A no-go for no-go theorems prohibiting cosmic acceleration in extra dimensional models

Abstract

A four-dimensional effective theory that arises as the low-energy limit of some extra-dimensional model is constrained by the higher dimensional Einstein equations. Steinhardt & Wesley use this to show that accelerated expansion in our four large dimensions can only be transient in a large class of Kaluza-Klein models that satisfy the (higher dimensional) null energy condition [1]. We point out that these no-go theorems are based on a rather ad-hoc assumption on the metric, without which no strong statements can be made.

Constraints and horizons for de Sitter with extra dimensions

“…While we are interested primarily in 4-dimensional dS space, [3,4] are interested more generally in accelerating cosmologies with a more general time-dependence, and specifically require integration over the internal dimensions in order to draw conclusions. A list of assumptions necessary to derive the results of [4] are conveniently summarized in [5]; in Table I, we briefly review some of the assumptions from [1,2] and [4] and how they compare to our assumptions. We see in general that our constraint requires fewer assumptions than previous no-gos.…”

“…In particular, there are a number of "no-go" theorems that detail how the matter content of the higher dimensional theory must violate an energy condition in the extra dimensions in order to obtain 4-dimensional dS [1][2][3][4][5][6][7]. Some separate related lines of investigation have been concerned with whether dS space can be explicitly realized in compactifications of string theory [8][9][10][11][12][13][14][15][16][17][18][19][20][21], see also [22][23][24], as well as the consistency of the model known as KKLT [25] as in [26][27][28][29][30][31][32][33].…”

Singularity theorems for warped products and the stability of spatial extra dimensions

“…Obviously, the assumptions are the same, so that only the equivalence of ( 3) with ( 21) must be justified. And this follows from the analysis in subsection 3.3, specifically from expression (20) (times f 4 ), which gives immediately the lefthand side of ( 21) together with the following straightforward calculation for the remaining term in (20), containing derivatives of f along γ, on using ( 19)…”

“…For instance, in [33] how accelerated expansion imposes strong constraints on compact extra dimensions was discussed, implying either (i) that both the gravitational constant G and the equation of state parameter w depend on time, or (ii) violation of the null convergence condition (1) in an inhomogenous way across the extra dimensions together with a sinchronyzed variation with the observable matter and "dark energy". Their conclusions were nevertheless criticized in [20], where one can find references to many other no-go and instability theorems [20].…”

“…The above displayed expression gives the righthand side of the condition (4) in Theorem 2.3. To find the lefthand side, it is enough to use (20). We thus have Theorem 3.2.…”