. The simplest theory that incorporates torsion is the Einstein-Cartan (EC) gravity [16–30]. The theory is based on the Einstein-Hilbert Lagrangian of GR but in Cartan geometry which is more general then the Rieman- nian one. The connection then splits up into the affine portion (the Levi-Civita symbol exclusive in GR) and a tensor involving the torsion T μ νσ = 1 2 (Γ μ νσ − Γ μ σν ) . That affine geometry with torsion has many applications [31–38]. In this letter we investigate EC gravity with a propagating torsion that produces a dark energy term and gives rise to a bouncing solution. * benidav@post.bgu.ac.il † guendel@bgu.ac.il ‡ venn@fias.uni-frankfurt.de § vasak@fias.uni-frankfurt.de ¶ struckmeier@fias.uni-frankfurt.de ** stoecker@fias.uni-frankfurt.de II. THE THEORY The action integral S =  √ −g (L G + L m ) d 4 x is based in this ansatz on the gravity Lagrangian L G = 1 2 R (Γ) − 4! 2m 2 μ ∂ [μ K ναβ] ∂ [μ K ναβ] , (1) that extends the Einstein-Hilbert term with a dynamical torsion. L G is the gravitational Lagrangian and L m is the matter fields Lagrangian. Here c =  =8πG =1, m μ is a constant with the dimension of mass, g is the determinant of the metric, and R (Γ) is the Ricci scalar. The metric signature is -2. The contortion tensor, K μνσ = 1 2 (T μνσ + T νσμ − T σμν ) , becomes identical with the torsion tensor if the latter is totally anti-symmetric which we assume in the following. The Ricci scalar then splits up into the Levi-Civita dependent part ¯ R and the torsional part: R (Γ) = ¯ R + (2) g μν ( K λ σλ K σ μν − K σ βν K β μσ + ¯ ∇ σ K σ μν − ¯ ∇ ν K λ μλ ) = ¯ R + T σ βν T νβ σ Almost all torsion dependent terms on the r.h.s. vanish since the torsion is taken to be totally anti-symmetric. With this ansatz for torsion applied in the action (1), the torsion acquires a mass term from the Ricci scalar, and retains the quadratic anti-symmetric derivative kinetic term: L G = 1 2 ( ¯ R + T σ βν T νβ σ ) − 4! 2m 2 μ ∂ [μ T ναβ] ∂ [μ T ναβ] . (3) arXiv:2109.01052v1 [astro-ph.CO] 2 Sep 2021