Download Chapter 6 The inflationary universe In this chapter, we encounter a

Chapter 6
The inflationary universe
In this chapter, we encounter a new version of the big bang model, the theory of cosmic inflation. We see how
inflation addresses three conundrums of the big bang model, the horizon problem, the flatness problem and the
problem of galaxy formation. The observational status of inflation is outlined and criticisms of the model
discussed.
We saw in the last chapter that despite its great successes, one shortcoming of the big bang
model is the problem of ‘fine tuning’; many parameters are not specified by the model, and
one must assume the universe was ‘born’ with certain characteristics such as the Hubble
constant, the initial density of matter and the baryon-to-photon ratio. As theorists analysed
the model further, additional problems emerged, notably the horizon and flatness problems.
The flatness problem
A study of the Friedmann-Lemaitre models led the Princeton physicist Bob Dicke to a strange
conundrum. As we saw earlier, relativity predicts that the curvature of the universe is
determined by the density of matter within it; in an expanding universe, the density of matter
decreases over time and hence the density parameter Ω also evolves (recall that Ω is the ratio
of the actual density of matter to the critical value required to close the universe). Analysing
the behaviour of this parameter over time, Dicke made a startling discovery; if the density of
matter in the first fractions of a second deviated from the critical value by even a minute
amount, this deviation would accelerate very rapidly with time, resulting in either a runaway
closed or a runaway open universe 1(see figure 8). As observation suggests that we live in
neither of these, Dicke’s analysis implied that the density of matter in the infant universe
must have been extremely close to the critical value (and the geometry of the very early
1
universe must have been the very special case of flat geometry). But there are two problems
with this conclusion. It seems an extraordinary coincidence that the infant universe was so
delicately tuned, i.e. so finely balanced between the energy of gravity and the energy of
expansion. And how to reconcile this prediction with empirical estimates of Ω ~ 0.3 in the
present universe (see chapter 5)? This conundrum became known as the flatness problem.
Figure 8; Calculations show that the slightest deviation from flatness in a universe 1 nanosecond old quickly
amplifies, resulting in a runaway closed or runaway open universe.
The horizon problem
Another problem concerned the homogeneity of the universe. We recall that the FriedmannLemaitre models assume Einstein’s cosmological principle – that the universe is both
isotropic and homogeneous on the largest scales. As astronomy progressed in the 1960s and
70s, this principle seemed to be supported more and more by the evidence. The Hubble
expansion appeared to be the same in every direction. Galaxy surveys also revealed a largescale uniformity; for any given epoch, the density of galaxies and galaxy clusters appeared to
2
be approximately constant. Most tellingly, there seemed to be no detectable variation in the
intensity of the cosmic background radiation, suggesting a high degree of homogeneity at the
time of recombination. Indeed, as studies of the cosmic microwave background progressed, it
became more and more obvious that the radiation was extremely smooth, indicating an
extremely homogeneous universe at least at the time of the formation of atoms2.
Unfortunately, it is not clear why the universe should be so homogeneous. In nature, such
equilibrium is only achieved by objects coming into thermal contact with one another and
exchanging energy until any inhomogeneties are balanced out (just as a hot cup of tea
eventually cools to the temperature of its environment). However, calculations showed that
the most distant regions of the universe could not have been in such contact; there simply
wasn’t enough time for light to travel from one such region to another during the lifetime of
the expanding universe. The limit of influence any section of space is set by the finite speed
of light and is called its horizon; hence this paradox is known as the horizon problem. The
conundrum was first articulated by the astronomer Charles Misner3 and it became widely
known. Simply put, the smoothness of the microwave background suggests that regions of
the universe separated by distances far greater than their respective horizons have nonetheless
been in thermal contact.
You might argue that no such paradox should apply to a universe that is assumed to have
originated in a tiny volume of space; surely all regions were originally in thermal contact? In
a way this is correct. The problem is one of backtracking, as we try to reconcile the
homogeneity of today’s universe with both its scale and its age. One way of thinking about
the horizon problem is simply that the size of the universe, as measured from the Hubble
graph, doesn’t seem to match its contents.
