Gluon condensate, modified gravity, and the accelerating Universe

It has been suggested recently to study the dynamics of a gravitating gluon condensate $q$ in the context of a spatially flat Friedmann-Robertson-Walker universe. The expansion of the Universe (or, more generally, the presence of a nonvanishing Ricci curvature scalar $R$) perturbs the gluon condensate and may induce a nonanalytic term $\tilde{h}(R,q)$ in the effective gravitational action. The aim of this article is to explore the cosmological implications of a particular nonanalytic term $\tilde{h}\propto \eta |R{|}^{1/2}|q{|}^{3/4}$. With a quadratic approximation of the gravitating gluon-condensate vacuum energy density ${\rho }_V(q)$ near the equilibrium value $q_0$ and a small coupling constant $\eta$ of the modified-gravity term $\tilde{h}$, an “accelerating universe” is obtained which resembles the present Universe, both qualitatively and quantitatively. The unknown component $X$ of this model universe (here, primarily due to modified-gravity effects) has an effective equation-of-state parameter ${\overline{w}}_X$ which is found to evolve toward the value $-1$ from above.

Figure 1

Numerical solution of ODEs (2.15), with vacuum energy density (2.18), Brans-Dicke scalar potential (2.20), and both relativistic matter (energy density $r_{M,1}$) and nonrelativistic matter (energy density $r_{M,2}$). The figure panels are organized as follows: the panels of the first column from the left concern the expansion factor $a(t)$, those of the second column the modified-gravity scalar $s(t)$, those of the third column the gluon-condensate vacuum variable $f(t)$, and those of the fourth column the matter energy densities $r_{M,n}$. The model parameters are $(\gamma ,{\eta }^2,w_{M,1},w_{M,2})=({10}^2,9\times {10}^{-7},1/3,0)$, with the resulting parameter $\kappa \equiv (3/32){\eta }^2/\gamma =8.4375\times {10}^{-10}$. The boundary conditions at $t_{\mathrm{start}}=0.1$ are $(a,h,s,v,1-f,r_{M,1},r_{M,2})=(1,4.082483,0.8,0.8164966,8.437500\times {10}^{-9},75.97469,24.02531)$; see Sec. 3d for details. The several energy-density parameters $\Omega$ and the effective “dark-energy” equation-of-state parameter ${\overline{w}}_X$ are defined in (3.4, 3.5), respectively. With $\gamma /{\eta }^2\gg 1$, the values of ${\Omega }_V$ are negligible compared to those of ${\Omega }_{\mathrm{grav}}$ for the time interval shown.