Tag Archives: Cosmology (general)

The theory of inflation

We have discussed the three main planks of evidence for the Big Bang model: the Hubble expansion graph (and consequent estimate of the age of the universe), the abundance of hydrogen and helium, and the cosmic background radiation. These leave little room for doubt that the basic model is correct. On the other hand, close examination of the model raises many questions – in particular the singularity, horizon and flatness problems (see posts below). Another problem is that it is not clear from the model how perturbations in the early universe led to the large scale structure of galaxies and galaxy clusters seen today.

A possible solution to these puzzles is the theory of inflation. First proposed by Alan Guth in 1981, inflation posits that in the very first fractions of an instant after the Bang, the young universe underwent an exponentially fast expansion (faster than the speed of light) – totally unike the Hubble expansion we see today. This does not violate principles of relativity, since relativity sets no constraints on the behaviour of spacetime itself.

An inflationary expansion of the very early universe offers a simple solution to the horizon problem: if the universe expanded arbitrarily fast, even the farthest flung points could once have been in thermal contact. In other words, the properties of distant points in the universe would not be determined by a competition between the finite speed of light and the finite age of the universe, as previously thought.

Inflation also offers a neat solution to the flatness problem: it was soon shown that, instead of deviations from flatness quickly leading to a runaway open or closed universe, deviations in an inflationary universe tend to be driven back towards flatness. The geometrical equivalent of this is to imagine a balloon being inflated to enormously large dimensions – of course the surface is driven towards flatness.

This is a simplified overview of the theory of inflation – the main point is that inflation offers a version of the Big Bang model in which the universe is driven towards the critical value of flatness/ mass density that exists today, far from accepting it as lucky coincidence.

What is most impressive about the theory is that, contrary to public perception, inflation was not originaly posited in order to address problems in Big Bang cosmology. In fact, the theory arose in an attempt to address certain puzzles in Grand Unified Theory (the branch of particle physics that seeks to unify the strong interaction with the electro-weak interaction). Guth’s proposal was at first treated with incredulity by the cosmological community – however, it was quickly realised that it offered an intriguing solution to the problems above.

As so often, the original model of inflation was found to contain a fatal mathematical flaw (the end of inflation was incompatible with the known universe). This flaw was soon overcome in a modified version of inflation by Linde and Steinhardt. Nowadays, many versions of inflationary models have been posited: which particular version is correct remains to be seen, but strong theoretical and experimental support for an inflationary universe has been forthcoming (more on this next day).


Filed under Cosmology (general), Cosmology 101

BB problem 3: the flatness problem

Another question concerning the Big Bang model concerns the geometry of the universe and has become known as the Flatness Problem.

Recall that a key prediction of general relativity is that matter distorts spacetime – i.e. the force of gravity is essentially a distortion of spacetime by mass. Hence the curvature of the spacetime of our universe will be determined by the density of matter in it. Assuming only that the universe is homogenous and isotropic, it can be shown from general relativity that three distinct types of universe are possible (first calculated by Alexander Friedmann).

If the universe has a high enough density of matter, gravity will triumph over the energy of expansion as time goes on and space will be pulled in on itself, much like a sphere (closed universe). On the other hand, if the universe has a low enough density of matter, gravity eventually loses the battle with the energy of expansion and space will curve outwards (open universe). A third but unlikely possibility is that the curvature of space caused by matter could be exactly balanced by the energy of expansion – in this case space would not be curved but have Euclidean geometry (flat universe).

Friedmann universes: 3 possibilities

We say the above mathematically by defining a flatness parameter Ω to be the ratio of the actual density of matter d to the critical density dc required for flatness i.e. Ω = d/dc . Hence we characterize an open,closed or flat universe as Ω < 1, Ω >1 and Ω = 1 respectively.

So what’s the problem? In the late 1960s, calculations by Bob Dicke showed that even the slightest deviation (1 in1015)  from flatness in the early universe would quickly lead to either a runaway closed or a runaway open universe.  As observation and mass calculations suggest that we live in neither of these, there is a clear implication that the geometry of the early universe must have been exactly flat. But why should the early universe have been so finely balanced between the energy of gravity and the energy of expansion? A curious example of fine tuning indeed…

This mystery has come back to the fore in recent years: measurements of the cosmic microwave background are strongly indicative of a universe that is exactly flat (at least to 1%) at the time of recombination. So now we also have experimental evidence of an exact balance between the density of matter in the universe and the energy of expansion. Again, why such a precise balancing act?