3
The theory of inflation
In the early 1980s, a new version of the big bang model was proposed that has had a major
impact on modern cosmology. This model arose from considerations in particle physics and
marked the beginning of an extremely fruitful alliance between the fields of particle physics
(the world of the extremely small) and cosmology (the world of the extremely large). The
core of the proposal was simple but startling; what if, during the first fractions of a second,
the infant universe underwent an extremely rapid expansion, after which it relaxed to the
much slower expansion we measure today? This idea was first associated with the young
American particle physicist Alan Guth and he named his proposal the inflationary universe 4.
Guth’s key insight was that a dramatic, exponential5 expansion of space almost at the very
beginning of time could provide an explanation for several big bang riddles. The idea of a
universe that expands exponentially was not entirely new (it arises in the context of the de
Sitter model and more modern versions were proposed by Andrei Linde , Yakov Zeldovich,
Alexei Starobinsky and others6); however, Guth was the first to realise that the postulate
offered a simultaneous solution to the horizon and flatness problems. Before we consider
Guth’s particular model, we consider how the concept of an exponential expansion in the
very early universe addresses these problems.
Inflation and the horizon problem
For the case of the horizon problem, we imagine a small region of space before inflation
begins; a uniform, homogeneous state is easily established in such a region because all points
are in thermal contact (it is causally connected). When inflation occurs, this region of space is
stretched exponentially; neighbouring points are driven apart to distances so large they cannot
communicate even by light signals. In the language of physics, the components of the region
4
are swept far beyond their particle horizons by inflation (recall that the horizon of each
component is set by the finite speed of light). The homogeneity of the region is preserved,
although it will appear quite mysterious to an observer! (see figure 10).
Figure 10 If a small, homgeneous region of space is stretched to a size larger than today’s observable universe,
its homogeneity is preserved
A common question is this; surely the rate of expansion of the early universe is constrained
by the speed of light? In fact, the rate of expansion of space can be arbitrarily large.
Relativity specifies that the speed of light in vacuum is a limiting speed for any massive
object in spacetime ; it sets no limits on the behaviour of spacetime itself. Note also that,
according to inflation, even a small region of space can be inflated to dimensions far larger
than the universe we observe today. This aspect of inflation forces one to consider that the
universe we measure may simply be the observable universe - a small patch of a much larger
entity!
Inflation and the flatness problem
The concept of inflation also offers a neat solution to Dicke’s flatness problem. If a region of
space is inflated in the first fractions of an instant to a size larger than the universe we
observe, its geometry will inevitably appear flat to us, just as the surface of an enormous
balloon appears flat to an insect on it. This solves the flatness paradox beautifully; instead of
5
deviations from flatness leading quickly to either a runaway open or runaway closed
universe, the inflationary model predicts that a universe is driven towards flatness (see figure
10). Indeed, the model makes a very clear prediction; the geometry of the observable universe
should be measured as exactly flat, to a high degree of accuracy. (This prediction seemed in
conflict with empirical estimates of Ω at the time – we shall see in chapter 7 that modern
measurements suggest that the geometry is indeed exactly flat).
Figure 10 Inflation drives the infant universe towards flatness
However, recall that a flat geometry implies a perfect balance between the competing forces
of expansion and gravity – why should the universe be so perfectly balanced? Or, in the
language of inflation, why is the early universe driven to this balance? In order to answer this
question, we must consider the physical mechanism of inflation. Before doing so, we
summarize by noting that inflation pushes the universe into a remarkably simple state since
all inhomogeneities and local curvatures of space are smoothed out by the enormous
expansion (except fluctuations in the inflationary field itself, as we shall see). Such a universe
is called a no-hair universe in analogy with Hawking’s no-hair theorem for black holes7.
6
The mechanism of inflation
What could cause the infant universe to undergo an exponential expansion in the first
fractions of an instant? In the language of physics, the answer is a repulsive force due to an
energy state known as the false vacuum; the latter is a state of high energy density that is
temporarily prevented from relaxing to the natural state of lowest energy (the true vacuum).
A good analogy is a marble sitting on top of an inverted bowl: the marble is stable, but will
easily fall to a lower energy state if nudged, as shown in figure 12a. An important property of
the false vacuum is that it exerts a negative pressure on its surroundings, not unlike suction.
Now it is a fundamental tenet of general relativity that pressure, like energy, has an
associated gravitational field; in particular negative pressure creates a repulsive gravitational
field. Hence, a false vacuum state can create a significant repulsive force.
