I'm studying the age of the universe for a universe dark energy dominated. Using Friedmann equation
$$
\left( \frac{\dot{a}}{a} \right)^2 = H_0^2 \left[ \Omega_R \cdot a^{-4} + \Omega_{NR} \cdot a^{-3} + \Omega_k \cdot a^{-2} + \Omega_{\Lambda} \right]
$$
with the conditions $\Omega_R=\Omega_{NR}=\Omega_k=0$, $\Omega_\Lambda=1$, I have found the following expression
$$
\frac{\dot{a}}{a} = H_0 \quad \to \quad \frac{da}{a} =H_0 \cdot dt
$$
If I integrate from the Big-Bang $(t=0, a(0)=0)$ to the present $(t=T, a(T)=1)$
$$
\int_0^1 \frac{da}{a} =H_0 \cdot \int_0^T dt
$$
but I have a singularity in the first integral. Does it mean that is impossible to find a universe full made of dark energy from the Big-Bang?
Cosmology – Can Dark Energy Dominate from the Big Bang?
big-bangcosmologydark-energyspace-expansion
Related Solutions
The Big Bang is a mathematical model of how the observable universe evolved , based on fitting astrophysical observations. Like all models it has its region of validity. When I read cosmology fifty years ago, the model included a singularity at the origin, because that is what the mathematical functions of the General Relativity solutions showed. The model at the time fitted the observed expansion of the universe, which showed clusters of galaxies receding from each other at a certain rate. In that model, all (x,y,z) points at t=0 were at (0,0,0) and it makes no sense to talk outside of this point within this model, a singularity. It was as if an explosion started from the origin, in this model.
It is well known though that when singularities appear in a physics model, it is a sign for the need of an extension of the model or a different model, as happened with the introduction of quantum mechanics for the microcosm of particles, which avoids singularities with the Heisenberg uncertainty principle.
The need for a quantum mechanical framework came from the new astrophysical observations of the last sixty years. The cosmic microwave background observations could not be reconciled with classical thermodynamics within General Relativity. The CMB showed great homogenization, with differences in the temperature map of order of 2*10^-5. This homogenization could not happen at that early time of photon separation, 380.000 years in the history, and quantum mechanics was introduced at the beginning of the universe to homogenize the system.
At the same time observations showed that the expansion of the universe is not constant, a bang and free tracks, but is accelerating and new physics is introduced, dark energy , to explain the phenomenon and still keep a shell of the original Big Bang General Relativity solution.
A representation of the evolution of the universe over 13.77 billion years. The far left depicts the earliest moment we can now probe, when a period of "inflation" produced a burst of exponential growth in the universe. (Size is depicted by the vertical extent of the grid in this graphic.) For the next several billion years, the expansion of the universe gradually slowed down as the matter in the universe pulled on itself via gravity. More recently, the expansion has begun to speed up again as the repulsive effects of dark energy have come to dominate the expansion of the universe. The afterglow light seen by WMAP was emitted about 375,000 years after inflation and has traversed the universe largely unimpeded since then. The conditions of earlier times are imprinted on this light; it also forms a backlight for later developments of the universe.
Now to your questions:
1). As I said, everything was at one point at t=0 in the theoretical model. In the present history there is a quantum mechanical uncertainty as to the region which projects to a point from the classical BB.
2). speculations there are many. The standard physics status is in the picture above
3).is answered by my exposition above
for flat universe ($\Omega_m + \Omega_{\Lambda}=1)$ $$H^2 = H_0^2(\Omega_ma^{-3}+\Omega_{\Lambda})$$
or $$\frac{\dot{a}^2}{a^2} = H_0(\Omega_ma^{-3}+\Omega_{\Lambda})$$
which becomes
$$\dot{a}^2 = H_0(\Omega_ma^{-1}+\Omega_{\Lambda}a^2)$$
taking square root $$\dot{a} = H_0\sqrt{\Omega_ma^{-1}+\Omega_{\Lambda}a^2}$$
let us write in the form of
$$\frac{da}{dt} = H_0\sqrt{\frac{\Omega_m}{a}+\Omega_{\Lambda}a^2}$$
$$\frac{da}{\sqrt{\frac{\Omega_m}{a}+\Omega_{\Lambda}a^2}} = H_0dt$$
$$\frac{da}{\sqrt{\Omega_{\Lambda}}\sqrt{(\frac{\Omega_m}{\Omega_{\Lambda}})(\frac{1}{a}) + a^2}}= H_0dt$$
set $$\frac{\Omega_m}{\Omega_{\Lambda}} = L = 0.44927$$
(I took $\Omega_{\Lambda} = 0.69$, $\Omega_m=0.31$)
By taking $a(t_{now})=1$
$$\int_0^{a(t_{now})=1}\frac{da}{\sqrt{\frac{L}{a}+a^2}} = \int_0^{t_{universe}} \sqrt{\Omega_{\Lambda}}H_0dt$$
by using wolfram the solution becomes,
$$\left.\frac{2}{3}log(a^{3/2} + \sqrt{a^2+L})\right|_0^1 = \sqrt{\Omega_{\Lambda}}H_0t_{uni}$$
or you can use https://www.integral-calculator.com to make a numerical calculation of the integral. In any case we have
$$\int_0^1\frac{da}{\sqrt{\frac{L}{a}+a^2}} = 0.793513$$
Hence,
$$t_{uni} = \frac{0.793513}{0.83066 \times H_0}$$
For $H_0 = 68km/s/Mpc$
$$t_{uni} = 0.9552801 \times H_0^{-1}= 0.9552801 \times 14.39~\text{Gyr} = 13.74 ~\text{Gyr}$$
Best Answer
Solving the differential equation for the scale factor $a$ in this case we have explicitly $$ \frac{da}{a} =H_0 \cdot dt\qquad \Longrightarrow\qquad a(t) \propto e^{H_0 t},$$ assumming $H_0$ is constant through time. From the expression above you can conclude for this model there is no point in time where the scale factor was exactly zero. You might then speak of infinite negative time or understand this as a model for some fraction of the cosmological history away from $a=0$.