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AN INTRODUCTION TO COSMOLOGICAL INFLATION
ANDREW R. LIDDLE Astronomy Centre, University of Sussex, Brighton BN1 9QJ, U. K. and Astrophysics Group, The Blackett Laboratory, Imperial Col lege, London SW7 2BZ, U. K. (present address)
An introductory account is given of the inflationary cosmology, which postulates a period of accelerated expansion during the Universe's earliest stages. The historical motivation is briefly outlined, and the modelling of the inflationary epoch explained. The most important aspect of inflation is that it provides a possible model for the origin of structure in the Universe, and key results are reviewed, along with a discussion of the current observational situation and outlook.

arXiv:astro-ph/9901124 v1 11 Jan 1999

1

Overview

One of the central planks of modern cosmology is the idea of inflation. Originally introduced by Guth 1 in order to explain the initial conditions for the hot big bang model, it has subsequently been given a much more important role as the currently-favoured candidate for the origin of structure in the Universe, such as galaxies, galaxy clusters and cosmic microwave background anisotropies. This article seeks to give an introductory account of the inflationary cosmology, with the focus aimed towards inflation as a mo del for the origin of structure. It begins with a quick review of the big bang cosmology, and the problems with it which led to the introduction of inflation. The modelling of the inflationary epoch using scalar fields is described, and then results giving the form of perturbations produced by inflation are quoted. Finally, the current observational situation is briefly sketched. 2 Big bang problems and the idea of inflation

The standard hot big bang theory is an extremely successful one, passing some crucial observational tests of which I'd highlight five. § The expansion of the Universe. § The existence and spectrum of the cosmic microwave background radiation. § The abundances of light elements in the Universe (nucleosynthesis). 1


§ That the predicted age of the Universe is comparable to direct age measurements of ob jects within the Universe. § That given the irregularities seen in the microwave background by COBE, there exists a reasonable explanation for the development of structure in the Universe, through gravitational collapse. In combination, these are extremely compelling. However, the standard hot big bang theory is limited to those epochs where the Universe is cool enough that the underlying physical processes are well established and understood through terrestrial experiment. It does not attempt to address the state of the Universe at earlier, hotter, times. Furthermore, the hot big bang theory leaves a range of crucial questions unanswered, for it turns out that it can successfully proceed only if the initial conditions are very carefully chosen. The assumption of early Universe studies is that the mysteries of the conditions under which the big bang theory operates may be explained through the physics occurring in its distant, unexplored past. If so, accurate observations of the present state of the Universe may highlight the types of process occurring during these early stages, and perhaps even shed light on the nature of physical laws at energies which it would be inconceivable to explore by other means. 2.1 A hot big bang reminder

To get us started, I'll give a quick review of the big bang cosmology. More detailed accounts can be found in any of a number of cosmological textbooks. One of my aims in this section is to set down the notation for the rest of the article. 2.2 Equations of motion

The hot big bang theory is based on the cosmological principle, which states that the Universe should look the same to all observers. That tells us that the Universe must be homogeneous and isotropic, which in turn tells us which metric must be used to describe it. It is the RobertsoníWalker metric ds2 = -dt2 + a2 (t) dr2 +r 1 - k r2
2

d2 + sin2 d2

.

(1)

Here t is the time variable, and ríí are (polar) coordinates. The constant k measures the spatial curvature, with k negative, zero and positive corresponding to open, flat and closed Universes respectively. If k is zero or negative, then the range of r is from zero to infinity and the Universe is infinite, while 2


if k is positive then r goes from zero to 1/ k . Usually the coordinates are rescaled to make k equal to -1, 0 or +1. The quantity a(t) is the scale-factor of the Universe, which measures its physical size. The form of a(t) depends on the properties of the material within the Universe, as we'll see. If no external forces are acting, then a particle at rest at a given set of coordinates (r, , ) will remain there. Such coordinates are said to be comoving with the expansion. One swaps between physical (ie actual) and comoving distances via physical distance = a(t) ½ comoving distance . (2)