So the key is a false vacuum state; but how did a small patch of the early universe get into
such a state? In his original paper, Guth postulated that the universe underwent a
phenomenon known as a supercooled phase transition. Phase transitions are extremely
important in physics. The term is used to describe the change that occurs when matter
transforms from one form to another –for example, when water cools enough to freeze (note
that this process releases energy). In the late 1970s, particle physicists became convinced that
phase transitions played an important role in the early universe. In particular, it was
hypothesized that two of the fundamental forces once comprised a single force (the electroweak interaction) in an earlier epoch8; this interaction split into two separate interactions as
the universe cooled. The mechanism for this splitting or symmetry breaking was hypothesized
to be a phase transition that occurs via the presence of a quantum field known as the Higgs
field. (This may all sound rather abstract, but the model was spectacularly vindicated by
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experiment in 1982, when exotic particles associated with the electro-weak interaction were
detected for the first time9.)
This success of electro-weak theory led particle physicists to conjecture that three of today’s
fundamental forces may have formed a larger, unified interaction at an earlier epoch, and that
this unified interaction also underwent a phase transition as the universe cooled. Such
theories are known as Grand Unified Theories (GUTs). However, one troubling aspect of
GUTs is that they predict a universe full of monopoles10, particles which have never been
observed. It was an attempt to resolve this paradox that led Guth and his colleague Henry Tye
to consider a particular type of phase transition in the early universe known as a supercooled
phase transition (such a phase transition occurs at much lower temperatures than normal;
water can be supercooled to -20 0C before it turns to ice). To Guth’s great surprise, his
calculations suggested that the process would lead to a state much like a false vacuum, which
would lead in turn to an exponentially expanding universe. This was an important result as it
offered a potential solution to the monopole problem – the density of monopoles created by
the phase transition would be enormously diluted by the inflationary process11. However,
Guth immediately realised that the real significance of his work was that an exponentially
expanding universe could have major implications for several cosmological puzzles, as we
have seen.
New inflation
Guth’s paper on inflation was ‘a shot around the world’, not least because he emphasized
how the model offered an intriguing solution to both the flatness and horizon problems12.
However, it was clear from the beginning that it contained a significant flaw. The theory did
not give a satisfactory account of how the inflationary phase could end, relaxing to the
familiar Hubble expansion (Guth postulated that the universe got out of its metastable state
8
by a process of quantum tunneling13, but calculations showed that this process gives rise to
huge inhomogeneities not observed in today’s universe). This problem became known as the
graceful exit problem.
The problem was overcome in 1982, when the Russian theorist Andrei Linde and the
American physicists Paul Steinhardt and Andy Albrecht independently published new
versions of the inflationary model14. In these models, the false vacuum state is less severe
than that of Guth (see figure 12b), and the phase transition to the state of lowest energy is a
much gentler process15. These models solved the graceful exit problem and became known as
new inflation (figure 13). It’s worth noting that new inflation did not specify a particular
quantum field, simply that the field should have an extremely flat potential energy and a slow
transition to the true vacuum16.
New inflation also gave a good description of the impotant phenomenon of reheating. During
the garangtuan expansion, one can expect that the universe underwent a tremendous
cooling.The new inflation models described an exit from the false vacuum state that resulted
in a huge release of energy in the form of incredibly hot radiation and particles, exactly the
‘initial state’ required by traditional big bang models. But the best was yet to come...
Figure 10: Old and new models of inflation
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A mechanism for galaxy formation
Many cosmologists were struck by the way inflation simultaneously addressed both the
horizon and flatnesss problems. With the problem of the ending of inflation being cleared up,
they became even more interested in the theory – in particular, they took a keen interest in
inflation as a potential mechanism for galaxy formation .
As we saw earlier, quantum physics predicts minute variations in the density of matter in the
infant universe (such fluctuations arise in any phenomena on the minute scale of quantum
physics due to a fundamental fuzziness of the quantum world17). However, detailed studies
of galaxy formation had long suggested that the large-scale structures we observe today could
not have arisen from such tiny perturbations in the context of the traditional big bang model –
the sums simply didn’t add up. Hence, one was forced to assume that the universe was ‘born’
with certain inhomogeneities. Inflation breathed new life into this question; if the infant
universe underwent an enormous expansion during the first fractions of a second, could
natural quantum fluctuations in density have given rise to the galaxies after all?
