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My CMB book is finally published!

Rather than do my usual Friday post about music, today I am giving an update on my Cosmic Microwave Background (CMB) book, which I am pleased to say has finally been published. I even got my own hardcopy yesterday, although I’ve had access to the electronic copy for several weeks.

My book “The Cosmic Microwave Background – how it changed our understanding of the Universe” is published by Springer

In the book I attempt to give the background to the accidental discovery of the CMB, and what studying it has taught us about the properties of the Universe. In chapters 1 and 2 I give the background to how we know that our Earth is not the centre of the Universe, and how we know that our Milky Way galaxy is just one of billions. In Chapter 3 I tell the story of the prediction and accidental discovery of the CMB, and in Chapter 4 I discuss the COBE satellite, which in 1992 discovered in the CMB the first seeds of what would grow into galaxies. In Chapter 5 I discuss ground-based observations to study the CMB in more detail, in Chapter 6 I discus the WMAP satellite’s findings. In the final chapter I discuss the latest results, including from the Planck satellite and also the BICEP2 results claiming to have found evidence for ‘cosmic inflation’.

I got my first hard copy of the book yesterday, Thursday the 12th of February.

I have tried to write the book at a level which I hope will make it accessible to anyone. Any physics knowledge required is explained when I introduce the particular idea, and I’ve included diagrams wherever possible to help illustrate and explain key concepts.

The book is available from the Springer website in both electronic and hardcopy versions,and also through Amazon and other booksellers.

This is the link to the Springer page for the book. On the Springer website you can look at the preface to the book.

And this is the link to the Amazon page. On the Amazon website you can read the first dozen or so pages of the book, including the table of contents and the opening pages of the first chapter.

I am now working on two other books, you can find more details on a Facebook page which I have recently created

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BICEP2 and Planck to share data

I thought it was about time I gave another update on currently the most important story in astrophysics – the BICEP2 team’s possible detection of B-mode polarisation in the cosmic microwave background. I have previously blogged about this story, for example here, here and here. But, just to quickly recap, in March the BICEP2 team announced that they had detected the B-mode polarisation in the cosmic microwave background (CMB), and argued that it was evidence of gravitational waves and cosmological inflation in the very early Universe.

Since then, controversy has been the order of the day as other astrophysicists and cosmologists have argued that the BICEP2 detection was not due to the CMB at all, but rather to emission from dust in our own Milky Way galaxy. BICEP2 on their own do not have sufficient data to rule out this possibility, something they concede in their published paper. However, it would seem that the European satellite Planck do, as it has not only observed the whole sky (including the part of the sky observed by BICEP2), but has done so at five different frequencies, compared to BICEP2’s single frequency measurement.

In the last few days, it has been announced that the BICEP2 team will formally collaborate and share data with the Planck team, which I think is good news in sorting out the controversy over the BICEP2 detection sooner rather than later.

The BICEP2 team and Planck team have announced that they will collaborate and share data to help clear up the controversy over the source of the B-mode polarisation detected by BICEP2.

Although the Planck measurements of the polarisation of dust in our Milky Way will presumably become public at some point (as is normal with publicly funded science projects), this would not be for many more months. By formally collaborating with Planck, the BICEP2 team will get not only earlier access to the Planck data, but just as importantly will get the experts in the Planck collaboration working with them to properly interpret the Planck measurements. It is hoped by all in the astrophysics and cosmology communities that this collaboration between BICEP2 and Planck will lead to the issue of the origin of the detected B-mode polarisation being sorted out in a timely fashion, possibly even by the end of this year.

We shall have to wait and see!

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What does a 1-sigma, a 3-sigma or a 5-sigma detection mean?

A few days ago, I blogged about the controversy over the BICEP2 result, and the possibility that their measured signal may actually be dominated by contamination from foreground Galactic dust. As Peter Coles’ blog mentions, their paper has now been published in Physical Review Letters. In the abstract to their paper, the BICEP2 team say

Cross correlating BICEP2 against 100 GHz maps from the BICEP1 experiment, the excess signal is confirmed with $3 \sigma$ significance and its spectral index is found to be consistent with that of the CMB, disfavoring dust at $1.7 \sigma$.

What does a phrase like “with $3 \sigma$ significance” actually mean? It is the significance with which scientists believe a result to be real as opposed to a random fluctuation in the background signal (the noise). In order to fully understand why scientists quote results to a particular $\sigma$, and what it means in detail, the first step is to understand something called the normal distribution.

You can read more about the BICEP2 result, and how its conclusions were withdrawn, in my book “The Cosmic Microwave Background – How it Changed Our Understanding of the Universe”. Follow this link for more details.

My book “The Cosmic Microwave Background – how it changed our understanding of the Universe” is published by Springer. Read more about it by following this link.

The Normal Distribution

If you have a large number of independent measurements, then their distribution will tend towards something called the normal distribution. This distribution looks like the following, where on the x-axis we have some variable (such as the the background noise in a signal), and the y-axis represents the frequency with which that variable occurs. Normal distributions are usually normalised so that the total probability (the area under the curve) is unity (1), as the sum of all probabilities is always equal to one. The curve is often referred to as a bell curve for obvious reasons.

The mathematical formula for the normal distribution is given by something called the Gaussian function (and so another name for a normal distribution is a “Gaussian distribution”) and has the form
$f(x,\mu,\sigma) = \frac{ 1 }{ \sigma \sqrt{ 2 \pi} } e^{ - \frac{ (x - \mu)^{2} }{ 2 \sigma^{2} } }$

where $x$ is the variable, $\mu$ is the mean of the distribution, and $\sigma$ is the standard deviation of the distribution. Usually in statistics we have a mean, a median and a mode, but for a normal distribution they are all equal. The standard deviation is related to the width of the curve. For example, in the figure below we show four normal distributions. The blue, red and orange curves all have the same mean (zero), but different standard deviations, which is related to the curve’s width (the diagram actually quotes the variance, which is just the square of the standard deviation). The green curve has a mean of -2 not 0, and it has a different standard deviation to the other three.

As can be seen from these diagrams, if the total probability under each curve is unity, then the probability of a value being measured depends on what the mean is and what the standard deviation is. The further a measurement is from the mean (i.e. towards either end of the bell curve), the less and less likely it is of being measured at random, or to put it another way the less and less likely the signal is of being due to a fluctuation in the background.

So what does a 3-sigma result mean?

We can work out the probability of a particular measurement once we know the mean and the standard deviation of a normal distribution. There are tables to do this, they give the area under the normal distribution function (which remember is related to probability) in terms of a parameter usually written as $Z$. Here is an example of such a table.

How do we use this table? The first thing to notice is that the normal distribution is symmetrical about the mean, so the probability from $-\infty$ up to the value of the mean is 0.5.

Suppose we have a normal distribution with a mean of $\mu = 2$ and a standard deviation of $\sigma = 0.5$. How would we use this table to calculate the probability of a value greater or equal to e.g. $3$ being real? (that is, any value greater and including 3).

The definition of $Z$ is

$Z = \frac{ | x - \mu | }{ \sigma }$

where the modulus in the numerator is so that $Z$ is always positive. With our example, $Z = (3 - 2)/0.5 = 2.0$. So, finding $Z = 2.0$ in the table gives the cumulative probability $P(Z)$ of the value $x$ being between $-\infty$ and $2$ being $P(Z=2) = 0.9772$. So the probability of a value of $x$ from $-\infty \text{ to } 3 \text{ is } 0.9772 \text{ or } 97.72 \%$.

If we are trying to work out the probability of measuring a value of $x > 3$ then we need to remember that the total probability is 1, so the probability of the value of $x > 3 \text{ is } 1-0.9772 = 0.0228$ or $2.28 \%$. Obviously, with our chosen value of $\sigma = 0.5$, a value of $x=3$ is 2-sigma away from the mean ($Z=2$), so a result quoted as a $2 \sigma$ result (or confidence) means that it has a $2.28 \%$ of being false, and a $97.72 \%$ of being real.

What would we get if we had chosen a value of 1-sigma from the mean, or in other words a value of $x = 2.5$? In this case, $Z = (2.5 - 2)/0.5 = 1$, and so using our table we find $P(Z) = 0.8413$. So the probability of $x$ being equal to or greater than 2.5 is $1 - 0.8413 = 0.1587$ or $15.87\%$. As you can see, a $84.13\%$ chance of a result being real (or a $15.87 \%$ chance of a result being false) is not very good, which is why a $1 \sigma$ detection of a signal is not usually considered good enough to be believed.

