Prerequisites: The standard model

See also: Cosmic inflation – The cosmic microwave background – Expansion of the universe, dark energy, quintessence – The cosmological constant

Introduction The big bang history of the universe Cosmological models The cosmological constant Nucleosynthesis Evidence for the big bang theory Open questions Recommended references: Web sites Recommended references: Magazine/journal articles Recommended references: Books

Introduction

So, what exactly is the big bang theory? One way to answer that is to consider the primary observational fact which suggested the theory in the first place. The fact is that the universe appears to be expanding in a uniform way.

This fact was established by Edwin Hubble in the late 1920s. He did this by comparing two sorts of observations – the "redshifts" in light from distant galaxies as determined by examining the spectra of light from those galaxies, and the distances of those galaxies from us as determined in a way that Hubble pioneered.

Let's consider redshift first. Star light, whether from our Sun or stars in distant galaxies, contains distinct features in its spectrum. In addition to the familiar continuous progression of colors from red to violet, the spectrum contains dark lines at specific wavelenghts, called "absorption lines" because they are the result of specific wavelengths of light absorbed by cooler gas in the star's atmosphere. These lines correspond to energy level transitions in orbital electrons of specific elements found in stellar atmospheres.

We know, from laboratory work, at exactly which wavelenghts such lines should occur. The lines, moreover, occur in a pattern of recognizable groupings. What we observe in light from distant galaxies is that the pattern of all lines we can identify is uniformly shifted to longer wavelengths, i. e. toward the red end of the spectrum. By "uniformly", we mean that every single line is observed at a wavelength which is the same constant multiple of what it should be. For example, sodium has an absorption line at a wavelength of 588.9 nm (nm = nanometers = 10^-9 meters), in the yellow part of the spectrum. In light from a distant galaxy we might instead observe this line at twice the expected wavelength (1177.8 nm, which is in the infrared part of the spectrum, so no longer visible to the eye). The redshift is defined as a number z which is 1 less than the factor by which wavelength is expanded. In other words,

(observed wavelength) / (emitted wavelength) = z + 1

z = [(observed wavelength) - (emitted wavelength)]/(emitted wavelength)

Redshifts in the spectra of stars and galaxies had been studied since Vesto Slipher began pioneering this area about 1912. By 1925 he had measured the spectral redshifts of about 40 galaxies. The natural assumption as to the cause of these redshifts was that they represented the Doppler effect due to motion away from us of stars and galaxies. For nearby objects, this explanation is entirely correct. In fact, we can deduce that nearby galaxies are rotating by observing that the spectrum of light from one side of the galaxy is redshifted, while the spectrum from the other side is blueshifted (since it is moving towards us). As it turns out, the redshift of distant galaxies is actually due to the expansion of space itself between us and the distant galaxy, rather than relative motion in the usual sense, so the redshift is not quite the same as a Doppler shift.

Now let's consider measuring distances to galaxies. This is the area Hubble was pioneering in the 1920s. Think first of determining how far away a single star is. If you know how bright a particular star is intrinsically and you know its apparent brightness as seen through a telescope, then the distance can be computed because the ratio of intrinsic to apparent brightness is proportional to the square of the distance.

The problem is being able to determine the intrinsic brightness of a given star. There's no good way to do this in general. But there is a certain type of star, called a Cepheid variable, whose brightness fluctuates periodically. The maximum brightness is proportional to the length of the star's period. The distance to nearby Cepheids can be measured by the method of parallax (applying trigonometry to the difference in the star's apparent position when the Earth is at opposite ends of its orbit). This makes it possible to compute the intrinsic brightness of nearby Cepheids.

Hubble was the first to observe a Cepheid in a galaxy other than the Milky Way – the Andromeda Galaxy, M31 – which enabled him to deduce that the galaxy was located well outside our own. This settled a very important open question of the time – the fact that spiral galaxies like M31 are huge aggregates of stars much like our own galaxy, instead of cloudlike "nebulae" inside our galaxy. Hence our galaxy isn't unique, but is only one of many. Hubble went on to find Cepheids in more distant galaxies.

Armed with distance data provided by the Cepheids, Hubble was then able to consider a new question: whether there is any relation between the amount of redshift associated with a galaxy and the distance of the galaxy. Hubble found that there was indeed a simple relationship – which is now known as Hubble's law.

Hubble's law

From these observations, Hubble stated the law that is named after him: the velocity of recession of a distant object is proportional to its distance, with constant of proportionality denoted by H. That is:

recession velocity = H × distance

Although other explanations for this redshift are conceivable, the one that eventually seemed compelling, in light of all the evidence, is that it is because the universe is expanding. If the universe is expanding homogeneously (i. e., at the same rate as measured from any point in the universe), the distant objects will appear to be receding from us at a rate that is proportional to the distance. To repeat, the redshift is not due to a traditional Doppler effect, but rather because the expansion of the universe causes the wavelength of each photon of light to be stretched. As noted, the end result is similar to the Doppler effect, though not quite the same. (And incidentally, objects held together by chemical or gravitational forces, such as molecules, planets, and galaxies, are not subject to this expansion, even though photons are.)

Of course, we are not at any special place in the universe. This is what is known as the cosmological principle – the universe looks the same no matter where it is observed from. Everything is receding from everything else according to the same Hubble law.

From this, we can draw a very interesting and rather disquieting conclusion, which leads directly to the "big bang" idea.

The conclusion is that, if we run the evolution of the universe backwards, there should be a time in the distant past when everything we can now see was a heck of a lot closer to everything else. It would have been very crowded at some point in time. While this does not prove that everything was bunched together in a state of very high density at some time long ago, it certainly suggests the possibility. Physicists like George Gamow who thought about such things made this and other inferences which, eventually, led to... the theory of the big bang – the notion that at some point in time about 14 billion years ago (by current reckoning), all matter in the observable universe was in a state of extremely high density and temperature, and subsequently "exploded", resulting in the expansion we still see today. ("Explosion" is used metaphorically. Exactly what did happen is still a major open question.)

How long ago, one can ask, might this event have occurred? Well, notice that if H is the Hubble parameter, then 1/H has the dimensions of time. We can rewrite Hubble's law as

distance = (1/H) × (recession velocity)

For many years there was a serious problem with that interpretation. It turns out that, because of the great difficulty of measuring cosmic distances, Hubble initially underestimated distances by nearly an order of magnitude. Since there was little uncertainty about the redshift, and hence the recession velocity, this made the "age" of the universe about a factor of 10 too small. Like, about a mere 2 or 3 billion years. Other estimates of the age of the solar system and the Earth pointed to an age of 4 or 5 billion years for our home planet. Oops.

Since Hubble's distance estimates were way off, this paradox was only apparent. In all fairness, measuring large cosmic distances, until recently, was pretty difficult. The process is, essentially, to take the actual luminosity of some object and compare it to the observed luminosity. Knowing that luminosity should fall off by the inverse square law, we can compute what the distance must be. However, stars and galaxies come in a wide range of brightnesses. Astronomers must rely on indirect means to estimate the actual luminosity of a particular object. In Hubble's case this involved measuring the periods of Cepheid variables in order to infer their luminosity.

The problem was that there are actually two different types of Cepheids. Hubble assumed a period-luminosity relationship appropriate for one type, but most of the Cepheids observed in distant galaxies were of the other type. Eventually, as these stars became better understood, and telescopes became more powerful, the estimated distances of remote galaxies were increased significantly. But even up until the 1990s, when Type 1a supernovae could be used to judge distances, there was a lot of uncertainty in the value of the Hubble parameter.

It might seem that a theory which held that the universe was in a process of rapid expansion from a dense lump of matter for no obvious reason might be a little hard for astronomers and others to swallow... except for one additional fact. Which is, that Einstein's fairly new, and even more newly verified, theory of general relativity predicted exactly this state of affairs. Even so, it was quite a stretch, which most cosmologists were reluctant to make. The idea of being able to extrapolate features of the whole universe from simple observations and principles seemed almost too good to be true, so for a long time astronomers and cosmologists didn't take their own predictions too seriously. This was true of the prediction of general relativity that the universe was not static, and there have been other important cases of the same thing. (In the case of "cosmic microwave background" radiation, for example.)

Einstein himself realized that his theory predicted the universe should not simply be static. It might be collapsing under the gravitational attraction of everything for everything else, or (for reasons less apparent) expanding. But it shouldn't just sit there, going nowhere, or levitating calmly in midair. Yet in 1915, when the theory was published, and a few years before Hubble came along, just sitting still quietly was what everyone assumed the universe was doing. To make this mathematically plausible, Einstein even added a new term to his equation of general relativity, the famous "cosmological constant". Then, around 1929 when Hubble showed that expansion was a fact, Einstein decided his cosmological constant was just a big mistake, and he scrapped it. (That may have been premature, as will be seen later.)

After Sir Arthur Eddington in 1919 confirmed a key prediction of the general theory of relativity, namely the bending of light by gravity, the overall theory began to be taken pretty seriously, even as unconventional as it was. Various people began coming up with solutions of Einstein's equation. Among them was an obscure Russian meteorologist named Alexander Friedmann. In the early 1920s he published solutions of the equation which implied several forms of an expanding universe, exactly consistent with Hubble's observations. Unfortunately, Friedmann's work was at first ignored, and later disputed, even by Einstein himself. Although Einstein quickly withdrew his objections, theorists remained reluctant to take such seemingly outrageous models entirely seriously.

The big bang hypothesis

What interested Gamow was the problem of nucleosynthesis. That is, the question of how nuclei of all the chemical elements came to be formed. This accompanied the question of how energy was generated inside of stars, because it became clear that the process of nuclear fusion – in which heavier elements are built up, ultimately, by protons (hydrogen nuclei) fusing together somehow – could explain the source of stellar energy. Gamow and his students made substantial contributions to both issues.

