Cascading Cosmology

TL;DR

This work constructs a covariant 5D proxy for 6D cascading gravity to study brane-world cosmology. By introducing a brane-bending scalar π and carefully handling boundary terms and junction conditions, the authors derive a self-consistent framework for a moving 3-brane in a static bulk. They show that strong- and weak-coupling regimes yield analytical and numerical insights, with π contributing to late-time acceleration but a brane embedding singularity preventing full degravitation. The results suggest degravitation-like behavior may be achieved in time-dependent or thickness-tiered extensions, offering a pathway toward an accelerating universe without a cosmological constant, albeit with caveats on stability and singularities.

Abstract

We develop a fully covariant, well-posed 5D effective action for the 6D cascading gravity brane-world model, and use this to study cosmological solutions. We obtain this effective action through the 6D decoupling limit, in which an additional scalar degree mode, π, called the brane-bending mode, determines the bulk-brane gravitational interaction. The 5D action obtained this way inherits from the sixth dimension an extra πself-interaction kinetic term. We compute appropriate boundary terms, to supplement the 5D action, and hence derive fully covariant junction conditions and the 5D Einstein field equations. Using these, we derive the cosmological evolution induced on a 3-brane moving in a static bulk. We study the strong- and weak-coupling regimes analytically in this static ansatz, and perform a complete numerical analysis of our solution. Although the cascading model can generate an accelerating solution in which the πfield comes to dominate at late times, the presence of a critical singularity prevents the πfield from dominating entirely. Our results open up the interesting possibility that a more general treatment of degravitation in a time-dependent bulk, or taking into account finite brane-thickness effects, may lead to an accelerating universe without a cosmological constant.

Figures (2)

• Figure 1: Evolution of the effective equation of state, $w_{\pi}=-1-(1/3){\rm d}\ln \rho_\pi/{\rm d}\ln a$, for the $\pi$-dependent modifications to the Friedmann equation (\ref{['rhopi']}). The numerical results are consistent with the analytical predictions for the large (upper panel) and small (lower panel) $m_6$ limits in the strong- ($a\ll1$) and weak-coupling ($a\gg1)$ regimes. Here we use the numerical values (in natural units $c=\hbar=1$): $H_{0} = 2.33 \times 10^{-4}$ Mpc$^{-1}$ (i.e. $H_0=70$ km s$^{-1}$Mpc$^{-1}$), (upper panel) $m_{6} = 10^{30}$ Mpc$^{-1}$ ($m_{6} \gg H$) and $m_{5} = 10^{-40}$ Mpc$^{-1}$ and (lower panel) $m_{6} = 10^{-15}$ Mpc$^{-1}$ ($m_{6} \ll H$) and $m_{5} = 10^{-30}$ Mpc$^{-1}$. The $\pi$ field is a subdominant component of the total energy density at all times, and late-time acceleration is driven by $\Lambda$.
• Figure 2: Example evolution histories in which no cosmological constant is present to drive cosmic acceleration. [Top panel] The deviation of the expansion history from that derived from standard matter (for which $3H^2/\rho_m=1$). The blue and red curves each show consistent solutions to the modified Friedmann equation (\ref{['staticjc']}): one solution (red, thick line) recovers the standard expansion history at early times and then undergoes accelerated expansion at late times; the other solution (blue, dotted line) has an expansion history entirely inconsistent with that of standard $\Lambda$CDM, with the $\pi$ field dominating the expansion at all eras, and undergoing heavily decelerated expansion at late times. [Center panel] The evolution of the effective fractional energy density, $\Omega_{\pi}=8\pi G\rho_\pi/3H^2$, for the two solutions discussed above. For the accelerating solution, the phantom-like behavior in the matter era allows the $\pi$ field to dominate and drive cosmic acceleration at late times. The model is not physical, however, since as $\Omega_\pi\rightarrow 2/3$ one finds $\dot{H}\rightarrow \infty$ and a singularity occurs. [Bottom panel] A comparison of the effective equation of state for the expansion, $w_{\rm eff} = -1-(2/3){\rm d}\ln H/{\rm d}\ln a$, for the accelerating $\pi$ (red, full line) and fiducial $\Lambda$CDM (black, dashed line) scenarios. For the $\pi$ driven expansion histories, we use the numerical values $H_{0} = 2.33 \times 10^{-4}$ Mpc$^{-1}$, $m_{6} = 3.5\times 10^{-18}$ Mpc$^{-1}$ and $m_{5} = 4.4\times 10^{-31}$Mpc$^{-1}$ for which the maximum singularity occurs just after $a=1$.