The expansion of the Universe is governed by the properties of material within it. This can be specified a by the energy density (t) and the pressure p(t). These are often related by an equation of state, which gives p as a function of ; the classic examples are p= Radiation , (3) 3 p=0 Non-relativistic matter . (4) In general though there need not be a simple equation of state; for example there may be more than one type of material, such as a combination of radiation and non-relativistic matter, and certain types of material, such as a scalar field (a type of material we'll encounter later which is crucial for modelling inflation), cannot be described by an equation of state at all. The crucial equations describing the expansion of the Universe are H2 = k 8 - 2 3 m2 l a P + 3H ( + p) = 0 Friedmann equation Fluid equation (5) (6)

where overdots are time derivatives and H = a/a is the Hubble parameter. The terms in the fluid equation contributing to have a simple interpretation; the term 3H is the reduction in density due to the increase in volume, and the term 3H p is the reduction in energy caused by the thermodynamic work done by the pressure when this expansion occurs. These can also be combined to form a new equation a è 4 ( + 3p) =- a 3 m2 l P
a

Acceleration equation

(7)

I follow standard cosmological practice of setting the fundamental constants c and ï equal h to one. This makes the energy density and mass density interchangeable (since the former is c2 times the latter). I shall also normally use the Planck mass mPl rather than the gravitational constant G; with the convention just mentioned they are related by G m-l2 . P

3


in which k does not appear explicitly. 2.3 Standard cosmological solutions

When k = 0 the Friedmann and fluid equations can readily be solved for the equations of state given earlier, leading to the classic cosmological solutions Matter Domination Radiation Domination p=0: a
-3 -4

p = / 3 : a

a(t) t

2/3 1/2

(8) (9)

a(t) t

In both cases the density falls as t-2 . When k = 0 we have the freedom to rescale a and it is normally chosen to be unity at the present, making physical and comoving scales coincide. The proportionality constants are then fixed by setting the density to be 0 at time t0 , where here and throughout the subscript zero indicates present value. A more intriguing solution appears for the case of a so-called cosmological constant, which corresponds to an equation of state p = -. The fluid equation then gives = 0 and hence = 0 , leading to a(t) exp (H t) . (10)

More complicated solutions can also be found for mixtures of components. For example, if there is both matter and radiation the Friedmann equation can be solved be using conformal time = dt/a, while if there is matter and a non-zero curvature term the solution can be given either in parametric form using normal time t, or in closed form with conformal time. 2.4 Critical density and the density parameter

The spatial geometry is flat if k = 0. For a given H , this requires that the density equals the critical density c (t) = 3 m2 l H 2 P . 8 (11)

Densities are often measured as fractions of c : (t) . c (12)

The quantity is known as the density parameter, and can be applied to individual types of material as well as the total density. 4


The present value of the Hubble parameter is still not that well known, and is normally parametrized as H0 = 100h km s-1 Mpc
-1

=

h Mpc 3000

-1

,

(13)

where h is normally assumed to lie in the range 0.5 h 0.8. The present critical density is c (t0 ) = 1.88 h2 ½ 10 2.5
-29

g cm

-3

= 2.77 h

-1

½ 10

11

M /(h

-1

Mpc)3 .

(14)

Characteristic scales and horizons

The big bang Universe has two characteristic scales § The Hubble time (or length) H § The curvature scale a|k |-
1/2 -1

.

. a(t), and having a scales; to The ratio uation we (15)

A crucial property of the big bang Universe is that it possesses horizons; even light can only have travelled a finite distance since the start of the Universe t , given by t dt . (16) dH (t) = a(t) a(t) t For example, matter domination gives dH (t) = 3t = 2H -1 . In a big bang Universe, dH (t0 ) is a good approximation to the distance to the surface of last scattering (the origin of the observed microwave background, at a time known as `decoupling'), since t0 td e c . 2.6 Redshift and temperature

The first of these gives the characteristic timescale of evolution of the second gives the distance up to which space can be taken as flat (Euclidean) geometry. As written above they are both physical obtain the corresponding comoving scale one should divide by a(t). of these scales actually gives a measure of ; from the Friedmann eq find H -1 . | - 1| = a|k |-1/2

The redshift measures the expansion of the Universe via the stretching of light 1+z = a(t0 ) a(t 5
emission

)

.

(17)


Redshift can be used to describe both time and distance. As a time, it simply refers to the time at which light would have to be emitted to have a present redshift z . As a distance, it refers to the present distance to an ob ject from which light is received with a redshift z . Note that this distance is not necessarily the time multiplied by the speed of light, since the Universe is expanding as the light travels across it. As the Universe expands, it cools according to the law 1 T . (18) a In its earliest stages the Universe may have been arbitrarily hot and dense. 2.7 The history of the Universe is dominated by non-relativistic matter, but because quickly with the expansion, this implies that at earwas radiation dominated. During the radiation era re related by t 1 s ec 10
10

Presently the Universe radiation reduces more lier times the Universe temperature and time a

K

2

T

.

(19)

The highest energies accessible to terrestrial experiment, generated in particle accelerators, correspond to a temperature of about 1015 K, which was attained when the Universe was about 10-10 sec old. Before that, we have no direct evidence of the applicable physical laws and must use extrapolation based on current particle physics model building. After that time there is a fairly clear picture of how the Universe evolved to reach the present, with the key events being as follows: § 10
-4

seconds: Quarks condense to form protons and neutrons.

§ 1 second: The Universe has cooled sufficiently that light nuclei are able to form, via a process known as nucleosynthesis. § 104 years: The radiation density drops to the level of the matter density, the epoch being known as matteríradiation equality. Subsequently the Universe is matter dominated. § 105 years: Decoupling of radiation from matter leads to the formation of the microwave background. This is more or less coincident with recombination, when the up-to-now free electrons combine with the nuclei to form atoms. § 10
10

years: The present. 6


3

Problems with the Big Bang

In this section I shall quickly review the original motivation for the inflationary cosmology. These problems were largely ones of initial conditions. While historically these problems were very important, they are now somewhat marginalized as focus is instead concentrated on inflation as a theory for the origin of cosmic structure. 3.1 The flatness problem

Taking advantage of the definition of the density parameter, and ignoring a possible cosmological constant contribution, the Friedmann equation can be written in the form |k | | - 1| = 2 2 . (20) aH During standard big bang evolution, a2 H 2 is decreasing, and so moves away from one, for example Matter domination: | - 1| t
2/3

(21) (22)

Radiation domination: | - 1| t

where the solutions apply provided is close to one. So = 1 is an unstable critical point. Since we know that today is certainly within an order of magnitude of one, it must have been much closer in the past. Inserting the appropriate behaviours for the matter and radiation eras (or if you like just assuming radiation domination all the way to the present) gives nucleosynthesis (t 1 sec) : electro-weak scale (t 10-11 sec) : | - 1| < O(10 | - 1| < O(10
-16 -27

) )

(23) (24)

That is, hardly any choices of the initial density lead to a Universe like our own. Typically, the Universe will either swiftly recollapse, or will rapidly expand and cool below 3K within its first second of existence. 3.2 The horizon problem

Microwave photons emitted from opposite sides of the sky appear to be in thermal equilibrium at almost the same temperature. The most natural explanation for this is that the Universe has indeed reached a state of thermal equilibrium, through interactions between the different regions. But unfortunately in the big bang theory this is not possible. There was no time for 7


those regions to interact before the photons were emitted, because of the finite horizon size, tdec t0 dt dt . (25) a(t) t tdec a(t) This says that the distance light could travel before the microwave background was released is much smaller than the present horizon distance. In fact, any regions separated by more than about 2 degrees would be causally separated at decoupling in the hot big bang theory. In the big bang theory there is therefore no explanation of why the Universe appears so homogeneous. In more recent years this problem has been brought into sharper focus through the improving understanding of irregularities in the Universe, as will be discussed later in this article. The same argument that prevents the smoothing of the Universe also prevents the creation of irregularities. For example, as we will see the COBE satellite observes irregularities on all accessible angular scales, from a few degrees upwards. In the simplest cosmological mo dels, where these irregularities are intrinsic to the last scattering surface, the perturbations are on too large a scale to have been created between the big bang and the time of decoupling, because the horizon size at decoupling subtends only a degree or so. Hence these perturbations must have been part of the initial conditions.b If this is the case, then the hot big bang theory does not allow a predictive theory for the origin of structure. While there is no reason why it is required to give a predictive theory, this would be a ma jor setback and disappointment for the study of structure formation in the Universe. 3.3 The monopole problem (and other relics)

Modern particle theories predict a variety of `unwanted relics', which would violate observations. These include § Magnetic monopoles. § Domain walls. § Supersymmetric particles such as the gravitino. § `Moduli' fields associated with superstrings.
b Note though that it is not yet known for definite that there are large-angle p erturbations intrinsic to the last scattering surface. For example, in a topological defect model such as cosmic strings, such perturbations could be generated as the microwave photons propagate towards us.

8


Typically, the problem is that these are expected to be created very early in the Universe's history, during the radiation era. But because they are diluted by the expansion more slowly than radiation (eg as a-3 instead of a-4 ) it is very easy for them to become the dominant material in the Universe, in contradiction to observations. One has to dispose of them without harming the conventional matter in the Universe. 4 The Idea of Inflation

Seen with many years of hindsight, the idea of inflation is actually rather obvious. Take for example the Friedmann equation as used to analyze the flatness problem |k | | - 1| = 2 2 . (26) aH The problem with the hot big bang model is that aH always decreases, and so is repelled away from one. In order to solve the problem, we will clearly need to reverse this state of affairs. Accordingly, define inflation to be any epoch where a > 0, an è accelerated expansion. We can rewrite this in several different ways INFLATION a>0 è d(H -1 /a) <0 dt p<- 3 (27) (28) (29)

The middle definition is the one which I prefer to use, because it has the most direct geometrical interpretation. It says that the Hubble length, as measured in comoving coordinates, decreases during inflation. At any other time, the comoving Hubble length increases. This is the key property of inflation; although typically the expansion of the Universe is very rapid, the crucial characteristic scale of the Universe is actually becoming smaller, when measured relative to that expansion. As we will see, quite a wide range of behaviours satisfy the inflationary condition. The most classic one is one we have already seen; when the equation of state is p = -, the solution is a(t) exp (H t) . (30)

Since the successes of the hot big bang theory rely on the Universe having a conventional (non-inflationary) evolution, we cannot permit this inflationary 9


period to go on forever -- it must come to an end early enough that the big bang successes are not threatened. Normally, then, inflation is viewed as a phenomenon of the very early Universe, which comes to an end and is followed by the conventional behaviour. Inflation does not replace the hot big bang theory; it is a bolt-on accessory attached at early times to improve the performance of the theory. 4.1 The flatness problem

Inflation solves the flatness problem more or less by definition (so that at least any classical, as opposed to quantum, solution of the problem will fall under the umbrella of the inflationary definition). From the middle condition, inflation is precisely the condition that is forced towards one rather than away from it. As we shall see, this typically happens very rapidly. A short period of such behaviour won't do us any good, as the subsequent non-inflationary behaviour (in particular the standard big bang evolution from nucleosynthesis onwards) will take us away from flatness again, but all will be well provided we have enough inflation that is moved extremely close to one during the inflationary epoch. If it is close enough, then it will stay very close to one right to the present, despite being repelled from one for all the post-inflationary period. Obtaining sufficient inflation to perform this task is actually fairly easy. A schematic illustration of this behaviour is shown in Figure 1. In the above discussion, I have ignored a possible cosmological constant contribution, but if present it modifies the Friedmann equation to | + - 1| = |k | , a2 H 2 (31)

and so it is + which is forced to one. In general, it is spatial flatness (k 0) that we are driven towards, not a critical matter density. 4.2 Relic abundances

The rapid expansion of the inflationary stage rapidly dilutes the unwanted relic particles, because the energy density during inflation falls off more slowly (as a-2 or slower) than the relic particle density. Very quickly their density becomes negligible. This resolution can only work if, after inflation, the energy density of the Universe can be turned into conventional matter without recreating the unwanted relics. This can be achieved by ensuring that during the conversion, known as reheating, the temperature never gets hot enough again to allow their thermal recreation. Then reheating can generate solely the things which 10


Log

Not to scale !!

