The flat Friedmann universes filled by stiff fluid and a nonminimally coupled material scalar field with polynomial potentials of the fourth degree are considered in the framework of the Einstein-Cartan theory. Exact general solution is obtained for arbitrary positive values of the coupling constant . A comparative analysis of the cosmological models with and without stiff fluid is carried out. Some effects of stiff fluid are elucidated. It is shown that singular models with a de Sitter asymptotic and with the power-law asymptotic at late times are possible. It is found that is a specific value of the coupling constant. It is demonstrated that the bouncing models without the particle horizon and with an accelerated expansion by a de Sitter law of an evolution at late times are admissible. 1. Introduction Recent cosmic observations [1–6] favor an isotropic spatially flat Universe, which is at present expanding with acceleration. The source of this expansion is an unknown substance with negative pressure called dark energy (DE). Establishment of the origin of DE has become an important problem. Different theoretical models of DE have been put forward (see, e.g., the reviews [7–10] and references therein). Among these models various modifications of general relativity (GR) were considered, the Einstein-Cartan theory (ECT) in particular [11–13]. This theory [14–17] is an extension of GR to a space time with torsion, and it reduces to GR when the torsion vanishes. The ECT is the simplest version of the Poincaré gauge theory of gravity (PGTG). It should be noted that the ECT contains a nondynamic torsion, because its gravitational action is proportional to the curvature scalar of the Riemann-Cartan space-time. In this sense, the ECT is a degenerate gauge theory [17–20]. This drawback is absent in the PGTG since its gravitational Lagrangian includes invariants quadratic in the curvature and torsion tensors. Nevertheless the ECT is a viable theory of gravity whose observational predictions are in agreement with the classical tests of GR, and it differs significantly from GR only at very high densities of matter [17, 21, 22]. The ECT finds applications in cosmology [23–28], particle theory [19, 29, 30], and the theory of strong interactions [31, 32]. From some time past, the interest to ECT has grown in connection with the fact that torsion arises naturally in the supergravity [33–35], Kaluza-Klein [36–38], and syperstring [39–41] theories. gravity with torsion has been developed [42–45] as one of the simplest extensions of the ECT. In  it has been
A. G. Riess, L.-G. Sirolger, J. Tonry et al., “Type Ia supernova discoveries at from the hubble space telescope: evidence for past deceleration and constraints on dark energy evolution,” The Astrophysical Journal, vol. 607, article 665, 2004.
E. Komatsu, J. Dunkley, M. R. Nolta et al., “Five-year Wilkinson microwave anisotropy probe observations: cosmological interpretation,” The Astrophysical Journal Supplement Series, vol. 180, no. 2, article 330, 2009.
W. M. Wood-Vasey, G. Miknaitis, C. W. Stubbs et al., “Observational constraints on the nature of dark energy: first cosmological results from the essence supernova survey,” Astrophysical Journal Letters, vol. 666, no. 2, pp. 694–715, 2007.
M. Tegmark, M. R. Blanton, M. A. Strauss et al., “The three-dimensional power spectrum of galaxies from the sloan digital sky survey,” Astrophysical Journal Letters, vol. 606, no. 2, pp. 702–740, 2004.
P. Baekler and F. W. Hehl, “Beyond Einstein-Cartan gravity: quadratic torsion and curvature invariants with even and odd parity including all boundary terms,” Classical and Quantum Gravity, vol. 28, no. 21, Article ID 215017, 2011.
I. L. Buchbinder and S. D. Odintsov, “The behaviour of effective coupling constants in “Finite” grand unification theories in curved space-time with torsion,” Europhysics Letters, vol. 8, no. 7, article 595, 1989.
A. O. Barvinsky, A. Y. Kamenshchik, C. Kiefer, A. A. Starobinsky, and C. Steinwachs, “Asymptotic freedom in inflationary cosmology with a non-minimally coupled Higgs field,” Journal of Cosmology and Astroparticle Physics, vol. 2009, article 003, 2009.
K. Bamba, S. Capozziello, S. Nojiri, and S. D. Odintsov, “Dark energy cosmology: the equivalent description via different theoretical models and cosmography tests,” Astrophysics and Space Science, vol. 342, no. 1, pp. 155–228, 2012.