Fractional action cosmology with an effective \(\wedge\) -term


  • Victor Shchigolev Department of Theoretical Physics, Ulyanovsk State University





Cosmological Models, Effective Cosmological Term, Exact Solutions, Fractional Einstein-Hilbert Action.


We continue studying the cosmological models derived from the fractional variational principle applied to the gravitational sector of the action functional. Within the frame of these models, the effective cosmological term could arise as a result of the non-zero Hubble parameter. At the same time, the continuity equation for the matter remains unchanged in its standard form. In this work, we are going to obtain some exact solutions for our model originating from the several kinematic assumptions. At that, we find the main cosmography parameters of the model and the corresponding equations of state of matter that fills the universe se. First, we proceed from an initially given law of evolution of the universe in some standard scenarios. Then, several exact solutions are obtained from the proposed earlier evolutionary laws for the effective cosmological term.


[1] Riess A G, et al. (1998), Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant. Astronomical Journal, 116: 1009.

[2] Perlmutter S, et al. (1999), Measurements of Omega and Lambda from 42 High-Redshift Supernovae. Astrophysical Journal, 517: 565.

[3] Netterfield C B, et al. (2002), A Measurement by Boomerang of Multiple Peaks in the Angular Power Spectrum of the Cosmic Microwave Background. Astrophysical Journal, 571: 604-614.

[4] Halverson N W, et al. (2002), Degree Angular Scale Interferometer First Results: A Measurement of the Cosmic Microwave Background Angular Power Spectrum. Astrophysical Journal, 568: 38-45.

[5] Spergel D N, et al. (2003), First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters. Astrophysical Jounal Supplement Series, 148: 175-194.

[6] Tegmark M, Strauss M A, et al. (2004), Cosmological parameters from SDSS and WMAP. Physical Review D 69: 103501.

[7] Allen S W, Schmidt R W, Ebeling H, et al. (2004), Constraints on dark energy from Chandra observations of the largest relaxed galaxy clusters. Monthly Notices of the Royal Astronomical Society, 353: 457-467.

[8] Peebles P J E, Ratra B (2003), the cosmological constant and dark energy. Reviews of Modern Physics 75: 559.

[9] Sahni V, Starobinsky A (2000), the Case for a Positive Cosmological -Term. International Journal of Modern Physics D 9: 373.

[10] Caldwell R R (2002), A phantom menace? Cosmological consequences of a dark energy component with super-negative equation of state. Physics Letters B 545: 23.

[11] Sen A (2002), Tachyon Matter. High Energy Physics 7: 65.

[12] Kamenshchik A, Moschella U, Pasquier V (2001), an alternative to quintessence. Physics Letters B 511: 265.

[13] Feng B, M. Li M, Piao Y S, Zhang X (2006), Oscillating quintom and the recurrent universe. Physics Letters B 634: 101.

[14] Horava P, Minic D (2000), Probable Values of the Cosmological Constant in a Holographic Theory. Physical Review Letters 85: 1610.

[15] Shchigolev V K (2011), Modelling Cosmic Acceleration in Modified Yang - Mills Theory. Gravitation and Cosmology 17: 272-275.

[16] Shchigolev V K, Orekhova G N (2011), Non-Minimal Cosmological Model in Modified Yang--Mills Theory. Modern Physics Letters a 26 (26): 1965-1973.

[17] Copeland E J, Sami M, Tsujikawa S (2006), Dynamics of dark energy. International Journal of Modern Physics D 15: 1753.

[18] Nojiri S, Odintsov S D, et al. (2007), Dark energy from modified F(R)-scalar-Gauss-Bonnet gravity. Physics Letters B 651: 224.

[19] Myrzakulov R (2012), Dark Energy in F(R, T) Gravity, arXiv: 1205.5266 [gr-qc].

[20] El-Nabulsi A R (2007), Cosmology with a Fractional Action Priciple. Romanian Reports in Physics 59 (3): 763.

[21] El-Nabulsi A R (2012), Gravitons in Fractional Action Cosmology. International Journal of Theoretical Physics 51 (12): 3978.

[22] El-Nabulsi A R (2013), Fractional derivatives generalization of Einstein’s field equations. Indian Journal of Physics 87 (2): 195.

[23] Sadallah M, Muslih S I, Baleanu D, Rabei E (2011), Fractional Time Action and Perturbed Gravity. Fractals 19 (2): 243.

[24] Shchigolev V K (2011), Cosmological Models with Fractional Derivatives and Fractional Action Functional. Communications in Theoretical Physics 56: 389.

[25] Shchigolev V K (2013a), Cosmic Evolution in Fractional Action Cosmology. Discontinuity, Nonlinearity, and Complexity 2 (2): 115.

[26] Shchigolev V K (2013), Fractional Einstein-Hilbert Action Cosmology. Modern Physics Letters A 28 (14): 1350056.

[27] Sahni V, Saini T D, Starobinsky A, Alam U (2003), Statefinder – a new geometrical diagnostic of dark energy. JETP Letters 77: 201.

[28] Debnath U, Jamil M, Chattopadhyay S (2010), Fractional Action Cosmology: Emergent, Logamediate, Intermediate, Power law Scenarios of the Universe and Generalized Second Law of Thermodynamics of dark energy. International Journal of Theoretical Physics 51: 812-837.

[29] Shchigolev V K (2015), Testing Fractional Action Cosmology, arXiv: 1512.04113 [gr-qc].

[30] Overduin J M, Cooperstock F I (1998), Evolution of the scale factor with a variable cosmological term. Physical Review D 58 (4): 043506.

[31] Brychkov Yu A, Prudnikov A P (2001), Whittaker function, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4.

View Full Article: