A ug 2 00 3 Accelerating expansion of the universe may be caused by inhomogenities

We point out that, due to the nonlinearity of the Einstein equations, a homogeneous approximation in cosmology leads to the appearance of an additional term in the Friedmann equation. This new term is associated with the spatial inhomogenities of the metric and can be expressed in terms of density fluctuations. Although it is not constant, it decays much slower (as t− 2 3 ) than the other terms (like density) which decrease as t−2. The presence of the new term leads to a correction in the scaling factor that is proportional to t2 and may give account of the recently observed accelerating expansion of the universe without introducing a cosmological constant. Measurements of the light curves of several hundred type Ia supernovae[1][4] and other independent observations[2]-[8] convincingly demonstrate that the expansion of the universe is accelarating. This unexpected result stimulated a number of theoretical investigations. The explanations suggested so far seem to belong to one of three categories: 1 assuming a nonzero cosmological constant [9], [10], assuming a new scalar field (“quitessence”) [11]-[21], or assuming new gravitational physics [22][27]. In the present paper we suggest a fourth approach, namely, we attribute the accelerated expansion to a consequence of the inhomogenities present in ordinary matter, according to the dynamics described by the usual Einstein equations. We assume a matter dominated and (on the average) flat universe (zero pressure, Ω = 1, k = 0), without cosmological constant or any other exotic constituent. It is natural and usual to apply a homogeneous approximation for the metric, since universe is indeed homogeneous on the large scale. Explicitly, this means that instead of the actual space dependent metric gik(t, r) one uses its spatial average, i.e. gik(t) = lim V→∞ 1 V ∫ V dr gik(t, r) (1) The actual, not precisely homogeneous metric gik(t, r) satisfies Einstein’s equations Rik({gik})− 1 2 gikR({gik}) = 8πGTik (2) Since these equations are nonlinear and since there are strong inhomogenities on smaller scales, the spatially averaged homogeneous metric will not satisfy this equation. In other words, inhomogenities (which, unlike a cosmological constant or quitessence, are unquestionably present) induce a correction term in the Friedmann equation. We demonstrate this below in the framework of second order perturbation theory. First order corrections are well known[28]. We write down the spatial average of the second order perturbation equations, which (as they are linear in the second order correction of the metric and the density) enable us to calculate the spatial average of the metric and the density. The metric is written in the form gjk = g (0) jk + g (1) jk + g (2) jk (3) where the upper, bracketed indices refer to the order of the perturbation. Henceforth we use comoving coordinates. Assuming isotropy, we have g00(t) = c 2 (4) g0α(t) = 0 (5) gαβ(t) = −R (t)δαβ (6) 2 where the bar over quantities means spatial average (cf. Eq.(f2)). Especially, we have g (i) αβ = −δαβ ( R )(i) (t) , i = 0, 1, 2 (7) The use of comoving coordinates imply that T 00 = ρ = ρ + ρ + ρ (8) T 0α = 0 (9) T αβ = 0 (10) First order corrections are given by [28] g (1) αβ = 4cη H 0 [