Invitation to a Contemporary Physics (2004)

Figure 10.2: Curves H, D, and A represent schematically a universe with a constant velocity, a universe which is slowing down, and a universe which is speeding up, respectively.

the random motion of the galaxies becomes negligible. When that happens the pressure P is practically zero, so only gravity is there to hold back the expansion of the universe. In that period a(t) grows a bit faster, as t2/3. If gravity were absent as well, then the velocity would be constant, and a(t) would be proportional to t. The persistent presence of gravity is what makes it grow slower than t, as t2/3. This also causes the present age of the universe to be (2/3)H0−1 rather than H0−1. The absence of pressure means there is no need to expend energy to do work, hence ρ(t)a3(t) remains a constant. This is the matter-dominated era.

The universe decelerates through both of these epochs, so the velocity v = ds(t)/dt = r(da(t)/dt) should be larger in the past than at the present. If no deceleration occured, then a(t) = t/t0. By di erentiation, we get back to Hubble’s law v = r/t0 = H0s(t0), with H0 = 1/t0. This is the straight line labeled H in Fig. 10.2. With deceleration, v should be larger in the past than at the present. Since light reaching us today was emitted some time ago, looking at a distant galaxy is like looking back in time. For example, the light we see today from a galaxy one billion light-years away was emitted one billion years ago. Consequently, the curve in the Hubble plot with deceleration taken into account should lie above the straight line H, as in D in Fig. 10.2.

10.3.2The Red-Shift Factor in an Expanding Universe

In Sec. 10.1.1 we used the Dopper shift formula z v/c to measure the velocity v of a nearby galaxy. This formula is derived for a static universe. How should it be modified when the universe is expanding?

For a nearby galaxy whose red-shift is small, there is no time for the size of the universe to change significantly before its light reaches us, so the static universe formula used in the original Hubble plot is valid.

For faraway galaxies when the red-shift is large, this is no longer the case. It will be shown in Sec. 10.3.5 that the correct formula is z+1 = a(t0)/a(t), the ratio of the

With the present value of H0, the critical mass density is ρc/c2 10−29 g/(cm)3, which is the density of five protons per cubic meter.

It is conventional to use the ratio Ω = ρ/ρc as a measure of the density ρ of the universe. If this is larger than 1, the universe is said to be closed. If it is less than 1, the universe is said to be open. If it is equal to 1, the universe is said to be critical.

A galaxy in an expanding universe possesses a kinetic energy. Being attracted by other galaxies it also possesses a gravitational potential energy. The kinetic energy is respectively larger, equal, and smaller than its potential energy when Ω > 1, = 1, < 1. A critical universe is spatially flat. Euclidean geometry is applicable there, so parallel lines never meet. A closed universe has a positive spatial curvature like the surface of a sphere. Similar to the longitude lines on earth which meet at the north and south poles, seemingly parallel lines do meet further on in a positively curved universe. An open universe has a negative curvature, seemingly parallel lines will get farther from each other further out.

The critical density is awfully small. The best vacuum that can be achieved in the laboratory contains about 2.5 × 106 molecules per cubic meter, which is at least some 500,000 times bigger than the critical density of the universe. Surely our universe must be closed?

It is not so, because there are lots of empty space in the universe. In fact, for years astronomers believed our universe to be open. Stars account for only about 0.35% of the critical density, but that by itself should not be taken as evidence for an open universe, because lots of matter do not reside in the living stars. By measuring the abundance of light elements made at the beginning of the universe, a subject which will be discussed in Sec. 10.6.4, one deduces that the total amount of ordinary matter, including the bright ones in shining stars, and the dark ones in dead stars, non-luminous clouds, planets, etc., accounts for about 4% to 5% of the critical density. On top of that, it has been known for some time that there must be additional dark matter around whose nature we do not know, because no such matter have yet been found through direct observation. We are aware of its presence only from its gravitational influence on stars and galaxies. We do know that most of it has to be non-relativistic in order for galaxies to cluster the way they do. For that reason this kind of dark matter is sometimes referred to as cold dark matter.

Since the universe is accelerating, we know that dark energy is present as well. Dark energy and matter of various kinds not only change the Hubble plot, they also a ect the cosmic background radiation (see Sec. 10.7), and the clustering of galaxies, so we can obtain their amounts from these observations as well. The result shows that our universe is compatible to being critical. About 70% of the critical density resides in the dark energy, and about 30% is contained in matter. As mentioned before, only about 4% to 5% of the critical density is in ordinary matter, so the cold dark matter outnumbers ordinary matter more than 5 to 1.

More than twenty years ago, when astronomers swore to an open universe, this possibility of a critical universe was already suggested by Alan Guth and others, to

explain some of the puzzles encountered in the classical Big Bang theory. We will come back to discuss this inflationary theory of the universe in the next section.

10.3.5The Dynamics

We study in this subsection how gravity and pressure quantitatively a ect the expansion of the universe. In particular, formulas used in the previous subsections will be derived. Readers not interested in such details may skip now to the next section.

We are interested in obtaining the equations governing the time variations of a(t), ρ(t), and P (t). To do so we need three equations, which can be obtained as follows.

Let us look at a nearby galaxy, represented by a black dot in Fig. 10.1. Suppose it has a mass m and a distance s(t) = a(t)r from the center.4 Its kinetic energy at any time is 12 m(ds/dt)2, and its potential energy is −GMm/s, where M is the mass inside the sphere and G is the Newtonian gravitational constant. Its total energy is a constant, which we shall denote by −21 mr2k. The total energy inside the sphere is Mc2 = (4π/3)s3ρ, and hence the equation expressing the total energy of the galaxy can be written as

In this form the equation is independent of the specific galaxy used to derive it. It is an equation governing the temporal change of the scale factor a(t). The ratio (da/dt)/a is the Hubble constant H at time t. This equation is known as the Friedmann equation.

A closed universe corresponds to k > 0, an open universe corresponds to k < 0, and a critical universe corresponds to k = 0. In the last case, the critical density ρc of the universe is related to the present Hubble constant H0 by the relation

ρc = 3c2H02 . 8πG

Using H0 = 72 km/sec/Mpc, we obtain ρc = 10−29g/(cm)3, as reported before.

To solve Eq. (10.1) for a(t) we must have an equation for ρ(t). This comes about by considering the work done by the sphere needed to overcome the surrounding pressure. To expand the sphere of radius s to s + ds, the amount of work required is P (4πs2)ds. The energy needed to do this work is supplied by the energy stored inside the sphere, and

By applying d/dt = (da/dt)(d/da) to Eq. (10.1) and then using Eq. (10.2), we obtain an equation for the acceleration:

4You can think of the center as where we are, or just any other fixed point in the universe.

400 Cosmology

Multiplying both sides by m r, this becomes the equation of motion for the galaxy with mass m, which shows that the e ective force on a galaxy is proportional to ρ + 3P , as previously claimed.

Since P is introduced in Eq. (10.2), we need a third equation in order to determine the three quantities a(t), ρ(t) and P (t). That equation depends on the structure of the fluid, and is known as the equation of state. It has nothing to do with the size of the universe.

At the beginning of the universe when it is hot, the galaxies acquire a highly relativistic random motion whose equation of state is given by P = ρ/3. Later on, when most of the matter becomes cool and non-relativistic, pressure is lost so that the equation of state is simply P 0. Dark energy has an equation of state P = wρ for some negative w.

Given an equation of state, Eq. (10.2) can be used to find out how ρ depends on a.

For P = ρ/3, we get ρ 1/a4. For P = 0, we get ρ 1/a3. And for P = wρ, it is

ρ a−3(w+1).

For the critical universe where k = 0, this information can be used to obtain an

where a(ti) is the initial size at time ti. For the three scenarios, relativistic matter, non-relativistic matter, and dark energy, the parameter p is respectively 2, 3/2, and (3/2)(1 + w).