Experienced cosmologists tackled the problem with gusto. A key stepping stone was to see if
quantum perturbations in an inflationary universe could give rise to the inhomogeneites in
the cosmic background radiation known to be necessary for the seeding of the galaxies. Such
inhomogeneities had not yet been observed experimentally; however, theoreticians had long
predicted the amplitude and shape of non-uniformities in density that could give rise to
today’s structures (known as the Harrison-Zeldovich spectrum). A feverish amount of
calculation followed, with analyses by Hawking, Guth, Linde, Steinhardt, Micheal Turner
and many others. After several false starts, a great deal of thought and three weeks of hard
10
calculation at a now-famous workshop in Cambridge University18, a stunning result was
announced - quantum fluctuations in an inflationary universe could indeed give rise to the
perturbations responsible for today’s large scale structures!19.
This was an exciting advance indeed – a theory that was posited to address the horizon and
flatness problems had also given the first working explanation for the seeding of the galaxies.
Best of all, the explanation arose from fundamental considerations of quantum physics,
opening up a new area of research – the synthesis of quantum theory, particle physics and
cosmology, a thriving field now known as particle cosmology. As we shall see in the next
chapter, increasingly precise measurements of the cosmic microwave background were to
offer further support for the analysis.
Inflation and the singularity problem
Cosmologists soon noticed another startling aspect of the hypothesis of inflation;it also spoke
to the problem of the singularity. This idea arises from a consideration of the Heisenberg
Uncertainy Principle , a fundamental tenet of quantum physics.
Acorrding to the uncertainty principle, the tiniest particles of matter can in principle appear
out of the vacuum if they disappear again quickly enough. It seems they can borrow energy to
exist, as long as that existence is extremely short. (The concept arises from a fundamental
indeterminacy in quantum entities)20. This phenomenon has long been known and the
existence of such virtual particles and anti-particles is routinely detected in accelerator
experiments19.
This ‘free lunch’ in the minute world of quantum physics was well known, but it did not seem
of much relevance to cosmology. However, inflation is hypothesised to occur to a tiny patch
of space over a timespan of fractions of a second. Although this is not quite on quantum
11
scales, one expects quantum processes to be relevant. This raises an intriguing possibility: if
virtual particles briefly appear in a minute region of space and that region undergoes
inflation, could they be blown up to become our entire universe? This scenario may seem
rather speculative but it is taken seriously by many theorists because it offers the first glimpse
of a possible explanation for the hardest question of all – how does something arise from
nothing? The concept of a universe ex nihio became extremely well-known when it was
popularized by the cosmologist Stephen Hawking22. We shall return to this question in
chapter 8.
Drawbacks of inflation
We note first that while it is often stated that the theory of inflation constituted a new
paradigm in cosmology, it is probably more accurate to say that it is an intriguing addition to
the existing big bang paradigm. After all, the theory simply superimposes an extremely brief
period of hyper-expansion on traditional models of the expanding universe. Inflation is
therefore a variant on the big bang model, a version of the theory that provides a natural
explanation for several ‘coincidences’ that are otherwise hard to explain – the homogeneity
of the universe, its geometry, and its large-scale structure (we shall see in chapter 7 that more
accurate measurements of all three of these parameters offer further support for the theory).
These are no mean accomplishments - scientists are always impressed by a theory that can
explain apparently special conditions in terms of general considerations.
However, there are drawbacks. Some physicists feel inflation is rather contrived and simply
too clever. For example, it is often pointed out that the flatness paradox was not widely
recognized as a problem before the advent of inflation23 . (I find this an odd argument;
12
irrespective of how much attention it received, it is clearly a conundum). A more serious
criticism is that of scale – is it reasonable to be talking about unimaginably large expansions
of space (of the order of 1050) ocurring over unimaginably short timespans (of the order of
10 -30 s)? It seems rather speculative and divorced from reality. How could such a theory ever
be tested directly?
