Onset of inflation in loop quantum cosmology

Using a Liouville measure, similar to the one proposed recently by Gibbons and Turok, we investigate the probability that single-field inflation with a polynomial potential can last long enough to solve the shortcomings of the standard hot big bang model, within the semiclassical regime of loop quantum cosmology. We conclude that, for such a class of inflationary models and for natural values of the loop quantum cosmology parameters, a successful inflationary scenario is highly improbable.

Figure 1

$D(q)$ is plotted as a function of $a$, with $a_{\star }=1$ and $l=0.75$. For $a\gg a_{\star }$, $D\to 1$, giving us the classical result.

Figure 2

$S(q_{\mathrm{G}})$ is plotted as a function of $a$, with $a_{\mathrm{G}}=1$. Notice that $S(q_{\mathrm{G}})<1$ for all $a$, and that for $a\gg a_{\mathrm{G}}$, $S\to 1$, giving us the $j=1/2$ result.

Figure 3

The probability measure is defined by integrating the $\mathbf{B}$ field over a constant surface in the 3-dimensional phase space produced by using the Hamiltonian constraint to eliminate one of the dynamical variables.

Figure 4

The function $[1-(n-m+1)\frac{q}{3D}(\frac{\partial D}{\partial q})]$ is plotted as a function of $a$, with $a_{\star }=1$ and $l=0.75$, for $n-m+1=-1$, $-4$, $-9$, $-19$ (small to large crosses, respectively). This is the term that determines the suppression of the probability of having $N$ e-foldings of inflation. Notice that for $n-m+1<-2$ we have two zeros, although this number is weakly dependent on $l$.

Figure 5

The function $[1-(n-m+1)\frac{q}{3D}(\frac{\partial D}{\partial q})]$ is plotted as a function of $a$, with $a_{\star }=1$ and $l=0.75$, for $n-m+1=1$, 3, 6, 11, 21 (small to large crosses, respectively). This is the term that determines the suppression of the probability of having $N$ e-foldings of inflation. Notice that for $n-m+1>0$ we have only one zero that is always close to $a_{\star }$.

Figure 6

The function $D^{n+1}[1-(n-m+1)\frac{q}{3D}(\frac{\partial D}{\partial q}){]}^{-1}$ is plotted as a function of $a$, with $a_{\star }=1$ and $l=0.75$, for $n-m+1=1$, 3, 5, 7, 11, 16 (small to large crosses, respectively). Also plotted is an example of $(a-a_{\mathrm{i}})/(a_s-a_i)$ for $a_i=a_{\star }=1$ and $a_s=2$, which allows us to calculate a lower bound on ${\varphi }_{\mathrm{i}}$, which appears as one of the coefficients of the probability.