Limits of extended quintessence

We use a low redshift expansion of the cosmological equations of extended (scalar-tensor) quintessence to divide the observable Hubble history parameter space in four sectors: A forbidden sector I where the scalar field of the theory becomes imaginary (the kinetic term becomes negative), a forbidden sector II where the scalar field rolls up (instead of down) its potential, an allowed “freezing” quintessence sector III where the scalar field is currently decelerating down its potential towards freezing and an allowed “thawing” sector IV where the scalar field is currently accelerating down its potential. The dividing lines between the sectors depend sensitively on the time derivatives of the Newton’s constant $G$ over powers of the Hubble parameter. For minimally coupled quintessence which appears as a special case for a constant $G$ our results are consistent with previous studies. Observable parameter ${\chi }^2$ contours based on current data (Supernova Legacy Survey data set) are also constructed on top of the sectors, for a prior of ${\Omega }_m=0.24$. By demanding that the observed $2\sigma$ ${\chi }^2$ parameter contours do not lie entirely in the forbidden sectors, we derive stringent constraints on the current second time derivative of Newton’s constant $G$. In particular, we find $\frac{\ddot{G}}{G}>-1.91H_0^2=-2\times {10}^{-20}{h}^2{\mathrm{yrs}}^{-2}$ at the $2\sigma$ level which is complementary to solar system tests which constrain only the first derivative of $G$ as $|\frac{\dot{G}}{G}|<{10}^{-14}{\mathrm{yrs}}^{-1}$ at $1\sigma$.

Figure 1

The $h_1-h_2$ sectors of EXQ: Sector I is the forbidden sector where the scalar field becomes imaginary. Sector II is also forbidden and corresponds to a scalar field that rolls up its potential. Sector III corresponds to freezing EXQ where the field is decelerating down its potential towards freezing. Sector IV corresponds to thawing EXQ where the scalar field is accelerating down its potential. The forbidden sector I shrinks for a Newton’s constant $G$ that decreases with time ($g_2>0$) while the “freezing sector” expands [Fig. 1a]. The boundaries of the sectors are provided by (2.20, 2.21, 2.31). The $1\sigma$, $2\sigma$ ${\chi }^2$ contours obtained from the SNLS SnIa data set for ${\Omega }_{0\mathrm{m}}=0.24$ using the CPL parametrization are also shown.

Figure 2

The sectors I–IV of Fig. 1 mapped on the $w_0-w_1$ parameter space. The allowed sectors extend to $w_0<-1$ for $g_2>0$.

Figure 3

The sectors I–IV of Fig. 1 mapped on the $d_{L2}-d_{L3}$ parameter space. Notice that in this case the forbidden sector I is on the right (large $d_{L2}$).

Figure 4

A $g_2>0$ implies a decreasing Newton’s constant ($\frac{G(t)}{G_0}\simeq 1+\frac{1}{2}{g}_2(H_0(t-t_0){)}^2$ around $t=t_0$).

Figure 5

Taking into account a $G$ variation consistent with the nucleosynthesis and solar system bounds introduces modifications to the $w_0-w_1$ ${\chi }^2$ contours which are not significant for the currently available SnIa data (for the dashed line $\alpha =0.2$ and the continuous $\alpha =0$).