= Solution
For a homogeneous canonical <scalar field>, $\rho_\phi=\dot\phi^2/2+V$ and $P_\phi=\dot\phi^2/2-V$. Potential-energy domination gives $P_\phi\simeq-\rho_\phi$, accelerated expansion and a decreasing <comoving Hubble radius>. <Hubble friction> can permit slow roll down a sufficiently flat potential. Enough inflation stretches an initially small causally connected region to encompass our observable universe, suppresses the relative spatial-curvature term and dilutes unwanted massive relics. <quantum fluctuations> supply possible seeds for cosmological structure; a successful model must also end inflation and create the hot post-inflationary universe. These statements do not imply that arbitrary initial conditions inevitably inflate.
Use the stated reduced-Planck units throughout the field calculation. Differentiating the Friedmann constraint and substituting the field equation gives
$$
2H\dot H=\frac13\dot\phi(\ddot\phi+m^2\phi)
=-H\dot\phi^2,\qquad
\boxed{\dot H=-\frac12\dot\phi^2}.
$$
Division assumes the nontrivial expanding branch $H>0$; the <vacuum> endpoint follows continuously.
During rapid <oscillations>, require $m\gg H$ and $|\dot A|\ll mA$. To leading order, $\dot\phi\simeq-mA\sin mt$, while the Friedmann constraint gives $H_m^2\simeq m^2A^2/6$. Thus
$$
\dot H\simeq-\frac{m^2A^2}{2}\sin^2mt
=-\frac32H_m^2(1-\cos2mt).
$$
Averaging over many <oscillations> on a timescale still short compared with $H^{-1}$ gives
$$
\boxed{\dot H_m=-\frac32H_m^2}.
$$
The <quadratic-inflaton Hubble oscillations> obey, to leading order in $H_m/m$,
$$
H_{\rm fluct}\simeq\frac{3H_m^2}{4m}\sin2mt,
\qquad \frac{|H_{\rm fluct}|}{H_m}=O(H_m/m)\ll1.
$$
Differentiating this expression reproduces the cosine term, with smaller corrections from the slowly varying amplitude. Instantaneously $H$ remains nonincreasing because its exact <derivative> is $-\dot\phi^2/2$; an oscillatory correction does not mean oscillatory growth of the total <Hubble parameter>.
The mean equation integrates to
$$
\frac1{H_m(t)}=\frac1{H_m(t_i)}+\frac32(t-t_i),\qquad
\boxed{H_m(t)=\frac{2}{3(t-t_*)}},\qquad
\boxed{a(t)\propto(t-t_*)^{2/3}}.
$$
The shift $t_*$ is set by matching to the earlier evolution, not necessarily the Big Bang origin of the full solution. Averaged kinetic and <potential energies> are equal, so $\langle P_\phi\rangle\simeq0$, $\langle\rho_\phi\rangle\propto a^{-3}$ and $A\propto a^{-3/2}$. \b[The coherent quadratic <oscillations> behave as <pressureless matter>, not <radiation>], even though they are a <scalar field> rather than a gas of cold particles. <radiation> would instead give $a\propto t^{1/2}$. The averaged approximation is controlled only after <oscillations> are fast compared with expansion.
\Image[/past-exam-of-the-particle-astrophysics-course-of-the-university-of-cambridge/2013/paper-1-inflaton-oscillations.png]
{title=Original dimensionless scalar-field integration showing a decaying oscillation envelope and a nonincreasing Hubble parameter with a matter-like average}
<reheating> requires interactions beyond the isolated field equations above. The oscillating <inflaton> can decay into other particles, whose scattering and further decays establish a thermal bath. The <perturbative inflaton reheating equations> are
$$
\dot\rho_\phi+3H\rho_\phi=-\Gamma\rho_\phi,\qquad
\dot\rho_r+4H\rho_r=\Gamma\rho_\phi.
$$
Expansion redshifts the components differently; transfer becomes efficient around $H\sim\Gamma$, and <radiation> domination follows when the transferred relativistic <energy> dominates. For suitable couplings, nonperturbative <preheating> by <parametric resonance> can precede thermalization, with backreaction and rescattering affecting its completion. Neither the decay rate nor thermalization is specified by $m$ and $V$ alone. Instantaneous <reheating> idealizes all this as immediate conversion of the end-of-inflation <energy> to <radiation>.<Towards the Theory of Reheating After Inflation by Lev Kofman and others>[Abstract and Introduction: perturbative decay, resonant preheating, backreaction and thermalization]{ref}
For <instantaneous-reheating horizon matching>, specify the comparison: take the initial patch to be of order $H_i^{-1}$, compare the same-shaped inflationary history with approximately fixed $H_i/H_{\rm end}$, and assume instantaneous <reheating> followed by <adiabatic expansion>. Ignoring changes in relativistic degrees of freedom, $a_0/a_{\rm reh}\propto T_{\rm reh}$, while <energy> conservation and the <Friedmann equation> give $H_{\rm end}\propto T_{\rm reh}^2$. The present size of this inflated patch therefore scales as
$$
\ell_0\sim H_i^{-1}e^N\frac{a_0}{a_{\rm reh}}
\propto\frac{e^N}{T_{\rm reh}}.
$$
At fixed present <Hubble radius>, \b[a higher instantaneous <reheating> scale requires more e-folds] in this standard comparison, approximately $\Delta N\simeq\ln(T_2/T_1)$, up to slow scale and entropy-degree corrections. Hotter <reheating> gives more subsequent expansion, but the associated initial Hubble patch is smaller by a stronger <power> of <temperature>. If instead one artificially holds the initial physical patch or $H_i$ fixed while varying the end <energy> independently, the post-inflationary geometric effect alone would give the opposite trend. <temperature> without an inflation-scale comparison convention does not fix that alternative problem; this is why the matching assumptions must be stated.<How long before the end of inflation were observable perturbations produced? by Andrew R. Liddle and Samuel M. Leach>[Section II, equations 2, 6–8: horizon-scale matching and its dependence on inflation and reheating energy scales]{ref}
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