What would we get if we had chosen a value of 3-sigma from the mean, or in other words a value of $x = 3.5$? In this case, $Z = (3.5 - 2)/0.5 = 3$, and so using our table we find $P(Z=3) = 0.9987$, so the probability of obtaining a value of equal to or greater than 3.5 is $1 - 0.9987 = 0.0013$ or $0.13\%$. So, when we say that a detection is made at the 3-sigma level, what we are saying is that it is $99.87\%$ certain, or that it has just a $0.13\%$ probability of being false.

Usually in science, a 3-sigma detection is taken as being the minimum to be believed, and quite often 5-sigma is chosen, which is essentially $0\%$ probability of the result being false.

Summary

The figure below summarises this graphically.

To translate between this figure and what we have calculated above, just note that the percentages to the left of the mean all add up to $50\%$, so if we wanted to work out the chance of a result being greater than $1\sigma$ above the mean we would work out $100\% - (50\% + 34.1\%) = 15.9\%$, just as we had above. For $3 \sigma$ we have $100\% - (50\% + 34.1\% + 13.6\% + 2.1\%) = 0.2\%$ (we got $0.13\%$ before, the difference is due to rounding).

And, here is a table summarising the significances, to two decimal places.

The significance of various levels of $\sigma$
$\sigma$ Confidence that result is real
$1 \sigma$ 84.13%
$1.5 \sigma$ 93.32%
$2 \sigma$ 97.73%
$2.5 \sigma$ 99.38%
$3 \sigma$ 99.87%
$3.5 \sigma$ 99.98%
$> 4 \sigma$ 100%

So, going back to the BICEP2 result, they state in their paper that their signal is in excess of the background (noise) signal by $3 \sigma$, which would mean that their signal is real with a $99.87\%$ certainty. But, of course, although there seems to be little doubt that their signal is real, what is still undecided and hotly disputed is whether the signal is nearly entirely due to the CMB or could be mainly due to foreground Galactic dust. We shall have to wait to find out the answer to that question!

***UPDATE***

In February 2015 the BICEP2 team withdrew their claim for having discovered primordial B-mode polarisation, and accepted that their detection was of Galactic dust. You can read far more about this fascinating story in my book “The Cosmic Microwave Background – How it Changed Our Understanding of the Universe”.

My book “The Cosmic Microwave Background – how it changed our understanding of the Universe” is published by Springer. Follow this link for more details.

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Is the BICEP2 result correct?

I blogged back in March about the announcement by a team of cosmologists that they had discovered evidence for gravitational waves in the Cosmic Microwave Background (CMB). The BICEP2 experiment, which is based at the South Pole, claimed to have detected the so-called “B-mode polarisation” in the CMB, and from the strength of the signal they argued that it was the best evidence yet of both gravitational waves in the very early Universe, and of the theory of Cosmic Inflation.

However, since that announcement there has been considerable controversy in the cosmology community as to whether their result is correct or not. I have re-blogged several other people’s blogs on this controversy, for example Peter Coles’ blog here and Matt Strassler here and here. As Peter and Matt’s blogs indicated, this controversy has been swirling around in the astronomical community for the last several months; but last Thursday (the 19th of June) it made it into the New York Times.

There now seems to be considerable doubt in the cosmological community about the BICEP2 result which was announced in March.

The main concern amongst the skeptics is that the BICEP2 team did not correctly subtract the effects of dust in our own Galaxy from their signal. Our Milky Way has a lot of dust in it, it is this dust which causes the dark clouds in the band of the Milky Way which are familiar to anyone who has looked at the Milky Way in any detail, even with the naked eye. Most of the dust is in the plane of the disk, but some is above and below the plane in what we refer to as “high Galactic latitudes”. The BICEP2 team chose their patch of sky to be well below the plane of the Milky Way to try to minimise the effects of dust.