1 Comment

Filed under Cosmology (general), Cosmology 101

BB problem 2: the horizon problem

A second question concerning the Big Bang model of the universe has become known as the Horizon Problem. Essentially, the very homogeneity of the universe, as measured from the cosmic background radiation (see previous post), requires some explanation.

The problem concerns the finite age of the universe versus the finite speed of light. When you do the math, it turns out that the furthest flung regions of the universe are further apart than light could have travelled in the age of the universe. A simple claculation shows that the furthest regions could never have been in thermal contact – yet they have the same temperature to 1 part in 100,000.

So we have a paradox: the homogeneity of the background radiation suggests that all of the observable universe was once in contact long enough to reach thermal equilibrium, while simple calculations based on rewinding the Hubble graph suggest that the universe is too big for this to have happened in the time available.

Artist’s impression of the horizon problem

What is the solution to the paradox? One interesting solution could be that the speed of light in the very early universe was different from what we measure today. A less drastic solution is that we have made an unjustified assumption – namely, by extrapolating the Hubble slope back to the very early universe, we have superimposed an expansion rate of one era on an earlier era we know nothing about.. more on this later.


Filed under Cosmology (general), Cosmology 101

The BB singularity: implications for a creator?

Before looking at further questions concerning the Big Bang model, let’s consider the possible implications of the singularity problem (see previous post) for religion, as this topic often arises in the media.

Our best description of gravity (the dominant force in the universe at large) is Einstein’s general theory of relativity. Applied to the cosmos, general relativity has made some spectacular predictions that have since been verified by experiment (the expanding universe in particular). However, it is true that the theory breaks down as we rewind the clock of the expanding universe all the way back to time zero (the equations blow up to infinity).

Some religiously-minded scientists consider this significant, suggesting that we may be coming to a limit to what the human mind may comprehend about the work of a creator and this is also the view of many theologians. I heard several interesting talks on this topic at Cambridge last year, see the July posts on Cosmology Day at the Faraday Institute here.

Implications for a creator?

However, it’s important to emphasize that most cosmologists would say that the uncertainty really arises from quantum physics. We already know that general relativity is incomplete because it doesn’t incorporate quantum theory (the theory that describes the world of the very small with extreme accuracy). Since the space and time of the infant universe will be of quantum dimensions, we will not be able to describe it until we have a successful quantum theory of gravity. In other words, our current lack of a clear picture of the birth of the universe simply arises from a well-known limitation of the theory of relativity. In time, theoreticians will hopefully find a quantum version of relativity and we will have a clearer picture – so no there are no real implications for religion, one way or the other. I made this point in a public talk on the Big Bang last year (see post on this here), and you can find my description of the lecture in the July issue of Physics World 2008.

Interestingly, one thing we do know from quantum theory is that, even with a successful theory of quantum gravity, some uncertainty will always apply to the first instant of the universe. This is because one of the central tenets of quantum physics is that an inherent fuzziness pertains to the properties of phenomena on the smallest scales (not just a fuzziness in measurement, as many philosophers wrongly believe). This is a direct consequence of the famous Heisenberg Uncertainty Principle. Applied to the Big Bang model, Heisenberg uncertainty predicts that there is no exact point in time called time zero – a trade off between time and energy ensures an in-built fuzziness around this (and any other) instant in time.

The Heisenberg Uncertainty Principle also has implications for the ‘something from nothing’ question. The trade off between time and energy of quantum theory predicts that particles of matter can ‘borrow energy’ to come into existence out of a vacuum, provided their lifetime is short enough. This bizarre behaviour of the smallest constituents of matter has been verified by experiment beyond a shadow of a doubt. Amazingly, it is actually possible in principle that the matter of the entire universe came into being this way, and then had its lifetime extended by a process known as inflation (more on inflation later).

In summary, most cosmologists would argue that there are no real implications for or against religion arising fom the singularity of the Big Bang model, particularly when quantum effects are considered. Indeed, quantum theory even predicts that the entire universe may be a free lunch!


Filed under Cosmology (general), Cosmology 101

BB problem 1: the singularity

While the evidence for the Big Bang model is very convincing (see posts below), many questions remain. The first and most obvious is the singularity problem. In essence, the problem is that while the BB model gives us much information on the evolution of the universe after the Bang, it gives little information about the Bang itself. What banged? What does time zero really mean?

The Big Bang model predicts that the matter, energy, space and time of our universe all came into being in a faction of the first instant and that the universe has been expanding and cooling ever since. The problem is that as we rewind the clock of an expanding universe and the intergalatic distances shrink to zero, relativity predicts that the temperature and density of the universe must increase to infinity (just as the function f(x) = 1/x approaches infinity as x approaches zero – known as a singularity in mathematics). What does this mean physically?

A singularity appears as the distance approaches zero

At first, it was thought that the singularity might be a facet of the simplfying assumptions used in applying the equations of general relativity to the universe. However, in the 1970s, Hawking and Penrose published a number of theorems showing that classical relativity predicts that an expanding universe must originate in a singularity, assuming only some very general conditions. (The most important general condition is a restriction on the behaviour of matter at high energy known as the strong energy condition – of course one way out of the problem is to assume that this condition does not apply, more on this later).

Hawking: an expanding universe must begin in a singularity

The most obvious solution to the singularity problem is to realise that when one is considering an extremely young universe (i.e. a universe of atomic dimensions), quantum effects will become important. However, these cannot be described as we do not have a quantum theory of gravity (i.e. we do not yet have a version of general relativity that takes quantum physics into account, hence the name classical general relativity). Until we do, we can say little of the universe in the time when it was of atomic dimensions or smaller.

In other words, the prediction of an initial singularity may well be a limitation of current theoretical physics, rather than a facet of reality. This incompleteness of the Big Bang model is well recognized, and it is the main reason one talks about the Big Bang model rather than the Big Bang theory. ..


Note on theory: one of the great challenges of modern physics is  to reconcile general relativity and quantum theory, the two great pillars of modern physics. Gravity is now the only one of the four interactions that we cannot currently describe as a quantum field theory. Most of the time, this doesn’t matter, as gravity typically deals with the world of the very large, while quantum theory deals with the world of the very small. However, when we when we attempt to describe gravity on small scales – (i.e.black holes or Big Bangs), the mutal incompatibility of the two theories becomes a major stumbling block…


Filed under Cosmology (general), Cosmology 101

Putting it all together: the Big Bang model

Putting the three planks of evidence together (see below), we are led to The Big Bang model : a model that posits that the universe began as a superhot, superdense singularity that has been expanding and cooling ever since. Although we don’t know much about the initial singularity , there is very strong evidence for the evolution of the universe from this state – so the name Big Bang is a bit of a misnomer! Let’s summarize the evidence once more before we examine the cracks:

1. The expansion of the universe: it’s been known since 1929 that far-away galaxies are receding away from each other at a speed proportional to their distance (Hubble’s Law). This is surprisingly easy to measure as the light emitted by moving galaxies is red-shifted by the Doppler effect. Following the famous Hubble graph back down to the origin led Georges Lemaitre to the idea of a Universe bursting out from a tiny volume (the primeval atom) . Nowadays, we say the Universe began as a superhot, superdense explosion of space, time , matter and radiation, expanding and cooling as time goes on.

Best of all, the slope of the Hubble graph gives an immediate estimate for the age of the universe: the modern value of this figure agrees exactly with that expected from independent estimates (the age of the oldest stars etc)

2. The composition of the elements: if the universe began as some sort of tiny dense fireball, it would have been too hot for atoms to form at first. Calculations by Gamow suggested that a universe made up of about 75% Hydrogen and 25 % Helium should have evolved after a certain time. Guess what – the figures match our universe (it was later realised that all the other elements are fused in dying stars and account for about 0.1%).

3. The cosmic backround radiation: Alpher, Herman and Gamow also suggested that radiation left over from the very early universe might still be observable. (The reason is once atoms began to form, the scattering of light becomes reduced and the universe becomes transparent). Such radiation would be hugely redshifted and freezing cold, but it should be there. Just such radiation was found by Penzias and Wilson in 1965, using the world’s first radiotelescope. Since the background radiation offers a real glimpse of the very early universe, much of modern cosmology is concerned with getting more and more accurate measurements of this ‘cosmic fossil’, using satellite telescopes such as COBE and WMAP.

And that’s the model..



Of course, the model is not perfect. The most obvious problem is the singularity itself – can the universe once really have been infinitely hot and infinitely dense, or is this a feature of our limited understanding of gravity on the quantum scale? There are other problems too, such as the horizon and flatness problems. More on this next day..