In particular, they made calculations in the late 1940s of what might come out of a very hot, dense "gas" of neutrons and protons – exactly the form that matter would take at a certain point in a big bang scenario. They succeeded in coming up with reasonable estimates, that only hydrogen and helium would be produced in substantial amounts, and more of the former than the latter. Although these calculations were largely ignored or forgotten for some time by the astrophysical community, they have more recently been much refined, to the point that they produce an excellent agreement with observation – and provide one of the key lines of evidence in support of the big bang model.

Gamow and his students also calculated that the ultrahot, ultradense state of matter in the early universe should have produced radiation which might still be observable today, at an apparent temperature of perhaps several tens of degrees above absolute zero. (These estimates, as well as the ones for nucleosynthesis, were somewhat off, but still remarkable.) Yet these calculations predicting a microwave background radiation were taken even less seriously. Even Gamow himself didn't pursue the idea very vigorously, when it seemed it would be very difficult to observe such background radiation.

Nevertheless, Gamow pushed on to fill out the general scenario of the evolution of the universe according to the big bang model. Throughout the 1950s and early 1960s it was locked in close competition with the "steady state" theory of Fred Hoyle and others as a model for the universe's evolution over time. The steady state theory was equally radical in some respects. For instance, it postulated the ongoing continuous creation of matter out of nothing. But there didn't seem to be enough observational evidence to tip the scales decisively towards either theory.

And then in 1964 Arno Penzias and Robert Wilson discovered, quite by accident and without looking for it, the cosmic microwave background radiation, largely as predicted by Gamow. (Penzias and Wilson got a Nobel Prize for this; Gamow never received the prize.)

The rest is history (which is, basically, what the "steady state" theory became).

The big bang history of the universe

Following the table, we will explain in more detail what was going on as the universe expanded and cooled.

Cosmological timeline
Name	Time	Temperature	Notes
Planck era	10-43 sec.	1032 K	Equality of all forces
Inflation begins	10-35 sec.	1028 K	GUT symmetry breaks
Inflation ends	10-32 sec.	1027 K	Strong & electroweak forces are separate
Electroweak symmetry breaks	10-12 sec.	1015 K	Separation of weak & electromagnetic force
Baryogenesis	10-6 sec.	1013 K	Quarks confined in hadrons, ending era of quark-gluon plasma
Quark confinement complete	10-5 sec.	3×1012 K	Only lightest hadrons remain (protons, neutrons); antimatter annihilated
Nuclear binding	1 sec.	1010 K	Nuclear binding energy exceeds photon energy; neutrinos decouple
Nucleosynthesis begins	300 sec.	9×108 K	2H, 3H, 3He, 4He form
Nucleosynthesis ends	35 min.	3×108 K	Mostly H and 4He left
Matter dominance	47,000 yrs.	8000 K	Matter density exceeds radiation density
Recombination begins	240,000 yrs.	3700 K	Electrons begin to bind into atoms of H and 4He
Last photon scattering	350,000 yrs.	3000 K	Universe becomes fully transparent, no matter/photon interactions
Reionization	∼108 yrs.	30 K	First stars
First galaxies	∼5×108 yrs.	10 K	Corresponds to redshift z = 8
Present	13.7×109 yrs.	2.725 K	Internet, Chinese food, etc.

In the beginning: the Planck era

One of the basic ideas of the standard model of particle physics is that there are four fundamental forces: gravity, the strong force, the weak force, and the electromagnetic force. However, at the very earliest time in the existence of the universe, all four forces were "unified" and indistinguishable. This is similar to the way we regard electromagnetism as a single "unified" force, even though at one time (before the work of James Clerk Maxwell) physicists regarded electric and magnetic force as distinct.

Contemporary physics is able to say almost nothing about the very first instants of the universe. This is because the strength of the gravitational force at the earliest times was the same as the strengths of the other physical forces. Therefore, both quantum mechanics and general relativity must be applied to describe the physics of this period. But unfortunately, this is not possible, since we do not currently have a workable theory of quantum gravity that would merge these two fundamental theories.

About all physicists can do is make certain educated guesses. As an example from more than 100 years ago, one very inspired insight was due to the German physicist Max Planck. His insight provided the foundations for quantum mechanics, and it also leads to a natural scale for the time and energy level of the universe during the brief instant when all four forces were unified.

Basic to all physical theories are certain fundamental quantities: mass, length, and time. Almost everything in physics can be described in terms of certain combinations of these units. For instance, speed is length divided by time (e. g. meters per second). If we denote these basic quantities with the letters M, L, T, then speed can be represented as L/T. Other physical quantities can be described as products of powers (including fractional and negative powers) of these basic ones.

In 1899 Planck was working on the theory of blackbody radiation, that is, the theory of how very hot objects such as molten rock or metal or the filament of an incandescent light give off visible (as well as invisible) light – or more generally, electromagnetic radiation. The theory was in big trouble, since as it existed, it predicted that the total amount of energy given off by a hot object could be infinite. Planck's insight, for which he won a Nobel Prize in 1918, was that the theory could be salvaged if energy did not occur in continuous increments, but necessarily occurred only in discrete units or quanta. This inspired guess is also the basis on which Planck is considered the inventor of quantum mechanics.

As part of his new theory, Planck introduced a new fundamental constant, which of course is called Planck's constant, denoted by the symbol ℎ. In symbols, Planck's result is that the energy of some amount of monochromatic light can be expressed as

E = n × ℎ × ν

Note that the actual value of the energy E depends on the units of measure in which the various quantities are expressed. Another brilliant idea Planck had was to use certain well known fundamental constants to derive what he considered to be "natural" units for quantities such as mass, length, time, energy, etc. – all relative to his constant ℎ. Two obvious fundamental constants were the speed of light, c (which has units L/T), and Newton's gravitational constant G (which has units L³/(T²M)). Planck also threw in a factor of 2π (so that frequency could be expressed in radians per second), to make the modified constant ℏ = ℎ/2π, and noted that the expression ℏG/c⁵ has units (ML²/T)(L³/(T²M))(T⁵/L⁵) = T². Therefore the quantity (ℏG/c⁵)^1/2 has units of time. This unit is called the Planck time unit, and it has a value of about 5.39×10^-44 seconds.

When he did this, Planck had no idea of what quantum mechanics would eventually become and the important role that his constant ℏ would ultimately play in the theory. Nevertheless, it turned out that the Planck time unit was the smallest quantity of time that was meaningful in the theory. If you multiply that unit by the speed of light (expressed in whatever units were originally used in defining ℏ), the result is the distance light can travel in one Planck time unit. This distance is known as the Planck length. It is about 1.616×10^-35 meters, and is the smallest unit of length that is meaningful to use – which at about 20 orders of magnitude smaller than the size of a proton is pretty darn small.

There is also a Planck energy, defined as (ℏc⁵/G)^1/2. Expressed in terms of temperature it is about 1.417×10³² K. (K stands for degrees Kelvin. This is just the ususal Celsius scale, having 100 degrees between the freezing and boiling points of water, except that it starts at 0 K, unlike Celsius, whose zero point, at which water freezes, is 273.15 K. Thus "absolute zero", or 0 K, is -273.15 C.)

The Planck time is the smallest amount of time that can meaningfully be discussed in quantum mechanics, so it is used to define the Planck era as the first instant after the big bang that we can hope to apply any laws of physics to the universe. That is the era in which all the fundamental forces – gravity, electromagnetism, the weak nuclear force, and the strong nuclear force – were essentially the same. As indicated above, the general theory of relativity cannot actually describe gravity at that time, and we don't know what laws governed the other forces either. All this is part of the unknown, hypothetical "theory of everything".

What is known is that at a certain distance scale an order of magnitude or two longer than the Planck length and hence (by Heisenberg's uncertainty principle) at a similar proportion less than the Planck energy, the gravitational force must have become distinct from, and much weaker than, the other three forces, which remained unified. Once gravitation becomes distinct from the other (still unified) forces, it can be described by laws separate from those of quantum mechanics. At that point, the theory of relativity and quantum mechanics are able to coexist, so physicists have at least the possibility of describing the universe with well-tested theories.

This possibility has not yet been achieved, however. There is still no consistent, unified theory of the strong force, the weak force, and electromagnetism. Physicists have tried very hard to construct such a theory, which is generally called a Grand Unified Theory, or GUT, even though it leaves out gravity. Nevertheless, by extrapolating the way the strong, weak, and electromagnetic forces grow with increasing energy levels, it is possible to predict that all three forces should have equal strength at a certain energy level three or four orders of magnitude less than the Planck scale.

There are various equivalent ways to specify energy levels. One is in terms of temperature. Although "temperature" is a concept from thermodynamics it can be related to energy levels, because a "black body" at a given temperature radiates photons over a spectrum of energies, but the maximum is at one specific energy level. This energy is taken to be representative of the "typical" photon energy emitted by the black body. One common unit of energy is the electron-volt (eV), the amount of energy required to move an electron across a one volt difference of electric potential. (A million electron volts is "MeV", and a billion is "GeV".) A conversion factor called the Boltzmann constant, denoted by k_B, relates temperature to the associated energy level. This constant is 8.619×10^-5 eV per degree K. 10^-4 eV per degree K is a good approximation, hence 10^-13 GeV per degree K. For example, the Planck temperature of 1.417×10³² K corresponds to the Planck energy of 1.221×10²⁸ eV = 1.221×10¹⁹ GeV.

Anyhow, three or four orders of magnitude below the Planck scale – where extrapolation indicates the strong, weak, and electromagnetic forces have equal strength – is 10¹⁵ GeV to 10¹⁶ GeV. That's about as close as physicists can estimate, because without an actual consistent GUT, there's no way to compute the value theoretically. We refer to this energy level as the GUT energy scale.

