A tight little universe

Take a scalar field, the inflaton, rolling on a potential $V(\varphi )$ in an expanding universe. Let it go from rest at some height ${\varphi }_0$ and watch where it goes. Now do it again from a different height.

Here is a question that sounds easy: given a point in the field’s phase space, is there exactly one release height ${\varphi }_0$ whose trajectory passes through it? Or can two different starts funnel to the same place, so that “where the field is and how fast it moves” forgets where it began? This is the question of whether a family of trajectories foliates-tiles phase space, one leaf through each point-or whether the leaves pile up. It has a crisp answer, and chasing it leads somewhere unexpected: contact geometry, a naive hunt for an overtwisted-disk, and a small surprise about how reassuringly rigid the inflaton’s phase space really is.

We work in the e-fold gauge that Part 2 singled out, the number of e-folds $N=\log a$ as the clock, velocity measured per e-fold, in which the equation of motion is

The friction term $3(1-{\varphi }^{\prime 2}/6)$ is the Hubble drag $3H\dot{\varphi }$ after the Hubble rate has been eliminated; hold onto it.

Plot solutions $\varphi (N)$ as curves over time and they cross constantly: in configuration space there is no foliation at all. The fix is the one every dynamicist reaches for: go to phase space $(\varphi ,{\varphi }^{\prime })$. A second-order equation is a first-order system on the two-dimensional plane of position and velocity,

The order of the equation is nothing but the dimension of the phase space. Now the Picard–Lindelöf theorem applies: as long as the right-hand side is smooth: here $V>0$, so that ${\partial }_{\varphi }\log V=V^{\prime }/V$ does not blow up: there is a unique solution through each phase point. Uniqueness is exactly what second order wants: Newton needs position and velocity to fix a trajectory, and that pair is a point in phase space. The velocity coordinate is precisely the room needed to un-cross the tangles of configuration space.

For an autonomous planar system, uniqueness has a beautiful corollary: orbits never cross. Two orbits sharing a point are the same orbit. So phase space is always foliated by orbits for any smooth potential.

If orbits never cross, what could go wrong with our release-from-rest question? Only this: a single orbit could touch the rest axis ${\varphi }^{\prime }=0$ twice. Then two release heights sit on one orbit, and downstream points are reached from both-the map from release height to phase point is no longer one-to-one. An orbit re-touching ${\varphi }^{\prime }=0$ means the velocity vanished, reversed, and vanished again: a turning point. So

the release-from-rest foliation breaks if and only if the field turns around.

When does a rolling field turn around? It must be pushed back: it needs a potential well to fall into and overshoot. And not just any well: an underdamped one. In the thick Hubble friction of an expanding universe a shallow well is approached monotonically; overshoot requires the well’s curvature to beat the friction,

A pure inflection point, a lone fold, the flattest inflaton potential there is, has no such well, and the field never reverses in the physical region (Fig. 1). To break the foliation you must add structure. The cleanest way is two inflections at once, which in catastrophe language is a cusp ($A_3$) resolved into two folds ($A_2$), with a genuine minimum between them.

Here it gets interesting. That friction term is no friend of a symplectic structure: a frictionless mechanical system lives on an even-dimensional, energy-conserving symplectic phase space, and dissipation breaks that. The natural home for a damped system is one dimension up: a contact manifold, odd-dimensional, the geometry of thermodynamics and of Hamilton–Jacobi theory, and the standard modern setting for dissipative dynamics [1, 2]. The reduced cosmological phase space $(N,\varphi ,{\varphi }^{\prime })$ is contact. This is the reduction Part 1 built, eliminating $H$ from the constrained system, and the friction is the signature.

Contact structures come in two flavors, and the distinction is deep [6]. A contact structure is tight if it is rigid and geometric. Otherwise it contains an overtwisted disk, a small embedded disk along whose boundary the contact planes make a full $2\pi$ turn. Overtwisted structures are flexible, classified entirely by algebraic topology [5], with no interesting geometry left; tight ones are where the geometry lives. So a natural, greedy question: if and when is the inflaton’s contact structure overtwisted? And if so, is an underdamped, spiralling well-exactly the thing that breaks our foliation-where the overtwisted disk hides? A spiral is rotation; rotation twists planes; enough twist is an overtwisted disk. The two germs seemed like the perfect place to look (Fig. 2).

