In the current model during inflation H remains nearly constant where $ H = \frac{ \dot{a}}{a} $ but the scale factor a grows exponentially and requires a large number of e - folds N where $ N = ln\frac{a(t_f)}{a(t_i)} $ but according to Loop Quantum Cosmology in super inflation a smaller number of e folds are required any one know of a published article that is accessible that shows how you need less e-folds the one I read which can be accessed here says you have a smaller N but it doesn't explain very well how Hubble rate $H$ increases rapidly while the scale factor a remains nearly constant and it gives no estimate for N. I need an article that I can cite. This is for class final project in my upper level undergrad course.


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I found an article by E. J. Copeland, D. J. Mulryne, N. J. Nunes, M. Shaeri that explains this, it's called Super-inflation in Loop Quantum Cosmology. This is part of the answer I wrote most the equations come from this article unless I cite otherwise.

According to Loop Quantum Cosmology, in super inflation a smaller number of e-folds are required. This is due to the equations which describe the inflation field in LQC, as well as their solutions where scale invariance occurs.

During inflation a modified Friedman equation is given by $ H = \frac{ \dot{a}}{a} = \frac{8\pi G}{3}S(\frac{\dot{\phi} }{2D} + V(\phi)) $ [eq. 1,abs/0708.1261] and they set $8\pi G = 1$. The variable $ \rho $ is defined to be $ \rho = \frac{\dot{\phi} }{2D} + V(\phi) $ .

A set of complex differential equations describe the behavior of the system. The derivations and solutions to these equations are outside the scope of my answer. The derivations are in the referenced article.

$D(a)$ is approximated by $D_* a^n$ where $n = \frac{3(3-l)}{1-l}$ and $S(a)$ and $D_* = \frac{}{} $ is approximated by $S_* a^r$ where $S_* = \frac{3}{2}a^{-3}$ both $S_* \approx D_* \approx 1$. $\alpha$ is defined as $\alpha = 1- \frac{n}{6} < 0$ , we define the variables $ V_\phi = \frac{dV}{d\phi}$ and $ \lambda = -\frac{\sqrt{D}}{\sqrt{S}}\frac{V}{V_\phi} $ [sec. A, abs/0708.1261].

The solution to the inflation field $\phi$ is given by $\phi = \frac{2\lambda}{\alpha(n-r)}\frac{\sqrt{D}}{\sqrt{S}} $, in units of energy [eq. 15,abs/0708.1261].

The field potential is defined as $V = V_0\phi{^\beta}$ where $\beta = \frac{-2\lambda^2}{\alpha(n-r)} > 0 $, in units of energy density [eq. 16, abs/0708.1261].

The scale factor during inflation is then expressed in the form $a(\tau) = (-\tau)^p$ "for an expanding universe $\tau$ is negative and increasing towards zero" [p. 3, eq 18, abs/0708.1261].

The CMB inflationary fluctuations have the property of being scale invariant so to reproduce them we must also produce scale invariant fluctuations in LQC [pg. 274 Introduction to Cosmology,Barbara Ryden].

Scale invariance occurs "whenever $p \to 0$, which, as we referred to, does indeed imply that $ \overline{\epsilon} \ll 1 $ and consequently $V < 0$" The parameter $ \overline{\epsilon}$ is given as $ \overline{\epsilon} = \frac{\lambda^2}{2} $ \citep[pg. 4]{abs/0708.1261} and $p$ is given as $ p = \frac{2\alpha}{2\overline{\epsilon} -\alpha(2 + r)} $ [eq. 9, abs/0708.1261].

$N = -68p$ and since $p$ must be small and negative to have scale invariance as stated above we thus have less e-folds required. [pg. 274, Introduction to Cosmology,Barbara Ryden]

$H$ in contrast to in standard inflation, is not constant $ \dot{H} \neq 0 $. The Hubble rate is related to the inflation field via $\dot{H} = -\frac{\dot{\phi}}{2}(1- \frac{\rho}{2\sigma}) $ [eq. 44, abs/0708.1261].

Where "$2\sigma$ represents the critical energy density arising from quantum geometry effects which leads to the scale factor undergoing a bounce as $\rho$ approaches it." [p. 6, abs/0708.1261] .

As we can see in order for LQC to reproduce the effects of inflation the scale factor must remain constant while the Hubble rate must vary during the inflationary phase of the universe.


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