Astrophysics & Cosmology - J. Garcia Bellido.pdf - ▲ ASTRONOMIA (KOSMOLOGIA) - adad.molo

ASTROPHYSICS AND COSMOLOGY

J. Garcıa-Bellido

Theoretical Physics Group, Blackett Laboratory, Imperial College of Science,

Technology and Medicine, Prince Consort Road, London SW7 2BZ, U.K.

Abstract

These notes are intended as an introductory course for experimental particle

physicists interested in the recent developments in astrophysics and cosmology.

I will describe the standard Big Bang theory of the evolution of the universe,

with its successes and shortcomings, which will lead to inationary cosmology

as the paradigm for the origin of the global structure of the universe as well as

the origin of the spectrum of density perturbations responsible for structure in

our local patch. I will present a review of the very rich phenomenology that we

have in cosmology today, as well as evidence for the observational revolution

that this eld is going through, which will provide us, in the next few years,

with an accurate determination of the parameters of our standard cosmological

model.

1. GENERAL INTRODUCTION

Cosmology (from the Greek: kosmos , universe, world, order, and logos , word, theory) is probably the

most ancient body of knowledge, dating from as far back as the predictions of seasons by early civiliza-

tions. Yet, until recently, we could only answer to some of its more basic questions with an order of mag-

nitude estimate. This poor state of affairs has dramatically changed in the last few years, thanks to (what

else?) raw data, coming from precise measurements of a wide range of cosmological parameters. Further-

more, we are entering a precision era in cosmology, and soon most of our observables will be measured

with a few percent accuracy. We are truly living in the Golden Age of Cosmology. It is a very exciting

time and I will try to communicate this enthusiasm to you.

Important results are coming out almost every month from a large set of experiments, which pro-

vide crucial information about the universe origin and evolution; so rapidly that these notes will proba-

bly be outdated before they are in print as a CERN report. In fact, some of the results I mentioned dur-

ing the Summer School have already been improved, specially in the area of the microwave background

anisotropies. Nevertheless, most of the new data can be interpreted within a coherent framework known

as the standard cosmological model, based on the Big Bang theory of the universe and the inationary

paradigm, which is with us for two decades. I will try to make such a theoretical model accesible to young

experimental particle physicists with little or no previous knowledge about general relativity and curved

space-time, but with some knowledge of quantum eld theory and the standard model of particle physics.

2. INTRODUCTION TO BIG BANG COSMOLOGY

Our present understanding of the universe is based upon the successful hot Big Bang theory, which ex-

plains its evolution from the rst fraction of a second to our present age, around 13 billion years later.

This theory rests upon four strong pillars, a theoretical framework based on general relativity, as put for-

ward by Albert Einstein [1] and Alexander A. Friedmann [2] in the 1920s, and three robust observational

facts: First, the expansion of the universe, discovered by Edwin P. Hubble [3] in the 1930s, as a reces-

sion of galaxies at a speed proportional to their distance from us. Second, the relative abundance of light

elements, explained by George Gamow [4] in the 1940s, mainly that of helium, deuterium and lithium,

which were cooked from the nuclear reactions that took place at around a second to a few minutes after

the Big Bang, when the universe was a few times hotter than the core of the sun. Third, the cosmic mi-

crowave background (CMB), the afterglow of the Big Bang, discovered in 1965 by Arno A. Penzias and

109

Robert W. Wilson [5] as a very isotropic blackbody radiation at a temperature of about 3 degrees Kelvin,

emitted when the universe was cold enough to form neutral atoms, and photons decoupled from matter,

approximately 500,000 years after the Big Bang. Today, these observations are conrmed to within a few

percent accuracy, and have helped establish the hot Big Bang as the preferred model of the universe.

2.1 Friedmann–Robertson–Walker universes

Where are we in the universe? During our lectures, of course, we were in Casta Papiernicka, in ‘the heart

of Europe’, on planet Earth, rotating (8 light-minutes away) around the Sun, an ordinary star 8.5 kpc 1

from the center of our galaxy, the Milky Way, which is part of the local group, within the Virgo cluster of

galaxies (of size a few Mpc), itself part of a supercluster (of size

Mpc), within the visible universe

Mpc), most probably a tiny homogeneous patch of the innite global structure of space-

time, much beyond our observable universe.

