Category Archives: Cosmology 101

The last 12 posts described each of the main topics of modern cosmology: from the predictions of general relativity to the three main planks of evidence for the Big Bang, from the theory of inflation to the most recent measurements of the cosmic microwave background. All of this theory and experiment has been put together to form what is known as the Standard Model of cosmology. Before we describe the Standard Model, let’s briefly review the main discoveries in chronological order (you can read a seperate post on each starting here)

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A Brief History of Cosmology

1.  The general theory of relativity (Einstein, 1916) and the prediction of a dynamic universe (Friedmann, 1919)

2. The observation of the expanding universe (Hubble’s Law, 1929)

3. Rewinding the Hubble graph: the ‘primeval atom’ and the calculation of the age of the universe (Lemaitre, 1929)

4. The Big Bang model and the problem of the singularity

5. The prediction of the abundances of H and He from the BB model (Gamow, 1945)

6. The predicton of the cosmic microwave background (CMB) from the BB model (Alphaer, Heuer and Gamow, 1949)

7. The detection of the cosmic microwave background (Penzias and Wilson, 1965)

8. The CMB puzzles of flatness, homogeneity and galaxy formation (1965 -)

9. The theory of inflation (Guth, Linde and Steinhardt, 1981-82)

10. The COBE study of the CMB –  support for the BB model and inflation (1992)

11. Dark energy – the supernova measurements of an accelerating universe (1998)

12. The WMAP study of the CMB – more support for inflation and dark energy (2005)

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I gave a talk on the Big Bang to our astronomy class at WIT last night, describing most of the above topics. It was great fun, with a lengthy question and answer session afterwards, fielded by both me and well-known astronomer Emmet Mordaunt who normally takes the class. You can find the slides for the talk here.

In 1998, a totally unexpected result from astronomy caused a dramatic rethink of the Big Bang model. Measurements of the light emitted by a certain supernova suggested that it was further away than predicted by the Hubble constant. In other words, the exploding star did not lie on the straight line of the Hubble graph! This is a startling discovery as it implies that the expansion of the universe is not constant – instead the expansion is currently accelerating (see a description of the experiment here).

Skeptics at first suggested the result might arise from an error in the measurement of stellar distance – however, a similar observation was reported by a different group within two years. Further, independent support for the result soon emerged from measurements of the cosmic microwave background (CMB). In 2002, precision measurements of the CMB by the WMAP satellite suggested a universe with geometry that is flat to within 1%. This result is completely inexplicable in the context of the known density of the matter of the universe (both ordinary and dark). The known density of matter points to a universe with Ω = 0.3, a long way from flatness (Ω =1). Hence the CMB measurements suggest that there is a great deal of matter/energy in the uiverse unaccounted for.

As a result, cosmologists now talk about a new phenomenon; a form of energy that is pushing the universe outward, causing the expansion to accelerate and the geometry to be flat. The phenomenon is labelled Dark Energy : the physical cause for dark energy is thought to be some sort of vacuum energy. However, it should be pointed out that the numbers don’t yet stack up – detailed calculations suggests that the postulated vacuum energy would cause an accelerated expansion many orders of magnitude greater than that observed…this is a major area of research at the moment.

Putting Dark Energy together with Dark Matter, cosmologists postulate that ordinary matter, dark matter and dark energy all add up to the critical density required for the geometry of the universe to be flat (as measured). In other words, the current model of the universe can be summed up by

Density ord matter (4%) + Dens dark matter (22%) + Dens dark energy (74%) = 100%

or Ω_M (0.04) + Ω_DM (0.22) + Ω_DE (0.74) = 1

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Technical notes:

1. Is Dark Energy compatible with relativity?

Yes, but note that an accelerating universe is not predicted by the Friedmann equation, i.e. does not feature in any of the Friedmann universes (see post on the expanding universe below). Going back to 1st principles, when one applies the equations of general relativity to the cosmos, an extra term must be added in order to account for the accelerating expansion. This is rather reminiscent of the manner in which Einstein himself , dismayed by the prediction of a dynamic universe, originally added a term to his equations in order to keep the universe static (the positive cosmological constant) . Now we apply a term to the other side of the equation for the opposite reason.