One might expect that "something" significant should happen as the temperature of the universe falls below this point, as it must, given that the universe is expanding rapidly. For one thing, the strong force separates off from the remaining weak and electromagnetic forces. (The latter two forces remain unified for a relatively much longer time, all the way down to an energy level of about 100 GeV, in fact. This unified force is called the electroweak force.)

The era of inflation

Another name for this process is the "collapse of the false vacuum", because although the universe had been in a temporarily stable state, it was not truly a state of minimal energy. What had appeared to be empty vacuum actually contained an enormous amount of latent energy which was released when the temperature dropped below a critical point, much as water releases heat energy when it freezes. (In the opposite direction, to melt ice, it is necessary to add heat.) Further, in both cases a state of relatively high symmetry undergoes an abrupt transition to a state of lower symmetry. In one case, the symmetry between the strong and electroweak forces was broken. In the other case, when water freezes, it goes from being an isotropic substance (in which all directions are equivalent) to one in which there are particular preferred directions: the axes of the ice crystals.

There are some very good reasons why this dramatic phase change followed by exponential inflation is thought to have occurred, even though the physical principles that explain it are still unknown. The idea of inflation was originated by Alan Guth in December 1979. (Anticipating the importance of the idea at the time, he described it as a "spectacular realization".) In 1979, although the big bang theory was widely favored over alternatives because of its success in predicting the existence of a uniform presence of microwave radiation at a blackbody temperature of about 2.725 K, cosmologists recognized that it had several troublesome problems:

• An equation known as the Friedmann equation – which we will later explain in detail – is used to describe how the universe expands with time. One of the important parameters in the equation is a constant that describes how much space itself is curved. In the form of the equation most often studied at the time, if the curvature is positive, the universe eventually stops expanding and begins to collapse. If the curvature is zero or negative, the universe continues expanding indefinitely. This was considered to be a very important distinction, as it predicted the ultimate "fate" of the universe. Curvature is related to another quantity, denoted by Ω, which reflects the density of matter and energy in the universe. The curvature is positive, zero, or negative according as &Omega is greater than 1, exactly 1, or less than 1. Unlike curvature, Ω varies over the life of the universe. In 1980 the density of matter in the universe, and hence Ω, could be measured only crudely, but the best estimates were Ω ≈ .01. However, this was a serious problem, because the Friedmann equation predicted that because the universe had expanded so much since the big bang, Ω would need to have been extremely close to 1 during the earliest moments (less than a millisecond after the big bang, say). Since as far as anyone knew, Ω could have any value at all, it seemed to be an extraordinary coincidence if Ω were extremely close to 1 at very early times. An explanation for this coincidence was needed. This was called the flatness problem, since Ω ≈ 1 is equivalent to having zero curvature – being completely flat, in other words.
• A second problem was that, observationally, the universe appears to look very much the same in all directions. The number of galaxies seems to be the same in all directions, as do the galactic redshifts which can be measured, when correlated with apparent distance. Since it would be a remarkable coincidence if this property, known as isotropy, were unique to our vantage point, it should be expected to exist around any vantage point. And in that case, the universe should actually be homogeneous – essentially the same everywhere, on a sufficiently large scale. The assumption of homogeneity wasn't merely attractive philosophically. It was in fact an essential ingredient (as we shall see) in deriving the Friedmann equation used to describe the evolution of the universe. However, homogeneity is also a big problem. The reason is that parts of the universe that we can now observe on opposite sides of the sky were, during the first instants after the big bang, too far apart from each other for light or any kind of information to travel between them. Therefore, it is very hard to understand how different parts of the universe which were not "causally connected" at very early times could have turned out to have identical statistical properties. This puzzle is known as the horizon problem, because at very early times most parts of the universe would have been "beyond the horizon" from each other.
• There was one additional problem, and it arises from most theories of the GUT type that attempt to describe the unification of the strong and electroweak forces. It turns out that in most such theories some freakish things are produced when the symmetry breaks. When water freezes into ice as the temperature falls below 0 C, ice crystals begin to form in many different regions at the same time. Eventually these all join together, but the ice crystals that form in each region all have different orientations. Therefore, discontinuities occur when ice from different regions come into contact. The same sort of thing should have occurred in the symmetry breaking predicted by GUT theories, and the resulting discontinuities of space itself are called topological defects. One particular type of topological defect is a magnetic monopole – a minuscule bit of magnetic material that has just one pole (north or south), unlike all magnets anyone has ever observed. Most GUT theories predict that magnetic monopoles should have been produced in vast quantities. However, magnetic monopoles have never been observed despite numerous attempts to find them. This is known as the monopole problem

How much expansion? It must have been (literally) at an exponential rate, that is, as a function of time, being proportional to 10^Ht, where H is a constant – essentially the Hubble parameter at that instant. The process can be thought of as a certain number N of cycles in each of which the universe expands by a factor of 10, so the total expansion would be a factor of 10^N. If the whole period of inflation lasted Δt seconds, then N = HΔt. If N = 100, N cycles would have inflated the universe by a factor of 10¹⁰⁰. Suppose inflation began at 10^-35 seconds after the big bang, and each cycle was that length of time. Then Δt = 10^-33 seconds, so that is about when the inflationary expansion would have ended. The Hubble parameter during this period would have been H = N/Δt = 10³⁵ (sec)^-1.

Although we can only guess at what the actual numbers might have been, an expansion factor of 10¹⁰⁰ would certainly have solved all of the problems listed above:

• The flatness problem is solved, because if you look at a curved piece of space at a magnification of 10¹⁰⁰, the piece looks very flat, just as the surface of the Earth does to us living on it.
• The horizon problem is solved, because all parts of the universe that we can see today would have originated, before inflation, in a single tiny speck of space that was causally connected. That is, every part of that speck could have communicated with every other part during the instant before inflation.
• The monopole problem is solved, because all the monopoles that could have been created in the GUT era would have become so diluted that it would be extremely unlikely that even one of them could exist in the whole universe observable today.

We will discuss inflation in much more detail elsewhere, and consider what sort of processes could have been responsible for such an immense expansion of space. For present purposes, we just want to note an important byproduct of this drastic event. At the GUT scale, just before the symmetry was broken, there was no essential distinction between one class of particles known as quarks, and another class, known as leptons. Quarks are the particles which combine in threesomes to make nucleons (protons and neutrons). Leptons are particles such as electrons and neutrinos. Each of these particle types comes in three distinct subtypes known whimsically as "flavors". The flavors are somewhat mutable, under the action of the weak force, but in our universe as it is presently, there is no way that a quark can turn into a lepton, or vice versa.

But this wasn't the case at the GUT scale. Since all the forces (except gravity) were equivalent, there were particle reactions which turned leptons into quarks and quarks into leptons. Fundamental forces are said to be "mediated" by a special kind of particle known as a boson. The effect of any force on a particle like a quark or a lepton is caused by an interaction between the particles in which the boson that mediates the force is exchanged. For example, the boson which mediates (or "carries") the electromagnetic force is simply the quantum of the electomagnetic field, namely the photon. The force of attraction (or repulsion) between charged particles of opposite (or the same) electric charge can be understood as an interaction between the particles and photons.

Similarly, at the GUT scale, before the GUT symmetry was broken, it is hypothesized that there were very massive bosons, which are usually known generically as X particles. When a quark interacts with an X it becomes a lepton, and when a lepton interacts with an X it becomes a quark. X particles must have been very massive, having mass-energy close to the GUT energy level of perhaps 10¹⁵ GeV. (Technically, since E = mc², the proper units for expressing mass are energy units divided by the square of the speed of light. So to be correct, masses should be expressed in terms of eV/c², GeV/c², and so forth. For simplicity, we will generally omit the c².)

Another way to think of the breakdown of GUT symmetry is to realize that when the universe cooled below a temperature equivalent to 10¹⁵ GeV, there would quickly cease to be any more X bosons around. This is for the simple reason that they would be unstable, and they could no longer be created by the pair production process out of photons (or any other particle), simply because there were no photons or other particles with a sufficiently large amount of energy. Since all X bosons would quickly disappear, there was eventually no way for quarks and leptons to change into each other. And thus the GUT symmetry was lost.

There are in fact a number of theoretical ideas about how inflation actually worked, but all produce roughly the same result. We will discuss some of them elsewhere. It is worth noting, however, that in many cases there will be subtle differences in the results of inflation that can potentially be used to discriminate between different possible inflationary mechanisms. One of the differences between different inflationary theories is in the mechanism that brings inflation to an end and in how the universe is reheated to almost the temperature it had before inflation began, in spite of the vast amount of expansion.

Although theories differ as to what ended inflation, it clearly did end. The result at the end of inflation is a universe which is potentially much easier to understand from current knowledge of particle physics. This is simply because the strong force is no longer unified with the electroweak force, so that physicists can work with two different forces (strong and electroweak). And this, in turn, is actually an advantage, because these two forces are somewhat understood. Since each of these forces is within the range of existing accelerators, actual experiments can be conducted that guide the development of the theory and allow physicists to separate fact from conjecture.

As exotic as it may be, the post-inflation era of the universe is a much more familiar environment than it was before inflation, though some very notable mysteries remain.

The era of particles and forces

For reasons we will explain shortly, there are no ways to produce new particles that have more mass-energy than is available given the prevailing temperature of the universe at a particular time. So, given that most particles ultimately decay, the diversity of particle species continually decreases. Although a vast menagerie of particles may have remained after inflation ended, we have little way to conjecture what most might be. For most purposes, at least as far as cosmology is concerned, such particles may as well have never existed.