The scale factor a(t) of the universe can be determined by measuring the red-shift of light leaving a galaxy at time t. The relation is

The right-hand side simply reduces to 1/a(t) if we normalize the magnitude of the presentday scale factor a(t0) to be 1. This formula can be derived as follows. By definition, z + 1 = νE/ν0 is the ratio of the frequency of light emitted at the galaxy (νE) to the frequency received on earth (ν0). Imagine now a pulse of light is sent out by the galaxy at the beginning of every period. This means that the first pulse and the second pulse are sent out a time 1/νE apart. The time di erence of the arrival of these two pulses on earth is the period, or 1/ν0. Now over a time period dt, light travels from the galaxy towards earth a distance ds(t) = a(t)dr = cdt. The comoving distance traveled by the first pulse of light between the galaxy and earth is therefore r = tt0 cdt/a(t). The second pulse of light is emitted at time t + 1/νE, and received at time t0 + 1/ν0. It travels the same comoving distance r. Equating these two equations, we obtain

tt0

Since the integration intervals on both sides are very small, each integrand stays practically constant, and hence νE−1/a(t) = ν0−1/a(t0). This is equivalent to Eq. (10.5).

10.3.6Summary

Expansion of the universe after the Big Bang is described by a scale factor a(t), whose time dependence is controlled by the energy density of the universe. Our universe is found to be critical, and hence spatially flat. The required energy density is equal to the rest energy of about 5 protons per cubic meter. About 70% of that resides in the form of dark energy, and 30% in the form of matter. Of the latter, about 4 to 5% of the critical energy is composed of ordinary hadronic matter, with the rest mostly in the form of cold dark matter, which has not yet been directly discovered. The origin of dark energy is not yet understood, but we do know that it causes the present universe to accelerate.

10.4Before the Big Bang

10.4.1Problems of the Classical Big Bang Theory

What caused the Big Bang?

A possible answer came to Alan Guth around 1980 when he investigated the problems arising from the standard Big Bang theory. He solved these problems by inventing a period of inflation preceeding the Big Bang, during which the universe expands exponentially fast. The end of inflation erupts into an explosion causing the Big Bang.

The inflation theory predicted our universe to be critical some fifteen years before its recent verification. It also predicted a power spectrum consistent with recent measurements of microwave background radiation and galactic distribution. To explain it, let us review the problems Guth was trying to solve.

The most serious is the horizon problem. This is the problem of understanding why di erent parts of the universe have such a uniform temperature, 2.725 K, as revealed by the cosmic microwave radiation (CMB), which will be discussed in Sec. 10.7. √

Right after the Big Bang, the expansion rate da/dt 1/ t is almost infinite, so galaxies were flying apart at that time faster than the speed of light. Even so, special relativity is not violated, because that merely asserts no particle and no information may propagate faster than the speed of light. Expansion of the universe comes from the stretching of space, and that cannot be used to propagate information or a particle from one place to another. However, that does make it di cult for information sent out from one galaxy at the beginning of the universe to reach another galaxy, because the other galaxy is running away faster than light can catch up. Gradually, as the expansion slows down, the speed of light is able to surpass the expansion rate, so information can begin to reach nearby galaxies. If we give it enough time, then eventually light emitted at the beginning of the universe from one galaxy can reach every other galaxy in the universe. However,

calculations show that by the time the CMB is emitted, this has not yet happened, so there are many parts of the universe which are not in causal contact with one another, meaning that light or information sent out from one cannot have reached the other. That makes it very hard to understand how the CMB emitted from di erent parts of the universe can carry the same temperature, as observed, unless they are already at the same temperature at the very beginning. But it is not in the nature of most explosions to produce the uniform temperature everywhere, so what happens?

This is called the horizon problem because two galaxies that are not causally connected are said to be outside each other’s event horizon.

The second problem is the flatness problem.

Even twenty odd years ago when astronomers believed the universe to be open, it was already known that the present density of the universe is not that di erent from the critical density ρc — a few percent or tens of percent of the critical density, but not orders of magnitude smaller. If we go back towards the beginning of the universe, then the ratio becomes very close to 1. This is so because at the beginning of the universe, the expansion rate is much greater, so both terms on the left-hand side of Eq. (10.1) are much bigger than their di erence, viz. the right-hand side. This means that the right-hand side becomes more and more negligible as we go back further and further in time. It also means that the actual density is getting closer and closer to the critical density, as the latter is obtained by setting the right-hand side of Eq. (10.1) equal to zero. In other words, the universe would be getting to be almost completely flat at the beginning. This is deemed to be ugly and unnatural, unless the universe is actually critical and completely flat.

