The Fabric of the Cosmos: Space, Time, and the Texture of Reality

by Brian Greene

INFLATION, QUANTUM JITTERS, AND THE ARROW OF TIME

Quantum Skywriting

How is it that there are galaxies, stars, planets, and other clumpy things in the universe?

If the universe is indeed smooth, uniform, and homogeneous on large scales—features that are supported by observation and that lie at the heart of all cosmological analyses—where could the smaller-scale lumpiness have come from?

According to inflationary cosmology, the initial nonuniformity that ultimately resulted in the formation of stars and galaxies came from quantum mechanics.

By essentially the same reasoning we used in its application to particles, the uncertainty principle implies that the more precisely the value of a field is determined at one location in space, the less precisely its rate of change at that location can be determined. (The position of a particle and the rate of change of its position—its velocity—play analogous roles in quantum mechanics to the value of a field and the rate of change of the field value, at a given location in space.)

If a field’s rate of change can’t be delineated with total precision, then we also can’t delineate what the value of the field will be, at any location, even a moment later.

The sudden burst of inflationary expansion stretched space by such an enormous factor that what initially inhabited the microscopic was drawn out to the macroscopic.

Through the enormous stretching of inevitable quantum fluctuations, inflationary cosmology provides an explanation: inflationary expansion stretches tiny, inhomogeneous quantum jitters and smears them clear across the sky.

Recall that inflationary expansion came to an end when the inflaton field’s value slid down its potential energy bowl and the field relinquished all its pent-up energy and negative pressure. We described this as happening uniformly throughout space—the inflaton value here, there, and everywhere experienced the same evolution—as that’s what naturally emerges from the governing equations. However, this is strictly true only if we ignore the effects of quantum mechanics. On average, the inflaton field value did indeed slide down the bowl, as we expect from thinking about a simple classical object like a marble rolling down an incline. But just as a frog sliding down the bowl is likely to jump and jiggle along the way, quantum mechanics tells us that the inflaton field experienced quivers and jitters.

But with inflation, space expanded at such a colossal rate, doubling in size every 10−37 seconds, that even a slightly different duration of inflation at nearby locations resulted in a significant wrinkle. In fact, calculations undertaken in specific realizations of inflation have shown that the inhomogeneities produced in this way have a tendency to be too large; researchers often have to adjust details in a given inflationary model (the precise shape of the inflaton field’s potential energy bowl) to ensure that the quantum jitters don’t predict a universe that’s too lumpy.

The Golden Age of Cosmology

Precision measurements, first accomplished in 1992 by COBE (the Cosmic Background Explorer satellite) and more recently by WMAP (the Wilkinson Microwave Anisotropy Probe), have determined that while the temperature might be 2.7249 Kelvin in one spot in space, it might be 2.7250 Kelvin in another, and 2.7251 Kelvin in still another.

Creating a Universe

the total energy carried by ordinary particles of matter and radiation drops because it is continually transferred to gravity as the universe expands.

the total energy embodied by the inflaton field increases as the universe expands because it extracts energy from gravity.

To summarize: as the universe expands, matter and radiation lose energy to gravity while an inflaton field gains energy from gravity.

Thus, instead of offering an explanation for where all the mass/energy currently inhabiting the universe originated, the standard big bang fights an unending uphill battle: the farther back the theory looks, the more mass/energy it must somehow explain.

As just explained, the inflaton field is a gravitational parasite—it feeds on gravity—and so the total energy the inflaton field carried increased as space expanded. More precisely, the mathematical analysis shows that the energy density of the inflaton field remained constant throughout the inflationary phase of rapid expansion, implying that the total energy it embodied grew in direct proportion to the volume of the space it filled.

the size of the universe increased by at least a factor of 1030 during inflation, which means the volume of the universe increased by a factor of at least (10 30)3 = 1090. Consequently, the energy embodied in the inflaton field increased by the same huge factor: as the inflationary phase drew to a close, a mere 10−35 or so seconds after it began, the energy in the inflaton field grew by a factor on the order of 1090, if not more. This means that at the onset of inflation, the inflaton field didn’t need to have much energy, since the enormous expansionit was about to spawn would enormously amplify the energy it carried. A simple calculation shows that a tiny nugget, on the order of 10 −26 centimeters across, filled with a uniform inflaton field—and weighing a mere twenty pounds—would, through the ensuing inflationary expansion, acquire enough energy to account for all we see in the universe today.

Inflation, Smoothness, and the Arrow of Time

The puzzle we encountered is to explain how this high-order, low-entropy starting point came to be.