However, there are a few apparent facts about the universe as we know it today that we have no good ideas how to explain. These facts may be effects left by particles that once existed but are no more, or that, at least, do not any longer interact significantly with the "ordinary" matter that makes up galaxies and stars and planets. The list of such mysterious facts includes:

• On the basis of well-understood physics, at the present time there should be roughly as many baryons (particles, such as protons and neutrons, made up of quarks) in the universe as there are photons. There are ways to estimate, from certain observations, how much matter does exist in the form of baryons. But instead of roughly equal numbers, there appear almost two billion times more photons than baryons. The process through which this particular relative abundance of baryons came about is known as baryogenesis.
• There is a symmetry in physics called charge symmetry that relates particles to antiparticles. It implies that there should be exactly as many particles of each kind as corresponding antiparticles. Plainly this couldn't have been true, and there must have been some asymmetry between matter and antimatter, because otherwise every particle would have annihilated with one of its antiparticles, and there would be no baryons (or leptons either) remaining. The charge symmetry must have been broken very slightly in order to leave just a sufficient excess of particles over antiparticles to account for the baryons that are present now, in a ratio of about one baryon per 1.8 billion photons. Physicists have ideas about why this could be, but the details are very hazy. However, there might be some relation to the dark matter issue.
• All visible matter, which makes up gas clouds, planets, stars, and galaxies, is composed of baryons. There must be some invisible dark matter that is also made of baryons, but observations strongly imply that there is much more matter in the form of nonbaryonic dark matter than exists in the form of both visible and dark "ordinary" baryonic matter. The ratio of nonbaryonic matter to baryonic matter, by mass, appears to be about five to one.
• The universe also appears to be full of something now called dark energy that plays a role in the evolution of the universe through a part of Einstein's equations of general relativity called the cosmological constant. This dark energy, in fact, has an effect on the universe that is more than twice as large as the effect of all matter, both dark and ordinary. The dark energy definitely seems to be around, but there are no good ideas why.

As already mentioned, this model consists of fundamental forces, a variety of different types of elementary particles, and a mathematical description of the behavior of those forces and particles. The forces, again, are gravity, the strong force, the weak force, and the electromagnetic force. At the very highest energy levels we can conceive of – at the Planck scale – all of the forces are "unified", which means they all have the same strength and obey the same equations. As the temperature of the universe decreases along with its expansion, this high state of symmetry is broken as the characteristics of the forces become more dissimilar. Gravity separates from the other three forces even before inflation begins. The subsequent separation of the strong force is thought to be associated with the phenomenon of inflation.

So after inflation is done, there are three distinct forces: gravity, the strong force, and the electroweak force – which comprises the still undifferentiated weak force and electromagnetic force. There are also a wide variety of elementary particles present. We'll say more about them in a moment, but they include photons, electrons, quarks, neutrinos, and far more exotic and massive particles as well.

We can associate an energy level with the universe at this time. There are several ways to express this energy level, as explained above. The temperature of the universe at the end of inflation is estimated to be about 10²⁷ K. Using Boltzmann's constant, this can also be expressed as 10¹⁴ GeV.

However expressed, that is still a huge amount. For comparison, the rest mass of a proton is .938 GeV – 14 orders of magnitude smaller. Even the heaviest particle that can be produced in present accelerators, the top quark, has a mass of only about 178 GeV. The importance of these energy levels is that we don't have the experimental tools to even come close to being able to study directly what physics is like at the time just after inflation ended. The gap is enormous. Not too surprisingly, physicists can't say very well what might actually going on at such energy levels.

Lacking experimental evidence, it's tempting to suppose there isn't a lot going on that really matters. For example, the electroweak force does not split into a separate weak force and electromagnetic force until the level of about 100 GeV, which is well within the energy range of present accelerators. And according to the still untested idea of supersymmetry, the lightest "supersymmetric" particles should weigh in under 1000 GeV (possibly a lot less). So there is an energy range of about 11 orders of magnitude above 1000 GeV which we know very little about, except that it seems to have relatively little effect on the physics we can actually study experimentally. Particle physicists sometimes call this energy range the "desert", since it doesn't seem to be all that interesting.

However, it's probably rather cavalier to suppose little important is going on at energy ranges above 1000 GeV, even though we can't presently study that range directly. And yet it's quite true that in the range of energies below 1000 GeV (i. e., 1 TeV) where experimentation is currently possible, the standard model has an extremely good fit with experimental results. It has proven extremely difficult to find physical effects which require "new physics" at high energies to explain.

Although we cannot presently perform experiments at very high energies – and probably won't ever be able to at the GUT scale – cosmology itself should be able to give us clues about what goes on, if we could only recognize and understand them. For example, while we can postulate many possible mechanisms by which inflation may have occurred, without understanding physics at the GUT scale, we can't determine what actual mechanism may have been involved. If inflation is in fact a valid hypothesis and if we can ever learn more about its effects, we may be able to place limits on what GUT-scale physics must be like.

Similarly, the other "mysteries" listed above certainly have their explanations rooted in physics that occurs above 1 TeV (trillion electron volts). For example:

• The facts that there does not appear to be any significant amount of antimatter left in the universe, that baryonic matter does exist, but that baryons are outnumbered by photons in a ratio of something like two billion to one are all related. The facts probably result from very slight asymmetries in particle reactions that occur at energies well above 1 TeV. We have no idea what those reactions might be or what detailed physical laws govern them. What we do know is that violations of both charge symmetry by itself and in combination with another symmetry (parity symmetry) must have been involved. The combination of charge and parity symmetry is called CP symmetry. We do know some reactions where CP is violated at accessible energy levels, but curiously only for interactions involving the weak force. It has been conjectured that certain as-yet undiscovered particles called axions may account for why we don't see violations of CP symmetry in interactions that involve the strong force. Even more tantalizingly, it is possible that axions make up some part of the nonbaryonic dark matter.
• A leading candidate for (nonbaryonic) dark matter is particles predicted by the theory of supersymmetry, particularly the lightest supersymmetric partner, or LSP (since that may be the only supersymmetric particle which is stable). The LSP is often conjectured to be a particle called the neutralino. If that accounts for most of the dark matter and if we could understand supersymmetry, we should be able to predict the actual ratio of nonbaryonic matter to baryonic matter, given conditions that existed at energy levels around 1 TeV or higher.
• The real mystery of dark energy, if it is represented by the cosmological constant, is that the latter is not vastly larger than it seems to be. Perhaps 120 orders of magnitude larger. We have no acceptable theory at all of what generates the cosmological constant. If we did it might be related to processes that occurred during the time period under discussion here, perhaps some very minuscule symmetry breaking like the very slight breaking of CP symmetry.

A little later, when we come to discuss nucleosynthesis – the process in which nuclei of elements such as helium and heavy hydrogen (deuterium) are formed – we will see a concrete example of how applying well-understood physical laws to conditions in the universe within a few minutes after the big bang make very testable predictions. The fact that these predictions have been confirmed is solid evidence for the big bang model, since the model tells us what the conditions of temperature and matter density must have been.

But that kind of reasoning can be turned around. If physicists come up with sufficiently detailed theories for things like supersymmetry and CP symmetry breaking that make predictions about particle interactions, then we can use the big bang model at higher energy levels to predict observable effects – such as the ratios of baryons to photons and baryonic to nonbaryonic matter, which have both already been measured.

There is, thus, an interplay between cosmology and very high energy physics. The early universe is the ultimate particle accelerator. Observational facts from cosmology put useful constraints on the physics. Conversely, anything we are able to learn about the physics of CP symmetry breaking or supersymmetry (for example) might suggest new phenomena to look for in cosmology.

One key connection between cosmology and particle physics has to do with simply counting numbers of particles of different types. Suppose p denotes any particle (not just a proton, as it usually does), and p′ denotes its antiparticle. Everyone knows that when a particle and its antiparticle interact, they annihilate each other and produce massless particles of energy – photons. This reaction can be written:

p + p′ ⇄ γ + γ

And therein lies the rub. As long as most photons have at least as much energy as the rest mass energy of a particular type of particle, the above reaction will go on constantly in both directions. Pairs of any type of particle will ceaselessly be produced out of photons, and will just as readily annihilate each other to give back photons. There's nothing to limit the reaction in the left to right direction, since photons can exist with any amount of energy however large or small.

However, the same is not true in the opposite direction. There is no such thing as a proton with an energy less than the proton rest mass of 938.3 MeV, though a proton can have an arbitrarily larger amount of energy in the form of kinetic energy. Therefore, as soon as the universe has cooled to the point where there are almost no photons left having that much energy, essentially no new protons will be created. Ever. (Well, hardly ever. Virtual particles of any kind can be created for very brief periods of time by virtue of the Heisenberg uncertainty principle. Virtual particles can even affect the "real" world, if they happen to interact with something else during their brief existence. A virtual X particle, for example, could possibly cause a proton to decay. But because the X is so massive its lifetime must be extremely short. Therefore, the probability of causing a proton to decay is extremely small, and the expected lifetime of a proton is extremely long.) Likewise, essentially no additional particles of any kind will be created once the universe has cooled beyond the point that essentially all photons have less energy than the rest mass of the particle. This includes neutrons (which have a rest mass of about 939.6 MeV), and that is crucial for the theory of nucleosynthesis, to be discussed later.

Of course, the same thing applies to heavier particles as well. Quarks are somewhat of a more complicated case, since below a certain energy level quarks apparently cannot exist except in bound states of two quarks (mesons) or three quarks (baryons). But all hadrons (either mesons or baryons) other than protons are unstable, and eventually decay spontaneously. Therefore, after they can no longer be created in the pair production process because photons of sufficient energy are not available, they will disappear from the universe. (Unless created in very unusual circumstances such as supernova events or in the vicinity of a black hole.)

This is relevant to one curious fact – the existence of any ordinary (baryonic) matter at all. The term baryogenesis refers to the sequence of events that leads to the residual baryonic matter. As previously noted, all different types of particles should exist in approximately equal numbers, at some sufficiently high temperature. This is simply because, at high enough energies, all particles, including photons, should be able to convert into each other, at least indirectly. For instance, a quark and an antiquark annihilate with each other, producing two photons. So quark plus antiquark → two photons. The two photons, in turn, can give rise (for example) to a muon plus antimuon. So pair production suggests that two photons should exist for each particle-antiparticle pair. (Photons are their own antiparticles, but you can arbitrarily count half of them as antiparticles.)