Filed under Cosmology (general), Cosmology 101

The third plank of evidence: cosmic radiation

The discovery of the expansion of the universe and the subsequent estimate of the age of the universe (once the distance scale was sorted out) was a key step in the development of the Big Bang model. Another step was the correct prediction of the Hydrogen and Helium content of the universe from Big Bang nucleosynthesis by Gamow (see post below). However, it was a third piece of evidence that was to prove the most convincing of all – the discovery of radiation left over from the hot young universe, now known as the cosmic microwave background (CMB).

Two of Gamov’s students, Ralph Alpher and Robert Herman, continued with his interest in the early universe. In particular, they calculated that as the early universe expanded and cooled, atoms would form after about 100,000 years (a process known as recombination). At this time, radiation left over from the cataclysmic origin of the universe that had been continually scattered by elementary particles would no longer be scattered. The two young scientists postulated that the universe would become transparent to radiation from this time onwards, and that this radiation might even be observable in today’s universe (like a cosmic fossil).  They calculated that it would be isotropic, homogeneous, of black body spectrum and extremely low temperature . Most intriguingly, the radiation would be Doppler shifted (by billions of years of universe expansion) from the extreme high-energy part of the electromagnetic spectrum all the way down to microwave frequencies, the least energetic part of the energy spectrum.

Sadly, no-one paid much attention to this prediction. (At the time, the Big Bang estimate of the age of the universe was way off and Gamov’s initial work on the nucleosynthesis of the heavier elements was also wrong). However, in 1965, Penzias and Wilson, two engineers at Bell Lab, detected an unexpected background noise in data they obtained with the world’s most sensitive radiotelescope. Having spent a year trying to eliminate it, they concluded that the source was extra-galactic and contacted the theoretician Bob Dicke at Princeton. Dicke was amazed. The Princeton group had been working on the theory of cosmic background radiation and drawing up plans for the construction of an experiment to search for it – now the Bell Lab  astronomers already had the  data  (“Boys, we’ve been scooped!”). The two groups published their findings, experiment and theoretical explanation side by side, in a famous issue of the Astrophysics Journal.

The world of science was stunned. This was convincing evidence indeed for the Big Bang model (there is no alternate explanation for the spectrum of the CMB) and the debate was effectively over. Penzias and Wilson were awarded the Nobel Prize and Dicke became instantly famous. (As a bonus, the redshift of the CMB was also extremely convincing evidence that space is indeed expanding, as the radiation isn’t going anywhere).

Penzias and Wilson with their giant antenna in the backgound

There is a sad postscript, however. Neither group was aware of the work of Alpher and Herman twenty years earlier and it was a long time before the work of the Gamow group was acknowledged. Ironic that the Nobel went to two astronomers who knew little of cosmology and that the great work of the theoreticians who predicted the CMB went unsung until recently.

Today, much of modern cosmology is concerned with the study of the CMB with ever more precision, using more and more sopisticated space telescopes. In particular, the COBE satellite  showed the blackbody spectrum of the radiation in precise detail in 1992, while the WMAP mission gave us priceless information on perturbations in the spectrum in 2006. More on this later….


Filed under Cosmology (general), Cosmology 101

The second plank of evidence: nucleosynthesis

The second plank of evidence for the Big Bang model lies in the area of nucleosynthesis. In English, this means that the abundance of the elements seen in the universe at large matches what one would expect in a universe that began in a state that was almost infinitely hot and infinitely dense.

The great Russian theoretician George Gamow was one of the first to take Lemaitre’s prediction of an early universe the size of an atom seriously (see post below). However, Gamow went even further. A specialist in nuclear physics, he calculated that the temperatures in the early universe would have been too high for atoms to form. Instead, matter would have existed as elementary particles, only gradually forming atoms as the universe expanded and cooled in the first few minutes.

Applying simple mathematics to the expanding universe led to a surprising result – that the matter of our universe should be composed almost entirely of the simplest two elements, Hydrogen and Helium (specifically, about 75% H and about 25% He). This prediction turned out to be correct – all the other elements of the Periodic Table account for about 0.1% of the matter of the universe!

The irrepressible George Gamow

This was an impressive early victory for the Big Bang model – however, it failed to explain where the other elements came from (Gamow’s own belief that the other elements are also made in the Big Bang was soon shown to be false, as the universe cools too quickly). It was later shown that all the other elements are ‘cooked in the stars’.

