Cosmological constant in spinfoam cosmology

We consider a simple modification of the amplitude defining the dynamics of loop quantum gravity, corresponding to the introduction of the cosmological constant, and possibly related to the $SL(2,\mathbb{C}{)}_q$ extension of the theory recently considered by Fairbairn-Meusburger and Han. We show that, in the context of spinfoam cosmology, this modification yields the de Sitter cosmological solution.

Figure 1

Numerical analysis of the transition amplitude Eq. (6). The only free parameter is $\lambda v_o$; here it is set equal to 0.3. The computation has been done truncating the sum over $j$ up to a maximum value $j_{\max }=200$. This choice is compatible with the maximal scale factor plotted.

Figure 2

Numerical analysis of the transition amplitude with the cosine replacing the exponential.