However, it may be that the amount of dust and its degree of polarisation where BICEP2 made their observations is greater than the BICEP2 team thought. If this is the case, then much of the polarised signal that BICEP2 measured may not be due to primordial gravitational waves, but instead may be dominated by this foreground contamination. As the New York Times story states, the BICEP2 team acknowledge that the foreground contamination may be greater than they assumed, but they are sticking to their claim that it is still small compared to the signal they detected.

We should know the answer to this burning issue within the next 6 to 12 months. The Planck satellite has done a detailed all-sky map of the strength of the polarised emission from dust in our Milky Way, far more detailed than any data currently available, and when these maps are released it should allow astronomers to correctly determine how much of BICEP2’s signal is due to foreground contamination. Planck will also do this at several different frequencies, and as Galactic dust is much warmer than the CMB the ratio of its signal at different frequencies will be different to that of the CMB, allowing for much better separation of the two effects.

In the meantime, we have a great insight into how science really works. Any result in science is closely scrutinised by the community, and is not accepted as being real until (a) it has been confirmed by other experiments and (b) that the community is satisfied that the interpretation of the measurement is correct, and that all other possibilities have been considered. As Carl Sagan once said

extraordinary claims require extraordinary evidence

So far, I think it is fair to say, most people in the astronomical and cosmological communities are treating the BICEP2 result with a good deal of caution, and that caution can only be allayed by further analysis and measurements.

If you wish to read more about the BICEP2 results and the surrounding controversy, an excellent place to start is Peter Coles’ blog here.

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Available in all good bookshops – soon!

Some of you may have noticed that I haven’t blogged much this last month. The reason is that I have been putting the finishing touches on a book – which has just been sent off to the publishers Springer. I am sure it will need some revision, but am also hopeful that it should be hitting the shelves / bookshops / electronic stores in the next few months.

The cover, even the title may change!

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To most of us, inflation is a nasty thing which sees the money in our pocket being worth less as prices go up. It’s a bad thing! But, in cosmology, a theory called cosmic inflation explains very neatly several key properties of the Universe. The theory of cosmic inflation was first suggested by Alan Guth in 1980, and yesterday (Monday the 17th of March 2014) a team led by John Kovak of Harvard University announced the first direct evidence that cosmic inflation did actually happen. There is also a Cardiff University involvement in this project.

The story on the confirmation of cosmic inflation as it appeared on the BBC science website.

What is cosmic inflation?

In 1980, particle physicist Alan Guth was pondering some of the observed properties of the Universe, and he came up with the idea of cosmic inflation. The observed properties he was hoping to explain with his theory were

• the “Horizon problem”
• the “Flatness problem”
• the “Magnetic-monopole problem”

The Horizon problem

When the cosmic microwave background radiation (CMBR) (the prediction of which I blogged about here) was discovered in 1964 it was recognised that it was most probably the “echo” of the Big Bang. By 1967 Bruce Partridge and David Wilkinson of Princeton University showed that the CMBR was the same from all parts of the sky down to a level of 0.1% of its 3 Kelvin temperature.

It was realised soon after this that this presented a problem, the so called “horizon problem”. It is actually perplexing that different parts of the sky should have the same CMBR temperature because when we look in different parts of the sky we are looking at parts of space which have not had the time to be in contact with each other in any way; they are simply too far apart. Therefore, a patch of sky in one direction with a particular CMBR temperature should have no knowledge of the CMBR temperature of a patch of sky in a different direction.

This is a little bit like switching on a heater in the centre of a large room. Everyone knows that it will take time for the whole room to come to the same temperature, and if the room were really really big you would not expect the corners which are far away from the heater to have the same temperature as the centre of the room next to the heater after just a few minutes. The heat just hasn’t had enough time to spread throughout the room. So, if you found that the whole room was at the same temperature, even though the heat hadn’t had enough time to spread throughout the room, it would be a bit of a puzzle. That is, in essence, the “horizon problem”.

The flatness problem

Einstein showed in his theory of gravity, the General Theory of Relativity, that gravity causes space to bend. A Universe with lots of matter in it will have a different geometry (shape) to a Universe with less matter in it. The so-called “critical density” of the Universe would be a density that would give it a flat geometry. It was realised since the 1960s that the density of the Universe seemed to be very close to the critical density. Why should this be, when it could have any value. It could be much much more or much much less? If you do the mathematics, for the density to be within about a factor of two of the critical density today means it had to have been incredibly close to the critical density in the earliest moments of the Universe. Close to about one part in $10^{60}$!! This is the “flatness problem”.

