Abstract
Recent astronomical observations of supernovae (SNIa) and barionic acoustic oscilations (BAO) indicate that the Universe is in an accelerated expansion period. Interpreted within the framework of general relativity (GR), the acceleration is explained by a positive cosmological constant or exotic matter models known in the literature as dark energy. However, there is an alternative approach to explain the acceleration without exotic matter models. Modifications of GR such as scalar–tensor gravity and high-order derivative gravity theories, naturally offer the explanation for the accelerated phase coming from the geometrical side. One of this higher-order theories is f(R) modified gravity. In this work, we use some mathematical results concerning to the Taylor expansions of tensor fields under the action of one-parameter families of diffeomorphism in the context of f(R) theories in the expanding universe. We mean gauge invariant in the sense of the second-kind gauge following the work exposed in Nakamura (Adv. Astr. 2010(576273):2010). We obtain the general gauge invariant at first-order equations in f(R) gravity. As an example, we write these first-order equations in f(R) gravity for a perturbed Friedmann–Lemaître–Robertson–Walker (FLRW) space-time. The gauge invariant scalar perturbations equations for perturbed FLRW are obtained explicitly in f(R) gravity.
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Notes
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like perturbation theory in quantum mechanics.
References
Astier, P. et al.: The Supernova Legacy Survey: measurement of \({\Omega}_M, {\Omega}_{\Lambda}\) and w from the first year data set. Astron. Astrophys. 447, 3–1 (2006)
Bean, R., Bernat, D., Pogosian, L., Silvestri, A., Trodden, M.: Dynamics of Linear Perturbations in f(R) Gravity. Phys. Rev. D 75, 06402–0 (2007). astro-ph/0611321v2
Bruni, M., Materrese, S., Mollerach, S., Sonego, S.: Perturbations of spacetime: gauge transformations and gauge invariance at second order and beyond. Class. Quantum Grav. 14, 2585–2606 (1997)
Clifton, T., Ferreira, P.G., Padilla, A., Skordis, C.. Modified gravity and cosmology. Phys. Rep. 513(1–3), 1–189 (2012)
de la Cruz-Dombriz, A., Dobado, A., Maroto, A.L.: On the evolution of density perturbations in f(R) theories of gravity. Phys. Rev. D 77, 12351–5 (2008). arXiv:0802.2999v2
Daly, R.A., Djorgovski, S.G.: A Model-Independent Determination of the Expansion and Acceleration Rates of the Universe as a Function of Redshift and Constraints on Dark Energy. Astrophys. J. 597, 9 (2003)
De Felice, A., Tsujikawa, S.: f(R) theories. Living Rev. Relativ. 13, 3 (2010)
Durrer, R.: The cosmic microwave background.Cambridge University Press Date Published: September 2008 isbn: 9780521847049.http://www.cambridge.org/9780521847049
Guarnizo, A., Castañeda, L., Tejeiro, J.M.: Boundary term in metric f(R) gravity: field equations in the metric formalism. Gen. Rel. Grav. 42, 2713–2728 (2010)
Hortua, J., Castañeda, L., Tejeiro, J.M.: Evolution of magnetic fields through cosmological perturbation theory. Phys. Rev. D 87, 10353–1 (2013)
Kodama, H., Sasaki, M.: Cosmological perturbation theory. Prog. Theor. Phys. Suppl. 78, 141–142 (1984)
Molano, D.: Teoría de perturbaciones cosmológicas en teorías de gravedad modificada f(R). M.Sc. thesis, Universidad Nacional de Colombia, Observatorio Astronómico Nacional (to be submitted)
Nakamura, K.: Second-order gauge-invariant cosmological perturbation heory: current status. Adv. Astr. 2010, ID 576273 1–26 (2010)
Padmanabhan, T.: Equipartition energy, Noether energy and boundary term in gravitational action. Gen. Rel. Grav. 44, 268–1 (2012)
Perlmutter, S. et al. [Supernova Cosmology Project Collaboration]: Measurements of Omega and Lambda from 42 high redshift supernovae. Astrophys. J. 517, 565 (1999)
Pogosian, L., Silvestri, A.: The pattern of growth in viable f(R) cosmologies. Phys. Rev. D 77, 02350–3 (2008). arXiv:0709.0296v3
Riess, A.G. et al. [Supernova Search Team Collaboration]: Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astron. J. 116, 1009 (1998)
Riess, A.G., et al.: Type Ia Supernova Discoveries at z > 1 from the Hubble Space Telescope: Evidence for Past Deceleration and Constraints on Dark Energy Evolution. Astrophys. J. 607, 665 (2004)
Schouten, J.A.: Ricci Calculus. Springer, Berlin (1954)
Sopuerta, C., Bruni, M., Gualtieri, L.: Non-linear N-parameter spacetime perturbations: Gauge transformations. Phys. Rev. D 70, 06400–2 (2004)
Stewart, J.M., Walker, M.: Perturbations of space-times in general relativity. Proc. R. Soc. Lond. A 341, 49–74 (1974)
Stewart, J.M.: Perturbations of Friedmann-Robertson-Walker cosmological models. Class. Quantum Grav 7 1169–1180 (1990)AQ7
Wald, R.M.: General Relativity. The University of Chicago Press, Chicago (1984)
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Molano, D., Castañeda, L. (2015). Some Mathematical Aspects in the Expanding Universe. In: Tost, G., Vasilieva, O. (eds) Analysis, Modelling, Optimization, and Numerical Techniques. Springer Proceedings in Mathematics & Statistics, vol 121. Springer, Cham. https://doi.org/10.1007/978-3-319-12583-1_20
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