Fourth order deformed general relativity

Whenever the condition of anomaly freedom is imposed within the framework of effective approaches to loop quantum cosmology, one seems to conclude that a deformation of general covariance is required. Here, starting from a general deformation we regain an effective gravitational Lagrangian including terms up to fourth order in extrinsic curvature. We subsequently constrain the form of the corrections for the homogeneous case, and then investigate the conditions for the occurrence of a big bounce and the realization of an inflationary era, in the presence of a perfect fluid or scalar field.

Figure 1

Line integral convolution plots showing trajectories in phase space for a scalar field with a potential $\lambda {\phi }^n/n$ with holonomy corrections. The hue at each point indicates the magnitude of the vector $(\dot{\phi },\ddot{\phi })$, with blue indicating low values. The trajectories do not extend outside of the region (3.36). The attractor solution is well approximated by (3.37), which corresponds to slow-roll inflation. The brightness of the trajectories is only a relic of the white noise image used to seed the convolution plot. We have used $\tilde{\lambda }=1$, $\tilde{\mathrm{\Delta }}=2\sqrt{3}\pi \gamma$, $\delta =0$, ${\gamma }^{\varnothing }=1$, and the plot is in Planck units. (a) Full phase space for $V\left(\phi \right)=\lambda {\phi }^2/2$. (b) Attractor solution for $V\left(\phi \right)=\lambda {\phi }^2/2$. (c) Full phase space for $V\left(\phi \right)=\lambda {\phi }^4/4$. (d) Attractor solution for $V\left(\phi \right)=\lambda {\phi }^4/4$.

Figure 2

Contour plots showing the region in scalar field phase space which satisfy the condition for accelerated expansion when holonomy corrections are included (3.38). The dashed line indicates the classical acceleration condition $w_a=-1/3$ and the dotted line indicates the boundary at which the bounce occurs. The white line indicates the slow-roll solution (3.37). The contours indicate the value of $w$ by their color, and the most blue contour is for $w\approx 0.2$. We have used $\tilde{\lambda }=1$, $\tilde{\mathrm{\Delta }}=2\sqrt{3}\pi \gamma$, $\delta =0$, ${\gamma }^{\varnothing }=1$, and the plot is in Planck units. (a) Accelerating values of $w$ for $V\left(\phi \right)=\lambda {\phi }^2/2$. (b) Accelerating values of $w$ for $V\left(\phi \right)=\lambda {\phi }^4/4$.