When the universe becomes sufficiently cool (at a different temperature for each particle type), it is no longer possible to produce any more new particles of each type by pair production. (As mentioned, particle-antiparticle pairs can still be created as virtual particles, but that's a different process, and the particles exist for only very brief instants.) After that point, all particles and antiparticles which can annihilate with suitable partners of the same kind will do so. If the number of particles and antiparticles is exactly the same, then either equal numbers of particles and corresponding antiparticles remain, or else neither remains at all. But many observations indicate that antimatter does not exist at all in significant amounts. Yet ordinary matter clearly does.

This is a problem, so the assumption of exactly equal numbers of particles and corresponding antiparticles must be wrong. Instead, there must have been slightly more quarks and leptons than antiquarks and antileptons. But why? Pair production and annihilation can't produce an imbalance between particles and antiparticles. So the excess must due to unknown processes other than pair production. And such other processes must exhibit some symmetry breaking, however slight.

Quarks and leptons must have existed from the earliest times when different particle types became distinguishable, after the breaking of GUT symmetry. They would remain in equilibrium with photons until the temperature level of the universe dropped to a certain point. For leptons, what happens is simple. As soon as the universe is cool enough that no new lepton-antilepton pairs of a given type are created by pair production, the reverse reaction proceeds until all antileptons are gone.

With quarks, however, the situation is a little more complicated. The two lightest quarks are called "up" and "down". They have rest masses of about 3 MeV and 6 MeV, respectively. A proton consists of two up quarks and one down quark, while a neutron consists of two down quarks and one up quark. It turns out that the strong force does not allow free quarks to exist below a certain energy level, somewhere around 1000 MeV. (Which happens to be about the rest mass of protons and neutrons.) So, for example, all up and down quarks become incorporated into protons and neutrons at that level. (The substantial excess rest mass of protons and neutrons over that of the constituent quarks is due to "binding energy".) This phenomenon is known as quark confinement.

Thus around the 1 GeV energy level all quarks become bound into either pairs of quarks (mesons) or groups of three (baryons). And therefore, long before the lightest quarks cease to be created by pair production, they become confined. But the same principles apply to the resulting hadrons, and there are simply too few photons with sufficient energy to create many baryons such as protons and neutrons by pair production after the baryons appear as a result of quark confinement.

In any case, all particles and antiparticles of the same type that can pair up annihilate with each other, yielding photons. Only a small unpaired excess of matter particles remains in the form of baryons. From calculations of the amount of deuterium that should be produced by nucleosynthesis, there must have been about one baryon per 1.8 billion photons in order to account for the amount of deuterium that is actually observed.

Nucleosynthesis

• The rest mass of a neutron is about 1.3 MeV more than that of a proton
• Neutrinos cease interacting with protons and neutrons and "fall out of thermal equilibrium" with the nucleons
• The ratio of neutrons to protons drops from 1:1 to about 1:5
• Electrons have a rest mass of about .5 MeV
• The binding energy of a deuterium nucleus (proton plus neutron) is about 1 MeV per nucleon.

• It becomes possible for atomic nuclei consisting of two or more protons and neutrons to exist, whereas at higher energies any that happened to form would quickly be blasted apart by energetic photons.
• The relative abundances of certain light nuclei (hydrogen, deuterium, helium-3, helium-4, and lithium-7) are determined based on various conditions prevailing at the time, and these abundances are fairly close to what can be measured today.

We are going to discuss the process of nucleosynthesis in much more detail later, because the big bang model successfully predicts certain key observational facts about the relative abundances of several very light nuclei. And this is therefore one of the strongest pieces of evidence in favor of the model. So we'll provide just a brief outline at this point.

As mentioned, stable atomic nuclei (other than hydrogen, a single proton) cannot exist at energies substantially higher than 1 MeV. But even when the temperature falls into this range, the probability that a nucleus will actually form depends on how fast a few critical reactions proceed. And these reaction rates depend on factors such as the temperature, the rate of expansion of the universe, the densities of baryons and photons, and the ratio of existing protons to neutrons.

The situation is further complicated by the fact that neutrons which are not bound into nuclei are unstable, with a half-life of 614 seconds. Once substantially all uncombined neutrons have decayed (into a proton plus an electron), nucleosynthesis stops. (In principle, two helium-4 nuclei, for example, could fuse to form beryllium-8, but such reactions are unlikely, mainly due to the electrostatic repulsion between positively charged nuclei.) Consequently, there is a race against the clock. All nuclei heavier than hydrogen that are going to form at this stage of the universe must form within the first half hour or so. Additional nuclei (from beryllium through iron in atomic number) form much later in stellar interiors. And still heavier nuclei can form only in supernova events.

Yet another impediment to nucleosynthesis is that most reactions add only one neutron at a time. And the first step is the hardest, because one proton plus one neutron yields deuterium. Unfortunately, deuterium is only barely stable and has a very small binding energy, about 1 MeV per nucleon. Consequently, deuterium cannot exist in significant amounts until the temperature drops by a factor of 10 from 10¹⁰ K (1 MeV). At this era, the energy of radiation (photons) governs the expansion of the universe and hence the rate that temperature decreases. Specifically, temperature is proportional to the square root of time (we'll show later why that is), so deuterium production can't begin until the universe is at least 100 seconds old. General nucleosynthesis doesn't really get going until more like 300 seconds, when there are no longer enough energetic photons to destroy deuterium, and enough deuterium nuclei have formed to combine with additional neutrons to make helium-3, helium-4, and lithium-7.

That, in a nutshell, is nucleosynthesis. Since the nuclear physics required to compute reaction rates is well understood, it is in principle simple to compute the relative abundances of different nuclei that result from this process. Computationally, it is rather more complex, since there are a number of interrelated factors involved. It's fairly easy to compute that most of the resulting nuclei are either hydrogen (protons) or helium-4. And further, given the ratio of protons to neutrons (which is about 8 to 1 when the process is well underway), the final result is that helium-4 makes up about 25% of all matter, by weight, and hydrogen makes up most of the rest. Everything else is only a trace. This prediction is verified by observation, which is important evidence in favor of the big bang model.

By taking into account how other factors affect other relative abundances, it is possible to infer certain other important facts, such as:

• The amount of deuterium produced is sensitively dependent on both the density of baryons (protons and neutrons) and the ratio of photons to baryons at the time of nucleosynthesis. If we take account of the measured abundances of deuterium, helium-4, and the other light elements, we can determine both the baryon density and the photon-baryon ratio. The result for baryon density is such that only at most one part in six of all dark matter can be in the form of baryons – a very important result. This value agrees very well with an independent estimate obtained from information in the cosmic microwave background (CMB). As for the photon-to-baryon ratio, it must be about 1.8×10⁹ to 1.
• From the baryon density and the photon-baryon ratio we can compute what the photon density was at the time of nucleosynthesis, and from this the present photon density. The result implies a temperature of the CMB very close to the value of 2.725 K that is observed. The correctness of this prediction is very good evidence for both the validity of nucleosynthesis calculations and the big bang model in general. But 15 years before the CMB was discovered, Gamow and his associates used a very rough estimate of the photon-baryon ratio to estimate the photon density. From this they predicted that photons having a current temperature less than 50 K or so would still exist. (The rough estimate of the photon-baryon ratio and an underestimate of the age of the universe account for most of the difference from the correct value.) Had this prediction been taken more seriously, the CMB would have been discovered earlier.
• The density of neutrinos present during nucleosynthesis has an effect on the amount of helium-4 produced. Since this latter amount has been determined by observation, we can estimate the neutrino density. But this in turn depends on the number of different types of neutrinos there are (i. e. whether there exist any other neutrinos besides those associated with electrons, muons, and tauons). The observed abundance of helium-4 implies that there cannot be more than three types of neutrinos, which is also a very important result.

After nucleosynthesis

We will explain the differing effects in much more detail in the next major section. But basically what happens is that as the universe expands, the density of energy in the form of radiation (photons) decreases faster than the density of energy in the form of matter. The reason for this is that the density of particles of matter decreases as the cube of the length scale of the universe, since the number of particles remains about the same (after the annihilation of antimatter), while the volume increases as the cube the linear dimension. The number density of photons also decreases as the cube of the length scale, but in addition the wavelength of every photon is also stretched by the expansion of the universe, and this decreases the energy of each photon by an additional factor of length, so that altogether the energy density of radiation decreases as the fourth power of length.

There was much more energy density in the form of radiation to start out with. But at approximately 47,000 years after the big bang, at a temperature of about 8000 K, the energy densities of matter and radiation became equal. After that point, the energy density of matter was higher. At that point we say that the universe went from a state of energy dominance to one of matter dominance. This change did not have any abrupt consequences. All that happened was that the universe began to expand more rapidly. More precisely, the length scale of the universe transitioned smoothly from increasing in proportion to the square root (one-half power) of the time to the two-thirds power of time. The temperature of the universe began to decrease more rapidly at this point, in a similar way.

This increase in the rate of expansion and temperature decline is important to keep track of for purposes of calculating how length scale and temperature vary. Apart from that, the universe wasn't greatly affected by the change from radiation dominance to matter dominance. Photons and matter continued to interact strongly with each other.

The next important transition, however, has much more important observational consequences. It is what gives rise to the cosmic microwave background which, if our eyes were sensitive enough to microwaves, would give the night sky a very distinctive pattern. This effect is the direct result of a change that occurred in how photons interact with matter when the temperature of the universe was in a specific range.

Although nuclei of a few lightweight elements were formed in the nucleosynthesis process, they were not yet atoms as we know them, since they were fully ionized – electrons still moved about freely instead of being bound to a nucleus. Since both nuclei and electrons carry electric charge, photons (which are carriers of electromagnetic force) interacted strongly with them. As a result, the still extrememly hot plasma was opaque to light. No telescope, however powerful, could see anything in the universe that occurred from the beginning to more than 300,000 years after the big bang.

