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Notes: Chapter 11
1. Among the leaders in the early 1980s in determining how quantum fluctuation would yield inhomogeneities were Stephen Hawking, Alexei Starobinsky, Alan Guth, So Young Pi, James Bardeen, Paul Steinhardt, Michael Turner, Viatcheslav Mukhanov, and Gennady Chibisov. Return to Text
2. Even with the discussion in the main text, you may still be puzzled regarding how a tiny amount of mass/energy in an inflaton nugget can yield the huge amount of mass/energy constituting the observable universe. [BI]How can you wind up with more mass/energy than you begin with?[/B] Well, as explained in the main text, the inflaton field, by virtue of its negative pressure, "mines" energy from gravity. This means that as the energy in the inflaton field increases, the energy in the gravitational field decreases. The special feature of the gravitational field, known since the days of Newton, is that its energy can become arbitrarily negative. Thus, gravity is like a bank that is willing to lend unlimited amounts of money — gravity embodies an essentially limitless supply of energy, which the inflaton field extracts as space expands.
The particular mass and size of the initial nugget of uniform inflaton field depend on the details of the model of inflationary cosmology one studies (most notably, on the precise details of the inflaton field's potential energy bowl). In the text, I've imagined that the initial inflaton field's energy density was about 10^82 grams per cubic centimeter, so that a volume of (10^-26 centimeters)^35= 10^-78 cubic centimeters would have total mass of about 10 kilograms, i.e., about 20 pounds. These values are typical to a fairly conventional class of inflationary models, but are only meant to give you a rough sense of the numbers involved. To give a flavor of the range of possibilities, let me note that in Andrei Linde's chaotic models of inflation (see note 11 of Chapter 10), our observable universe would have emerged from an initial nugget of even smaller size, 10^-33 centimeters across (the so-called Planck length), whose energy density was even higher, about 10^94 grams per cubic centimeter, combining to give a lower total mass of about 10^-5 grams (the so-called Planck mass). In these realizations of inflation, the initial nugget would have weighed about as much as a grain of dust. Return to Text
3. See Paul Davies, "Inflation and Time Asymmetry in the Universe," in Nature, vol. 301, p. 398; Don Page, "Inflation Does Not Explain Time Asymmetry," in Nature, vol. 304, p. 39; and Paul Davies, "Inflation in the Universe and Time Asymmetry," in Nature, vol. 312, p. 524. Return to Text
4. To explain the essential point, it is convenient to split entropy up into a part due to spacetime and gravity, and a remaining part due to everything else, as this captures intuitively the key ideas. However, I should note that it proves elusive to give a mathematically rigorous treatment in which the gravitational contribution to entropy is cleanly identified, separated off, and accounted for. Nevertheless, this doesn't compromise the qualitative conclusions we reach. In case you find this troublesome, note that the whole discussion can be rephrased largely without reference to gravitational entropy. As we emphasized in Chapter 6, when ordinary attractive gravity is relevant, matter falls together into clumps. In so doing, the matter converts gravitational potential energy into kinetic energy that, subsequently, is partially converted into radiation that emanates from the clump itself. This is an entropy-increasing sequence of events (larger average particle velocities increase the relevant phase space volume; the production of radiation through interactions increases the total number of particles — both of which increase overall entropy). In this way, what we refer to in the text as gravitational entropy can be rephrased as matter entropy generated by the gravitational force. When we say gravitational entropy is low, we mean that the gravitational force has the potential to generate significant quantities of entropy through matter clumping. In realizing such entropy potential, the clumps of matter create a non-uniform, non-homogeneous gravitational field — warps and ripples in spacetime — which, in the text, I've described as having higher entropy. But as this discussion makes clear, it really can be thought of as the clumpy matter (and radiation produced in the process) as having higher entropy (than when uniformly dispersed). This is good since the expert reader will note that if we view a classical gravitational background (a classical spacetime) as a coherent state of gravitons, it is an essentially unique state and hence
has low entropy. Only by suitably coarse graining would an entropy assignment be possible. As this note emphasizes, though, this isn't particularly necessary. On the other hand, should the matter clump sufficiently to create black holes, then an unassailable entropy assignment becomes available: the area of the black hole's event horizon (as explained further in Chapter 16) is a measure of the black hole's entropy. And this entropy can unambiguously be called gravitational entropy. Return to Text
5. Just as it is possible both for an egg to break and for broken eggshell pieces to reassemble into a pristine egg, it is possible for quantum-induced fluctuations to grow into larger inhomogeneities (as we've described) or for sufficiently correlated inhomogeneities to work in tandem to suppress such growth. Thus, the inflationoty contribution to resolving time's arrow also requires sufficiently uncorrelated initial quantum fluctuations. Again, if we think in a Boltzmann-like manner, among all the fluctuations yielding conditions ripe for inflation, sooner or later there will be one that meets this condition as well, allowing the universe as we know it to initiate. Return to Text
6. There are some physicists who would claim that the situation is better than described. For example, Andrei Linde argues that in chaotic inflation (see note 11, Chapter 10), the observable universe emerged from a Planck-sized nugget containing a uniform inflaton field with Planck scale energy density. Under certain assumptions, Linde further argues that the entropy of a uniform inflaton field in such a tiny nugget is roughly equal to the entropy of any other inflaton field configuration, and hence the conditions necessary for achieving inflation weren't special. The entropy of the Planck-sized nugget was small but on a par with the possible entropy that the Planck-sized nugget could have had. The ensuing inflationary burst then created, in a flash, a huge universe with an enormously higher entropy — but one that, because of its smooth, uniform distribution of matter, was also enormously far from the entropy that it could have. The arrow of time points in the direction in which this entropy gap is being lessened.
While I am partial to this optimistic vision, until we have a better grasp on the physics out of which inflation is supposed to have emerged, caution is warranted. For example, the expert reader will note that this approach makes favorable but unjustified assumptions about the high-energy (transplanckian) field modes — modes that can affect the onset of inflation and play a crucial role in structure formation. Return to Text
2. Even with the discussion in the main text, you may still be puzzled regarding how a tiny amount of mass/energy in an inflaton nugget can yield the huge amount of mass/energy constituting the observable universe. [BI]How can you wind up with more mass/energy than you begin with?[/B] Well, as explained in the main text, the inflaton field, by virtue of its negative pressure, "mines" energy from gravity. This means that as the energy in the inflaton field increases, the energy in the gravitational field decreases. The special feature of the gravitational field, known since the days of Newton, is that its energy can become arbitrarily negative. Thus, gravity is like a bank that is willing to lend unlimited amounts of money — gravity embodies an essentially limitless supply of energy, which the inflaton field extracts as space expands.
The particular mass and size of the initial nugget of uniform inflaton field depend on the details of the model of inflationary cosmology one studies (most notably, on the precise details of the inflaton field's potential energy bowl). In the text, I've imagined that the initial inflaton field's energy density was about 10^82 grams per cubic centimeter, so that a volume of (10^-26 centimeters)^35= 10^-78 cubic centimeters would have total mass of about 10 kilograms, i.e., about 20 pounds. These values are typical to a fairly conventional class of inflationary models, but are only meant to give you a rough sense of the numbers involved. To give a flavor of the range of possibilities, let me note that in Andrei Linde's chaotic models of inflation (see note 11 of Chapter 10), our observable universe would have emerged from an initial nugget of even smaller size, 10^-33 centimeters across (the so-called Planck length), whose energy density was even higher, about 10^94 grams per cubic centimeter, combining to give a lower total mass of about 10^-5 grams (the so-called Planck mass). In these realizations of inflation, the initial nugget would have weighed about as much as a grain of dust. Return to Text
3. See Paul Davies, "Inflation and Time Asymmetry in the Universe," in Nature, vol. 301, p. 398; Don Page, "Inflation Does Not Explain Time Asymmetry," in Nature, vol. 304, p. 39; and Paul Davies, "Inflation in the Universe and Time Asymmetry," in Nature, vol. 312, p. 524. Return to Text
4. To explain the essential point, it is convenient to split entropy up into a part due to spacetime and gravity, and a remaining part due to everything else, as this captures intuitively the key ideas. However, I should note that it proves elusive to give a mathematically rigorous treatment in which the gravitational contribution to entropy is cleanly identified, separated off, and accounted for. Nevertheless, this doesn't compromise the qualitative conclusions we reach. In case you find this troublesome, note that the whole discussion can be rephrased largely without reference to gravitational entropy. As we emphasized in Chapter 6, when ordinary attractive gravity is relevant, matter falls together into clumps. In so doing, the matter converts gravitational potential energy into kinetic energy that, subsequently, is partially converted into radiation that emanates from the clump itself. This is an entropy-increasing sequence of events (larger average particle velocities increase the relevant phase space volume; the production of radiation through interactions increases the total number of particles — both of which increase overall entropy). In this way, what we refer to in the text as gravitational entropy can be rephrased as matter entropy generated by the gravitational force. When we say gravitational entropy is low, we mean that the gravitational force has the potential to generate significant quantities of entropy through matter clumping. In realizing such entropy potential, the clumps of matter create a non-uniform, non-homogeneous gravitational field — warps and ripples in spacetime — which, in the text, I've described as having higher entropy. But as this discussion makes clear, it really can be thought of as the clumpy matter (and radiation produced in the process) as having higher entropy (than when uniformly dispersed). This is good since the expert reader will note that if we view a classical gravitational background (a classical spacetime) as a coherent state of gravitons, it is an essentially unique state and hence
has low entropy. Only by suitably coarse graining would an entropy assignment be possible. As this note emphasizes, though, this isn't particularly necessary. On the other hand, should the matter clump sufficiently to create black holes, then an unassailable entropy assignment becomes available: the area of the black hole's event horizon (as explained further in Chapter 16) is a measure of the black hole's entropy. And this entropy can unambiguously be called gravitational entropy. Return to Text
5. Just as it is possible both for an egg to break and for broken eggshell pieces to reassemble into a pristine egg, it is possible for quantum-induced fluctuations to grow into larger inhomogeneities (as we've described) or for sufficiently correlated inhomogeneities to work in tandem to suppress such growth. Thus, the inflationoty contribution to resolving time's arrow also requires sufficiently uncorrelated initial quantum fluctuations. Again, if we think in a Boltzmann-like manner, among all the fluctuations yielding conditions ripe for inflation, sooner or later there will be one that meets this condition as well, allowing the universe as we know it to initiate. Return to Text
6. There are some physicists who would claim that the situation is better than described. For example, Andrei Linde argues that in chaotic inflation (see note 11, Chapter 10), the observable universe emerged from a Planck-sized nugget containing a uniform inflaton field with Planck scale energy density. Under certain assumptions, Linde further argues that the entropy of a uniform inflaton field in such a tiny nugget is roughly equal to the entropy of any other inflaton field configuration, and hence the conditions necessary for achieving inflation weren't special. The entropy of the Planck-sized nugget was small but on a par with the possible entropy that the Planck-sized nugget could have had. The ensuing inflationary burst then created, in a flash, a huge universe with an enormously higher entropy — but one that, because of its smooth, uniform distribution of matter, was also enormously far from the entropy that it could have. The arrow of time points in the direction in which this entropy gap is being lessened.
While I am partial to this optimistic vision, until we have a better grasp on the physics out of which inflation is supposed to have emerged, caution is warranted. For example, the expert reader will note that this approach makes favorable but unjustified assumptions about the high-energy (transplanckian) field modes — modes that can affect the onset of inflation and play a crucial role in structure formation. Return to Text