Quantum geometry adds one term to the Friedmann equation, and the Big Bang singularity turns into a bounce
The previous lesson ended in an honest impasse. The full dynamics of the theory is hard: the Hamiltonian constraint can be turned into a finite operator but not a unique one, and the sum over spin foams is not under control. So try the oldest trick in physics. Throw away almost every degree of freedom, keep only the ones you need, and see whether the surviving problem is small enough to solve.
For cosmology the reduction is brutal and familiar. Assume the universe is homogeneous and isotropic, so a single number, the scale factor, describes all of geometry. Then quantise that one number using loop variables rather than the usual ones. The resulting subject is loop quantum cosmology, started by Bojowald around 2001 and put on firm footing by Ashtekar, Pawlowski and Singh in 2006, and it is where this theory makes its most concrete statement about the world.
Be clear about what that statement rests on. This is the quantisation of a model that was reduced first, not a solution of the full theory that happens to be homogeneous. Whether reducing and then quantising gives the same answer as quantising and then reducing is a genuinely open question, and it is the standing objection to everything in this lesson. What follows is worth taking seriously anyway, because the reduced model is solvable in closed form and the thing it says is not subtle.
The scale factor against time, for a universe filled with a massless scalar field. One curve is the classical solution and drives into zero at a finite time. The other is the same expression with a single 1 added inside a bracket. Drag the time marker across the bounce and watch the density climb, level off at the critical value, and come back down.
Drag anywhere on the plot to move the marker. Watch the ρ/ρ_c readout: on the quantum branch it climbs to exactly 1 at the bounce and comes back down, while the classical value beside it runs off to infinity. The two curves are indistinguishable a few Planck times either side of zero and disagree completely within one, which is the whole content of loop quantum cosmology. Note that moving ab only rescales the vertical axis: the size of the universe at the bounce depends on how much matter it contains, but the density there is always ρ_c — 0.409 Planck densities, or 2.11e96 kg/m³, at the γ you have selected.
The reason the loop quantisation behaves differently from the one Wheeler and DeWitt wrote down is the same reason the third lesson gave for using holonomies at all. In the old canonical quantisation you promote the connection itself to an operator, which gives you a differential operator in the volume, and a differential equation has no trouble marching straight into zero volume and stopping there. In the loop quantisation the connection is not an operator at all. Only holonomies are, and a holonomy is an exponential of the connection, which acts by shifting the volume by a finite step rather than differentiating it.
So the quantum constraint is a difference equation, and the step is set by the area gap from the fifth lesson. That is the whole mechanism, and it is worth noticing how economical it is. Nobody added a repulsive force, a bouncing potential, or an exotic fluid. The discreteness that came out of the kinematics is what keeps the recursion from having a last step. The wavefunction simply continues through to negative volume, which is to say to a universe on the other side.
The remarkable part is how well the quantum evolution is tracked by a purely classical-looking equation. Take the Friedmann equation and multiply the right-hand side by one factor, one minus the density over a critical density. Below that critical value the correction is unmeasurable. At it, the bracket vanishes, the expansion rate is zero, and a contracting universe has no choice but to turn around. The critical density works out to about 0.41 Planck densities, which is roughly a Planck mass in a Planck volume, or two times ten to the ninety-six kilograms per cubic metre.
The expansion rate plotted against density. The classical line climbs forever; the quantum curve peaks and returns to zero. Then switch views and see the same correction plotted across every density anyone has ever measured, where it is smaller than one part in ten to the seventy-nine. This is not an effect we might have missed through carelessness.
In the first view the two curves are indistinguishable near the origin and part company only in the last stretch before ρ_c, where the quantum one turns over and comes back to zero. That return is the bounce: a vanishing expansion rate at finite density instead of an infinite one at zero size. The second view is the reason nobody noticed. Sliding γ moves ρ_c, which is the only place the Barbero-Immirzi parameter shows its face in cosmology, and it moves the turnover without changing anything at densities we can reach.
A singularity is where a theory stops being able to answer questions, and this model removes one. That is a real result and it should not be undersold: the quantity that classically diverges is bounded, the evolution continues, and nothing has to be put in by hand to make it happen. It is also a much smaller claim than it sounds like, and three qualifications matter.
The first is the one from the opening. This is a reduced model, and the reduction happened before the quantisation. The second is that removing the singularity does not explain the initial conditions. The observable universe began in a state of extraordinarily low entropy, and a bounce inherits that puzzle rather than solving it: something has to explain why the contracting phase arrived in such a special configuration. Turning a beginning into a middle relocates the question without answering it.
The third is the sharpest, and the third demo carries it. The bounce depends on how the holonomy step is taken to scale as the universe grows. Early work used a fixed step and got a bounce at a density that depended on the arbitrary fiducial cell you drew around the universe, which is not physics. The field converged on the so-called improved dynamics, in which the step is tied to the physical area, and that version puts the bounce at a fixed physical density. The reasoning was good and the result is much better behaved. It was still a choice among prescriptions rather than something the theory forced.
The massless scalar is the case everyone quotes, which invites the suspicion that the turnaround is an artefact of it. Switch the matter content to dust or radiation and the turnaround survives, because the general case solves in closed form too. Note the caveat panel: this is robustness within one prescription, not across all of them.
Every equation of state bounces, so the result is not an artefact of the massless scalar. But this is robustness within one quantisation prescription, not across all of them. The bounce depends on how the holonomy step is taken to scale with the volume, and other prescriptions give other answers. The field settled on the so-called improved dynamics because it puts the bounce at a fixed physical density rather than one that depends on the arbitrary fiducial cell you drew around the universe. That was a well-motivated choice. It was still a choice.
Widen the time window and the solid curve settles onto the dashed one: far from the bounce the quantum universe is the classical universe, exactly. Narrow it and the two part company only in the last Planck time or so, where the dashed curve runs into zero and the solid one refuses to. Stiff matter dilutes fastest, so it makes the sharpest turn and the slowest subsequent growth.
The bounce happens at the Planck density, which is not a regime any experiment will reach. But the universe is an unusual laboratory: it stretched whatever was happening then across the entire sky. The modes we observe at the largest angular scales in the cosmic microwave background are the ones that left the horizon earliest, which makes them the closest thing we have to a fossil of the pre-inflation era. Loop quantum cosmology predicts modifications to the primordial power spectrum precisely there.
The trouble is that the largest angular scales are also where the data are worst. There are only so many independent patches that big on a single sky, so cosmic variance sets a floor on the error bars that no better telescope can lower. Current observations neither confirm nor exclude the predicted deviations. The final lesson of this module takes the observational question seriously and at length, including what would actually have to be seen.