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## Cosmic Microwave Background

May 30, 2015 1 comment

The Cosmic Microwave Background (CMB) radiation is a very faint but observable form of radiation that is coming to us (and to other places too) from all directions. (By ‘radiation’ here is meant photons of light, or electromagnetic waves, from a wide range of possible frequencies or energies.) In today’s standard model of cosmology, this radiation is believed to emanate from about a time 200,000 to 400,000 years after the Big Bang – a timeframe known as ‘last scattering’ because that was when superheavy collisions between photons of light and other particles (electrons, protons, neutrons, etc) eased off to a degree that photons can ‘escape’ into the expanding space. At the time of last scattering, this radiation was very hot, around ${3000}^{\circ}$K (in the Kelvin scale). And in time, as space expanded, radiation cooled to its currently observed value of $2.726^\circ$K.

$\displaystyle I(\nu) = \frac{8\pi h}{c^3} \cdot \frac{\nu^3}{e^{h\nu/kT} - 1}.$

In this formula, $\nu$ is the frequency variable (a positive real number that gives the number of cycles per second of a wave) and $I(\nu)$ is the energy density as a function of frequency $\nu$.

The other variables are: $T$ is the temperature of the black body which is assumed to be in equilibrium (so the temperature is uniformly constant throughout the body of radiation), $c$ is the speed of light in vacuum, $h$ is Planck’s constant, and $k$ is Boltzmann’s constant from statistical mechanics.

If you plot the graph of this energy density function (against $\nu$) you get a curve that looks like a skewed ‘normal distribution’. Here are some examples I hijacked from the internet:

Various Planck radiation density graphs depending on temperature T.

Here you see various plots of Planck’s function for different temperatures $T$. The horizontal axis labels the frequency $\nu$, and the vertical gives the energy density $I(\nu)$ per frequency. (Please ignore the rising black dotted curve.)

You’ll notice that the graphs have a maximum peak point. And that the lower the temperature, the smaller the frequency where the maximum occurs. Well, that’s what happened as the CMB radiation cooled from a long time ago till today: as the temperature T cooled (decreased) so did the frequency where the peak occurs.

To those of us who know calculus, we can actually compute what frequency $\nu$ gives maximum density and give our answer in terms of the temperature $T$. All we do is compute the derivative of $I(\nu)$ with respect to $\nu$ and set it to zero and solve the resulting equation for $\nu$. You will get an equation whose solution isn’t so trivial to solve, so we’ll need some software or a calculator to approximate it. Anyway, I worked it out (and you can check my answer) and obtained the following:

$\displaystyle\nu_{\max} = 2.82 \frac{kT}{h}.$

(The equal sign here is an approximation!)

The $\nu_{\max}$ is the frequency that gives maximum density and as you can see it is a straight linear function of temperature. The greater the temperature, the proportionately greater the max frequency. The colder the temperature gets the smaller the max-frequency $\nu_{\max}$ gets, and from Planck’s energy equation $E_{\max} = h\nu_{\max}$, so also does the energy of the radiation drop.

Now plug in the observed value for the temperature of the background radiation, which is $T = 2.726$ (degrees Kelvin), and working it out we get (approximately)

$\displaystyle\nu_{\max} = 160.2 \text{ GHz}$

This frequency lies inside the microwave band which is why we call it the microwave radiation! (Even though it does also radiate in other higher and lower frequencies too but at much less intensity!)

Far back in time, when photons were released from their collision `trap’ (and the temperature of the radiation was much hotter) this max frequency was not in the microwave band.

Homework Question: what was the max-frequency $\nu_{\max}$ at the time of last scattering? What frequency band does it belong to? In the visible range? Infrared? Ultraviolet? Higher still? (I’m dying to know! 😉 )

(It isn’t hard as it can be figured from the data above.)

Anyway, I thought working these out was fun.

The CMB radiation was first discovered by Penzias and Wilson in 1965. According to their measurements and calculations (and polite disposal of the pigeons nesting in their antenna!), they measured the temperature as being $3.5^\circ$ K plus or minus 1 Kelvin. (So the actual value that was confirmed later, namely 2.726, fits within their range.) The frequency of radiation that they detected, however, was not the maximum yielding one but was (as they had it in the title of their paper) 1080 Mc/s — which is ‘mega cycles per second’ and is the same as MHz (megahertz). The wavelength value corresponding to this is 7.35 cm. To do the conversion between wavelength $\lambda$ and frequency $\nu$ for electromagnetic waves we use the simple formula

$\displaystyle\lambda\nu = c$

where c is the speed of light (in vacuum).

And that’s the end of our little story for today!

Cheers, Sam Postscript.

The sacred physical constants:

Planck’s constant $h = 6.6254 \times 10^{-27} \text{ erg sec}$

Boltzmann’s constant $k = 1.38049 \times 10^{-16} \text{ erg/K}$

Speed of light $c = 2.9979 \times 10^{10} \text{ cm/sec}$.

## The 21 cm line of hydrogen in Radio Astronomy

This has been a wonderful discovery back in the 1950s that gave Radio Astronomy a good push forward. It also helped in mapping out our Milky Way galaxy (which we really can’t see very well!).

It arose from a feature of quantum field theory, specifically from the hyperfine structure of hydrogen. (I’ll try to explain.)

You know that the hydrogen atom consists of a single proton at its central nucleus and a single electron moving around it somehow in certain specific quantized orbits. It cannot just circle around in any orbit.

That was one of Niels Bohr’s major contributions to our understanding of the atom. In fact this year we’re celebrating the 100th anniversary of his model of the atom (his major papers written in 1913). Some articles in the June issue of Nature magazine are in honor of Bohr’s work.

Normally the electron circles in the lowest orbit associated with the lowest energy state – usually called the ground state (the one with n = 1).

It is known that protons and electrons are particles that have “spin”. (That’s why they are sometimes also called ‘fermions’.) It’s as if they behave like spinning tops. (The Earth and Milky Way are spinning too!)

The spin can be in one direction (say ‘up’) or in the other direction (we label as ‘down’). (These labels of where ‘up’ and ‘down’ are depends on the coordinates we choose, but let’s now worry about that.)

When scientists looked at the spectrum of hydrogen more closely they saw that even while the electron can be in the same ground state – and with definite smallest energy – it can have slightly different energies that are very very close to one another. That’s what is meant by “hyperfine structure” — meaning that the usual energy levels of hydrogen are basically correct except that there are ever so slight deviations from the normal energy levels.

It was discovered by means of quantum field theory that this difference in ground state energies arise when the electron and proton switch between spinning in the same direction to spinning in opposite directions (or vice versa).

When they spin in the same direction the hydrogen atom has slightly more energy than when they are spinning in opposite direction.

And the difference between them?

The difference in these energies corresponds to an electromagnetic wave corresponding to about 21 cm wavelength. And that falls in the radio band of the electromagnetic spectrum.

So when the hydrogen atom shows an emission or absorption spectrum in that wavelength level it means that the electron and proton have switched between having parallel spins to having opposite spins. When the switch happens you see an electromagnetic ray either emitted or absorbed.

It does not happen too often, but when you have a huge number of hydrogen atoms — as you would in hydrogen clouds in our galaxy — it will invariably happen and can be measured.

Now it’s a really nice thing that our galaxy contains several hydrogen clouds.  So by measuring the Doppler shift in the spectrum of hydrogen — at the 21 cm line! — you can measure the velocities of these clouds in relation to our location near the sun.

These velocity distributions are used together with other techniques to map out the hydrogen clouds in order to map out and locate the spiral arms they fall into.

That work (lots of hard work!) showed astronomers that our Milky Way does indeed have arms, just as we would see in some other galaxies, such as in the picture shown here of NGC 1232.

The one UNKNOWN about the structure of our Milky Way is that we don’t know whether it has 2 or 4 arms.

References:
[1] University Astronomy, by Pasachoff and Kutner.
[2] Astronomy (The Evolving Universe), by Michael Zeilik.

(These are excellent sources, by the way.)