But when gravity matters, as it does when considering the entire universe, a uniform distribution of matter is a rare, low-entropy, highly ordered configuration, because gravity drives matter to form clumps. Similarly, a smooth and uniform spatial curvature also has very low entropy; it is highly ordered compared with a wildly bumpy, nonuniform spatial curvature.

Why did the early universe have a low-entropy (highly ordered) uniform distribution of matter instead of a high-entropy (highly disordered) clumpy distribution of matter such as a diverse population of black holes?

And why was the curvature of space smooth, ordered, and uniform to extremely high accuracy rather than being riddled with a variety of huge warps and severe curves, also like those generated by black holes?

When gravity matters, ordinary, unremarkable, high-entropy configurations are lumpy and bumpy.

The enormous outward push of repulsive gravity drove space to swell so swiftly that initial bumps and warps were stretched smooth, much as fully inflating a shriveled balloon stretches out its creased surface.

Thus, although attractive gravity causes clumps of matter and creases of space to grow, repulsive gravity does the opposite: it causes them to diminish, leading to an ever smoother, ever more uniform outcome.

The outcome we’ve reached via inflation —a smooth, uniform spatial expansion populated by a nearly uniform distribution of matter—was exactly what we were trying to explain. It’s exactly the low-entropy configuration that we need to explain time’s arrow.

Entropy and Inflation

You see, by the end of the inflationary phase, space was stretched smooth and so the gravitational contribution to entropy—the entropy associated with the possible bumpy, nonordered, nonuniform shape of space—was minimal. However, when the inflaton field slid down its bowl and relinquished its pent-up energy, it is estimated to have produced about 1080 particles of matter and radiation. Such a huge number of particles, like a book with a huge number of pages, embodies a huge amount of entropy. Thus, even though the gravitational entropy went down, the increase in entropy from the production of all these particles more than compensated. The total entropy increased, just as we expect from the second law.

Overall entropy increased during inflation, but by a paltry amount compared with how much it could have increased. It’s in this sense that inflation generated a low-entropy universe: by the end of inflation, entropy had increased, but by nowhere near the factor by which the spatial expanse had increased.

And so inflationary cosmology gives a direction to time’s arrow by generating a past with exceedingly low gravitational entropy; the future is the direction in which this entropy grows.

That is, physicists bristle at the standard big bang’s reliance on finely tuned homogeneous initial conditions that, while observationally motivated, are theoretically unexplained. It feels deeply unsatisfying for the low-entropy state of the early universe simply to be assumed; it feels hollow for time’s arrow to be imposed on the universe, without explanation.

What are the conditions necessary for inflation?

Boltzmann Redux

Thus, rather than assuming or simply declaring that conditions in the early universe were right for inflationary expansion to take place, in this way of thinking about things an ultramicroscopic fluctuation weighing a mere twenty pounds, occurring within an ordinary, unremarkable environment of disorder, gave rise to the necessary conditions.

But all that matters to us is that there was one nugget that yielded the space-smoothing inflationary burst that provided the first link in the low-entropy chain, ultimately leading to our familiar cosmos.

From the get-go, inflation gave the universe an amazing deal. A jump to lower entropy within a tiny nugget of space was leveraged by inflationary expansion into the vast reaches of the cosmos. And, of utmost importance, the inflationary stretching didn’t just yield any old large universe. It yielded our large universe—inflation explains the shape of space, it explains the large-scale uniformity, and it even explains the “smaller”-scale inhomogeneities such as galaxies and temperature variations in the background radiation.

Inflation and the Egg

Where does the arrow of time that we all experience come from?

Through a chance but every so often expectable fluctuation from an unremarkable primordial state with high entropy, a tiny, twenty-pound nugget of space achieved conditions that led to a brief burst of inflationary expansion.

it is this early state of order—the absence of severe bumps or warps or gargantuan black holes—that primed the universe for the subsequent evolution to higher entropy and hence provided the arrow of time we all experience.

With our current level of understanding, this is the most complete explanation for time’s arrow that has been given.

The Fly in the O...

But in telling this story, we’ve made a pivotal assumption that’s as yet unjustified. To assess the likelihood of inflation’s being initiated, we’ve had to specify the characteristics of the preinflationary realm out of which inflationary expansion is supposed to have emerged. The particular realm we’ve envisioned—wild, chaotic, energetic—sounds reasonable, but delineating this intuitive description with mathematical precision proves challenging. Moreover, it is only a guess. The bottom line is that we don’t know what conditions were like in the supposed preinflationary realm, in the fuzzy patch of Figure 10.3, and without that information we are unable to make a convincing assessment of the likelihood of inflation’s initiating; any calculation of the likelihood depends sensitively on the assumptions we make.

IV

ORIGINS AND UNIFICATION

The World on a String

THE FABRIC ACCORDING TO STRING THEORY