0

Time
Start of inflation End of inflation Present day distant future

Figure 1: A possible evolution of . There may or may not be evolution before inflation, shown by the dotted line. During inflation is forced dramatically towards one, and remains there right up to the present. Only in the extremely distant future will it begin to evolve away from one again.

we want. Such successful reheating allows us to get back into the hot big bang Universe, recovering all its later successes such as nucleosynthesis and the microwave background. 4.3 The horizon problem and homogeneity

The inflationary expansion also solves the horizon problem. The basic strategy is to ensure that tdec t0 dt dt , (32) a(t) t tdec a(t) so that light can travel much further before decoupling than it can afterwards. This cannot be done with standard evolution, but can be achieved by inflation. An alternative way to view this is to remember that inflation corresponds to a decreasing comoving Hubble length. The Hubble length is ordinarily a good measure of how far things can travel in the Universe; what this is telling us is that the region of the Universe we can see after (even long after) inflation is much smaller than the region which would have been visible before inflation started. Hence causal physics was perfectly capable of pro ducing a large smooth thermalized region, encompassing a volume greatly in excess 11


Hubble length

COMOVING

start

now end

smooth patch

Figure 2: Solving the horizon problem. Initially the Hubble length is large, and a smooth patch forms by causal interactions. Inflation then shrinks the Hubble length, and even the subsequent expansion again after inflation leaves the observable Universe within the smoothed patch.

of our presently observable Universe. In Figure 2, the outer circle indicates the initial Hubble length, encompassing the shaded smooth patch. Inflation shrinks this dramatically inwards towards the dot indicating our position, and then after inflation it increases while staying within the initial smooth patch.c
c Although this is a standard description, it isn't totally accurate. A more accurate argument is as follows.2 At the beginning of inflation particles are distributed in a set of modes. This may be a thermal distribution or something else; whatever, since the energy density is finite there will be a shortest wavelength occupied mode, e.g. for a thermal distribution max 1/T . Expressed in physical coordinates, once inflation has stretched all modes including this one to be much larger than the Hubble length, the Universe becomes homogeneous. In comoving coordinates, the equivalent picture is that the Hubble length shrinks in until it's much smaller than the shortest wavelength, and the Universe, as before, appears homogeneous.

12


Equally, causal processes would be capable of generating irregularities in the Universe on scales greatly exceeding our presently observable Universe, provided they happened at an early enough time that those scales were within causal contact. This will be explored in detail later. 5 Mo delling the Inflationary Expansion

We have seen that a period of accelerated expansion -- inflation -- is sufficient to resolve a range of cosmological problems. But we need a plausible scenario for driving such an expansion if we are to be able to make proper calculations. This is provided by cosmological scalar fields. 5.1 Scalar fields and their potentials

In particle physics, a scalar field is used to represent spin zero particles. It transforms as a scalar (that is, it is unchanged) under coordinate transformations. In a homogeneous Universe, the scalar field is a function of time alone. In particle theories, scalar fields are a crucial ingredient for spontaneous symmetry breaking. The most famous example is the Higgs field which breaks the electro-weak symmetry, whose existence is hoped to be verified at the Large Hadron Collider at CERN when it commences experiments next millennium. Scalar fields are also expected to be associated with the breaking of other symmetries, such as those of Grand Unified Theories, supersymmetry etc. § Any specific particle theory (eg GUTS, superstrings) contains scalar fields. § No fundamental scalar field has yet been observed. § In condensed matter systems (such as superconductors, superfluid helium etc) scalar fields are widely observed, associated with any phase transition. People working in that sub ject normally refer to the scalar fields as `order parameters'. The traditional starting point for particle physics models is the action, which is an integral of the Lagrange density over space and time and from which the equations of motion can be obtained. As an intermediate step, one might write down the energyímomentum tensor, which sits on the righthand side of Einstein's equations. Rather than begin there, I will take as my starting point expressions for the effective energy density and pressure of a homogeneous scalar field, which I'll call . These are obtained by comparison 13


of the energyímomentum tensor of the scalar field with that of a perfect fluid, and are


= =

p

1 2 1 2

2 + V () 2 - V () .