Indeed, the next major event occurred when electrons finally began to bind with nuclei. This is called the recombination era, even though it was in fact the first time that nuclei and electrons combined. The process of recombination began only when the temperature dropped far enough that a "typical" photon no longer had enough energy to break the electrostatic attraction (also known as Coulomb force) between an electron and a nucleus. This process happened gradually, because not all photons had the same energy. They were distributed in a way characteristic of blackbody radiation, where the energy range of photons is spread out on either side of a peak value. The range is especially large on the high end, so there are always a few very energetic photons around even after the "typical" photon has an energy close to the peak of the curve.

The recombination process began around 240,000 years after the big bang, when the temperature had dropped to very low levels, relatively speaking – about 3700 K. By comparison, the surface of the Sun has a temperature of about 6000 K. The peak of the spectrum of a blackbody radiator at 6000 K is at roughly 500 nm (nanometers) – photons with that wavelength correspond to yellow light, the color of the Sun. At 3700 K the blackbody spectrum is flatter, but has a peak around 700 nm, on the boundary between red and infrared. Remember, though, that there would still be many photons with much higher energies, so the probability was still high that an electron bound to a nucleus would interact with a photon.

3700 K is the temperature at which half of the nuclei should have a full complement of electrons, while the other half was fully or partially ionized. A photon could still interact with an electron bound to a nucleus without stripping the electron from the nucleus; it might simply raise the energy level of the electron rather than ejecting it from the atom.

The universe continued to expand, so the density of photons and matter kept dropping. The simultaneous decrease of both density and temperature made it increasingly less likely that a photon would interact with matter. The probability of interaction can be expressed in terms of the mean free path of a photon, which is the expected distance that a photon can travel before interacting with matter. When this mean free path became longer than the Hubble length (defined as the speed of light divided by the Hubble parameter at the time – which is the distance light could travel in the approximate age of the universe at the time), the probability of interaction became essentially nil. The time that this occurred is called the time of photon decoupling. That was about 350,000 years after the big bang, when the temperature was about 3000 K.

At the time of photon decoupling, the universe first became transparent, for all practical purposes – light was essentially no longer scattered by matter. This is the origin of cosmic background radiation, also known as the cosmic microwave background (CMB). (The "microwave" here is misleading – that is the part of the spectrum in which the radiation is observed today, over 13 billion years later. At the time of photon decoupling, the radiation was mostly in the near infrared.)

Of course, the probability of a photon interacting with matter is never quite zero. Photons from the CMB are still interacting with matter today, such as the matter in one of our microwave antennas. That is how we are able to "see" the CMB. The fact that we can actually "see" the CMB is one of two important features about it. It is important, because the existence of the CMB, at an equivalent temperature of about 2.725 K, is a very strong piece of evidence supporting the big bang model. The model predicts its existence, so we must see it (as we do) to confirm the prediction. It is also a feature which is very implausible to be found in other proposed cosmological theories, such as the "steady state" theory.

The other reason that the CMB is so important is that it carries a wealth of information about the universe at the time of decoupling, and much earlier as well. Some of what we have learned from very careful measurements of the CMB includes:

• The CMB is almost (but not quite) perfectly homogeneous. The homogeneity, with temperature variations no more than 1 part in 10,000, is what we expect from the hypothesis that the universe itself is homogeneous. The small amount of inhomogeneity is exactly what we expect from slight pressure waves that reverberated in the universe at the time of decoupling.
• The size of inhomogeneities (measured in angular size rather than temperature variation from the mean) is just what we would expect on the assumption the universe is nearly "flat", with no measurable curvature. This is strong evidence for the hypothesis of cosmic inflation. In addition, flatness is a very important property influencing the evolution of the universe, as we will explain at length in the next section.
• The full size spectrum of inhomogeneities also has implications for the relative amounts of baryonic and nonbaryonic dark matter, as well as for the way that matter later clumped together (after about half a billion years) to form the earliest galaxies.
• The lack of repetitive patterns of any kind observable in the temperature variations in the CMB implies that the universe as a whole has a very simple topology (shape), rather than something more complicated such as a torus.

We have an extensive discussion of the CMB elsewhere, so that should be enough about it for now.

The period after photon decoupling is called, simply, the period of the "early universe". The universe was a very boring place then. Light could move freely, but (at first) there was nothing to see – no stars, no galaxies. The period is sometimes also called the "dark age" – partly because the only light source was the CMB, which was actually infrared at the time, hence not all that bright, and partly because we know very little about the period.

It can be calculated that the existing hydrogen and helium gas should have been able to condense into the first generation of stars within 100 million years after the big bang, perhaps even earlier. It can also be calculated that the very first stars would have been rather different from typical stars today. They would have been much more massive. But like very massive stars today, they would have converted matter to energy in thermonuclear reactions quite rapidly, and so they would have been very bright, with very high surface temperatures. The stars would have been so hot that the photons they emitted would be energetic enough to reionize lots of interstellar gas, producing glowing gas clouds such as still exist around very hot stars. Accordingly, this period is known as the period of reionization.

Still later, maybe around 500 million years after the big bang, stars began to cluster together in galaxies. This could not have occurred so early without the presence of vast amounts of nonbaryonic dark matter. It is supposed that there were regions of relatively higher and lower densities of such dark matter. The regions of higher density were massive enough to cause existing (baryonic matter) stars and gas to fall into them – making galaxies. Some of these very first galaxies may actually have been imaged by the Hubble space telescope, at a redshift of 8 or more.

We will discuss elsewhere in more detail the processes that went on in the early universe to produce stars and galaxies, so this is an appropriate time to bring our overview of the first few hundred million years of the universe to a close.

Cosmological models

In order to give a more detailed explanation of how the universe evolves under the big bang scenario, we need to look at the general features of any cosmological model. And to do that, we need to say a few words about the use of "models" in science.

A model in science is really quite simple. It typically consists of a set of concepts that either implicitly or explicitly refer to some aspect of the "real world". In addition to the concepts, a model explicitly contains some assumptions about properties of the real world things to which the concepts refer. Generally, these assumptions make some sort of intuitive sense, or may even seem fairly self-evidently true – but not necessarily. A good case in point would be Einstein's special theory of relativity. Two of the key assumptions there are that the speed of light (in a vacuum) is always the same and that nothing, neither physical objects nor information, can travel faster than the speed of light. The first assumption seems reasonable enough, though it certainly requires experimental confirmation. The second assumption is not so intuitively obvious, and it certainly isn't easy to imagine how it could be definitely proven by experiment. Nevertheless, the theory actually works very well: it makes testable predictions, and its predictions have survived all experimental tests – so far.

What's the difference between a "model" and a "theory", then? Nothing, really, other than semantics. The term "theory" carries with it various differing connotations. Sometimes the term is used as in "It's only a theory", implying that the theory has not been rigorously confirmed, and may in fact be somewhat doubtful. However, nothing in science has in fact been absolutely confirmed beyond any possibility of doubt. Science doesn't claim to be able to do that. It claims only, at best, to be able to confirm some theories very, very well, beyond any "reasonable" doubt. Quite a few "theories" are actually in this category, such as Newton's "theory" of gravity (at least as an approximation – some of its details are known to be not 100% accurate), and the special theory of relativity.

With a model, however, one doesn't worry excessively whether all of its assumptions seem intuitively true or can even be confirmed directly. The only thing that really matters is that the model has testable predictions, and that its predictions, when tested, range from "pretty accurate" to "correct up to the limits of measurement". The "theory of quantum electrodynamics", for example, which describes the behavior of light and electric charge, is often cited as one of the most exact theories in science, and its predictions have been confirmed to more than 10 decimal places.

But many times, a scientific model may be far less accurate and definitive. Physicists, for instance, often make various simplifying assumptions about a situation in order to make numerical predictions feasible. From this arise such known impossibilities as a "perfect gas" or a "frictionless plane". Part of the folk humor of the physics community is the story of the physicist who, when asked by a dairy farmer for advice, began by saying, "Consider a spherical cow..."

Often a model will contain many somewhat arbitrary adjustable parameters and arbitrary mathematical equations that relate parts of the model. Typically, the equations can't be solved exactly, and the whole model is programmed into computer code. You run the code, and if the results are a good match with details that can be measured empirically, the model is considered a success. The better the match, and the wider and more general the set of circumstances covered, the more successful the model is considered to be. There are a vast number of such models used in science today, from the "standard model" of particle physics to weather and climate models.

But this doesn't really have anything to do with computers and certainly isn't new in science. It's actually just an instance of what used to be called "inductive reasoning". If you make a series of observations and consequently are able to formulate a set of rules which accurately describe the results, then you have a "theory" or a "model" or whatever you want to call it. For example, between 1609 and 1618 Johannes Kepler enunciated three "laws of planetary motion". These laws accurately described mathematically the motion of planets in the solar system. Kepler was an astronomer (actually, an astrologer) who began as an assistant to Tycho Brahe. He devised his laws inductively, based on the extensive observations made by Tycho and himself. The result was quite a good model of planetary motion. Kepler could not explain or justify why his laws worked. It was good enough that they did. But of course, about 50 years later, Isaac Newton came along and demonstrated mathematically why Kepler's laws worked – provided you accepted Newton's theory of gravity and his more general "laws of motion". Newton's theory is considered to be "deductive", even though his own laws of motion themselves are essentially just plausible assumptions which are verifiable and which, when analyzed mathematically, imply Kepler's empirical laws among their predictions.

Newton's theory of gravity itself was eventually superceded by Einstein's general theory of relativity. And this theory, like Newton's before it, starts by making various sweeping but plausible assumptions. One of these was the special theory of relativity. Others concerned the mathematical form that a theory of gravity should have. But what eventually resulted was not only a theory which generalized Newton's theory, but which was also able to make correct predictions where Newton's theory failed (such as the orbit of Mercury), and to make entirely new and surprising predictions, such as the bending of light by matter.