Cosmology studies the universe as we see it. Due to our inherent inability to experiment with it,

its origin and evolution has always been prone to wild speculation. However, cosmology was born as a

science with the advent of general relativity and the realization that the geometry of space-time, and thus

the general attraction of matter, is determined by the energy content of the universe [6],

(1)

These non-linear equations are simply too difcult to solve without some insight coming from the sym-

metries of the problem at hand: the universe itself. At the time (1917-1922) the known (observed) uni-

verse extended a few hundreds of parsecs away, to the galaxies in the local group, Andromeda and the

Large and Small Magellanic Clouds: The universe looked extremely anisotropic. Nevertheless, both Ein-

stein and Friedmann speculated that the most ‘reasonable’ symmetry for the universe at large should be

homogeneity at all points, and thus isotropy . It was not until the detection, a few decades later, of the

microwave background by Penzias and Wilson that this important assumption was nally put onto rm

experimental ground. So, what is the most general metric satisfying homogeneity and isotropy at large

scales? The Friedmann-Robertson-Walker (FRW) metric, written here in terms of the invariant geodesic

distance , -/. ,10

in four dimensions, 2 3 3 314

, see Ref. [6], 2

(2)

characterized by just two quantities, a scale factor

, which determines the physical size of the universe,

and a constant

, which characterizes the spatial curvature of the universe,

(3)

Spatially open, at and closed universes have different geometries. Light geodesics on these universes

behave differently, and thus could in principle be distinguished observationally, as we shall discuss later.

Apart from the three-dimensional spatial curvature, we can also compute a four-dimensional space-time

curvature,

(4)

Depending on the dynamics (and thus on the matter/energy content) of the universe, we will have different

possible outcomes of its evolution. The universe may expand for ever, recollapse in the future or approach

an asymptotic state in between.

1 One parallax second (1 pc), parsec for short, corresponds to a distance of about 3.26 light-years or ‘+acbed fhg cm.

110

(

2 I am using iBjkb everywhere, unless specied.

2.1.1 The expansion of the universe

In 1929, Edwin P. Hubble observed a redshift in the spectra of distant galaxies, which indicated that they

were receding from us at a velocity proportional to their distance to us [3]. This was correctly interpreted

as mainly due to the expansion of the universe, that is, to the fact that the scale factor today is larger

than when the photons were emitted by the observed galaxies. For simplicity, consider the metric of a

spatially at universe, , - . , 6 . 7 . 9 6 :S,Ql0 . (the generalization of the following argument to curved

gives physical size to the spatial coordinates l

(5)

is the present value of the scale factor. Since the universe today is larger than in the past, the

observed wavelengths will be shifted towards the red, or redshifted , by an amount characterized by

, the

redshift parameter.

In the context of a FRW metric, the universe expansion is characterized by a quantity known as the

Hubble rate of expansion, v 9 6 : _

, whose value today is denoted by v t

(6)

, we can safely keep only the linear term, and thus the recession

velocity becomes proportional to the distance from us, €‚ u ƒ v t , y , the proportionality constant

u~}

. This expression constitutes the so-called Hubble law, and is spectacularly

conrmed by a huge range of data, up to distances of hundreds of megaparsecs. In fact, only recently

measurements from very bright and distant supernovae, at

, were obtained, and are beginning to

probe the second-order term, proportional to the deceleration parameter

, see Eq. (22). I will come

back to these measurements in Section 3.

One may be puzzled as to why do we see such a stretching of space-time. Indeed, if all spatial

distances are scaled with a universal scale factor, our local measuring units (our rulers) should also be

stretched, and therefore we should not see the difference when comparing the two distances (e.g. the two

wavelengths) at different times. The reason we see the difference is because we live in a gravitationally

bound system, decoupled from the expansion of the universe: local spatial units in these systems are not

stretched by the expansion. 4 The wavelengths of photons are stretched along their geodesic path from

one galaxy to another. In this consistent world picture, galaxies are like point particles, moving as a uid

in an expanding universe.

2.1.2 The matter and energy content of the universe

So far I have only discussed the geometrical aspects of space-time. Let us now consider the matter and

energy content of such a universe. The most general matter uid consistent with the assumption of ho-

mogeneity and isotropy is a perfect uid, one in which an observer comoving with the uid would see the

universe around it as isotropic. The energy momentum tensor associated with such a uid can be written

as [6]

(7)

3 The subscript Œ refers to Luminosity, which characterizes the amount of light emitted by an object. See Eq. (61).

111

space is straightforward). The scale factor

, and the

expansion is nothing but a change of scale (of spatial units) with time. Except for peculiar velocities , i.e.

motion due to the local attraction of matter, galaxies do not move in coordinate space, it is the space-time

fabric which is stretching between galaxies. Due to this continuous stretching, the observed wavelength

of photons coming from distant objects is greater than when they were emitted by a factor precisely equal

to the ratio of scale factors,

where

. As I shall deduce later,

it is possible to compute the relation between the physical distance , y and the present rate of expansion,

in terms of the redshift parameter, 3

At small distances from us, i.e. at

being the Hubble rate, v t

4 The local space-time of a gravitationally bound system is described by the Schwarzschild metric, which is static [6].

where

and

are the pressure and energy density of the uid at a given time in the expansion, and