Finally, there is the monopole problem. Some speculative but widely accepted theories of particle physics suggest that plenty of magnetic monopoles should be produced in the early universe, but none of them have been found. The same is true for many other hypothetical particles.

10.4.2The Theory of Inflation

If for some reason the density ρ of a patch of the universe remains constant for a su ciently long period of time, then a(t) expands exponentially during that period, giving,5 a(t) = a(0) exp(λt).

Calculations show that if the expansion undergoes a factor of 1025 or more, then the problems mentioned in the last subsection will disappear. The flatness problem is solved because the curvature becomes almost zero after such a huge inflation. This is like living on a spherical earth without noticing its curvature because it is so large. The horizon problem disappears because the patch that inflated was originally very small, so everywhere inside can easily reach a uniform temperature

5This follows from Eq. (10.1) if k = 0 and ρ is time independent. In that case λ = 8πGρ/3c2. If the universe has an initial curvature, then the inflation solution is more complicated, but for large time, this is still the solution, with an e ective k = 0.

404 Cosmology

particles were absent at the beginning, they can be created by collisions, provided the thermal energy present is su cient to allow this to take place. At the end, every particle of every energetically available species will have the same average energy. This basic fact of statistical mechanics is known as the principle of equipartition of energy. It means, for example, that at the beginning of the universe when the temperature is high, we have equal numbers of electrons, positrons, protons, antiprotons, neutrons, antineutrons, photons, neutrinos, antineutrinos and other particles,6 and they all have the same average energy.

The average kinetic energy of a non-relativistic particle7 is 3kBT/2, where the Boltzmann constant kB = 1 eV/(1.16 × 104 K) is simply the conversion factor8 between the Kelvin scale of temperature and the electron-volt scale of energy.

This is not to say that every particle has the same kinetic energy 3kBT/2. Only the average is the same. Individual particles may have any energy, though the probability of having an energy E kBT is damped by the Boltzmann factor e−E/kBT . In particular, a particle with mass m may disappear if mc2 kBT , whenever there is a reaction to change it into something lighter. For example, electrons (e−) and positrons (e+) of mass m can annihilate each other and produce two photons (γ) through the reaction e+ + e− → γ + γ, but the inverse reaction γ + γ → e+ + e− cannot take place e ciently for kBT mc2, when there is not a su cient amount of thermal energy around. Since mc2 = 0.5 MeV, electrons and positrons in the initial universe will largely disappear when the temperature cools down to kBT 1 MeV. We will come back to this phenomenon in Sec. 10.6.2.

10.5.2Stefan–Boltzmann and Wien’s Laws

Photons are always relativistic and their number can fluctuate. Their average energy density ργ is given by the Stefan–Boltzmann law to be9 ργ = (π2/15)(kBT )4/( c)3. The average number density nγ is equal to nγ = (2.402/π2)(kBT/ c)3. The average energy of a photon is therefore ργ/nγ = 2.7kBT .

There is a spread in photon energies as well. The spread is once again determined by the Boltzmann factor, which when incorporated correctly gives rise

6Please consult Chapter 9 for information on elementary particles.

7A particle of mass m is relativistic if its total amount of energy is far greater than its rest energy mc2. It is non-relativistic if the opposite is true. A photon is always relativistic, and a massive particle with mc2 kBT is non-relativistic.

8The absolute temperature T in the Kelvin (K) scale is equal to 273 plus the temperature in the Celsius scale. The eV (electron volt) is an energy unit equal to the amount of energy gained by

an electron dropping through 1 volt. It is numerically equal to 1.6 × 10−19 joules.

9Other than the numerical factor π2/15 which can be obtained only by a detailed calculation, the other factors in ργ can be understood in the following way. The photon has no mass, so its energy density can depend only on temperature T and fundamental physical constants such as the

Planck’s constant = h/2π, the Boltzmann constant kB, and the speed of light c. c is equal to 0.2×10−6 (eV) m (m = meter). If we measure ργ in eV/m3, then ( c)3ργ is a quantity measured in (eV)4. Since this must depend on T , and kBT is measured in eV, ( c)3ρ must be proportional to (kBT )4, or else the units on the two sides of the equation would not be the same. Similarly, ( c)3nγ is the number density measured in (eV)3, so it must be proportional to (kBT )3.