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THE FABRIC of the COSMOS, Brian Greene, 2004
```(annotated and with added bold highlights by Epsilon=One)
```(annotated and with added bold highlights by Epsilon=One)
Chapter 10 - Deconstructing the Bang
Inflation and the Flatness Problem
A second problem addressed by inflationary cosmology has to do with the shape of space. In Chapter 8, we imposed the criterion of uniform spatial symmetry and found three ways in which the fabric of space can curve. Resorting to our two-dimensional visualizations, the possibilities are positive curvature (shaped like the surface of a ball), negative curvature (saddle-shaped), and zero curvature (shaped like, an infinite flat tabletop or like a finite-sized video game screen). Since the early days of general relativity, physicists have realized that the total matter and energy in each volume of space — the matter/energy density — determine the curvature of space. If the matter/energy density is high, space will pull back on itself in the shape of a sphere; that is, there will be positive curvature. If the matter/energy density is low, space will flare outward like a saddle; that is, there will be negative curvature. Or, as mentioned in the last chapter, for a very special amount of matter/energy density — the critical density, equal to the mass of about five hydrogen atoms (about 10^-23 grams) in each cubic meter — space will lie just between these two extremes, and will be perfectly flat: that is, there will be no curvature.
Now for the puzzle.
The equations of general relativity, which underlie the standard big bang model, show that if the matter/energy density early on was exactly equal to the critical density, then it would stay equal to the critical density as space expanded. 17 But if the matter/energy density was even slightly more or slightly less than the critical density, subsequent expansion would drive it enormously far from the critical density. Just to get a feel for the numbers, if at one second ATB, the universe was just shy of criticality, having 99.99 percent of the critical density, calculations show that by today its density would have been driven all the way down to .00000000001 of the critical density. It's kind of like the situation faced by a mountain climber who is walking across a razor-thin ledge with a steep drop off on either side. If her step is right on the mark, she'll make it across. But even a tiny misstep that's just a little too far left or right will be amplified into a significantly different outcome. (And, at the risk of having one too many analogies, this feature of the standard big bang model also reminds me of the shower years ago in my college dorm: if you managed to set the knob perfectly, you could get a comfortable water temperature. But if you were off by the slightest bit, one way or the other, the water would be either scalding or freezing. Some students just stopped showering altogether.)
For decades, physicists have been attempting to measure the matter/ energy density in the universe. By the 1980s, although the measurements were far from complete, one thing was certain: the matter/energy density of the universe is not thousands and thousands of times smaller or larger than the critical density; equivalently, space is not substantially curved, either positively or negatively. This realization cast an awkward light on the standard big bang model. It implied that for the standard big bang to be consistent with observations, some mechanism — one that nobody could explain or identify — must have tuned the matter/energy density of the early universe extraordinarily close to the critical density. For example, calculations showed that at one second ATB, the matter/energy density of the universe needed to have been within a millionth of a millionth of a percent of the critical density; if the matter/energy density deviated from the critical value by any more than this minuscule amount, the standard big bang model predicts a matter/energy density today that is vastly different from what we observe. According to the standard big bang model, then, the early universe, much like the mountain climber, teetered along an extremely narrow ledge. A tiny deviation in conditions billions of years ago would have led to a present-day universe very different from the one revealed by astronomers' measurements. This is known as the flatness problem.
Although we've covered the essential idea, it's important to understand the sense in which the flatness problem is a problem. By no means does the flatness problem show that the standard big bang model is wrong. A staunch believer reacts to the flatness problem with a shrug of the shoulders and the curt reply "That's just how it was back then," taking the finely tuned matter/energy density of the early universe — which the standard big bang requires to yield predictions that are in the same ball park as observations — as an unexplained given. But this answer makes most physicists recoil. Physicists feel that a theory is grossly unnatural if its success hinges on extremely precise tunings of features for which we lack a fundamental explanation. Without supplying a reason for why the matter/energy density of the early universe would have been so finely tuned to an acceptable value, many physicists have found the standard big bang model highly contrived. Thus, the flatness problem highlights the extreme sensitivity of the standard big bang model to conditions in the remote past of which we know very little; it shows how the theory must assume the universe was just so, in order to work.