Let us now write the equations of motion of such a uid in an expanding universe. According to

general relativity, these equations can be deduced from the Einstein equations (1), where we substitute

the FRW metric (2) and the perfect uid tensor (7). The 2 Ž

9 6;:

component of the Einstein equations

constitutes the so-called Friedmann equation

(8)

as a different component from matter. In fact, it can

be associated with the vacuum energy of quantum eld theory, although we still do not understand why

should it have such a small value (120 orders of magnitude below that predicted by quantum theory), if it

is non-zero. This constitutes today one of the most fundamental problems of physics, let alone cosmology.

The conservation of energy (

), a direct consequence of the general covariance of the

theory (

), can be written in terms of the FRW metric and the perfect uid tensor (7) as

(9)

where the energy density and pressure can be split into its matter and radiation components,

. Together, the Friedmann

and the energy-conservation equation give the evolution equation for the scale factor,

, with corresponding equations of state,

(10)

in units of

100 km s š‚› Mpc š & › , in terms of which one can estimate the order of magnitude for the present size and

I will now make a few useful denitions. We can write the Hubble parameter today v t

(11)

(12)

4B œ

(13)

The parameter

has been measured to be in the range

for decades, and only in the last few

years has it been found to lie within 10% of

. I will discuss those recent measurements in the

next Section.

One can also dene a critical density

, that which in the absence of a cosmological constant would

correspond to a at universe,

(14)

š .p ® # ¯

(15)

corresponds to approximately 4

protons per cubic meter, certainly a very dilute uid! In terms of the critical density it is possible to dene

the ratios ‡B· ˆŠ · µ w Š” ‹

g is a solar mass unit. The critical density

, for matter, radiation, cosmological constant and even curvature, today,

(16)

(17)

112

is the comoving four-velocity, satisfying ‹

where I have treated the cosmological constant

age of the universe,

where °˜ – *¥£ £ EpE

We can evaluate today the radiation component ‡ “

, corresponding to relativistic particles, from the

, which

gives ‡ „ ” ‚ » * ™ š”À œ š . . Three massless neutrinos contribute an even smaller amount. Therefore,

we can safely neglect the contribution of relativistic particles to the total density of the universe today,

which is dominated either by non-relativistic particles (baryons, dark matter or massive neutrinos) or by

a cosmological constant, and write the rate of expansion v“. in terms of its value today,

(18)

An interesting consequence of these redenitions is that I can now write the Friedmann equation today,

, as a cosmic sum rule ,

(19)

today. That is, in the context of a FRW universe, the total fraction of matter

density, cosmological constant and spatial curvature today must add up to one. For instance, if we measure

one of the three components, say the spatial curvature, we can deduce the sum of the other two. Making

use of the cosmic sum rule today, we can write the matter and cosmological constant as a function of the

scale factor (

)

7 tK

(20)

(21)

É ¨

, all matter-dominated FRW universes can be de-

scribed by Einstein-de Sitter (EdS) models ( ‡ ‚ 3 ‡ •

). 5

On the other hand, the vacuum energy

will always dominate in the future.

Another relationship which becomes very useful is that of the cosmological deceleration parameter

today,

, in terms of the matter and cosmological constant components of the universe, see Eq. (10),

(22)

and requires a

precise cancellation: ‡ ‘Ì ‡†• . It represents spatial sections that are expanding at a xed rate, its scale

factor growing by the same amount in equally-spaced time intervals. Accelerated expansion corresponds

and comes about whenever ‡ ‘ ' ‡Q• : spatial sections expand at an increasing rate, their scale

factor growing at a greater speed with each time interval. Decelerated expansion corresponds to

and occurs whenever ‡ ‘ ˝ ‡Q• : spatial sections expand at a decreasing rate, their scale factor growing

at a smaller speed with each time interval.

2.1.3 Mechanical analogy

It is enlightening to work with a mechanical analogy of the Friedmann equation. Let us rewrite Eq. (8) as

(23)

is the equivalent of mass for the whole volume of the universe. Equation (23) can

be understood as the energy conservation law Ò $ ’ ÔÓ

for a test particle of unit mass in the central

potential

(24)

5 Note that in the limit ÕÖ 8d the radiation component starts dominating, see Eq. (18), but we still recover the EdS model.

113

density of microwave background photons,

where we have neglected ‡ “

This implies that for sufciently early times,

which is independent of the spatial curvature. Uniform expansion corresponds to

where °

Astrophysics & Cosmology - J. Garcia Bellido.pdf

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika: