Cosmic Evolution in a Cyclic Universe

TL;DR

The paper proposes a detailed cyclic cosmology in which the Universe undergoes endless cycles of expansion and contraction, connected by a non-singular big crunch–big bang transition mediated by brane dynamics and a four-dimensional scalar field φ. A key mechanism is the inter-brane potential V(φ) with a negative ekpyrotic region that generates a scale-invariant density perturbation spectrum during contraction, and a positive plateau that drives late-time cosmic acceleration to reset initial conditions before each bounce. The authors argue that the cyclic solution is a robust attractor with a wide basin of attraction, largely independent of initial conditions, and discuss implications for fundamental physics, including the nature of dark energy, the cosmological constant problem, and potential consequences for supersymmetry and string theory. They also address how observable features such as homogeneity, flatness, and perturbations arise, and how the bounce can be consistently matched within a higher-dimensional framework using Israel matching and energy–momentum conservation. Overall, the cyclic model provides an integrated, low-energy alternative to inflation that links past, present, and future cosmic evolution through brane cosmology and ekpyrotic dynamics.

Abstract

Based on concepts drawn from the ekpyrotic scenario and M-theory, we elaborate our recent proposal of a cyclic model of the Universe. In this model, the Universe undergoes an endless sequence of cosmic epochs which begin with the Universe expanding from a `big bang' and end with the Universe contracting to a `big crunch.' Matching from `big crunch' to `big bang' is performed according to the prescription recently proposed with Khoury, Ovrut and Seiberg. The expansion part of the cycle includes a period of radiation and matter domination followed by an extended period of cosmic acceleration at low energies. The cosmic acceleration is crucial in establishing the flat and vacuous initial conditions required for ekpyrosis and for removing the entropy, black holes, and other debris produced in the preceding cycle. By restoring the Universe to the same vacuum state before each big crunch, the acceleration insures that the cycle can repeat and that the cyclic solution is an attractor.

Figures (6)

• Figure 1: The interbrane potential $V(\phi)$ versus $\phi$, whose value ($-\infty<\phi<\phi_{\infty}$) determines the distance between branes. The shaded circle represents the maximum positive value of $\phi$ during the cycle. The various stages are: (1) quintessence/potential domination and cosmic acceleration (duration $\ge$ trillion years); (2) $\phi$ kinetic energy becomes non-negligible, decelerated expansion begins (duration $\sim$ 1 billion years); (3) $H=0$, contraction begins; (4) density fluctuations on observed scales created ($(t_0 t_R)^{1/2} \approx 1$ ms before big crunch); (5) $\phi$ kinetic energy domination begins ($t_{min} \sim 10^{-30}$ s before big crunch); (6) bounce and reversal from big crunch to big bang; (7) end of $\phi$ kinetic energy domination, potential also contributes ($t_{min} \sim 10^{-30}$ s after big bang); (8) radiation dominated epoch begins $t_R \sim 10^{-25}$ s after big bang); (9) matter domination epoch begins ($\sim 10^{10}$ s after big bang). As the potential begins to dominate and the Universe returns to stage (1), the field turns around and rolls back towards $-\infty$.
• Figure 2: Schematic plot of the scale factor $a(t)$, the modulus $\phi(t)$, and $H_5 \equiv {2\over 3} d( {\rm exp}(\sqrt{3/2} \phi)/dt$ for one cycle, where $t$ is Einstein frame proper time. The scale factor starts out zero but expands as $t^{1\over 3}$, and the scalar field grows logarithmically with $t$, in the scalar kinetic energy dominated early regime. Then, when radiation begins to dominate we have $a \propto t^{1\over 2}$, and the scalar field motion is strongly damped. This is followed by the matter era, where $a \propto t^{2\over 3}$, and a potential dominated phase in which $a(t)$ increases exponentially, before a final collapse on a timescale $H_0^{-1}$, to $a=0$ once more. $H_5$ is proportional to the proper (five dimensional) speed of contraction of the fifth dimension. To obtain a cyclic solution, the magnitude of $H_5$ at the start of the big bang, $H_5({out})$, must be slightly larger than the value at the end of the big crunch, $H_5({in})$. This is the case if more radiation is generated on the negative tension brane (see Appendix).
• Figure 3: Schematic plot of the $a_0$-$a_1$ plane showing a sequence of cycles of expansion and contraction (indicated by tick marks). The dashed line represents the "light-cone" $a_0 =a_1$ corresponding to a bounce ($a=0$). Each cycle includes a moduli kinetic energy, radiation, matter and quintessence dominated phase and lasts an exponentially large number of e-folds. The insert shows the trajectory near the big crunch and bounce. The potential energy $V(\phi)$ assumed takes the form shown in Fig. 1.
• Figure 4: The cyclic trajectory in the $(H_5,\phi)$-plane for the case where no matter and radiation are produced at the bounce ($\chi=0$). The grey region which corresponds to negative energy density, is forbidden. The solid (dashed) line represents the trajectory during an expanding (contracting) phase. Expansion turns to contraction and vice versa when the trajectory hits the zero energy surface (the rightmost tip of grey region in this case).
• Figure 5: Trajectories in the $(H_5,\phi)$-plane for the case where there is radiation. The solid (dashed) curves represent the trajectory during an expanding (contracting) phase. The thin lines illustrate undershoot solutions and the heavy line represents an overshoot solution.