value of $\phi$ at the maximum of the full potential), and the slope to the
right of the plateau to be given by $k=V(\phi_{\rm min},T_{\rm
h})/(\Lambda-\phi_{\rm max})$. Given that for $\phi>\phi_{\rm max}$,
$\partial^{2}/\partial\phi^{2}V(\phi,T)>0$ we have that $V_{\rm
app}(\phi)<V(\phi,T)$ for all $\phi>\phi_{\rm min}$. Further, we can estimate
the value of $\phi_{\rm max}$ by noting that the barrier in the full potential
will occur around where the high temperature approximation breaks down

$\phi_{\rm max}/\Lambda\sim T_{\rm h}/m_{\text{int}}~{},$ (86)

(i.e. $m_{\psi}^{\rm eff}\sim T_{\rm h}$, but still $\phi_{\rm
max}\ll\Lambda$).

Figure 8:  The finite temperature corrected potential of $\phi$, $V_{\rm
T}(\phi)$, and the approximation we make to obtain a lower bound on the
tunnelling rate, $V_{\rm app}(\phi)$. We also indicate the zero temperature
potential, $V_{\rm exp}(\phi)$, and the metastable minimum $\phi_{\rm min}$
and maximum $\phi_{\rm max}$ of $\phi$’s corrected potential. Note that the
plot is not to scale for a realistic theory that can lead to an era of Dark
Energy domination (for which $\phi_{\min}/\Lambda$, the height of the barrier
and $T_{\rm h}^{4}/\rho_{\rm de}$ would all be much smaller).

Quantum tunnelling in the no-barrier potential eq. (85) has been analysed in
Lee and Weinberg (1986). The bounce solution to the Euclidean equations of
motion

$\phi^{\prime\prime}+\frac{3}{s}\phi^{\prime}=\frac{\partial
V}{\partial\phi}\,,$ (87)

that satisfies the boundary conditions $\phi^{\prime}(0)=0$ and
$\raisebox{2.15277pt}{\scalebox{0.8}{$\displaystyle\lim_{s\rightarrow\infty}\;$}}\phi(s)=\phi_{\rm
min}\approx 12\Lambda\rho_{\text{de}}/(m_{\text{int}}^{2}T_{\rm h}^{2})\approx
0$, and continuity across $s_{\rm max}$, where $\phi=\phi_{\rm max}$, is found
to be

$\displaystyle\phi(s)=\begin{cases}\phi_{\rm max}\left(2-\frac{s^{2}}{s_{\rm
max}^{2}}\right)&\textrm{for }s<s_{\rm max}\\\ \phi_{\rm max}\frac{s_{\rm
max}^{2}}{s^{2}}&\textrm{for }s>s_{\rm max}\,,\end{cases}$ (88)

with $s_{\rm max}=\sqrt{8\phi_{\rm max}/k}$. The Euclidean action evaluated on
this $O(4)$ bounce gives

$S_{4}=\frac{32\pi^{2}}{3}\frac{\phi_{\rm max}^{3}}{k}\,,$ (89)

leading to a quantum tunnelling rate

$\frac{\Gamma_{4}}{\mathcal{V}}\sim v^{4}\frac{T_{\rm
h}^{6}}{m_{\text{int}}^{6}}\frac{\Lambda^{8}}{\rho_{\rm
de}^{2}}e^{-\frac{32\pi^{2}}{3}\frac{\Lambda^{4}}{\rho_{\rm de}}\frac{T_{\rm
h}^{3}}{m_{\text{int}}^{3}}}\,.$ (90)

The run-away part of $\phi$’s finite temperature-corrected potential can also
be reached via thermal fluctuations that roll $\phi$ up and over its barrier,
with an associated decay rate per unit volume $\mathcal{V}$ given
approximately by Affleck (1981); Turner  _et al._ (1992)

$\frac{\Gamma_{3}}{\mathcal{V}}=T_{\rm h}^{4}\left(\frac{S_{3}}{2\pi T_{\rm
h}}\right)^{3/2}e^{-S_{3}/T_{\rm h}}\,.$ (91)

This process is dominated by the $O(3)$ bounce solution with an associated
action $S_{3}$. The corresponding Euclidean equation of motion is

$\phi^{\prime\prime}+\frac{2}{r}\phi^{\prime}=\frac{\partial
V}{\partial\phi}\,,$ (92)

with boundary conditions $\phi^{\prime}(0)=0$ and
$\raisebox{2.15277pt}{\scalebox{0.8}{$\displaystyle\lim_{r\rightarrow\infty}\;$}}\phi(r)=\phi_{\rm
min}\approx 0$, and $\phi$ continuous across $r_{\rm max}$. The solution in
this case is

