Traversable wormholes and energy conditions with two different shape functions in f (R) gravity

Traversable wormholes, tunnel-like structures introduced by Morris and Thorne [Am. J. Phys. 56 (1988) 395], have a significant role in connection of two different spacetimes or two different parts of the same spacetime. The characteristics of these wormholes depend upon the redshift and shape functions which are defined in terms of radial coordinate. In literature, several shape functions are defined and wormholes are studied in f (R) gravity with respect to these shape functions [F. S. N. Lobo and M. A. Oliveira, Phys. Rev. D 80 (2009) 104012; H. Saiedi and B. N. Esfahani, Mod. Phys. Lett. A 26 (2011) 1211; S. Bahamonde, M. Jamil, P. Pavlovic and M. Sossich, Phys. Rev. D 94 (2016) 044041]. In this paper, two shape functions (i) b(r) = r(0) log(r+1) / log(r(0)+1) and (ii) b(r) = r0(r/r0)(gamma), 0 < gamma < 1, are considered. The first shape function is newly defined, however, the second one is collected from the literature [M. Cataldo, L. Liempi and P. Rodriguez, Eur. Phys. J. C 77 (2017) 7481. The wormholes are investigated for each type of shape function in f (R) gravity with f(R) = R + alpha R-m - beta R-n , where m, n, alpha and beta are real constants. Varying the parameter alpha or beta, f(R) model is studied in five subcases for each type of shape function. In each case, the energy density, radial and tangential pressures, energy conditions that include null energy condition, weak energy condition, strong energy condition and dominated energy condition and anisotropic parameter are computed. The energy density is found to be positive and all energy conditions are obtained to be violated which support the existence of wormholes. Also, the equation-of-state parameter is obtained to possess values less than -1, that shows the presence of the phantom fluid and leads toward the expansion of the universe.