By contrast, physicists long for theories whose predictions are insensitive to unknown quantities such as how things were a long time ago. Such theories feel robust and natural because their predictions don't depend delicately on details that are hard, or perhaps even impossible, to determine directly. This is the kind of theory provided by inflationary cosmology, and its solution to the flatness problem illustrates why.
The essential observation is that whereas attractive gravity amplifies any deviation from the critical matter/energy density, the repulsive gravity of the inflationary theory does the opposite: it reduces any deviation from the critical density. To get a feel for why this is the case, it's easiest to use the tight connection between the universe's matter/energy density and its curvature to reason geometrically. In particular, notice that [B]even if the shape of the universe were significantly curved early on, [B/]after inflationary expansion a portion of space large enough to encompass today's observable universe looks very nearly flat. This is a feature of geometry we are all well aware of: The surface of a basketball is obviously curved, but it took both time and thinkers with chutzpah before everyone was convinced that the earth's surface was also curved. The reason is that, all else being equal, the larger something is, the more gradually it curves and the flatter a patch of a given size on its surface appears. If you draped the state of Nebraska over a sphere just a few hundred miles in diameter, as in Figure 10.4a, it would look curved, but on the earth's surface, as just about all Nebraskans concur, it looks flat. If you laid Nebraska out on a sphere a billion times larger than earth, it would look flatter still. In inflationary cosmology, space was stretched by such a colossal factor that the observable universe, the part we can see, is but a small patch in a gigantic cosmos. And so, like Nebraska laid out on a giant sphere as in Figure 10.4d, even if the entire universe were curved, the observable universe would be very nearly flat. 18
Figure 10.4 (a) A shape of fixed size, such as that of the state of Nebraska, appears flatter and flatter when laid out on larger and larger spheres. In this analogy, the sphere represents the entire universe, while Nebraska represents the observable universe — the part within our cosmic horizon.
It's as if there are powerful, oppositely oriented magnets embedded in the mountain climber's boots and the thin ledge she is crossing. Even if her step is aimed somewhat off the mark, the strong attraction between the magnets ensures that her foot lands squarely on the ledge. Similarly, even if the early universe deviated a fair bit from the critical matter/energy density and hence was far from flat, the inflationary expansion ensured that the part of space we have access to was driven toward a flat shape and that the matter/energy density we have access to was driven to the critical value.
Now for the puzzle.
The equations of general relativity, which underlie the standard big bang model, show that if the matter/energy density early on was exactly equal to the critical density, then it would stay equal to the critical density as space expanded. 17 But if the matter/energy density was even slightly more or slightly less than the critical density, subsequent expansion would drive it enormously far from the critical density. Just to get a feel for the numbers, if at one second ATB, the universe was just shy of criticality, having 99.99 percent of the critical density, calculations show that by today its density would have been driven all the way down to .00000000001 of the critical density. It's kind of like the situation faced by a mountain climber who is walking across a razor-thin ledge with a steep drop off on either side. If her step is right on the mark, she'll make it across. But even a tiny misstep that's just a little too far left or right will be amplified into a significantly different outcome. (And, at the risk of having one too many analogies, this feature of the standard big bang model also reminds me of the shower years ago in my college dorm: if you managed to set the knob perfectly, you could get a comfortable water temperature. But if you were off by the slightest bit, one way or the other, the water would be either scalding or freezing. Some students just stopped showering altogether.)
For decades, physicists have been attempting to measure the matter/ energy density in the universe. By the 1980s, although the measurements were far from complete, one thing was certain: the matter/energy density of the universe is not thousands and thousands of times smaller or larger than the critical density; equivalently, space is not substantially curved, either positively or negatively. This realization cast an awkward light on the standard big bang model. It implied that for the standard big bang to be consistent with observations, some mechanism — one that nobody could explain or identify — must have tuned the matter/energy density of the early universe extraordinarily close to the critical density. For example, calculations showed that at one second ATB, the matter/energy density of the universe needed to have been within a millionth of a millionth of a percent of the critical density; if the matter/energy density deviated from the critical value by any more than this minuscule amount, the standard big bang model predicts a matter/energy density today that is vastly different from what we observe. According to the standard big bang model, then, the early universe, much like the mountain climber, teetered along an extremely narrow ledge. A tiny deviation in conditions billions of years ago would have led to a present-day universe very different from the one revealed by astronomers' measurements. This is known as the flatness problem.
Although we've covered the essential idea, it's important to understand the sense in which the flatness problem is a problem. By no means does the flatness problem show that the standard big bang model is wrong. A staunch believer reacts to the flatness problem with a shrug of the shoulders and the curt reply "That's just how it was back then," taking the finely tuned matter/energy density of the early universe — which the standard big bang requires to yield predictions that are in the same ball park as observations — as an unexplained given. But this answer makes most physicists recoil. Physicists feel that a theory is grossly unnatural if its success hinges on extremely precise tunings of features for which we lack a fundamental explanation. Without supplying a reason for why the matter/energy density of the early universe would have been so finely tuned to an acceptable value, many physicists have found the standard big bang model highly contrived. Thus, the flatness problem highlights the extreme sensitivity of the standard big bang model to conditions in the remote past of which we know very little; it shows how the theory must assume the universe was just so, in order to work.
By contrast, physicists long for theories whose predictions are insensitive to unknown quantities such as how things were a long time ago. Such theories feel robust and natural because their predictions don't depend delicately on details that are hard, or perhaps even impossible, to determine directly. This is the kind of theory provided by inflationary cosmology, and its solution to the flatness problem illustrates why.
The essential observation is that whereas attractive gravity amplifies any deviation from the critical matter/energy density, the repulsive gravity of the inflationary theory does the opposite: it reduces any deviation from the critical density. To get a feel for why this is the case, it's easiest to use the tight connection between the universe's matter/energy density and its curvature to reason geometrically. In particular, notice that [B]even if the shape of the universe were significantly curved early on, [B/]after inflationary expansion a portion of space large enough to encompass today's observable universe looks very nearly flat. This is a feature of geometry we are all well aware of: The surface of a basketball is obviously curved, but it took both time and thinkers with chutzpah before everyone was convinced that the earth's surface was also curved. The reason is that, all else being equal, the larger something is, the more gradually it curves and the flatter a patch of a given size on its surface appears. If you draped the state of Nebraska over a sphere just a few hundred miles in diameter, as in Figure 10.4a, it would look curved, but on the earth's surface, as just about all Nebraskans concur, it looks flat. If you laid Nebraska out on a sphere a billion times larger than earth, it would look flatter still. In inflationary cosmology, space was stretched by such a colossal factor that the observable universe, the part we can see, is but a small patch in a gigantic cosmos. And so, like Nebraska laid out on a giant sphere as in Figure 10.4d, even if the entire universe were curved, the observable universe would be very nearly flat. 18
Figure 10.4 (a) A shape of fixed size, such as that of the state of Nebraska, appears flatter and flatter when laid out on larger and larger spheres. In this analogy, the sphere represents the entire universe, while Nebraska represents the observable universe — the part within our cosmic horizon.
It's as if there are powerful, oppositely oriented magnets embedded in the mountain climber's boots and the thin ledge she is crossing. Even if her step is aimed somewhat off the mark, the strong attraction between the magnets ensures that her foot lands squarely on the ledge. Similarly, even if the early universe deviated a fair bit from the critical matter/energy density and hence was far from flat, the inflationary expansion ensured that the part of space we have access to was driven toward a flat shape and that the matter/energy density we have access to was driven to the critical value.