Invitation to a Contemporary Physics (2004)

Air is stationary, but the universe is expanding, so to use the expression we must use a distance and time which are not a ected by the expansion. We saw before that the appropriate distance is the comoving distance r, related to the expanding distance s(t) and the scale factor a(t) by r = s/a. With the comoving distance, the correct time scale to go with it — to ensure that c is the speed of light — is the conformal time τ. It is defined by the equations ds = a(t)dr = c(dt) and dr = c(dτ).

A di erent wavelength at decoupling time subtends a di erent angle in the sky today. (See Fig. 10.6.) The inner sphere in the picture is the sphere of last scattering at decoupling (conformal) time τ . The outer sphere indicates the farthest reach in space, from which light emitted at the beginning of time has just reached us. Its conformal time is 0. The center of the spheres indicate our present position, at a conformal time τ0. The comoving distance of the inner sphere from us is c(τ0 − τ ). Since τ0 τ , we may approximate this by cτ0. The angle subtended

occur at values of 172n. A more detailed calculation shows that the first peak (n = 1) actually occurs at an slightly larger than 200, rather than 172.

There is one implicit assumption in this estimate, which can be seen in Fig. 10.6. We are estimating the angle θ using Euclidean geometry. Namely, we are assuming the universe to be flat. This is the case if Ω = 1. On the other hand, if Ω < 1, it is closed like a sphere. Seemingly parallel lines will intersect at a distance, so the observed angle θ is seemingly larger than that in a flat universe, or smaller. In an open universe with Ω > 1, it is the other way around, with a larger . The fact

λ

Figure 10.6: Wavelength at conformal decoupling τ and the angle θ it subtends today. Flat geometry is assumed.

416 Cosmology

that the observed first peak in Fig. 10.4 is so close to 200 is further independent support for the flatness, or the criticality, of the universe.

The various peaks in Figs. 10.4 and 10.5 are not of the same height, contrary to the simple argument given above. There are at least two reasons for that. First, gravity compresses the cosmic fluid, and unlike photons, matter provides no pressure to counteract this compression. Hence, the amplitudes of the compression phases of the wave (odd n’s) are larger than the decompression phases (even n’s). This is borne out by the observation that the height of the n = 2 peak is lower than that of the first or third peak. Secondly, electromagnetic coupling is not all that strong, so photons are not completely trapped. The leakage makes the fluid less bouncy

For very long wavelengths, or very small wave numbers k, cos(kcτ / 3 ) 1 at decoupling time. Thus, the quantum fluctuations at these wavelengths are completely frozen between the inflationary era and the decoupling time. This o ers the exciting possibility of using the CMB to probe directly into the inflationary era. Most inflationary models predict a near scale-invariant fluctuation, which is verified by the CMB observation, thus providing further support for the inflationary origin of the universe.

At the other end of the wavelength scale, gravitational influence becomes very important and that leads to the formation of clusters and galaxies after the decoupling. The size and distribution of the clusters are sensitive to the amount of light and heavy matter involved. One can vary these amounts and compare the result of computer simulations directly with the observed distributions through galaxy surveys. The resulting content of matter is again consistent with the amount obtained by other methods. So all in all, the general composition of the universe, with about 70% dark energy and 30% dark matter, of which only a small fraction is the ordinary hadronic matter, seems to be well supported by di erent observations.

10.7.3Summary

The Cosmic Microwave Background (CMB) radiation provides us with very detailed information about the universe, at decoupling time and before. The observed mean temperature of T0 = 2.725 K today allows us to determine various cosmological parameters. Its dipole component tells us the velocity of the solar system, and its higher multipole fluctuations provides evidence for the acoustic wave in the cosmic fluid generated by quantum fluctuations in the inflationary universe. These fluctuations also provide seeds for cluster and galaxy formation in the later universe.

Detailed cosmological parameters can be obtained from the CMB data alone. They are completely consistent with those obtained in di erent ways and from a di erent epoch of the universe.

10.8 Further Reading

Useful websites

•http://lambda.gsfc.nasa.gov/m uni.html

•http://www.damtp.cam.ac.uk/user/gr/public/cos home.html

•http://casa.colorado.edu/ (contains links to many other webpages)

•http://www.galacticsurf.com/cosmolGB.htm (contains links to many other webpages)

•http://map.gsfc.nasa.gov/index.html (the home page of CMB satellite WMAP)

10.9Problems

10.1A galaxy lying on the Hubble curve has a red-shift z = 0.1. The Hubble constant is H0 = 72 km/sec/Mpc.

1.What is its recessional velocity v in km/sec?

2.What is its distance s in light-years?

3.If its absolute magnitude is −20, what is its apparent magnitude?

10.2Assume the present critical universe to consist of 2/3 dark energy and 1/3 non-relativistic matter. Assume further that the equation of state for the dark energy is P = −0.8ρ.

1.What is the scale factor a when the deceleration of the universe turns into acceleration?

2.What is the red-shift z at that time?

10.3Suppose kBT right after the Big Bang is 1015 GeV, and the number of species at that time is g = gU . Suppose further that the universe has inflated 1025 times before it reaches the Big Bang. What is the original size of the infla-

tionary region before the inflation?

10.4Take the present temperature of the universe to be T0 = 3 K.

1.What is the scale factor a(t) of the universe when kBT = 0.2 MeV?

2.What is the corresponding red-shift factor a(t)?

3.How long after the Big Bang does it take to reach this temperature?

10.5Suppose a darkened room on a summer’s day has a temperature of 27◦C.

1.What is the photon number density nγ in the room?

2.What is the peak wavelength λm of these photons?

3.With so many photons in the room, why is the room dark?

10.6Elements in the universe.

1.What is the most abundant element in the universe?

2.Which of the following elements are produced in Big Bang nucleosynthesis?

(a)4He

(b)D

(c)12C

(d)235U

3.Which of these four is the most abundant in the universe?

10.7The solar system moves with a velocity v = 371 km/s in the background of the cosmic microwave radiation. This causes the sky in front to appear slightly hotter than the sky behind.

1.Compute the Doppler shifts of the cosmic microwaves in front and behind.

2.Use Wien’s displacement formula to estimate the temperature di erence in front and behind.

Figure A.1: Scales of length, mass and time of the physical universe.

numerical values of things; they epitomize the fruits of labor by several generations of scientists and represent important signposts for e orts by new generations.

We characterize size, mass and time as ‘fundamental’ because all other physical quantities can be considered as derived quantities, expressible in units that are various combinations of powers of the base units. Among the more familiar examples,

force is measured in newton (N = m kg/s2), pressure in pascal (Pa = kg/m s2) and energy in joule (J = kg m2/s2) in SI units; or, respectively, in dyne (10−5 N), dyne per square-cm (0.1 Pa) and erg (10−7 J) in cgs units.

In atomic and subatomic physics, one often prefers a smaller, more appropriate unit of energy, the electron volt (eV), defined as 1 eV = 1.60 × 10−19 J. The temperature is a measure of heat, and heat is a form of energy (Appendix C). Thus, the unit of temperature (degree Kelvin or K) may be considered a derived unit; it is related to the unit of energy by a numerical factor which one may identify with a constant called the Boltzmann constant k.

The Boltzmann constant is one of the many fundamental (or universal) constants which recur again and again in physics. They are considered universal, i.e., unchanging in any physical circumstances and, of course, constant in time. Their numerical values are the results of many delicate and laborious experiments (and therefore subject to experimental errors) and are as yet unexplainable by theories. Some of the more important physical constants are given in Table A.1.

The popular belief that scientific information doubles every seven years may well be a myth, but it is nevertheless true that a vast body of facts has been accumulated by mankind ever since the ancient Greeks started pestering each other with endless discourses and questions about earth and heaven, thus embarking on a great intellectual adventure that modern scientists are still pursuing. But facts alone do not necessarily give understanding and knowledge by itself is not science. Explanations of specific phenomena in terms of known facts must be generalized into rules that apply to larger classes of phenomena, and rules must further be distilled into a few universal brief statements, or laws that cover not only observed cases, here and now, but also unobserved cases, anywhere and at any time. Finally, laws are further organized into a coherent and logical structure, called theory.

In physics, existing theories can be divided into three broad categories: mechanics, the theories of matter and interactions and the physics of large systems. Mechanics, the study of motion, plays a unique role central to all of physics. On the basis of the oldest of its three components, classical mechanics, modern physicists have constructed quantum mechanics and special relativity, two superb theories that underpin all contemporary physics and serve as guiding principles for further research. Loosely speaking, the fundamental dynamical theories deal with forces.

Table A.1: Fundamental physical constants (SI units).

But, in one of the greatest developments of this century, it was discovered that forces (or interactions) are in fact intimately related to matter. All matter is composed of a few species of fundamental, or elementary, particles, and these particles interact via four fundamental forces: the gravitational force, the electromagnetic force, the strong force and the weak force. Finally, when we deal with macroscopic bodies, the presence of a very large number of component parts requires specialized techniques. The study of the behavior of such systems is the domain of both thermodynamics (a theory based on a set of four general rules deduced from observations of controlled experiments on macroscopic bodies) and statistical mechanics (a statistical treatment of the mechanics of a very large number of particles).

In this appendix, we give a very brief discussion of some general aspects of classical mechanics. Quantum mechanics will be the subject of Appendix B, while thermal physics and statistical mechanics will be considered in Appendix C. Other more specialized parts of physics are dealt with at appropriate places in the main body of the book.

A.2 Matter and Motion

Space and Time

Whenever physicists describe their observations or formulate their theories, they cannot fail to make use of certain quantities called space and time, two basic physical concepts that are intuitively evident to us all and yet cannot be formally defined in terms of any simpler entities. Since space and time cannot be defined, ways of measuring them must be devised — a distance in space is measured by comparing it with some unit length and a time interval is measured in terms of the period of some cyclical phenomenon. The concepts of distance and time are ultimately defined by the operations carried out in making the measurement. This measurement then is equivalent to the missing formal definition.2 Such measurements, absolutely essential to the development of physics, may appear trivial to us now, but had not always been possible in the past. It was perhaps not a coincidence that the first detailed study of the pendulum and the first systematic experiments on motions were carried out by the same person, Galileo Galilei.

It is a fact of experience that the space in which we live is three-dimensional and, to a very good approximation, Euclidean. That our space is three-dimensional (i.e., any point can be located relative to another by specifying three and only three coordinates) is intuitively clear. The assertion that the geometry of our space is ‘flat,’ or Euclidean (i.e., the postulates of Euclid are valid for our world) is less evident but nevertheless true. A simple way to check it is to measure the sum of the interior angles of a planar triangle; it is always found to be 180◦ to within the

2As P. W. Bridgman, physicist and philosopher, once wrote: ‘The true meaning of a term is to be found by observing what a man does with it, not what he says about it.’

measurement error. Euclidean geometry is valid on or near the earth but breaks down in the immediate vicinity of very massive bodies (e.g., the sun or a neutron star).

Day-to-day experience tells us that time flows in one direction only, from past to present to future. This is indeed confirmed by experiment; time reversal never occurs in the macroscopic world. Time has another important property: it is absolute, that is, it flows at a rate that does not depend on position or velocity. Suppose we have two identical well-made clocks, initially at rest and perfectly synchronized. Suppose one of them is transported in motion along some path at varying moderate speeds, then brought back to its initial position. If we can verify that the two clocks are still synchronized, we say that time is absolute. Experiments show that time is indeed absolute to a very high precision in ordinary circumstances on earth. But this notion of absolute time is not exact in any situation; in particular, it fails when very rapid motions or gravitational e ects are involved: clocks moving at high velocities or exposed to strong gravity are found to lose or gain time relative to stationary clocks.

Classical (or Newtonian) physics is based on the assumption that space is Euclidean and time absolute. Because Newton’s theory rests on postulates that do not exactly hold in all situations, we admit that it is not an exact theory. It is only an approximation, but nevertheless an excellent approximation to the real world, quite adequate in most circumstances we encounter on earth or in space. Furthermore, its formulation is restricted to certain special frames of reference, called the inertial reference frames, i.e., those frames that are at rest or, at most, moving at a constant velocity relative to distant stars. Thus, a reference frame fixed on the ground is an inertial frame, but one fixed on an accelerating train is not. As any good theory, Newtonian physics has few, simple and plausible assumptions, which are general enough to let it ‘grow’ to maturity and in various directions, and eventually blend itself with new, improved theories.

Three Laws of Motion

With the notions of space and time understood, one can define velocity and acceleration in the usual way. The only other concept we need is that of force. Again, this concept comes naturally to us, as part of our everyday experience. We can instinctively perceive a force wherever it exists and can gauge its strength. We can feel it as the airplane carrying us on board is taking o or landing; we know the amount of push or pull we must exert to set an object at rest into motion. So, we can consider force as a basic entity and define it by measuring the e ect it would have on some standard device (for example, the amount of stretching of a standard spring). With those elementary definitions on hand, we can state Newton’s three laws of motion, which together form the basis of classical mechanics and dynamics.

First Law: A body at rest remains at rest and a body in a state of uniform linear motion continues its uniform motion in a straight line unless acted on by an unbalanced force.

This statement (often referred to as the law of inertia) simply means that a body persists in its state of rest or motion unless or until there exists an unbalanced force (i.e., when the sum of forces acting on it does not vanish). What happens then is described by the second law:

Second Law: An unbalanced force (F ) applied to a body gives it an acceleration (a) in the direction of the force such that the magnitude of the force divided by the magnitude of the acceleration is a constant (m) independent of the applied force. This constant, m, is identified with the inertial mass of the body.

Stated algebraically, this law reads: F = ma, which is referred to as ‘Newton’s equation of motion’ because it describes, for a given force, the motion of the body via its acceleration. Three points should be noted. First, it is a vectorial equation; it relates the three components of a vector (force) to the three components of another vector (acceleration). Second, it is general, applicable to any force, independently of its nature. On the other hand, to solve the equation for the body’s acceleration, one must know the exact algebraic expression of the force, which is a separate problem. Finally, the Second Law plays a dual role: it gives a law of motion and also states a precise definition of mass by indicating an operational procedure to measure it. At this point, turning things around, one could choose mass as a basic quantity, define a mass unit and use the Second Law to derive the force.

Third Law: If a body exerts a force of any kind on another body, the latter exerts an exactly equal and opposite force on the former.

That is, forces always occur in equal and opposite pairs in nature. We walk by pushing backward with our foot on the ground, while the ground pushes the sole of our foot forward. The pull of the sun on the earth is equal to the earth’s pull on the sun, and the two forces are along the same line but opposite in direction. Similarly, the positively charged nucleus of the hydrogen atom exerts an attractive force on the orbiting electron, just as the electron exerts an equal and opposite force on the nucleus.

As long as we are describing the kinematics of an isolated body, we just need to know how its velocity changes. But when we deal with the dynamics of the particle, we must take into account its mass as well. It is then more meaningful to deal with mass and velocity together in a single combination, called the momentum, p. The momentum of a body is given by the product of the mass, m, and the velocity, v, of the particle. It is a vector pointing in the same direction as the velocity. Since the acceleration of a particle is the rate at which its velocity changes, the