Fundamental assumptions

The main assumption that is made in the model is what is known as the cosmological principle. This says that the universe looks, approximately, the same from any vantage point. More specifically, the assumption is, first, that the universe is homogeneous – essentially the same everywhere. And second, the universe is isotropic – it looks essentially the same in any direction. We hedge a little, with words like "approximately" and "essentially", because clearly on a small scale the assumptions are not true. Our part of the universe, near a medium size star in a rather average spiral galaxy, is certainly not like the interior of a black hole or the empty space between galaxies. However, the assumption is that on very large scales the universe has no preferred location and no preferred direction.

The two parts of the assumption are not redundant. The universe could be isotropic when viewed from one location without being homogeneous. For instance, it could be much denser (on a large scale) at some distances than others. And it could be homogeneous without being isotropic if, for example, all galaxies were lined up parallel to each other. But it is true that if the universe is isotropic everywhere, it must in fact be homogeneous. These two conditions amount to the universe having certain types of symmetry: everything about the universe is symmetric (at large scales) under the symmetry operations of spatial translation and rotation.

One consequence of the homogeneity condition is that the big bang event (if such existed, which isn't in fact absolutely required in all versions of the model) didn't happen at one particular point in space. It must have happened "everywhere". And although the universe certainly appears to be expanding away from us in all directions, the same must be true from all other vantage points as well. This is the familiar analogy to spots on an inflating balloon.

Note that the universe is not assumed to have such symmetry with respect to time, even though spacetime is still assumed to be 4-dimensional just as described in special relativity. The symmetries pertain strictly to the 3 spatial dimensions. It is in fact definitely assumed that the universe must have been very different at times sufficiently far back in the past, as well as (probably) far in the future. The so-called "steady state" cosmological model which assumes time symmetry – that the universe has always looked about the same and always will – seems to conflict in a number of ways with what we actually see around us. For instance, there are several compelling reasons to believe that the universe must have been much hotter and denser in the past than it is now.

This brings us to another assumption: that the universe is not spatially static but is in fact expanding, in a certain sense that we will make more precise. As we noted above, this assumption is founded on the observations first made by Edwin Hubble of galactic redshifts, and which, since his time, have been confirmed in many ways. Yet in some sense, the assumption of an expanding universe remains "just" an assumption. There are still astronomers and cosmologists today who seek alternative explanations for the observed redshifts. If one of these alternatives were correct, it might not be necessary to assume the universe is expanding, as it appears to be. However, if the universe is not in fact expanding, then a variety of other things we can observe, such as the cosmic microwave background (CMB) have no obvious explanation. So we make the assumption of expansion not only because that's the simplest explanation of Hubble's observations, but also because it yields a consistent model that explains many other things too – such as the apparent age of the oldest stars, the chemical composition of the universe, and the CMB.

It's also worth noting that Einstein himself, shortly after developing the general theory of relativity, assumed the universe was not expanding, since before Hubble almost everyone assumed the universe was static. Einstein was in fact much annoyed because his theory strongly implied that the universe shouldn't be static, and ought to be either expanding or contracting. After all, an object thrown into the air from the surface of the Earth doesn't just hang in midair. At first it rises, and then it falls back under the force of gravity. It would be very surprising if the universe, under the force of gravity, didn't behave similarly. Einstein even went so far as to modify his theory to allow for the universe to be static by adding the cosmological constant to it. As it turned out, after Hubble proved the universe wasn't static, it seemed that adding the cosmological constant was a mistake. (Though it has returned, with a vengeance, as a result of fairly recent observations, as we will explain at some length.)

General relativity, geometry, coordinate systems

The mathematical way to describe space and curvature is a subject known as Riemannian geometry, after Berhnard Riemann (1826-66), who developed the necessary mathematics about 50 years before Einstein availed himself of it. In modern terminology, the theory is all about objects known as Riemannian manifolds. The key feature of such an object is that it possesses something known as a metric, which provides a way of measuring distances within the manifold, independently of any external geometrical constructs, such as a global Cartesian coordinate system. For example, it is possible to describe distances and all other geometric aspects of the 2-dimensional surface of a sphere (such as the surface of the Earth), without referring to how the surface is embedded in 3-dimensional space. Likewise, the 4-dimensional spacetime that figures in the special and general theories of relativity can be described completely in terms of a metric, and things like curvature can be defined rigorously, even though it is not easy to visualize just how a 4-dimensional space is able to curve.

From our assumption that the universe is homogeneous, we can deduce that the same sort of local coordinate system can be used everywhere. Further, the metric used to define distances much have the same form everywhere. This isn't to say that one single coordinate system applies everywhere, only the same sort of coordinate system.

We can make this more concrete. Imagine any point within the universe, such as the current location of the Earth. You can make this the origin of a spherical coordinate system, involving radial distance and two angular coordinates – essentially radial distance from the center of the Earth, plus latitude and longitude. (This is only a way to describe 3-dimensional space. For the moment, don't worry about the time dimension of spacetime.)

By virtue of the assumed homogeneity of the universe, it simply does not matter where we place the center of the coordinate system. It could be here on Earth, or it could be somewhere a few billion light-years away. Any description of the universe should be essentially the same no matter where we place the center.

There is one special type of coordinate system which is particularly useful. It is called a comoving coordinate system. Still assuming a spherical type of coordinate system as well, given some arbitrary point as a center, we can identify any other point in the universe by a 3-tuple of numbers: (x, θ, &phi). The quantity x is still the distance from the center to the other given point, and θ, φ are the two angular coordinates. The thing about a comoving coordinate system is that, with both the center and the other point fixed, the 3-tuple of coordinates never changes as a result of deformations (such as expansion and contraction) of space itself. You could imprint the coordinate system onto space, and the numbers would never change. Of course, an actual particle momentarily located at a particular point can move around within space, but the points through which the particle moves keep the same coordinates no matter what space does. That is, x, θ, and φ are constants.

However, we are still assuming that space actually is expanding (or maybe contracting). Therefore, the "real" distance r (sometimes called proper distance) from the origin to the selected point will change as space expands or contracts. r = r(t) is actually a function of time alone. By virtue of our assumption that the universe is isotropic (on a large scale), the "real" angles θ and φ of the selected point never change. This means that we assume the universe isn't expanding at different rates in different directions that we might look. (r(t), θ, φ) is the time-dependent vector that defines the real location of the selected point as time passes. It follows from our assumptions of homogeneity and isotropy that r(t) = a(t)x for some function a(t) that doesn't depend on the origin of the coordinate system, x, θ, or φ – only on time.

Note that a(t) is a ratio of distances, and so it is a dimensionless quantity. However, we are free to pick the time at which the proper distance is measured. It's only natural to pick this to be the present time, which means that right now real distance = comoving distance, and so a(t_now) = 1. This fixes a(t) completely (up to uncertainty of measurement).

a(t) is the most important mathematical quantity in the big bang model. It is called the scale factor. a(t) describes how space is expanding or contracting everywhere (by virtue of homogeneity) solely as a function of time, t, measured since the instant of the big bang. For some values of t, space may be expanding (a(t) is increasing) and at other values space may be contracting (a(t) is decreasing). If we knew a(t), then we'd know, in principle, exactly what space is doing at any point in time. Other interesting quantities one might care about, such as the rate of expansion (i. e., the Hubble "constant"), the density of matter and energy, etc., can be related to a(t).

A simple way to think of the scale factor a(t) is that it is a kind of ruler one may use to measure distances in the universe. This ruler has the useful property that, regardless of when you measure the distance between two particular distantly-separated objects, like galaxies, the distance never changes. (The objects should be far enough apart that actual relative motions are small enough to be negligible in comparison to the expansion of space.) So this ruler measures comoving distance. But it is a peculiar sort of ruler, which itself expands (or contracts) just as space does.

Since the scale factor measures the expansion of space, it will affect the wavelength of all photons. Therefore it must be related to redshift z, which was defined so that z + 1 is the ratio of a photon's observed wavelength to its wavelength at the time of emission. This ratio is the same as the ratio between the present scale factor, which is 1, and the scale factor when the photon was emitted. Therefore z + 1 = 1/a(t). So a is about 1/z for large z (i. e., early in the history of the universe).

There is another, equivalent, way to conceptualize a(t). It is based on the mathematical forumlation of general relativity. General relativity can be summarized in a single equation:

R^μ_ν - g^μ_νR/2 = (8πG/c⁴)T^μ_ν

From this, assuming a particular metric tensor, if a distribution of matter and energy is specified, the curvature of space can (in principle) be calculated. Conversely, given complete information about curvature, the distribution of matter and energy can be calculated. One way to use the Einstein equation is to assume a metric tensor of a specific form and some distribution of matter and energy, and from these derive simpler differential equations that can be solved, at least approximately. So, as promised, the equation relates the curvature of space to the distribution of matter and energy. (Note that energy as well as matter affects how space curves.)

The first step in applying the equation is to put some constraints on the form of the metric tensor. For purposes of cosmology, we want to have a metric which corresponds to a spacetime that is spatially homogeneous and isotropic. It can be shown that the metric must have the form:

ds² = -c²dt² + a(t)² [dr²/(1-kr²) + r² (dθ² + sin²θ dφ²)]

Usually this metric isn't worked with directly in cosmology. Instead, one uses the metric and the Einstein equation to derive simpler differential equations, most importantly one called the Friedmann equation (after Alexander Friedmann), which involves quantities of more direct cosmological relevance. The Friedmann equation is rigorously derivable from Einstein's equation and the Robertson-Walker metric, but in fact it can be derived, with a little hand-waving, from Newton's original theory of gravity. (Unfortunately, we have to assume you've been exposed to the mathematical form of some of Newton's laws involving energy, as well as a tiny bit of calculus in order to fully understand this part.)

