Scalar field cosmology model: what should be some "realistic" values? I'm considering a "classical" model of scalar field cosmology:  A simple real scalar field minimally coupled to gravity, with a quartic Higgs-like field potential:
\begin{equation}\tag{1}
\mathcal{V}(\phi) = \frac{\lambda}{4} (\phi^2 - v^2)^2,
\end{equation}
where $v$ is the "true vacuum" field.  We can derive this formula:
\begin{equation}\tag{2}
v = \pm \: \frac{m}{\sqrt{2 \lambda}},
\end{equation}
where $m$ is the mass of the scalar field $\phi$, and $\lambda$ is the auto-coupling of the field.  Take note that $\phi \sim \mathrm{L}^{-1}$, $m \sim \mathrm{L}^{-1}$ and $\lambda$ is a dimensionless number.
Now, the Friedmann-Lemaître equations and the scalar field equation are the following ($a(t) \sim \mathrm{L}$ is the cosmological scale factor, and dots are the usual cosmological time derivatives.  $G \equiv \ell_P^2 \sim \mathrm{L}^2$):
\begin{gather}\tag{3}
\frac{\dot{a}^2}{a^2} + \frac{k}{a^2} = \frac{8 \pi G}{3} \big( \frac{1}{2} \; \dot{\phi}^2 + \mathcal{V}(\phi) \Big), \\[12pt] \tag{4}
\frac{\ddot{a}}{a} = -\: \frac{8 \pi G}{3} \Big( \dot{\phi}^2 - \mathcal{V}(\phi) \Big), \\[12pt] \tag{5}
\ddot{\phi} + 3 \frac{\dot{a}}{a} \dot{\phi} + \mathcal{V}^{\prime} = 0.
\end{gather}
The initial conditions that I need to use are these ($t = 0$ is the present time, i.e. our human epoc):
\begin{align}
a(0) &= a_0,
& \dot{a}(0) &= H_0, \; \text{(Hubble's constant)} \\[12pt]
\phi(0) &= \phi_0, \; \text{(any value)}
& \dot{\phi}(0) &= \psi_0. \; \text{(any value)}
\end{align}
I can numerically solve these equations (after a scale transformation to remove all units), and get some very nice graphical output.  I solve the second order equations (4) and (5) using the initial conditions defined above.  Equation (3) is only used to find the curvature parameter $k$.
So the question is the following.  I have 4 parameters as input to the numerical simulation:


*

*The field mass $m$.

*The field coupling constant $\lambda$.

*The field initial value (at present time): $\phi_0$.

*The field initial derivative (at present time): $\psi_0$.



I use an initial slow roll condition for simplicity: $\psi_0 = 0$. 
  But what should be the "typical" realistic values of the three
  remaining parameters?

Currently, to get my nice graphical output, I had to use some very fantaisist and extravagant input:


*

*$m \approx 10^{- 68} \text{kg}$ (wow!)

*$\lambda \approx 10^{-121}$ (wowee!)

*$\phi(0) \sim \frac{1}{\ell_P}$, i.e. inverse Planck length (a very large field, to compensate the small values above).



EDIT :  To numerically solve equations (4) and (5), we need to make a scale transformation to remove all units.  I use these new variables :
\begin{align}
\tau &= H_0 \, t, \text{(dimensionless time, in units of the Hubble's time)} \tag{6} \\[12pt]
\Phi &= \sqrt{\frac{8 \pi G}{3}} \; \phi, \text{(dimensionless scalar field, in units of the Planck lenght since $G \equiv \ell_P^2$)} \tag{7} \\[12pt]
\tilde{m} &= \frac{m}{H_0}, \text{(dimensionless mass)} \tag{8} \\[12pt]
\tilde{\lambda} &= \frac{3 \lambda}{8 \pi G H_0^2}, \text{(dimensionless coupling)} \tag{9}
\end{align}
Then, equations (4) and (5) become these (the prime is the derivative with respect to dimensionless time $\tau$) :
\begin{align}
\frac{a^{\prime \prime}}{a} = - \Phi^{\prime \, 2} + \mathcal{V}(\Phi), \tag{10} \\[12pt]
\Phi^{\prime \prime} + 3 \frac{a^{\prime}}{a} \Phi^{\prime} + \frac{d \mathcal{V}}{d\Phi} = 0. \tag{11}
\end{align}
The typical input I used for my numerical simulation are $\tilde{m} \sim 1$, $\tilde{\lambda} \sim 1$ and $\Phi_0 \sim 1$ (not too small or too large numbers).  From the scale transformation (6)-(9), this gives the fantaisist values cited above, for $m$, $\lambda$ and $\phi_0$.
 A: The coupling constant $\lambda$ gives the overall energy scale of inflation, but otherwise does not effect any of the current observables directly.
As soon as we find Primordial gravity waves we will have a clear indication for the actual energy scale. Until that time you can set $\lambda$ to whatever is convenient in the sense of numerical evaluation. Physically people kind of guess it should be in the GUT energy scale, for numerous reasons. 
$\phi_0$ should be very close to whatever value you have for $\phi$ at the end of inflation, since we see a very subtle accelerated expansion these days. So I think you have to set $\phi_0 \simeq \pm v$.
Now as for the value of $v$: the combination $\sqrt{\lambda} v^2$ should give an energy scale which is at the very least higher than the Electroweak energy scale (or 14TeV -current LHC energy scales), since we know physics up to theses scales and haven't found the inflaton yet. That is unless you think the Higgs is the inflaton, but it is probably not because of many different reasons.
EDIT:
you can work out the actual units you need to use by reintroducing $\hbar,c,K_B$, etc.
