Formation and evolution of a 2-brane structure in multidimensional $f(R)$ gravity

TL;DR

This work investigates multidimensional $f(R)$ gravity with compact extra dimensions that naturally produce a dynamical two-brane structure. It shows that two 4D branes nucleate at the highest energy scales with a Planck-Scale inter-brane separation that grows as the universe cools (lower $H$), while the effective 4D Planck mass $m_4$ and the Higgs VEV $v_H$ vary with the Hubble parameter, the latter differing between branes. In the low-energy regime, the effective cosmological constant satisfies $\Lambda_4 \approx 3H^2$, and Higgs physics on our brane matches the observed $v_H=246$ GeV, whereas on the hidden brane the Higgs VEV is exponentially larger, potentially yielding enormous fermion masses there. The analysis reveals a barrier-induced decoupling of Higgs fluctuations between branes, justifying independent brane dynamics and motivating extensions to multi-brane setups with rich phenomenology.

Abstract

It has been previously shown that multidimensional $f(R)$ gravity {can lead} to a two-brane structure. In this paper, we analyze such a model with a spatially flat 4D de Sitter (dS) cosmology {whose Hubble parameter $H$ determines the universal energy scale}. We show that the two-brane metric is nucleated at the highest energies. The distance between the branes grows gradually as the energy decreases, tending to a finite value at zero energy density. It is stated that the physical parameters such as the 4D Planck mass, the Higgs vacuum expectation value, and vacuum energy density vary with the evolving universal energy scale, even on the classical level. We also show that the Higgs vacuum expectation value is different on different branes.

Figures (5)

• Figure 1: Example of a solution to the equations of motion for $n=2$, $f(R) = 300R^2 +R +0.002$ and $H=0$. Left panel: the extra-dimensional radius $r(u)$, middle: the warp factor ${\,\rm e}^{\gamma(u)}$, right: the Ricci scalar of extra dimensions. The initial conditions: $r(0)=50, \ \gamma(0) =1, \ r'(0) = 1.6, \ \gamma'(0) = 0, \ R'(0) =-10^{-5}$, and $R(0)$ is determined by \ref{['cons']}.
• Figure 2: Left panel: the distance between branes, right panel: variation of the Planck mass $m_4$ with the Hubble parameter. The solution parameters are: $n=2, f(R) = 300R^2 +R +0.002$. The curves are parametrized by the conditions $r(0)=50, \ \gamma(0) =0, \ r'(0) = 1.6, \ \gamma'(0) = 0, \ R'(0) =-10^{-5}$, $R(0)$ is determined by \ref{['cons']}.
• Figure 3: The Higgs field distribution along the internal coordinate $u$ is specific to each brane. The background metric functions are presented in Fig. \ref{['sol']}. The parameters are $n=2, f(R) = 300R^2 +R +0.002$, $H=0, \ \nu=0.1, \lambda=0.1$. The curve is parametrized by the conditions $r(0)=50,\ \gamma(0) =0,\ r'(0) = 1.6,\ \gamma'(0) = 0,\ R'(0) =-10^{-5}$, $R(0)$ is determined by \ref{['cons']}, $U_h(0)=1,\ U_h'(0)=1 \cdot 10^{-8}$.
• Figure 4: A barrier between the two branes prevents an exchange of Higgs field fluctuations between them. The parameters are $n=2, f(R) = 300R^2 +R +0.002, H=0$. The curve is parametrized by the conditions $r(0)=50, \ \gamma(0) =0, \ r'(0) = 1.6, \ \gamma'(0) = 0, \ R'(0) =-10^{-5}$, $R(0)$ is determined by \ref{['cons']}, $\nu=0.1$.
• Figure 5: The Higgs VEV near brane-1 as a function of $H$. The parameters are $n=2, f(R) = 300R^2 +R +0.002$, $\nu=0.1, \lambda=0.1$. The curve is parametrized by the conditions $r(0)=50, \ \gamma(0) =0, \ r'(0) = 1.6, \ \gamma'(0) = 0, \ R'(0) =-10^{-5}, \ R(0)$ is determined by \ref{['cons']}, $U_h(0)=0.7 , \ U_h'(0)=-1 \cdot 10^{-7}$.