The spiral is real. Numerically the well becomes a spiral focus exactly when $R>3/2$: the same threshold (3) that breaks the foliation. One condition, two hats: turning point and spiral focus are the same event.

But there is a wall, and it is called Darboux’s theorem [6]: every contact structure looks locally like the one standard model. Overtwistedness is never a local property: you cannot find an overtwisted disk by staring at a neighborhood of the spiral, no matter how tightly it winds. If a disk exists it is a global object, built from how the contact planes are glued across the whole two-germ landscape. To settle it, we need the global contact form.

Writing the global form runs into the friction: the obvious form $\alpha =dz-{\varphi }^{\prime }d\varphi$ is not preserved by the flow, because the state-dependent coefficient $1-{\varphi }^{\prime 2}/6$ injects terms it cannot absorb. That coefficient is really a gauge artifact: the velocity looks bounded, , only because we clock by e-folds.

So change the velocity variable. Write it as a rapidity, ${\varphi }^{\prime }=\sqrt{6}\tanh \theta$. Then $1-{\varphi }^{\prime 2}/6={\mathrm{sech}}^2\theta$ exactly, and in the equation of motion it cancels:

Two things happen. The friction factor vanishes from the velocity equation. And the kination boundary ${\varphi }^{\prime }=\pm \sqrt{6}$, where the contact structure degenerated, maps to $\theta \to \pm \mathrm{\infty }$: it was a coordinate horizon of the e-fold clock, and rapidity sends it to infinity where it belongs. (Both facts check to eight digits.)

Now take the slow-roll heart of the dynamics, $\theta$ small, where inflation happens. The system becomes, back in $(\varphi ,{\varphi }^{\prime })$,

and this is exactly a contact-Hamiltonian system [1]:

One checks $\dot{\varphi }={\partial }_p\mathscr{H}={\varphi }^{\prime }$ and $\dot{p}=-{\partial }_{\varphi }\mathscr{H}-p{\partial }_z\mathscr{H}=-3{(\log V)}^{\prime }-3{\varphi }^{\prime }$; it matches the full system wherever $|{\varphi }^{\prime }|$ is small, the discarded ${\varphi }^{\prime 2}/6$ living only on the fast-roll plunge.

Look at where the potential went in (6): $V$ appears only in the Hamiltonian $\mathscr{H}$, never in the contact form $\alpha$. The contact structure is

the standard contact structure on ${ℝ}^3$, which Bennequin proved is tight [4]. It is the same $\xi$ for every potential: one germ, two germs, a well, a spiral, the whole ADE zoo. The potential shapes the flow inside a fixed, universal, tight geometry; it does not, and cannot, twist the structure.

So the honest answer to the hunt is no. The two-germ well hosts no overtwisted disk, because overtwistedness is a property of $\xi$, and $\xi$ is blind to $V$. The spiral we found is a focus of the flow, not a twist of the structure-a perfectly tight geometry is entirely happy to carry a spiralling contact-Hamiltonian flow. We went looking for exotic topology and found the tightest, most standard geometry there is.

One loophole I will not paper over: this is proven where the contact structure is exact, the slow-roll bulk. The full flow, with its fast-roll friction, is not contact-Hamiltonian for this $\alpha$; the obstruction lives out near the kination horizon ${\varphi }^{\prime }\to \sqrt{6}$. Whether some exotic global form lurks there and is overtwisted, we have not ruled out, but that is the gauge boundary, outside inflation, with no positive reason to expect it. That boundary, where the field kinates and the reduction genuinely comes apart, is exactly where Part 4 goes.

The detour hands back a clean way to say what the cosmological contact structure is: FLRW scalar-field dynamics is contact-Hamiltonian flow on the standard tight ${ℝ}^3$, with the entire potential, inflection points, catastrophe unfoldings, all of it, packed into $\mathscr{H}=\frac{1}{2}{\varphi }^{\prime 2}+3\log V(\varphi )+3z$, and the friction coefficient $3$ set by the three spatial dimensions. The geometry is universal and rigid; every scrap of physics is in the Hamiltonian.

There is a smaller lesson, too. The condition that breaks the trajectory foliation ($R>3/2$), the condition that makes a spiral focus, and the condition that governs whether a field oscillates after inflation are one condition, $m_{\mathrm{eff}}/H>3/2$, seen from three sides. And the boundary where the tidy contact picture frays is not a real edge of physics: it is the horizon of the clock we chose to read. Mind your gauge.