Essentially, what happens is this: a young star burns Hydrogen as fuel, fusing it into Helium, a process that balances the intense inward gravitational force on the star. As the star ages, it runs out of fuel to burn, becomes denser and the temperature increases. Helium then fuses into Lithium, Lithium into the next element, and so on. This process continues right down to Iron. At this point, some stars lose the battle with gravity, turning into neutron stars or even black holes.

Others have a different fate. By a process of accretion of a nearby star, they go supernova –  a cataclsmic process at extreme temperatures during which all the heavy elements of the Periodic Table are fused in turn, and then all spat out in one great explosion…

And that’s where you and I, and all the little children and everything we see about us comes from….stardust!


Note: an interesting sidenote is that the formation of carbon in the fusion chain of stars was long a stumbling block. The puzzle was eventually solved by the British astrophysicist Fred Hoyle, by postulating the existence of a new, unkown isotope of carbon. He badgered an American experimental group to look for this isotope and they soon found it. The riddle of the formation of heavier elements was solved….but Hoyle was to become a controversial figure despite this great success. He was bitterly opposed to the Big Bang model on philosophical grounds – as the evidence mounted, he concocted ever more convoluted alternative models, refusing to concede defeat right up until his death. More on this later…

Note 2: the above is a simplified version of Big Bang nucleosynthesis. In fact, traces of Lithium and Berellium were also formed in the Bang, you can read the details here


Filed under Cosmology 101

The first plank of evidence: Hubble’s law

This week, our 1st years finally get to the first plank of evidence for the Big Bang in their introductory course in cosmology. Having covered the work of Kepler, Galileo, Newton and Einstein, and the theoretical debate between Einstein’s static universe and Friedmann’s dynamic one, they have reached the point where it’s time to introduce the first piece of experimental evidence in modern cosmology – Hubble’s law.

As every schoolgirl knows, Hubble discovered that distant galaxies are moving away from us (or any other point) with a velocity that is proportional to their distance. This is the crux of the evidence for the expanding universe and a major piece of the evidence for the Big Bang.

The law arose from Hubble’s observation of distant galaxies and is usually written as

v = Hd

where v is the recessional velocity of a galaxy, d is the displacement of the galaxy from us and H is the Hubble ‘constant’, or the slope of the graph.

The velocity of a galaxy is measured as a Doppler redshift of the light it emits and is relatively easy to measure. However, the measurement of the distance of a given galaxy is trickier (it is done using stars known as Cepheid variables.). In fact, Hubble had a major systematic error in his distance calculations, an error that was not corrected for decades.

Hubble’s graph was an important advance, for it settled the debate between a static and expanding universe (leading Einstein to label his cosmological constant, introduced to the equations of GR to force a static universe, ‘his greatest blunder’). The graph also immediately hinted at a universe that might once have been very much smaller – perhaps even as small as an atom, as first suggested by the mathematician Georges Lemaitre. (To see this, trace your finger down the slope of the graph towards the origin). This is the model now known as the Big Bang model of the origin of the universe.

However, one problem with the new model immediately emerged. It is easily shown that the age of such a universe can be calculated as the inverse of the slope of the Hubble graph (1/H): unfortunately this calculation gave an answer that was far younger (not older) than the known age of some stars! This observation severely weakened the Big Bang hypothesis and it was only years later that it was discovered that the error lay in Hubble’s estimation of stellar distances (the x-axis). Nowadays, estimates of the age of the universe from updated and expanded Hubble graphs agree exactly with the age calculated from several independent phenomena.

Hubble’s graph contains many subtleties so here are a few other points:

1. Of course the galaxies aren’t really moving at all. General relativity predicts that it’s really space that’s expanding and the galaxies ride the wave.

2. A question arises when Hubble’s law is applied to one galaxy only – surely any object that has a velocity that is proportional to its displacement it must be accelerating? (since its displacement is changing). I wrote a post on this very question a while ago, and got some very interesting answers…see post on Hubble Puzzle

3. Not all galaxies are moving away from one another – for example, our nearest galaxy is approaching us, due to local gravitation effects. Woody Allen has a famous skit on this in the film ‘Annie Hall’, you can see the clip on YouTube here (thanks Dave!)

Comments Off on The first plank of evidence: Hubble’s law

Filed under Cosmology 101