Expanding cosmologies in brane geometries

Five dimensional gravity coupled, both in the bulk and on a brane, to a scalar Liouville ﬁeld yields a geometry conﬁned to a strip around the brane and with time dependent scale factors for the four geometry. In various limits known models can be recovered as well as a temporally expanding four geometry with a warp factor falling exponentially away from the brane. The effective theory on the brane has a time dependent Planck mass and ‘‘cosmological constant.’’ Although the scale factor expands, the expansion is not an accel-eration.

There is considerable interest in theories of gravity and of cosmology with extra dimensions where our world is confined to a four dimensional space-time subspace or 3-brane. All of our known fields, with the exception of gravity, are confined to the brane. The extra dimensions may have compact toroidal topologies ͓1͔ or be unbounded with a scale factor, warp, depending on the ''distance'' from the brane ͓2,3͔. Branes meandering in the internal space, with the brane metric dependent on the internal coordinates, were treated in Ref. ͓4͔. Less singular metrics we obtained in situations where the branes were thickened ͓5͔. As many of these works concerned themselves with the hierarchy problem they restricted themselves to Minkowski metrics on the brane; specifically, the metrics were time independent. Extending these concepts to cosmology requires the introduction of time dependent scale factors on the brane and possibly in the extra dimensions. Solutions in which one posits various stress tensors on the brane and a general review of cosmology restricted to a brane in extra dimensions may be found in Ref. ͓6͔.
In this work we look at a five dimensional bulk whose dynamics is governed by a scalar Liouville field coupled to gravity in the usual way. In addition there is a coupling to the scalar field and to the tension on a thin 3-brane. As in previous works the brane tension is finely tuned to parameters of the bulk action. The general form of the metric we obtain is In the extra dimension the bulk geometry is confined to a finite strip, yрy 0 , around the brane of interest. Although y 0 may be scaled away ͑set equal to one͒, we keep it for the convenience of limiting procedures discussed further on. It will turn out that for у1/2 we may ignore singularities at the edges of the strip; for Ͻ1/2 these singularities force us either to identify the opposite edges of the strip and place a regulator brane at yϭy 0 or place two branes at yϭϮy 0 . As the metric vanishes on these extra branes they do not support any physics.
In addition to the trivial solution with a(t) and b(t) in Eq. ͑1͒ being constant in time, the ansatz a(t)ϭa 0 (t/t 0 ) ␣ , b(t) ϭb 0 (t/t 0 ) ␤ yields a solution provided These satisfy the relation ␤ϭ1Ϫ3␣, reminiscent of one of the Kasner ͓7͔ conditions. For the upper solution, ␣ ranges from 1/2 to 1/3 and ␤ from Ϫ1/2 to 0 as goes from 0 to infinity; for the lower solution ␣ goes from 0 to 1/3 and ␤ from 1 to 0. There are various interesting limits. In addition to being able to recover the geometry of Refs. ͓2,3͔ we can obtain a cosmology where the four metric represents an expanding universe with a warp factor decreasing exponentially as we move away from the brane This limit is interesting as we recover an effective four dimensional cosmology with a time dependent Planck mass. For ϭ0 we recover the Kasner solutions. We shall return to a discussion of these metrics further on. The solution ͑1͒ is obtained from the action for the metric tensor and for a scalar Liouville field with contributions from the bulk and from one or two branes. Five dimensional theories with bulk scalar fields have been previously considered. A massive scalar field can determine the size of the internal dimension ͓8͔ and with intricate self couplings can thicken the branes ͓9,10͔.
The contribution from one of the branes, presumably the one we are on, will be indicated explicitly while the one from the other brane or branes will be left for later elaboration: *Electronic address: mbander@uci.edu SϭS bulk ϩS brane , where M 5 is the five dimensional Planck mass. is a free parameter and although is included for convenience it can be scaled by any positive number through a shift in the field . In S brane n is a spacelike vector normal to the brane and the product ␦(n x )ͱn n g is independent of the magnitude of this vector. Varying the combination ͱgͱn n g with respect to g yields terms proportional to (Ϫg ϩn n /n ␣ n ␤ g ␣␤ ), namely depending only on the metric along the brane; this procedure leads to the same results as one would get by using the Israel junction conditions ͓11͔. As in all previous works we will take n to be along x 5 for which we will use the symbol y. Note that the coupling in S brane is /2 as opposed to in S bulk . The magnitude of the brane tension, determined by h, as in previous works, is related to bulk parameters; our solutions require this restricts у0 for 2 р8/3 and Ͻ0 otherwise. The solution for the equations of motion obtained by varying Eq. ͑4͒ with respect to g and we seek have a metric given in Eq. ͑1͒ and the scalar field of the form