(33) (34)

One can think of the first term in each as a kinetic energy, and the second as a potential energy. The potential energy V () can be thought of as a form of `configurational' or `binding' energy; it measures how much internal energy is associated with a particular field value. Normally, like all systems, scalar fields try to minimize this energy; however, a crucial ingredient which allows inflation is that scalar fields are not always very efficient at reaching this minimum energy state. Note in passing that a scalar field cannot in general be described by an equation of state; there is no unique value of p that can be associated with a given as the energy density can be divided between potential and kinetic energy in different ways. In a given theory, there would be a specific form for the potential V (), at least up to some parameters which one could hope to measure (such as the effective mass and interaction strength of the scalar field). However, we are not presently in a position where there is a well established fundamental theory that one can use, so, in the absence of such a theory, inflation workers tend to regard V () as a function to be chosen arbitrarily, with different choices corresponding to different models of inflation (of which there are many). Some example potentials are V () = 2 - M
1 2 22

Higgs potential Massive scalar field Self-interacting scalar field

(35) (36) (37)

V () = m V () = 4

22

The strength of this approach is that it seems possible to capture many of the crucial properties of inflation by looking at some simple potentials; one is looking for results which will still hold when more `realistic' potentials are chosen. Figure 3 shows such a generic potential, with the scalar field displaced from the minimum and trying to reach it. 5.2 Equations of motion and solutions

The equations for an expanding Universe containing a homogeneous scalar field are easily obtained by substituting Eqs. (33) and (34) into the Friedmann and 14


V()


Figure 3: A generic inflationary potential.

fluid equations, giving H
2

=

8 3 m2 P

l

1 V () + 2 , 2

(38) (39)

è + 3H =

-V () ,

where prime indicates d/d. Here I have know that by definition it will quickly b This is done for simplicity only; there is Since a > 0 p < - è 3

ignored the curvature term k , since we ecome negligible once inflation starts. no obstacle to including that term. 2 < V () (40)

we will have inflation whenever the potential energy dominates. This should be possible provided the potential is flat enough, as the scalar field would then be expected to roll slowly. The potential should also have a minimum in which inflation can end. The standard strategy for solving these equations is the slow-roll approximation (SRA); this assumes that a term can be neglected in each of the equations of motion to leave the simpler set H
2

8 V 3 m2 l P -V 15

(41) (42)

3H


If we define slow-roll parameters () = m2 l P 16 V V

3 2

;

() =

m2 l V P , 8 V

(43)

where the first measures the slope of the potential and the second the curvature, then necessary conditions for the slow-roll approximation to hold are d 1 ; | | 1. (44)

Unfortunately, although these are necessary conditions for the slow-roll approximation to hold, they are not sufficient, since even if the potential is very flat it may be that the scalar field has a large velocity. A more elaborate version of the SRA exists, based on the HamiltoníJacobi formulation of inflation,4 which is sufficient as well as necessary.5 Note also that the SRA reduces the order of the system of equations by one, and so its general solution contains one less initial condition. It works only because one can prove 4,5 that the solution to the full equations possesses an attractor property, eliminating the dependence on the extra parameter. 5.3 The relation between inflation and slow-rol l

As it happens, the applicability of the slow-roll condition is closely connected to the condition for inflation to take place, and in many contexts the conditions can be regarded as equivalent. Let's quickly see why. The inflationary condition a > 0 is satisfied for a much wider range of beè haviours than just (quasi-)exponential expansion. A classic example is powerlaw inflation a tp for p > 1, which is an exact solution for an exponential potential 16 . (45) V () = V0 exp - p mP l We can manipulate the condition for inflation as a è = H + H2 > 0 a H - 2 <1 H 2 m2 l V P <1 16 V


d



Note that

is positive by definition, whilst can have either sign.

16


where the last manipulation uses the slow-roll approximation. The final condition is just the slow-roll condition < 1, and hence Slow-roll = Inflation Inflation will occur when the slow-roll conditions are satisfied (sub ject to some caveats on whether the `attractor' behaviour has been attained.5 ) However, the converse is not strictly true, since we had to use the SRA in the derivation. However, in practice <1 Inflation = Prolonged inflation = < 1 The last condition arises because unless the curvature of the potential is small, the potential will not be flat for a wide enough range of . 5.4 The amount of inflation


The amount of inflation is normally specified by the logarithm of the amount of expansion, the number of e-foldings N , given by N ln a(tend ) a(tini