Cosmology, inflation and the physics of nothing.pdf

COSMOLOGY, INFLATION, AND THE PHYSICS

OF NOTHING

William H. Kinney

Institute for Strings, Cosmology and Astroparticle Physics

Columbia University

550 W. 120th Street

New York, NY 10027

kinney@physics.columbia.edu

Abstract These four lectures cover four topics in modern cosmology: the cos-

mological constant, the cosmic microwave background, ination, and

cosmology as a probe of physics at the Planck scale. The underlying

theme is that cosmology gives us a unique window on the “physics of

nothing,” or the quantum-mechanical properties of the vacuum. The

theory of ination postulates that vacuum energy, or something very

much like it, was the dominant force shaping the evolution of the very

early universe. Recent astrophysical observations indicate that vacuum

energy, or something very much like it, is also the dominant component

of the universe today. Therefore cosmology gives us a way to study

an important piece of particle physics inaccessible to accelerators. The

lectures are oriented toward graduate students with only a passing fa-

miliarity with general relativity and knowledge of basic quantum eld

theory.

1. Introduction

Cosmology is undergoing an explosive burst of activity, fueled both by

new, accurate astrophysical data and by innovative theoretical develop-

ments. Cosmological parameters such as the total density of the universe

and the rate of cosmological expansion are being precisely measured for

the rst time, and a consistent standard picture of the universe is be-

ginning to emerge. This is exciting, but why talk about astrophysics

at a school for particle physicists? The answer is that over the past

twenty years or so, it has become evident that the the story of the uni-

verse is really a story of fundamental physics. I will argue that not only

should particle physicists care about cosmology, but you should care a

lot. Recent developments in cosmology indicate that it will be possible

to use astrophysics to perform tests of fundamental theory inaccessible

to particle accelerators, namely the physics of the vacuum itself. This

has proven to be a surprise to cosmologists: the old picture of a uni-

verse lled only with matter and light have given way to a picture of

a universe whose history is largely written in terms of the quantum-

mechanical properties of empty space. It is currently believed that the

universe today is dominated by the energy of vacuum, about 70% by

weight. In addition, the idea of ination postulates that the universe at

the earliest times in its history was also dominated by vacuum energy,

which introduces the intriguing possibility that all structure in the uni-

verse, from superclusters to planets, had a quantum-mechanical origin

in the earliest moments of the universe. Furthermore, these ideas are

not idle theorizing, but are predictive and subject to meaningful exper-

imental test. Cosmological observations are providing several surprising

challenges to fundamental theory.

These lectures are organized as follows. Section 2 provides an intro-

duction to basic cosmology and a description of the surprising recent

discovery of the accelerating universe. Section 3 discusses the physics of

the cosmic microwave background (CMB), one of the most useful obser-

vational tools in modern cosmology. Section 4 discusses some unresolved

problems in standard Big-Bang cosmology, and introduces the idea of in-

ation as a solution to those problems. Section 5 discusses the intriguing

(and somewhat speculative) idea of using ination as a “microscope” to

illuminate physics at the very highest energy scales, where eects from

quantum gravity are likely to be important. These lectures are geared

toward graduate students who are familiar with special relativity and

quantum mechanics, and who have at least been introduced to general

relativity and quantum eld theory. There are many things I will not

talk about, such as dark matter and structure formation, which are in-

teresting but do not touch directly on the main theme of the “physics

of nothing.” I omit many details, but I provide references to texts and

review articles where possible.

2. Resurrecting Einstein’s greatest blunder.

2.1 Cosmology for beginners

All of modern cosmology stems essentially from an application of the

Copernican principle: we are not at the center of the universe. In fact,

today we take Copernicus’ idea one step further and assert the “cos-

mological principle”: nobody is at the center of the universe. The cos-

mos, viewed from any point, looks the same as when viewed from any

other point. This, like other symmetry principles more directly famil-

iar to particle physicists, turns out to be an immensely powerful idea.

In particular, it leads to the apparently inescapable conclusion that the

universe has a nite age. There was a beginning of time.

We wish to express the cosmological principle mathematically, as a

symmetry. To do this, and to understand the rest of these lectures,

we need to talk about metric tensors and General Relativity, at least

briey. A metric on a space is simply a generalization of Pythagoras’

theorem for the distance ds between two points separated by distances

dx = (dx, dy, dz),

ds 2

=|dx| 2

= dx 2 + dy 2 + dz 2 .

(1)

We can write this as a matrix equation,

ds 2

ij dx i dx j ,

(2)

i,j =1 , 3

where ij is just the unit matrix,

1 0 0

0 1 0

0 0 1

ij =

(3)

The matrix ij is referred to as the metric of the space, in this case a

three-dimensional Euclidean space. One can dene other, non-Euclidean

spaces by specifying a dierent metric. A familiar one is the four-

dimensional “Minkowski” space of special relativity, where the proper

distance between two points in spacetime is given by

ds 2

= dt 2 −dx 2 ,

(4)

corresponding to a metric tensor with indices µ, = 0, . . . , 3:

@ 1 0 0 0

µ =

0−1 0 0

0 0 −1 0

0 0

(5)

0 −1

In a Minkowski space, photons travel on null paths, or geodesics, ds 2 = 0,

and massive particles travel on timelike geodesics, ds 2 > 0. Note that

in both of the examples given above, the metric is time-independent,

describing a static space. In General Relativity, the metric becomes a

dynamic object, and can in general depend on time and space. The

fundamental equation of general relativity is the Einstein eld equation,

G µ = 8GT µ ,

(6)

where T µ is a stress energy tensor describing the distribution of mass

in space, G is Newton’s gravitational constant and the Einstein Tensor

G µ is a complicated function of the metric and its rst and second

derivatives. This should be familiar to anyone who has taken a course

in electromagnetism, since we can write Maxwell’s equations in matrix

form as

@ F µ

J µ ,

(7)

where F µ is the eld tensor and J µ is the current. Here we use the

standard convention that we sum over the repeated indices of four-

dimensional spacetime = 0, 3. Note the similarity between Eq. (6)

and Eq. (7). The similarity is more than formal: both have a charge on

the right hand side acting as a source for a eld on the left hand side.

In the case of Maxwell’s equations, the source is electric charge and the

eld is the electromagnetic eld. In the case of Einstein’s equations, the

source is mass/energy, and the eld is the shape of the spacetime, or the

metric. An additional feature of the Einstein eld equation is that it is

much more complicated than Maxwell’s equations: Eq. (6) represents

six independent nonlinear partial dierential equations of ten functions,

the components of the (symmetric) metric tensor g µ (t, x). (The other

four degrees of freedom are accounted for by invariance under transfor-

mations among the four coordinates.)

Clearly, nding a general solution to a set of equations as complex as

the Einstein eld equations is a hopeless task. Therefore, we do what

any good physicist does when faced with an impossible problem: we

introduce a symmetry to make the problem simpler. The three simplest

symmetries we can apply to the Einstein eld equations are: (1) vacuum,

(2) spherical symmetry, and (3) homogeneity and isotropy. Each of

these symmetries is useful (and should be familiar). The assumption of

vacuum is just the case where there’s no matter at all:

T µ = 0.

(8)

In this case, the Einstein eld equation reduces to a wave equation, and

the solution is gravitational radiation. If we assume that the matter dis-

tribution T µ has spherical symmetry, the solution to the Einstein eld

equations is the Schwarzschild solution describing a black hole. The third

case, homogeneity and isotropy, is the one we will concern ourselves with

in more detail here [1]. By homogeneity, we mean that the universe is

invariant under spatial translations, and by isotropy we mean that the

universe is invariant under rotations. (A universe that is isotropic every-

where is necessarily homogeneous, but a homogeneous universe need not

be isotropic: imagine a homogeneous space lled with a uniform electric

eld!) We will model the contents of the universe as a perfect uid with

density and pressure p, for which the stress-energy tensor is

T µ =

@ 0 0 0

0−p 0 0

0 0 −p 0

0 0

(9)

0 −p

While this is certainly a poor description of the contents of the universe

on small scales, such as the size of people or planets or even galaxies, it

is an excellent approximation if we average over extremely large scales

in the universe, for which the matter is known observationally to be very

smoothly distributed. If the matter in the universe is homogeneous and

isotropic, then the metric tensor must also obey the symmetry. The

most general line element consistent with homogeneity and isotropy is

ds 2 = dt 2 −a 2 (t)dx 2 ,

(10)

where the scale factor a(t) contains all the dynamics of the universe, and

the vector product dx 2 contains the geometry of the space, which can

be either Euclidian (dx 2 = dx 2 + dy 2 + dz 2 ) or positively or negatively

curved. The metric tensor for the Euclidean case is particularly simple,

@ 1

g µ =

0−a(t)

(11)

0 −a(t)

which can be compared to the Minkowski metric (5). In this Friedmann-

Robertson-Walker (FRW) space, spatial distances are multiplied by a

dynamical factor a(t) that describes the expansion (or contraction) of

the spacetime. With the general metric (10), the Einstein eld equations

take on a particularly simple form,

− k

a 2 ,

(12)

where k is a constant that describes the curvature of the space: k = 0

(at), or k =±1 (positive or negative curvature). This is known as the

Friedmann equation. In addition, we have a second-order equation

=− 4G

( + 3p) .

(13)

Note that the second derivative of the scale factor depends on the equa-

tion of state of the uid. The equation of state is frequently given by a

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