A third problem concerns the mechanism of inflation. To this day, it is not known what
specific field could give rise to the phenomenon (Guth’s original postulate of a Higgs field
was ruled out by new inflation). A great many models of inflation have emerged and it is not
clear how to decide between them – this is not a situation physicists enjoy (a similar criticism
pertains to string theory). Finally, there is the problem of fine tuning; although inflation
neatly obviates many of the special initial conditions required by traditional big bang models,
the ‘no-hair’ inflationary universe requires a few initial conditions of its own – namely a
certain type of quantum field and a certain type of phase transition. Hence it can be argued
that inflation has simply replaced one set of initial conditions with another.
Above are the technical drawbacks of the model of an inflationary universe. However, it is
the philosophical implications of the theory that are most disturbing. Recall that inflation
posits that a small region of space could have been inflated to our observable universe – this
immediately raises the possibility that the universe we observe is just a fraction of a much
larger, unobservable ensemble. Worse, one has to consider the possibility that other regions
of this ensemble could have been inflated. As we could never detect the existence of such
regions (they are far beyond our horizon) they would effectively become parallel universes.
This idea, that we may live in one particular universe of a multiverse24 of unobservable
parallel universes is not attractive to scientists (despite its popularity in wider culture), but it
has proved hard to rule out.
13
Further, the theorist Andrei Linde has shown that it is unlikely that the inflation field was
exactly the same everywhere – hence we can expect other universes to have different
properties to our own (this theory is known as chaotic inflation.) If this is right, chaotic
inflation offers an intriguing insight into the fine tuning problem – in a multitude of
universes with vastly different properties , perhaps it is not so unlikely that at least one
universe should have the right conditions for life to emerge!25
All in all, it can be seen that inflation raises some uncomfortable philosophical questions. A
succinct statement of the problem was recently supplied by the Archbishop of Canterbury,
when he stated on the BBC that “I find it disappointing that, in order to explain the properties
of one observable universe, scientists are now postulating the existence of an infinite number
of unobservable ones”!.
However, we note in defence of inflation that far from being a contrived put-up job, the
theory arose from considerations of particle physics, not cosmology. (Indeed, Guth's proposal
was recognized as an intriguing advance by the particle physics community before it was
accepted by the cosmologists26). In addition, the model was not proposed as a solution to the
riddle of galaxy formation – that was an impressive bonus that came a year later. My own
view is that inflation exposes an unexamined assumption, the assumption that one can
extrapolate the expansion we observe today (Hubble graph) all the way back to the epoch of
the very early universe. Indeed, our estimates of the size, age and geometry of the cosmos
were all based on this assumption. In retrospect, there is no real justification for this
extrapolation – we have no way of knowing the conditions that pertained in the very early
universe. Finally, one should not lose sight of the fact that inflation makes one clear
prediction – that the universe should exhibit a flat geometry – that has been verified in recent
years.
14
Alternatives to inflation
Finally, it is often said that inflation ‘wins by default’ i.e. is unsatisfactory but is retained
because of lack of alternatives. In fact, quite a number of alternative theories have been
proposed since 1982, but few have survived the increasingly accurate measurements of the
cosmic microwave background.
One interesting concept is the variable speed of light (VSL) hypothesis. This idea revisits an
old problem we met in chapter 2 – if the universe has a beginning, is it ‘born’ obeying the
laws of physics or at what point do the laws of physics come into play? More specifically, at
what point do the constants of nature (the gravitational constant, the speed of light etc) take
on their present values? In 1993, the Danish theoretician John Moffat postulated that the
speed of light may have had a much larger value in the first fractions of a second. He then
showed that this hypothesis could offer a solution to the horizon, flatness and monopole
problems without invoking inflation27. However, this work did not become very well known,
and an independent version of the same idea was developed in 1999 by João Magueijo and
Andreas Albrecht as a specific alternative to inflation28.
The VSL hypothesis received a lot of attention in 2001 when astronomical observations
suggested a variation in the fine structure constant; however, this data was later found to be in
error. The importance of the hypothesis is that it exposes another possibly unjustified
assumption – that the laws of nature behaved in the first fractions of a second the same way
they do today. That said, VSL theories have suffered because they make few predictions that
distinguish them from inflation, at least with present technology. More importantly, the
hypothesis assumes that both the principle of relativity and the principle of the conservation
of energy are violated in the early universe. Most physicists are reluctant to give up such
cherished principles unless convincing evidence is forthcoming...
