Abstract. Recently, a Hamilton-Jacobi method beyond semiclassical approximation in black hole physics was developed by Banerjee and Majhi [29]. In this paper, we generalize their analysis of black holes to the case of Friedmann-Robertson-Walker (FRW) universe. It is shown that all the higher order quantum corrections in the single particle action are proportional to the usual semiclassical contribution. The corrections to the Hawking-like temperature and entropy of apparent horizon for FRW universe are also obtained. In the corrected entropy, the area law involves logarithmic area correction together with the standard inverse power of area term.
PACS. 04.70.DyInspired by black hole thermodynamics [1,2], it was realized that there is a profound connection between gravity and thermodynamics. In [3], Jacobson first showed that the Einstein equation can be derived from the proportionality of entropy to the horizon area, together with the Clausius relation δQ = T dS. Here δQ and T are the energy flux and Unruh temperature detected by an accelerated observer just inside the local Rindler causal horizons through spacetime point. Jacobson's derivation has also been applied to f (R) theory [4] and scalar-tensor theory [5], where the non-equilibrium thermodynamics must be taken into account. For other viewpoint see [6].With the spirit of Jacobson's derivation of Einstein field equation, one is able to derive Friedmann equations of a FRW universe with any spatial curvature by applying the Clausius relation to apparent horizon of the FRW universe. For FRW universe [7], after replacing the event horizon by the apparent horizon of FRW universe and assuming that the apparent horizon has an associated entropy S BH and a temperature T 0one can turn the first law of thermodynamics, dE = T 0 dS BH , to the Friedmann equations. Hereh, A, andr A are the Planck constant, area of the apparent horizon, and radius of the apparent horizon, respectively. Here it should be noted that the entropy S BH and temperature T 0 are both the semiclassical results. The first law of thermodynamics not only holds in Einstein gravity, but also in GaussBonnet gravity, Lovelock gravity, and various braneworld a Email: zhut05@lzu.cn scenarios [8,9,10]. The fact that the first law of thermodynamics holds extensively in various spacetime and gravity theories suggests a deep connection between gravity and thermodynamics. (Some other viewpoints and further developments in this direction see [11,12,13,14,15,16] and references therein.) Since we can view a FRW thermodynamical system, like as black holes [17], it is of great interest to ask that whether there is a Hawking-like temperature associated with the apparent horizon of FRW universe. Recently, the scalar particle and fermion's Hawking-like radiation from apparent horizon of FRW universe were investigated by using the semiclassical tunneling method [18,19]. The Hawking-like temperature T 0 =h/2πr A , which associated with the apparent horizon of FRW universe, was recovered.The semiclassical tunneling proc...
