Notes

Preface

1. To account for the uniformity of the universe on large scales and its enormous size, the theory of inflation proposes that the early universe underwent a period of super-fast expansion, where it repeatedly doubled in size. This inflationary epoch explains how the Big Bang expansion of the universe got started.

Chapter 1: Hubble Discovers the Universe

1. Here is what Einstein’s equation looks like: R_μν − ½g_μνR = 8πT_μν. I just wanted you to see it! Needless to say, it requires a lot of unpacking of mathematical terms to understand it in detail, but putting it simply, the “stuff” of the universe (matter, mass energy, pressure, etc.) causes space and time to curve. The right-hand side of the equation uses the stress-energy tensor T_μν to describe the mass-energy density and pressure of stuff, plus the momentum flux and stress associated with stuff at a point. The left-hand side of the equation tells how space and time are curved at that point. The equation as a whole describes exactly how the stuff of the universe (matter, radiation) causes space and time to curve.

The terms in the equation are tensors—mathematical objects that can be translated from one coordinate system to another. A vector is a tensor of rank 1. Vectors are used to show velocities; a velocity has a particular magnitude and a particular direction. It has components in each of the different directions. A vector on a two-dimensional surface has two components; wind on the surface of Earth, for example, has a North-South component and an East-West component. If one changes coordinate systems (say to Magnetic North), the components of the vector will change in a particular way, specified by the equations relating the two coordinate systems. The stress-energy tensor T_μν is a tensor of rank two that has 4 × 4 = 16 components in four-dimensional spacetime. In a local coordinate system where the observer is at rest: T_tt = mass-energy density, T_xx = pressure in the x-direction, T_yy = pressure in the y-direction, T_zz = pressure in the z direction. T_tx = T_xt, T_ty = T_yt, T_tz = T_zt represent components of momentum flux in the x-, y-, and z-directions, and T_xy = T_yx, T_xz = T_zx, T_zy = T_yz represent components of stress in different directions. Importantly, pressure gravitates as well as energy density.

The laws of tensor arithmetic tell us how the components of the stress-energy tensor change when we change coordinate systems. The term R_μν is a tensor derived by summing components of the even more complicated Riemann curvature tensor R_μναβ (which has 4 × 4 × 4 × 4 = 256 components describing how spacetime is curved at a particular point). The term −½g_μνR is composed of the metric tensor g_μν multiplied by a quantity R derived by summing components of R_μν. The metric tensor measures separations between nearby events.

It took Einstein 8 years of hard work to derive this equation; it is one of the triumphs of human thought. I’m showing it to you so you might appreciate its complexity and yet its ultimate elegance and simplicity. It embodies a simple idea, that the stuff in the universe causes spacetime to curve—and that is what is responsible for gravity. Amazing.

2. To produce a static universe, Einstein added a new term to his field equation, making it read: R_μν − ½g_μνR + Λg_μν = 8πT_μν. The new term Λg_μν was called the cosmological constant term. It is the metric tensor multiplied by a constant Λ (capital lambda). R is a measure of the overall curvature and can vary throughout spacetime, but Λ is a universal constant that is constant throughout spacetime. As pointed out by Lemaître, the term may be moved to the opposite side of the equation: R_μν − ½g_μνR = 8πT_μν − Λg_μν, where it may be interpreted as a vacuum energy density: R_μν − ½g_μνR = 8π[T_μν + (T_μν)_vac], with (T_μν)_vac = −Λg_μν/8π. The structure of the metric tensor (which locally can be written as ds²= −dt²+ dx²+ dy²+ dz² or g = −1, g_xx = 1, g_yy = 1, g_zz = 1) with opposite signs between the time and space components ensures that when multiplied by −Λ/8π, it gives opposite signs for the vacuum energy and the vacuum pressure: a positive vacuum energy (T_tt)_vac must be accompanied by a negative vacuum pressure in the x-, y-, and z-directions: (T_xx)_vac, (T_yy)_vac, (T_zz)_vac. This negative pressure has a gravitationally repulsive effect that is larger than the gravitational attractive effect of the energy density in the vacuum by a factor of 3 because there are 3 dimensions of space and only 1 dimension of time. Thus, the overall effect of the vacuum energy is repulsive, and can balance the gravitational attraction of the galaxies for each other to produce a static universe.

3. Einstein introduced the cosmological constant to allow a static cosmological model. Interestingly, if Einstein had been a Flatlander (living in a universe with only two spatial dimensions instead of three), he wouldn’t have needed to invent the cosmological constant. Mark Alpert and I (Gott and Alpert 1984) showed that Einstein’s field equation (without a cosmological constant) in this case had a static cosmological solution: a universe of uniform density, shaped like the surface of a sphere whose radius did not change with time (Sphereland). Point masses in Flatland do not attract each other gravitationally and so a static solution is possible. A Flatlander Einstein could have gotten the static geometry he wanted without having to introduce the cosmological constant.

