Cosmic perturbations through the cyclic ages

We analyze the evolution of cosmological perturbations in the cyclic model, paying particular attention to their behavior and interplay over multiple cycles. Our key results are: (1) galaxies and large scale structure present in one cycle are generated by the quantum fluctuations in the preceding cycle without interference from perturbations or structure generated in earlier cycles and without interfering with structure generated in later cycles; (2) the ekpyrotic phase, an epoch of gentle contraction with equation of state $w\gg 1$ preceding the hot big bang, makes the universe homogeneous, isotropic and flat within any given observer’s horizon; and (3) although the universe is uniform within each observer’s horizon, the structure of the cyclic universe on very large scales is more complex, owing to the effects of superhorizon length perturbations, and cannot be described globally as a Friedmann-Robertson-Walker cosmology. In particular, we show that the ekpyrotic contraction phase is so effective in smoothing, flattening and isotropizing the universe within the horizon that this phase alone suffices to solve the horizon and flatness problems even without an extended period of dark energy domination (a kind of low energy inflation). Instead, the cyclic model rests on a genuinely novel, noninflationary mechanism (ekpyrotic contraction) for resolving the classic cosmological conundrums.

Figure 1

An example potential $V(\varphi )$. This plot shows where $\varphi$ is on its potential at each stage in a cycle. The equation of state parameter of the background solution is denoted by $w$.

Figure 2

A wheel diagram indicating the behavior of both the background solution (inner, solid, line) and perturbations (outer, dashed and dotted, lines) in the cyclic model. “Start” marks the point in the cycle at which our perturbation analysis in Sec. 3 begins.

Figure 3

A plot of the power spectrum $P_{\Phi }$ of quantum-generated fluctuations in the Newtonian potential $\Phi$ going into the crunch. A horizontal line corresponds to scale invariance. $k_{\mathrm{end}}$ indicates the wave number of the last modes to go ultralocal during ekpyrotic contraction. $k_0$ indicates the scale corresponding to our horizon today. The parameters chosen in Eq. (4) are $b=0.1$ and $c=30$.

Figure 4

A timeline of the cyclic universe showing the behavior of key quantities over the course of a cycle. The labels “$k_{\mathrm{tran}}$ exits” and “$k_{\mathrm{end}}$ exits” show the times where these two modes start to follow ultralocal evolution (i.e. when spatial gradients become negligible).

Figure 5

A plot of the power spectrum $P_{\Phi }$ of the Newtonian potential $\Phi$ (in arbitrary units) at various points in a cosmic cycle. $k=1$ corresponds to the horizon at the start of the calculation in the radiation era, and $\ln k\approx -6.6$ corresponds to the horizon at matter-radiation equality. A horizontal line corresponds to scale invariance. The curve A indicates the power at the end of the matter epoch. Curve B indicates the power at turnaround ($H=0$). Curve C indicates the power some way into ekpyrotic contraction, and curve D indicates the power further into the ekpyrotic phase.