$\displaystyle\phi(r)=\begin{cases}\frac{1}{2}\phi_{\rm
max}\left(3-\frac{r^{2}}{r_{\rm max}^{2}}\right)&\textrm{for }r<r_{\rm max}\\\
\phi_{\rm max}\frac{r_{\rm max}}{r}&\textrm{for }r>r_{\rm max}\,,\end{cases}$
(93)

where $r_{\rm max}=\sqrt{3\phi_{\rm max}/k}$. The Euclidean action evaluated
on the $O(3)$ bounce is

$S_{3}=\frac{8\sqrt{3}\pi}{5}\frac{\phi_{\rm max}^{5/2}}{k^{1/2}}\,,$ (94)

thus giving a thermal decay rate

$\frac{\Gamma_{3}}{\mathcal{V}}\sim\frac{\Lambda^{9/2}T_{\rm
h}^{25/4}}{m_{\text{int}}^{15/4}\rho_{\text{de}}^{3/4}}e^{-\frac{8\pi\sqrt{3}}{5}\frac{\Lambda^{2}}{\rho_{\rm
de}^{1/2}}\frac{T_{\rm h}^{3/2}\Lambda}{m_{\text{int}}^{5/2}}}\,.$ (95)

Note that the action obtained from the full potential must be greater than the
action obtained from the approximate no-barrier potential and, therefore, the
actual decay rates will be slower than the estimates provided here. This can
be seen by noting that the full Euclidean action evaluated on its full bounce
solution must be greater than the approximate no-barrier Euclidean action
evaluated on the full bounce solution (since the full potential is always
greater than or equal to the approximate no-barrier potential). Moreover, the
approximate no-barrier Euclidean action evaluated on the full bounce solution
must be greater than the approximate no-barrier Euclidean action evaluated on
the no-barrier bounce solution because the no-barrier bounce solution
minimises the no-barrier Euclidean action. Therefore, the conditions in eq.
(83), using the approximations eqs. (90) and (95), guarantee that the actual
models are viable. The limits quoted in the main text, eqs. (44a) and (44b)
are obtained simply by insisting that $\Gamma_{3}$ and $\Gamma_{4}$ are
strongly exponentially suppressed, which is sufficient precision for our
purposes.

##  Appendix C More details on the Quintessence assisted scenario

In Figure 9 left panel, we show the evolution of the fields $\phi$ and $\psi$
in a theory of Quintessence assisted Dark Energy in which $\phi$ has a
potential with an exponential run-away. The dynamics of the theory are similar
to the early stages of a Dark Matter assisted Dark Energy theory, e.g. as
plotted in Figure 6, except that $\phi$ sources Dark Energy while $\psi$ is
still frozen by Hubble friction. Once $\psi$ is unfrozen, at around
$a/a_{0}\simeq 10$, $\phi$ is no longer trapped at $\phi/\Lambda\simeq 0$ by
the $\phi$-$\psi$ interaction term and rolls down its run-away potential (for
the parameters of the theory shown, there is no era in which $\phi$ is trapped
while $\psi$ behaves as matter). In the right panel we plot the equation of
state parameter of the same theory. Prior to $a/a_{0}\simeq 10$ this matches
the $\Lambda{\rm CMD}$ prediction closely. Soon after $\phi$ is released from
$\phi/\Lambda\simeq 0$ the theory is non-linear and $w$ evolves in a
complicated way. However, eventually the system will approach the standard
tracker solution for an exponential potential Copeland  _et al._ (1998),
which, for the value of $\Lambda/M_{\rm Pl}$ used, leads to non-accelerating
expansion.

Figure 9:  _Left:_ The evolution of $\phi$ and $\psi$ in a theory of
Quintessence assisted Dark Energy, i.e. two field quintessence, in which
$\phi$ has the exponential run-away potential of eq. (3). The Lagrangian
parameters are $m_{\psi}=H_{0}/5$, $m_{\rm int}=40H_{0}$,
$\Lambda=M_{\text{Pl}}/10$ and $\lambda=0$ and initial field values $\phi_{\rm
i}=\Lambda$, $\psi_{\rm i}=2M_{\text{Pl}}/5$. The dynamics are similar to the
Dark Matter assisted scenario shown in Figure 6, except that because of its
small mass $\psi$ does not roll down its potential until after the present-
day. While $\psi$ is frozen, it first drives and then traps $\phi$ at a field
value $\ll\Lambda$ where it sources Dark Energy. Once $\psi$ rolls to
sufficiently small values, $\phi$ rolls down its run-away potential to large
field values (for the particular parameters chosen, there is not an era of
Dark Matter assisted Dark energy after $\psi$ starts oscillating). Eventually
the system will reach the standard tracker solution discussed in the main
text. _Right:_ The equation of state parameter of the theory plotted in the
left panel $w_{\rm tot}$ compared to the $\Lambda$CDM prediction,
$w_{\Lambda{\rm CDM}}$. Until $a/a_{0}\simeq 10$, the theory matches the
$\Lambda$CDM result, and subsequently the era of accelerated expansion ends.

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