The Friedmann equation

Our plan for deriving the equation is to consider a particle of matter, a "test particle", having mass m and coordinates (r, θ, φ) in our chosen coordinate system. The test particle may be at an arbitrarily large distance r from the origin of the coordinate system, even billions of light years. Therefore, a sphere centered at the origin will contain a certain amount of matter. Because we are assuming that space is homogeneous, we don't need to specify where each individual lump of matter is, only that all matter is distributed uniformly with average density of ρ (mass per unit volume, expressed in units compatible with those used for r and m). It is perfectly true that this idealized situation is not exactly correct in the real universe, but – like such things as "frictionless planes" – it is close enough that we actually get a useful result that is a good approximation for our purposes when we make the simplifying assumptions.

Now, a basic theorem of Newton's theory of gravitation is that the motion of the test particle depends only on the matter located inside a sphere about the origin whose radius r is the same as the particle's distance from the origin, so that the particle lies on the surface of that sphere. In other words, all matter farther from the origin than the test particle is irrelevant to how the particle moves (as long as all that matter is distributed uniformly). Moreover, the particle's motion is the same as what it would be if the entire mass inside the sphere were located at the origin. This reduces the whole problem to a situation involving only two particles.

Let M be the mass of the matter inside the sphere, so that M = 4πρr³/3, since the volume of the sphere is 4πr³/3. Newton's law then says that the force acting on the test particle is

F = GMm/r² = 4πGρrm/3

Most of the time when one wants to derive equations for the motion of a particle, it is done by appealing to the principle of energy conservation – the total energy of the system is constant in time. The total energy is made up of two parts: kinetic energy and potential energy – energy due to motion and energy due to gravitational force. The mass M at the origin is assumed not to move (in the chosen coordinate system), so it has no kinetic energy. Since the test particle moves as though only one other particle is involved, all its motion is in the radial direction, and its velocity is dr/dt = r′, so its kinetic energy is mr′²/2.

The potential energy is defined in terms of the two masses together, and is given by

V = -GMm/r = -4πGρr²m/3

E = mr′²/2 - 4πGρr²m/3

We can now switch to a co-moving coordinate system where x (which is related to r(t) by r(t) = a(t)x) is the unvarying distance of the test particle from the origin. Taking derivatives, r′ = a′x (since x = x(t) is constant). Substituting this into the above gives

E = m(a′x)²/2 - 4πGρ(ax)²m/3

(a′/a)²mx²/2 = 4πGρx²m/3 + E/a²

(a′/a)² = 8πGρ/3 + (2E/mx²)/a²

(a′/a)² = 8πGρ/3 - kc²/a²

All the terms in the equation have dimensions of (time)^-2. In particular, a′/a has dimensions (time)^-1. But what is it? Well, if you go back to the equation r(t) = a(t)x, then r′ = a′x = (a′/a)r. That should ring a bell. It says that the rate that two points in space are becoming separated is proportional to the separation between the points. Hubble's law! And the "constant" of proportionality (which is actually a time-dependent quantity) is simply Hubble's parameter, i. e. H(t) = a′(t)/a(t).

If we were to derive the Friedmann equation in full generality from Einstein's equation of general relativity, we would have to take into account energy as well as matter. In particular, it is clear that the universe at present is filled with a great deal of energy as well as matter, in the form of the cosmic microwave background photons. Indeed, from the relation E = mc² we know that matter itself is just a form of energy, though there's no way to account for this in Newton's physics. However, just as we can replace m with E/c², we can also replace matter density, ρ, with a more general quantity involving "energy density", denoted by ε. Specifically, we should replace ρ with ε/c², to get the equation

(a′/a)² = (8πG/3)ε/c² - kc²/a²

The fluid and acceleration equations

dE + PdV = TdS

These derivatives are easy to calculate. Consider a sphere with radius 1 in comoving coordinates. Its volume is V = 4πa³/3, and so V′ = 4πa²a′. The mass-energy inside the sphere is E = εV = 4πεa³/3. The density ε is (like a) a function of time, so the derivative E′ = (4π/3)(a³ε′ + 3a²a′ε). Plugging into E′ + PV′ = 0 yields 4π(a³ε′/3 + a²a′ε + Pa²a′) = 0. Dividing through by 4πa³ gives ε′/3 + (a′/a)ε + P(a′/a) = 0, hence

ε′ + 3(ε + P)(a′/a) = 0

ρ′ + 3(ρ + P/c²)(a′/a) = 0

Taking another derivative of the Friedmann equation with ρ gives

2(a′/a)[(aa″-a′²)/a²] = (8πG/3)ρ′ + 2kc²a′/a³

a″/a - (a′/a)² = -4πG(ρ+P/c²) + kc²/a²

a″/a = -(4πG/3)(ρ + 3P/c²)

a″/a = -(4πG/3c²)(ε + 3P)

Applying the equations

In order to talk about the evolution of the universe, we have to consider various scenarios about what the universe is made of. Generically, it is "energy" that the universe is made of. However, different forms of energy behave differently in an expanding universe.

Matter is certainly one form of energy. But some care is required in how matter is defined. It turns out that we need to define "matter" as largely what we normally think of it as, namely a collection of particles such as baryons and leptons, provided they are not moving at or near the speed of light. In contrast, photons are never "matter", as they always move at the speed of light. Likewise, neutrinos, which are leptons, are not considered to be matter, even though they may have a small rest mass, since their velocity is always near the speed of light. We define massless particles and particles such as neutrinos which move at relativistic speeds (i. e. near the speed of light) to be radiation instead of matter. Baryons and leptons other than neutrinos are usually considered to be matter. However, in the very early universe when all particles had extremely high energy and so moved near the speed of light, baryons and leptons must be considered to be radiation also, since their kinetic energy was far in excess of their rest-mass energy.

One key distinction betwee matter and radiation is that matter does not contribute in a significant way to pressure, because its kinetic energy is relatively low. However, all forms of radiation, including photons, neutrinos, and relativistic particles do contribute to pressure, since their energy is mostly kinetic.

Another form of energy we will consider a little later is "dark energy", also known as the "cosmological constant". But to begin with, we will only consider models that involve matter, radiation, or both. Whenever we make some assumptions about the presence or absence of matter and radiation in the universe, or about their relative importance, we are specifying part of one particular model of the universe. This will affect the pressure parameter, P, in our equations. Another type of assumption we might make is about the parameter k, at least as to the three possibilities of whether it is positive, negative, or zero.

We haven't talked much about curvature before. If k is zero, we say that space is flat. Geometrically, that means the axioms of Euclidean geometry apply. For example, the three interior angles of a plane triangle on a flat 2-dimensional surface add up to exactly 180 degrees. If k is positive, space is said to have positive curvature. In the 2-dimensional case, the interior angles of a triangle add up to more than 180 degrees. Space with a positive curvature (in two dimensions) is like the surface of a sphere. On the other hand, if k is negative, space has a negative curvature, and the angles of a triangle add up to less than 180 degrees. The standard example is a saddle shape. The spacetime we live in is 4-dimensional, and curvature is nearly impossible to visualize or characterise in terms of triangles, but the distinctions between flat, positively curved, and negatively curved are still valid and important. A choice of positive, negative, or zero curvature is another key feature that applies to a particular model.

Let's first consider a universe that consists (mainly) of matter. In that case, we assume P = 0. The pressure is (essentially) zero, because constituent particles have low average velocity and seldom interact with each other. In this case, the fluid equation says ρ′ + 3ρ(a'/a) = 0. We want to know how the functions a(t) and ρ(t) behave, since that tells us how the universe is expanding (or contracting) and how density evolves with time.

Note that d(ρa³)/dt = ρ′a³ + 3ρa²a′. But this is just the fluid equation multiplied by a³, so the indicated derivative is 0, and ρa³ must be a constant, i. e. ρ = K/a³. In this case, the density of matter declines like 1/a³ with increasing scale factor, which is exactly what we would expect, since the amount of matter is constant, but volume grows like a³. Another way to symbolize this without mentioning a specific constant is ρ ∝ 1/a³.

The same thing is true of ε – it really doesn't matter what form of density we consider. What's important is how pressure P behaves for energy in various forms. This is described by an equation called the equation of state. When we have only matter with no radiation or dark energy, the equation of state is simply P = 0.

Notice that it doesn't matter here what k is, since k doesn't appear in the fluid equation. So ρ ∝ 1/a³ for any value of k in the "matter only" case. We can similarly ignore k when considering the acceleration a″ in the acceleration equation. When P = 0, this becomes a″/a = -4πGρ/3, so a″ = -4πGρa/3. It follows that a″ is always negative, so the universe is always decelerating when it consists only of (non-relativistic) matter, no matter what k is. However, since ρ ∝ a^-3, we have a″ ∝ 1/a². Thus the deceleration becomes very small in magnitude when a is large. But so far, we don't know how a(t) itself behaves – it might be decreasing for large values of t, even though it is increasing at the present time.

In order to investigate how a(t) varies with t we have to use the Friedmann equation, and so (for simplicity) we first assume the universe is flat, k = 0. Then the Friedmann equation says (a′/a)² = Cρ for some constant C. (C has nothing to do with the speed of light c, and we will use C as a "generic" constant, which might be different each time it is mentioned, yet does not vary with time.) We know ρ ∝ a^-3 for any k, so a′² = C/a, hence a′ = Ca^-1/2. A solution to this differential equation is given by a = Ct^2/3, as is easily verified since (d/dt)(t^2/3) = Ct^-1/3 = Ca^-1/2. In other words, we have a ∝ t^2/3, provided k = 0. Thus a(t) is a monotonically increasing function of time. So the universe in this model is always expanding, but from the discussion in the previous paragraph, the rate of expansion is decreasing to zero as t → ∞.

We can also find the Hubble parameter H(t) = a′/a in this model