Chapter 3: How Clusters Form and Grow

1. This distance to the cosmic microwave background of 13.8 billion light-years is known as the lookback-time distance. The photons we see from the cosmic microwave background were last scattered by electrons 13.8 billion years ago and have been traveling through curved spacetime for 13.8 billion years; therefore, they have traveled 13.8 billion light-years. This is the distance from us back to those electrons off which the photons last scattered. But we are seeing these electrons where they were in the past, 13.8 billion years ago, just 380,000 years after the Big Bang. Where are those same electrons now? They have been expanding outward with the expansion of the universe and are, at the current epoch, about 46 billion light-years away. This is called the comoving distance. It marks the radius of the visible universe at the present epoch. How did those electrons that last scattered the cosmic microwave background photons get out to a distance of 46 billion light-years from us in just 13.8 billion years? This occurs because the space between us and them has been expanding faster than the speed of light, something allowed by Einstein’s theory of General Relativity.

Usually we shall be using the lookback-time distance when we are talking about the distance to a galaxy. But in the Map of the Universe (Chapter 9), which presents a snapshot of the universe at the current epoch—specifically, where objects were when the map was made on August 23, 2003—we will use the comoving distance. In that map we are interested in what distances the objects will have attained by the current epoch, not where they were in the past when they emitted the light we are seeing now. We also use comoving distance in most maps of large-scale structure, including Figure 9.6, where it is mentioned explicitly. In these pictures we are interested in making a snapshot of the universe at the current epoch.

2. Peebles’ calculation went as follows. The collapse time of a cluster T_c is proportional to (δρ/ρ) ^−3/2 in the initial conditions. In a cluster today, ρ is proportional to 1/T². So ρ today is proportional to (δρ/ρ)³ in the initial conditions or proportional to M^−3/2 for Poisson initial conditions. And, of course, today—by definition—ρ ~ M/r³. Thus, M^−3/2 is proportional to M/r³, making M^5/2 proportional to r³ and M proportional to r^6/5, so ρ (~M/r³) surrounding a galaxy in the hierarchy out to a radius r is proportional to M^−3/2 and r^−1.8.

Chapter 5: Inflation

1. Thermal radiation gravitationally decelerates the worldlines of particles participating in the expansion twice as much as Newton could have predicted because of the gravitationally attractive effects of the radiation pressure that Einstein’s theory predicts. Including this effect was essential in the Gamow, Herman, and Alpher calculations of the formation of the light elements in the first 3 minutes after the Big Bang. The fact that modern nucleosynthesis results predict abundances for these light elements in agreement with what we observe supports Einstein’s conclusion that positive pressure is gravitationally attractive.

2. The vacuum state in inflation has a positive energy density and a negative pressure. These have opposite signs, because time and space have opposite signs in special relativity. The metric tensor that measures distances between events has the form ds²= −dt²+ dx²+ dy²+ dz² in special relativity. The last two terms represent the Pythagorean theorem from Euclidean plane geometry: ds²= dy²+ dz²(the square of the hypotenuse is equal to the sum of the squares of the two sides—along the coordinate directions dy and dz). The last three terms represent the Pythagorean theorem generalized to give the usual 3D result from Euclidean solid geometry. The minus sign on the dt² term guarantees that the separation in time [in years] (dt) and space [in light-years] (√[dx²+ dy²+ dz²]) between two events connected by a light ray (where ds²= 0) are equal, ensuring that all observers will observe light traveling at the same speed through empty space—a fundamental postulate of special relativity. It is that little minus sign in front of the dt² term that makes all the difference between time and an ordinary spatial dimension such as width. In general relativity, energy density is associated with the time dimension, while the pressure in the x-, y-, and z-directions is associated with the x, y, and z spatial dimensions. The vacuum state is proportional to the metric tensor and, therefore, has opposite signs for its energy density and pressure (in the x-, y-, and z-directions).

Chapter 6: A Cosmic Sponge

1. Many years later, I was asked to serve as a judge for the Westinghouse Science Talent Search and then as chair of the judges. I was chair of the judges for 14 years and saw the contest into its first year as the Intel Science Talent Search. Glenn Seaborg was a member of my judging panel. The students and I met with Presidents George Herbert Walker Bush and Bill Clinton during this time. One of our first-place winners was Jacob Lurie, who had a project in surreal numbers in mathematics. He would go on to become a professor at Harvard and win the inaugural Breakthrough Prize in Mathematics in 2014 ($3 million), as well as a MacArthur “genius grant.” He is just one example of the many wonderful students who passed through the contest during those years and have gone on to great careers in science.