Revised Friedmann graphs of the evolution of the universe

2. Is Dark Energy compatible with inflation?

Yes, for two reasons:

1. The fact that the expansion is accelerating now makes the suggestion of an exponential expansion in the first instants a lot less fanciful.

2. While the current accleration is many orders of magnitude less than that of inflation, it may be that the cause is some energy left over from inflation – more on this later.

The posts below constitute a brief introduction to the Big Bang model: the three planks of evidence, the problems of singularity, horizon and flatness, and the theory of inflation. Before we go on to discuss the Standard Model of cosmology (yes, there is such a thing), two further concepts are necessary: the old puzzle of Dark Matter and the new puzzle of Dark Energy.

Dark matter is thought to make up about 70% of the matter of the universe. Although we can’t see it, we presume it exists because of its gravitational effect on visible matter. Put differently, we don’t insist that all matter be ‘visible’ i.e. interact with the electromagnetic force. Instead, we include the possibility that some matter may be seen only by its gravitational effect on other matter.

DM was first postulated by Fritz Zwicky in the 1930s to account for a discrepancy between the calculated velocity of spiral galaxies and that observed. Nowadays, it has been proposed to account for the motion of many astrophysical phenomena from the smaller scales to the largest e.g. local stellar dynamics, galaxy rotation, galaxy cluster dynamics, X-ray halos, gravitational lensing and cluster streaming.

Calculations for galaxy rotation based on ordinary matter (curve A) and experimental points (curve B)

A second pointer of evidence for Dark Matter comes from cosmology, in particular from the cosmic microwave background (see post below). In order to to relate the miniscule variations in temperature seen in the CMB to galaxy formation in the early universe, all current models invoke the existence of DM. Even more importantly, the existence of DM is necessary to provide enough gravity to explain the flatness of the universe, as measured from the CMB (in conjunction with the postulate of dark energy – see next week).

It should be pointed out that not everyone agrees with the postulate of Dark Matter. Skeptics point to the possibility that our laws of gravity (both Newtonian and Einsteinian) may be failing at the largest scales – a theory known as modified Newtonian dynamics or MOND. However, most cosmologists now consider this possibility unlikely, due to the astrophysical and cosmological evidence above.

Best of all, the first hint of direct evidence DM was reported in a study of galaxy collision in 2007. If Dark Matter really exists, one might expect to observe ‘galaxy splitting’ in the case of galaxy collision. This is because the DM of each galaxy should interact little with the other, while the ordinary matter of each will interact strongly (just as a couple crossing a crowded room soon become separated if one is more social than the other!). Researchers at the University of Arizona are pretty sure this is exactly what they observed (see here). A similar result was reported by NASA in September 2008.

The famous bullet cluster collision (2007)

What could Dark Matter be made up of? Clearly, DM particles must be weakly interacting (otherwise we would see them) and possibly massive – i.e. weakly interacting massive particles or WIMPs. It is currently thought that the most likely candidates might be supersymmetric particles. (As we saw before, the theory of supersymmetry (SUSY) arises out of attempts to unify three of the fundamental forces – the theory postulates that every normal particle has a heavier supersymmetric partner). It turns out the most likely candidiate for DM is the neutralino, the lightest SUSY particle which cannot decay further.

Many groups around the world have been constructing experiments to look for particles that might be candidates for Dark Matter – you can find a post on a lecture on this subject by Tim Sumner of the Zeplin III experiment here and there is a very good overview of the Zeplin experiment itself here . However, this is straying into the area of particle physics; for cosmologists, establishing the existence of DM unequivocally is the real challenge.

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).

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 d_c required for flatness i.e. Ω = d/d_c . 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?

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.

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!

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. ..

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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…