Astrophysics Tutorial 8 - Further Cosmology

(Only IB students need to know this, but everyone is welcome.)


Cosmic Scale Factor

The Cosmic Scale Factor and Temperature

Cosmological Principle

Rotation Curves

Fluctuation in CMB

Observing the Fluctuations

What the satellites saw

Cosmological Origin of Redshift



Cosmic Scale Factor (IB Students only)

We know that the Universe is expanding.  We can quantify this using a parameter called the Cosmic Scale Factor, physics code R.  This quantity has no units.  It is a relationship between time and distance.


Distances between objects in the universe are increasing as the universe expands. Consider two galaxies A and B that are a certain distance apart:


The Galaxy A is at coordinates (0, 1) while Galaxy B is at coordinates (4, 4).  We can work out the distance between them using Pythagoras:


distance2 = 42 + 32 = 25 units2

distance = 5 units


We notice that due to expansion, the 5 units at the present time t0 is longer than the 5 units at the past time t1.


Since the relative coordinates are the same in both diagrams, we can describe them as co-moving coordinates.  To keep things simple, we can assume that the galaxies always have the same coordinates in distance and time.


Let us look at the Cosmic Scale Factor between these two diagrams, using the red line between Galaxy A and Galaxy B.  We will assume that the expansions of space and time are uniform.  The size of the red line in the left hand diagram is 4.40 cm upwards and 5.76 cm across.  Similarly the sizes for the right hand diagram is 6.19 cm upwards and 8.10 cm across.  These figures are from the properties given by PowerPoint , which I use to generate my graphics.



Smaller line:  Length2 = (4.40 cm)2 + (5.76 cm)2 = 52.54 cm2

                  Length = 7.25 cm.


Larger line: Length2 = (6.19 cm)2 + (8.10 cm)2 = 103.9 cm2

                Length = 10.19 cm.


The scale factor in this case is 10.19 cm 7.25 cm = 1.41


The distance between the co-moving coordinates remains the same, 5 units.  It will be the same regardless of whether we observed 1000 million years ago, observe now, or will observe 1000 million years from now.


We describe the proper distance x0 as the distance we would measure right now.  In terms of the cosmic scale factor, we can write a simple relationship for the proper distance.



R0 is the cosmic scale factor at the present time.  The term r is the radial distance, the distance between the co-moving coordinates.  If the observation were made at a time t, we can write the equation as:



The cosmic scale factor can be related to the red-shift by this equation:

where z is the Doppler Shift.  From above, we know that:


We also know that:

Worked Example

A chemical element gives a spectral line that has a wavelength of 520 nm.  A far distant galaxy contains the element, but its spectral line is found to be 544 nm.

(a) Calculate the Doppler Shift.

(b) Work out the speed of recession.

(c) Work out the size of the Universe compared with what it is now.


(a) Change in wavelength = 520 nm - 544 nm = -24 nm

     z = -24 nm 520 nm = -0.0462

(b) v = -0.0462 3.0 108 m s-1 = -1.38 107 m s-1 (i.e. receding)

(c) -0.0462 = Rt - 1  (R0 = 1, as it's the cosmic scale factor as seen now)

      Rt = 1 - 0.0462 = 0.954

      This means that the universe was 95 % of its present size when the light left that galaxy.


We can relate the Cosmic Scale Factor to the Hubble Constant.  We know that:

v = H0d


We also know that:

v = zc


So we can write:

zc = H0d


Which becomes:

So we can write:


Note that the value of H0 is 65 km s-1 Mpc-1.  Therefore we need to have the distance in Parsec (pc).  Since we are using km s-1, we need to have the speed of light in km s-1.  Therefore we need to use:


c = 3.0 105 km s-1


Of course you knew that, didn't you?


Question 1

Light left a particular galaxy at a time when the Universe was 86 % of its present size.  Calculate the distance in parsecs to that distant galaxy.  The cosmic scale factor at the present time is 1.

 (H0 = 65 km s-1 Mpc-1.)



The Universe is said to be going through three phases:

Further discussion of these phases requires complex models to describe them, which you will encounter at second or third year at university.


The Cosmic Scale Factor and Temperature

In Astrophysics 7, we saw the time line for the Universe since the Big Bang.  We can link this with the cosmic scale factor.



Temperature /K

Scale Factor

Compared with now (R = 1)


Time / s

Strong force freezes out


2.7 10-27

10-35 s


Weak force freezes out


2.7 10-15

10-10 s


Protons, neutrons formed


2.7 10-11

10-4 s


Neutrinos decouple

3 1010

9.1 10-11

1 s


Electrons freeze out

6 109

4.5 10-10

100 s


Formation of 2H and 4 He nuclei

9 108

3.0 10-9

2 - 15 min

120 - 900

Simple atoms form


9.2 10-4

377000 yr

1.2 1013

First stars



109 yr

3.2 1016




1.4 1010 yr

4.4 1017



The term freezing out refers to the temperature that physics phenomena become effective.  For example, the strong force becomes effective at 1027K.