͑6͒
It is straightforward to check that for y 0 Ͼ͉y͉ these are indeed solutions provided the overdot represents differentiation with respect to time.
With the choice of metric in Eq. ͑1͒, of the twenty five equations for the components of the Einstein tensor, R Ϫ•••ϭ0 and the equation of motion for the field only five are nontrivial and independent. These may be chosen to be the equation of motion for and for the tt, ty, yy and any of the diagonal space-space component of the Einstein tensor along the brane. The relations between A, and in Eq. ͑7͒ solve the ty equation while Eq. ͑8͒ takes care of the other four. That, in the bulk, these four equations yield only the three conditions in Eq. ͑8͒ is not surprising as the equation of motion for the field and the Einstein equations for the metric are related by the conservation of the energy-momentum tensor. What is pleasant is that all the four independent equations on the brane, the ones involving ␦(y) terms, where the energy-momentum tensor is not conserved, are also satisfied.
We now turn to possible singularities at ͉y͉ϭy 0 . For Ͼ1/2 or equivalently Ͻͱ3/8 we can restrict the bulk to the strip ͉y͉рy 0 as the solutions discussed above may be continued to the end points. For р1/2 singularities develop at these points and the solutions are no longer valid there. As in many previous discussions of bulk-brane geometries the cure consists of either identifying yϭy 0 with yϭϪy 0 ͑orbifold-ing͒ and introducing a brane at ͉y͉ϭy 0 or introducing independent branes at yϭϮy 0 . In the first case, the action on the yϭy 0 brane is

͑9͒
If one wishes to place branes at both ends of the strip, the action contributed is one half that of Eq. ͑9͒ on each of the two branes. Since the four-metric in Eq. ͑1͒ vanishes at ͉y͉ ϭy 0 , these branes or brane cannot support any physics. The explicit forms for a(t) and b(t) are given in Eq. ͑2͒. For у2 the edge ͉y͉ϭy 0 is at the horizon in that it takes an infinite time to reach it. Certain limits of these solutions are interesting. The case ϭ0 corresponds to Kasner's ͓7͔ solutions. For the a(t) and b(t) constant case the limit →ϱ with y 0 ϭ/(2k) yields the Randall-Sundrum solution ͓2,3͔. Equation ͑5͒ is equivalent to their relation between the bulk cosmological constant and brane tension. In the same limit, but with ␣ and ␤ given by either solution in Eq. ͑2͒ we obtain the metric ͑3͒ and ͑t,y ͒ϭ2 ln͑t ͒

͑10͒
The above solution may be obtained independently from the action SϭS bulk ϩS brane , where

͑11͒
How well gravity on the brane is described by ds 2 ϭdt 2 Ϫa(t) 2 dx 2 depends on solution of the equation with h(y,t) describing fluctuations around the metric. Fluctuation equations for nonzero m are quite complex ͑see, for example, Refs. ͓12,13͔͒ and here we shall restrict our study to mϭ0; it is necessary to have an acceptable solution for this case and it is easy to exhibit such a solution Four dimensional gravity on the brane appears after integrating the action ͑4͒ with the metric over y; (4) g i j (x) is the four metric on the brane. This may be accomplished by using the ADM reduction with b(t) playing the role of the lapse function and conformally transforming the resulting four dimensional metric by the factor (1 Ϫ͉y͉/y 0 ) Ϫ . The result is

͑15͒
The four dimensional theory has a time dependent Planck mass In cosmologies with small extra dimensions, when known physics is not restricted to a brane, time dependence of the internal dimensions is severely restricted by limits on the temporal variations of all fundamental constants ͓14͔; in contrast for theories with most of known physics restricted to a brane, only limits on the time evolution of the Planck mass may come into play. The solutions discussed here can accommodate any such limits as by choosing small and equivalently ␤ small we can make this variation as soft as necessary. The most stringent present limit on Ġ /Gр8 ϫ10 Ϫ12 ͓15͔ translates into a limit on ␤ of ␤р0.1. Temporal variations of the cosmological constant, or more generally of the dark energy are coming into consideration ͓16͔. The solutions presented have ␣р1/2 and thus represent decelerating cosmologies. In line with recent observations ͓16͔ we would like to accommodate an accelerating, expanding scale factor. Having a time dependent scale factor for the external dimensions circumvents some no-go theorems ͓17,18͔ and such cosmologies have been found in M theories can achieve accelerating scale factors by analytically continuing the solutions to negative . This, however, corresponds to an imaginary exponent in the Liouville action. Whether this difficulty can be circumvented is under investigation. Difficulty in finding accelerating solution was noted in Refs. ͓18,19͔.