The magnetic monopole problem

In electricity, we are all familiar with positive and negative charges. James Clerk Maxwell showed in the mid 1800s that electricity and magnetism are part of the same force, electromagnetism. And yet, you never find a magnetic monopole, you always find magnetic poles come in pairs, they always have both a north and south pole. Theoretically there is no reason why one shouldn’t find just e.g. a north pole on its own, without a south pole. This is the “magnetic monopole problem”.

What is cosmic inflation?

Alan Guth’s idea of cosmic inflation suggested that when the Universe was incredibly young, some $10^{-36}$ seconds old, it went through a brief period of very rapid expansion. This period ended when the Universe was about $10^{-33} \text{ or } 10^{-32}$ seconds old, but in this incredibly brief period Guth argued that the Universe grew from being much smaller than a proton to something about the size of a marble. After this brief period of very rapid expansion (inflation), the expansion of the Universe settled down to the more sedate rate of expansion that we see today.

How does cosmic inflation solve these three problems?

The horizon problem is solved by inflation because the very rapid expansion which inflation proposes would allow parts of the Universe which are now too far apart to have ever communicated with each other to have been close enough together before inflation. So, going back to my analogy with the room being heated, it is as if the room started off really small, so small that all parts of it could come to the same temperature, then it suddenly expanded so that the room we are now looking at is much much bigger.

The flatness problem is solved by cosmic inflation by drawing the analogy between the geometry of the Universe and a curved surface. If a curved surface is large enough, then on a local scale it is always going to look flat. An easy analogy to understand this is the surface of our Earth. We all know it is spherical, but on a local scale it appears flat. If the Universe underwent a period of cosmic inflation, then we are seeing such a small part of it that the small part we see is always going to appear flat, no matter what the overall geometry.

The magnetic monopole problem is solved by cosmic inflation in the following manner. The idea is that magnetic monopoles were created in large quantities before the period of cosmic inflation. They should still exist today, but because the Universe expanded so rapidly during cosmic inflation, their number density (how many there are per unit volume) is so tiny that we haven’t found any in the part of the Universe which we are able to observe.

The discovery made by BICEP2

Until yesterday, there had been no direct evidence of anything that cosmic inflation predicted, only agreement between the theory and things which had already been observed. One prediction of the theory is that the CMBR should be polarised in a particular way with a particular amount of polarisation (you can think of polarisation as a particular twisting of radiation, instead of vibrating in all directions it only vibrates in particular directions). The BICEP2 experiment (“Background Imaging of Cosmic Extragalactic Polarization”, the “2” indicates it is the second generation of this experiment) has been using the South Pole Telescope which is, as the name implies, at the Earth’s south pole, and has been looking for a particular signature in the CMBR – the “B-mode polarisation” as it is called.

Yesterday the team announced that they had, for the first time, detected this B-mode polarisation, which is the most direct evidence yet that the theory of cosmic inflation is correct. This polarisation comes about due to gravitational waves in the very very early Universe, so the detection of the B-mode polarisation is also direct evidence of gravitational waves, which were predicted by Einstein but have never been directly detected before.

If you want to read the actual announcement paper you can find the pre-print by following this link here. Here is a screen capture of the first page of the paper.

The first page of the paper announcing the detection of evidence for cosmic inflation. Notice that Cardiff University has an involvement with Peter Ade being the first author in the alphabetical list.

Superimposed on the variations in the temperature of the cosmic microwave background (red and blue blobs) is the evidence for the B-mode polarisation (the small black swirls).

This is very exciting news for cosmology and our understanding of the earliest moments of the Universe. It suggests that our model of the early Universe, including the theory of cosmic inflation, is correct (or at least is on the right tracks). Little by little, astronomers are unfolding the mysteries of the very earliest moments of creation!

If you want to read a more technical (but still non-specialist) explanation, then this story in Sky & Telescope is pretty good. Or, you may prefer this from Sean Carroll’s blog.

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