From what I've learned so far, it appears that all models that attempt to explain the expansion of the universe are either based on Lambda-CDM or quintessence. The former support a big bang with rapid expansion, then deceleration of the expansion and then expansion again (non accelerated expansion) with $w=-1$. The latter (quintessence) do not support big bang, but support accelerated expansion with $w<-1$. The two schools of thought appear to box you in one way or the other, depending whether $w=-1$ or $w<-1$.
Why doesn't Lambda-CDM have a model that explain an accelerated expansion (i.e. $w < -1$) ? Or do they have one? Does Lambda-CDM maintain that $\Lambda$ has to be constant and so you're stuck with quintessence whenever $w<-1$? If that is the case, why couldn't $\Lambda$ increase with time?
In summary, is there any model that support a universe with:
Big Bang
Inflationary period with rapid expansion
Deceleration of expansion
Linear Expansion
Future acceleration of expansion?
That is, we should be able to see in that model that $H_t > H_0$ for any $t_i >> t_0$ when $w<-1$.
Answer
(Disclaimer: this is a follow-up question to Equation for Hubble Value as a function of time)
You still have a few misconceptions:
First, a model with a cosmological constant does lead to accelerated expansion. Look at the second derivative of $a(t)$ in my post: $$ \ddot{a}(t) = \frac{1}{2}H_0^2\left(-2\,\Omega_{R,0}\,a(t)^{-3} - \Omega_{M,0}\,a(t)^{-2} +2\,\Omega_{\Lambda,0}\,a(t)\right). $$ You see that $\ddot{a}(t)>0$ if $a(t)$ is sufficiently large. In particular, with the values $$ H_0 = 67.3\;\text{km}\,\text{s}^{-1}\text{Mpc}^{-1},\\ \Omega_{R,0} \approx 0,\qquad\Omega_{M,0} = 0.315,\qquad\Omega_{\Lambda,0} = 0.685,\qquad\Omega_{K,0} = 0, $$ you can work out that $\ddot{a}(t) > 0$ for $a(t)> 0.6$, which corresponds with $t > 7.7$ billion years. That is, the expansion began to accelerate when the universe was 7.7 billion years old, and will continue to do so. See also this post for more details: Can space expand with unlimited speed?
Second, a model with quintessence does have a big bang. The most general equation for $t(a)$, which I hadn't posted in my answer to your other question, is $$ \begin{align} t(a) &= \frac{1}{H_0}\int_0^a \frac{a'\,\text{d}a'}{\sqrt{\Omega_{R,0} + \Omega_{M,0}\,a' + \Omega_{K,0}\,a'^2 + \Omega_{\Lambda,0}\,a'^{1-3w}}}. \end{align} $$ This is a well-behaved function when $a\rightarrow 0$, because $\Omega_{R,0}>0$. This means that in the early universe, radiation was the dominant factor (the other terms in the denominator of the integrand go to zero for $a\rightarrow 0$), and we get $t(0)=0$ or conversely $a(0)=0$.
In your previous question, you used a simplified model with $\Omega_{R,0} = \Omega_{M,0} = \Omega_{K,0} =0$. Those models don't have a big bang, because then the integrand becomes infinite for $a\rightarrow 0$. But of course, those toy models do not correspond with our actual universe. You need the general model.
So, both a model with a cosmological constant and one with quintessence produce a universe with a big bang and an accelerated expansion. Does the data suggest a non-constant dark energy? It's too soon to tell, the error-bars are still too large; a cosmological constant is still consistent with the data. It's obvious that quintessence provides a better fit, because it has the extra parameter $w$, but that itself doesn't mean that this extra parameter is necesary to explain the observations. But time will tell...
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