2. Tom Banchoff was an instructor in math at Harvard when he encouraged me to publish my paper on pseudopolyhedrons. He went on to become a professor at Brown University. He famously became a friend of Salvador Dali, who was interested in learning more about developments in higher-dimensional geometry for possible use in his paintings. Banchoff recently provided commentary on higher-dimensional geometry on the DVD of the charming 2007 animated film Flatland: The Movie, based on the famous 1884 novel by Edwin A. Abbott, describing what life in a world with only two spatial dimensions would be like.

3. The N-body simulations assumed periodic boundary conditions on the cubical sample. The computer used a very efficient fast Fourier algorithm to calculate the forces on the particles as if the cubical sample was a video game in which: if you exited out the top, you reentered at the bottom; if you exited out the right side, you reentered at the left; and if you exited out the front, you reentered at the back. This makes it possible for the computer to handle the maximum number of particles, in the least amount of computer time. This is standard for large N-body simulations today. For comparative purposes, in our paper we treated our observational cubical CfA sample the same way. Thus, we were effectively assuming that the universe was an infinite warehouse filled with cubical boxes that were copies of our CfA sample. We found that our median density contour, which divided the universe into a high-density half and a low-density half by volume, was spongelike and all in one connected piece. In Figure 6.8a the triangular-shaped piece of the median density contour visible at the top front of the cube facing you is connected to the rest of the median density contour at the top of the back-right face of the cube because of the periodic boundary conditions. In this periodic universe the highdensity regions are all in one connected spongelike piece and the low-density regions are all in one connected, interlocking, spongelike piece. [The N-body computer simulation cubes portrayed later in the book all have periodic boundary conditions, but in subsequent observational data sets, where many structures are present and the samples have a non-cubical shape, the periodic boundary conditions were dropped and the sample was analyzed without assuming it lived in a universe with many other copies of itself.

The Voronoi honeycomb universe simulation in Figure 6.11 also has periodic boundary conditions, and yet here the median density contour is broken up into separate pieces enclosing isolated voids. In the 50%-low picture you can see one large void extending out the top of the survey and reentering at the bottom, for example. This is a Swiss cheese topology, and the periodic boundary conditions do not affect that.

4. Euler’s formula may be applied to calculate the genus of spongelike polyhedrons. For example, the pseudopolyhedron having squares, 5 around a point (see Figure 6.4) consists of two planes of squares connected by a periodic array of cubical holes. It can be constructed from a repeated arrangement of 8 vertices, 20 edges, and 10 faces having an Euler characteristic (V − E + F) = −2. Every time we add 8 more vertices and their associated edges and faces to the structure, we add one to the genus and create a new cubical hole. The structure is infinite and so has an infinite number of holes. One can prove that calculating the genus by adding up the angle deficits (or excesses) at vertices, and calculating the genus by using the Euler characteristic (V − E + F) always give the same answer.

Chapter 8: Park’s Simulation of the Universe

1. We can estimate the physical size of a galaxy (from, for example, the internal velocity dispersion of its stars). Comparing its physical size with its angular size in the sky allows us to calculate its distance. That distance tells us via Hubble’s law what we expect its radial velocity away from us to be. Subtracting that from its actual radial velocity gives us its individual or peculiar velocity in the radial direction relative to that expected due to the average Hubble expansion of the universe. This gives us only the component of the peculiar velocity in the radial direction (pointing toward Earth). This velocity is due to the component in the slope of the gravitational potential in the direction of Earth. If we integrate such slopes along an entire line of sight, we can map the gravitational potential along the entire line of sight. Repeat this along different lines of sight spread over the entire sky, and we can construct a 3D map of the gravitational potential throughout space. Then we can measure the slopes in this gravitational potential topography map in 3D, giving us the peculiar velocities of galaxies in 3D. This is the clever technique developed by Bertschinger and Dekel. It allows us to turn peculiar velocities in the radial direction only into a 3D map of the peculiar velocities showing their components in other directions as well.

Chapter 10: Spots in the Cosmic Microwave Background

1. Interestingly, in 3D the genus (number of donut holes − number of isolated regions in a density-contour surface) is equal to −1/4π times the Gaussian curvature integrated over the contour surface, while the 2D genus (number of hot spots − number of cold spots) is equal to 1/2π times the curvature encountered driving around the 2D contour line. In both cases the genus is directly related to curvature integrated over the boundary.

