Oxidised cosmic acceleration

TL;DR

This work establishes no-go theorems showing that flat four-dimensional cosmic acceleration from warped extra dimensions necessarily requires violation of higher-dimensional energy conditions (NEC or SEC). By classifying internal manifolds as curvature-free or curved and by treating time-dependent compactifications, it derives thresholds for the four-dimensional equation-of-state parameter $w$ and explicit bounds on the number of e-foldings, using an optimal breathing-mode reduction and a hierarchy of metric-averaging schemes. For curvature-free $\mathcal{M}$, acceleration below a critical $w$ becomes transient or NEC-violating; for curved $\mathcal{M}$, SEC-violation bounds generalize to non-de Sitter cases, with warped de Sitter scenarios constrained by a bounded-average condition that can force NEC violation. The results connect finite-resolution $w$ measurements to constraints on higher-dimensional physics, offering observational handles to rule out large classes of extra-dimensional models while clarifying when NEC/SEC-violating ingredients (e.g., branes, warping) are essential. $

Abstract

We give detailed proofs of several new no-go theorems for constructing flat four-dimensional accelerating universes from warped dimensional reduction. These new theorems improve upon previous ones by weakening the energy conditions, by including time-dependent compactifications, and by treating accelerated expansion that is not precisely de Sitter. We show that de Sitter expansion violates the higher-dimensional null energy condition (NEC) if the compactification manifold M is one-dimensional, if its intrinsic Ricci scalar R vanishes everywhere, or if R and the warp function satisfy a simple limit condition. If expansion is not de Sitter, we establish threshold equation-of-state parameters w below which accelerated expansion must be transient. Below the threshold w there are bounds on the number of e-foldings of expansion. If M is one-dimensional or R everywhere vanishing, exceeding the bound implies the NEC is violated. If R does not vanish everywhere on M, exceeding the bound implies the strong energy condition (SEC) is violated. Observationally, the w thresholds indicate that experiments with finite resolution in w can cleanly discriminate between different models which satisfy or violate the relevant energy conditions.

Figures (8)

• Figure 1: Left panel: Higher-dimensional $\rho+P=T_{00} + T_{\alpha\beta}$ for breathing-mode solutions, with the $y$-axis in arbitrary units, and vertical lines are critical $w_k$ (\ref{['e:CriticalWk']}). The cases $k=1$ (solid) and $k=6$ (dashed) are shown. The NEC is satisfied if $\rho+P > 0$, which only happens for the "$+$" branch when $w_k < w < 1$, and for the "$-$" branch when $w > 1$. Right panel: critical $w_k$ for various dimensions, which is the $w$ that arises from compactification of a positive cosmological constant on a $k$-torus, and also the lower bound for $w$ that can be obtained by breathing-mode dynamics satisfying the NEC.
• Figure 2: The solid curves bound the region $[-v_F,v_F]$ outside of which the NEC is violated. For fixed $w$, trajectories with the maximal number of e-foldings begin with $v$ at the bottom solid curve, and move vertically upwards. The dashed curves show zeros $v_0^\pm$ of the right-hand side of (\ref{['eq:vEq']}), representing fixed points of the evolution of $v$. The solid vertical lines denote the value $w_\times$ at which the fixed point moves into the region $[-v_F,v_F]$. The dashed vertical lines at $w_\Delta$ separate regions where $\Delta$ is real (to the left) and imaginary (to the right). For $k=1,2,3$ any value of $w$ to the left of this line allows $v$ to reach the top curve in finite time, while for values of $w$ to the right $v$ reaches a fixed point and is trapped. For $k=4$, the relevant limit is $w_\Delta$ which is slightly less than $w_\times$.
• Figure 3: A summary of the constraints on curvature-free models. The upper curve denotes the maximal possible exclusion region for no-go theorems, since explicit eternally accelerating models can be constructed with these values of $w$. Below the lower lines, models are forced to have transient acceleration or violate the NEC in the higher-dimensional theory. Below the lower solid line, acceleration must be transient, and below the dashed lines the number of e-foldings is bounded as shown. The $k=4$ case is a continuation of the $k<4$ cases.
• Figure 4: Summary of e-folding constraints for curvature-free $\mathcal{M}$, showing the maximum number of e-folds allowed at a given value of $w$. Left panel: from top to bottom curve, $k=1$ to 4. Right panel: $k=5$ to 15 from left to right, dotted curves for $k=5-9$, and solid curves are for $k \ge 10$.
• Figure 5: Left panel: Value of $w_\Delta$ at which $\Delta_A$ becomes imaginary, as a function of $A$, plotted for $k=5$. The maximum value of $w_\Delta$ is always equal to $w_k$. Right panel: The solid curve shows the lower bound on $A$ coming from positivity requirements in (\ref{['eq:NonPosWarpCond']}): $A$ must lie above this line. After $k=10$, we can choose $A=4$ which gives an optimal average. The long-dashed line shows $A_*$, the value of $A$ for which $w_\Delta$ is largest. The short-dashed line shows $A_{\rm min}$, and for $A < A_{\rm min}$$\Delta$ is always imaginary.