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Notes
1 See Dicke (1969). The calculations show that this scenario results if Ω differs from unity at t = 1 nanosecond
by even by a factor of (1 in10-15)
2. This observation is usually attributed to the data from satellite measurements; in fact the homogeneity of the
universe was observed empirically long before this
3 See Misner (1969)
4. Guth’s first major paper on the subject was specifically titled Inflationary Universe; A Possible Solution to
the Horizon and Flatness problems (Guth 1981)
5. An exponential growth is extremely fast; it is written as en where e =2.718 and n can be any number. In
Guth’s proposal, the universe is proposed to have grown by a factor of about 1050 during a timespans of about
10 -30 s
6. A very good history of the development of the theory can be found in Kolb and Turner (1988) chapters ...
7. See Hawking (1988) chapter ..
8. The four fundamental forces are; gravity, electromagnetism , the strong nuclear force and the weak nuclear
force. Gravity is the odd one out as it is by far the weakest, but is the only force that acts on all matter. It is now
thought that all four forces originally constituted a single interaction, with the strong, the weak, and the
electromagnetic force splitting off (in that order) as the universe cooled.
9 The detection of W and Z particles at CERN in 1982 was a major vindication of electro-weak theory
10. A monopole is a magnet with one pole only. Most Grand Unified Theories predict that many should exist in
today’s universe but none have ever been observed
11 An observable universe that is but a fraction a much larger universe could explain the low density of
monopoles
12 See note 4. It’s interesting that the monople problem is not included in the title, although it is addressed in
the paper.
13 Qantum tunneling is the process whereby quantum particles penetrate a barrier that should be too high for
them to cross, according to classical physics
14 Linde had independently developed his own model of inflation, so he was quick to appreciate the problem.
As a tongue-in-cheek tribute to Guth he titled his paper A New Inflationary Universe Scenario; A Possible
Solution of the Horizon, Flatness, Isotropy, and Primordial Monopole Problems (Linde, 1982)
15 Technically speaking, new inflation posits a second-order phase transition, rather than the first-order phase
transition of the Guth model
16There is an excellent description of the difference between new and old inflation in chapters 11 and 12 of
Guth’s book (1997)
17 The fuzziness is specified by the Heisenberg Uncertainty Principle, a theorem that predicts a fundamental
ambiguity in the properties of quantum entities. For example, a particle cannot possess an exact position and
momentum simultaneously; the more definite its position, the less defined its momentum. (It is not a problem of
measurement, as is often stated).
16
18 The occasion was the 1982 Nuffield Workshop on Cosmology in Cambridge (see Gibbons, Hawking and
Silos 2003). There is a superb description of the process of discovering this result in chapter 13 of Guth’s book
(Guth 1997)
19 This is a slight simplification; in fact the work showed that inflation could give rise to the shape of the
Harrison –Zeldovich spectrum (i.e. its scale invariance). The amplitude of the perturbations depended on the
nature of the inflationary field which was not known
20 See note 15. The Heisenberg Uncertainty Principle also applies to the variables energy and time. Hence an
extremely well-defined time corresponds to a certain fuzziness in energy – and a tiny perturbation in energy is
possible if it occurs over an extremely short timeframe. Note that the energy borrowed is equivalent to mass as
predicted by E = mc2
21 The effect is detected as a perturbation in the masses of known particles (known as loop perturbations). Note
that particles are only created with their corresponding anti-particles, by conservation of energy.
22 See Hawking (1988) Chapter ....
23 See Lightman and Brower (1990), p 24. My own view is that scientists like to publish solutions, not
problems. Once Dicke had defined the problem, there wasn’t much to say on the subject until inflation appeared.
24 It doesn’t help that the concept of parallel universes is often confused with ‘parallel worlds’ – a particular
interpretation of quantum physics that has nothing to do with cosmology
25 This isn’t strictly true as a lotto player has exactly the same chance of winning even if he is the only entrant;
more on this later
26 See (Guth, 1997) chapter 10
27 See Moffat (1993)
28 See Albrecht and Mageuijo (1999). A more recent technical review of the Magueijo-Albrecht theory can be
found in Magueijo (2003). He has also written an account of the development theory for a popular audience that
is a great introduction to cosmology (Mageuijo 2003).
29 The word ‘extremely’ is an understatement here; the universe is proposed to have grown by a factor of 1050
during a timespansof the order of 10 -30 s
17