2. Standard slow-roll inflation (a simple field rolling down a hill) predicts values of the non-Gaussian parameter f_NL from 10⁻² to 10⁻¹ (close to zero and undetectably different from zero with current data) according to calculations by Juan Maldacena and others. The COBE results set 68% confidence limits of −1,500 < f_NL < 1,500. Using the higher-resolution WMAP data in 2007, Wes Colley and I and our colleagues, using our genus topology technique, were able to improve these limits to −101 < f_NL < 107 at the 95% confidence level. The WMAP team found −58 < f_NL < 134 with an independent analysis. After 6 more years of taking data, the WMAP team was able to narrow these limits to −3 < f_NL < 77. All these ranges are consistent with f_NL near zero. The Planck satellite, drawing upon a still-higher resolution map and testing for random phases with a different (bispectrum) method that uses temperature correlations found between triples of points in the map, has recently found −3.1 < f_NL < 8.5 (at 68% confidence), again consistent with f_NL near zero and the predictions of standard inflation.

Chapter 11: Dark Energy and the Fate of the Universe

1. I argued in a Nature paper in 1993 that we are not likely to find ourselves in a special position among intelligent observers. I described this as an application of the Copernican principle: that our location should not be special among those locations occupied by intelligent observers. Vilenkin calls this the principle of mediocrity. Same concept.

2. Paul Steinhardt and Neil Turok (2002) have argued that the “inflationary state” at the beginning of our universe was really the accelerated dark-energy expansion at the end of a previous universe. This inflation would occur at low energy (leading to a prediction of no significant gravity waves) and the universe would reheat by a collision of membranes in an 11-dimensional spacetime derived from M-theory. This scenario would also produce Gaussian random-phase initial conditions for our universe. Linde envisions the multiverse as an infinite, never-ending fractal tree of inflating universes with the branching produced by quantum fluctuations in the inflating state. In this model our universe is born as just one pocket universe on one of the branches. The question of where the original trunk came from is discussed in my book Time Travel in Einstein’s Universe. Ideas range from quantum tunneling from nothing (where a de Sitter waist, like that at the bottom of Figure 5.4, simply pops into existence and inflates from there (Vilenkin 1982; Hartle and Hawking 1983)) to time loops allowing the multiverse to be its own mother (Gott and Li-Xin Li 1998).

3. Given the current best estimates of the mass of the Higgs boson, and the mass of the top quark, the Higgs vacuum could decay into negative-vacuum-energy bubbles in about 10⁴² years in the slow-roll dark energy model. Because the expansion is not exponential in the end but only linear or slower, the negative-energy bubbles will percolate, colliding with each other and blocking all ways to the future. Rather than an eternally fizzing champagne, the journey to the future would be more like a voyage upward in a glass of beer, where you eventually find a froth of colliding bubbles (the head on the beer) blocking your way. Once you enter a negative-energy bubble, the insides of the bubble begin sharply contracting, ending with a Big Crunch singularity. The bubbles are hitting each other, filling up space completely, and the universe is ending in a series of Big Crunch singularities. If the Higgs vacuum state is stabilized by higher-energy physics—as Arkani-Hamed thinks likely—the negative-energy bubbles would form less frequently, on a time scale of 10∧10∧34 years within the observable universe, perhaps, and you would likely be hit first by another expanding sibling bubble universe like ours left over from the early inflationary epoch (on a time scale of 10^1,300 years), as we will discuss.

4. How much more of the cosmic web we will be able eventually to see depends critically on how many doubling times of exponential expansion in the past occur during inflation within our universe versus how many doubling times will occur in the future during the exponential expansion occurring in dark energy today. If the universe doubles in size many more times during inflation than during the dark energy–dominated epoch starting now, we will see vast new regions of the cosmic web.

A large number of doublings in size during inflation within our bubble after our universe forms shields us from impending sibling bubble universes by effectively pushing them further away. That allows us enough time to recover the galaxies in our universe we have said goodbye to during the dark energy phase and say hello to many new ones.

The calculation in the text assumed bubbles form at a rate of one per 10^1,000 causal horizon volumes per doubling time during inflation in the multiverse. Quantum tunneling to form a bubble universe is a rare event because you must tunnel through a high mountain range to get from the high valley you start in to a place where you can roll down to the nearest low-altitude valley in the landscape. The numbers could, in principle, be even more dramatic. If the mountains in the landscape are high enough to produce a vacuum energy equal to the Planck density, we could have 10⁷ doublings inside our bubble and a tunneling rate of 1 per (10∧10∧5) causal volumes per doubling time in the inflationary multiverse, for example. In that case, a factor of 10∧10∧5 more structures in the cosmic web than we can see now would come into view.

We won’t know exactly how many doubling times our universe has inflated after its formation or how rare bubble formation really is until we have obtained a “theory of everything” based, hopefully, on some version of M-theory. Only then will we be able to know the landscape well enough around our particular valley to calculate just how many doubling times of inflation we are likely to see within our bubble and how long it will be before another bubble universe or pocket universe is likely to hit us. Suffice it to say that in the slow-roll dark energy scenario, there is the possibility for vast new regions of the distant cosmic web to come into view in the future. This does not occur in the w₀ = −1 or w₀ < −1 (phantom energy) models.