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Kasner and Mixmaster behavior in universes with equation of state w \ge 1

Joel K. Erickson, Daniel H. Wesley, Paul J. Steinhardt, Neil Turok

Abstract

We consider cosmological models with a scalar field with equation of state $w\ge 1$ that contract towards a big crunch singularity, as in recent cyclic and ekpyrotic scenarios. We show that chaotic mixmaster oscillations due to anisotropy and curvature are suppressed, and the contraction is described by a homogeneous and isotropic Friedmann equation if $w>1$. We generalize the results to theories where the scalar field couples to p-forms and show that there exists a finite value of $w$, depending on the p-forms, such that chaotic oscillations are suppressed. We show that $Z_2$ orbifold compactification also contributes to suppressing chaotic behavior. In particular, chaos is avoided in contracting heterotic M-theory models if $w>1$ at the crunch.

Kasner and Mixmaster behavior in universes with equation of state w \ge 1

Abstract

We consider cosmological models with a scalar field with equation of state that contract towards a big crunch singularity, as in recent cyclic and ekpyrotic scenarios. We show that chaotic mixmaster oscillations due to anisotropy and curvature are suppressed, and the contraction is described by a homogeneous and isotropic Friedmann equation if . We generalize the results to theories where the scalar field couples to p-forms and show that there exists a finite value of , depending on the p-forms, such that chaotic oscillations are suppressed. We show that orbifold compactification also contributes to suppressing chaotic behavior. In particular, chaos is avoided in contracting heterotic M-theory models if at the crunch.

Paper Structure

This paper contains 14 sections, 56 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A "Cosmic No--Hair Theorem" for Contracting Universes
  3. The Curvature--Free Case
  4. $w < 1$:
  5. $w = 1$:
  6. $w > 1$:
  7. Curvature and Chaos
  8. $w < 1$:
  9. $w = 1$:
  10. $w > 1$:
  11. Coupling to p-forms and Chaos
  12. Time-varying equation of state and chaos
  13. Extra Dimensions, Orbifolds and Chaos
  14. Conclusions

Figures (2)

  • Figure 1: The Kasner plane $p_1+p_2+p_3=1$ and its intersections (the Kasner circles) with various spheres $p_1^2 + p_2^2 + p_3^2 = 1 -q^2$ where $q^2=\tfrac{2}{3}(1-\Omega_\sigma)$; see (\ref{['eq:q2']}). The vacuum solution corresponds to $\Omega_\sigma=1$ (the outermost circle). The inner circles are relevant to the case where $w=1$ and $\Omega_\sigma<1$. In the white regions, the Kasner exponents are all positive (corresponding to contraction); in gray regions, one exponent is negative (expanding). If the spatial curvature is non-zero, points along the circles in the white region (thick parts of circles) are stable but points in the gray regions (dashed parts of circles) are unstable, jumping to new values after a short period of contraction. If a model (i.e. a circle) has an open set of stable points (the three innermost circles but not the outermost circle), the contracting phase does not exhibit chaotic mixmaster behavior.
  • Figure 2: The four dimensional electric and magnetic couplings $\lambda$ as a function of the critical equation of state for $p=0,1,2$. The upper and lower three curves represent the critical electric and magnetic exponents, respectively. A form with given $p$ and $\lambda$ is stable in a universe with equation of state $w$ if the point $(w,\lambda)$ lies between the two curves for the given $p$.