We can use these data to plot a graph of the log10 (Cosmic Scale Factor) against the log10 (Temperature):



The temperature of the universe varies inversely with the scale factor:


Question 2

Use the graph above to show that the variance above is correct.



The relationship tells us that when the scale factor is 2, the temperature will be be 1.37 K.


We can also use these data to show the scale factor with time:



We can see that there is a straight-line progression between 1 s (log10 t = 0) and 1018 s (log10 t = 18).  In the first second of the Big Bang, the data are not that precise.  This reflects that the size of the Universe inflated from less than 10-35 m to 1018 m in that first second.



Cosmological Principle (IB students)

The Cosmic Density derivation in Astrophysics 7 leads us to the Cosmological Principle, which states:


When viewed on a sufficiently large scale, the properties of the universe are the same for all observers.


In other words we see a fair and representative sample of the Universe and that the laws of physics apply throughout.


Wherever observers may be in the universe, all observations will be the same.  This is called homogeneity.  The universe has the same properties at all points.   Also the same observational evidence is available, whichever direction the universe is observed.  The word for this is isotropy.


This all sounds a bit confusing, but the implications are that:


Distant galaxies contain a lower than expected proportion of elements heavier than lithium, indicating that nucleosynthesis of heavier elements did not occur due to the Big Bang.  Instead these elements have their sources in the remains of dead stars.


Some cosmologists have discovered inconsistencies with the cosmological principle.  There exist giant structures in the universe such as:



Rotation Curves

Stars in a galaxy rotate about the centre of a the galaxy, just like planets rotate about a star.  A rotation curve (or velocity curve) is a plot of the orbital speed against the radial distance from the centre of the galaxy.  The graph below shows the idea:



You can see that the graph derived from the observed data is rather different to what would be expected from data that have been worked out using theoretical models.  The explanation centres of the presence of dark matter


We cannot use the normal models such as Kepler III to study the star movement about the centre of a galaxy. 



Kepler III depends on a central mass, such as a star.  The further away a planet (or moon) is away from the star, the longer the orbital period.  Therefore the orbital speed is lower.


This does not apply in a galaxy.  Most stars tend to orbit a galaxy at constant speed.  Some stars that are further out have a greater orbital speed.  The observations are not consistent with what has been predicted.  The explanation offered for this is that dark matter is involved and that its is distributed from the centre of the galaxy to outside the visible galaxy to form a halo.  The halo extends a considerable way beyond the visible limits of the galaxy and contains much of the mass.


The halo is a theoretical explanation of how dark matter is distributed.


Question 3

Explain why disk galaxies often have spiral arms.





Fluctuation in CMB

The cosmic microwave background (CMB) was predicted in 1948 and its existence was proved in 1964 (Astrophysics Tutorial 7).  It is often described as the "echoes of the big bang".  It has these properties (depending on what model is used):

It also gives the universe an overall temperature of about 2.7 K.  This is cold, but not absolute zero.


Originally the cosmic microwave background was thought to be uniform, but more sophisticated analysis revealed small variations in the CMB.  Dipole anisotropy was detected.  Anisotropy is a word that means how a property of a system changes with direction.  Dipole anisotropy refers to difference of observed temperature as a result of red-shift and blue-shift of the CMB depending on the motion of the Earth.  This corresponds to a temperature change of about 18 mK.


The picture below shows a false colour images of the CMB fluctuations as observed by two different satellites:


Image by NASA

The top pair of images show the map of the temperature where blue is 0 K and red is 4 K.  The temperature is shown as green (2.7 K).  The temperature is shown as uniform, because its variation is much smaller than the range 0 - 4 K.

The second pair show a much narrower range of temperatures, where blue is 2.721 K, green shows 2.725 K, and red shows 2.729 K.  Note that the blue region shows red-shift, and the red region shows blue-shift.  The second pair has this pattern due to the dipole anisotropy as a result of the movement of the Earth (and Sun) relative to the cosmic microwave background.

The third pair show the fluctuations when the dipole anisotropy is compensated for. In this pair of maps the red regions are 0.0002 K (20 mK) hotter than the cold region.  The red band across the middle is due to the Milk Way.


Observing the Fluctuations

This was carried out using orbiting satellites:


What the satellites saw

COBE was looking at the microwave radiation between 1 mm and 1 cm across the universe.  The astrophysicists analysed the extent of the anisotropy to a precision of 0.005 %.  There were found to be very slight variations in temperature that reflected the tiny density variations within the early universe.  These density variations are thought to be the areas in which the earliest galaxies started to form.


Image from NASA


The blue regions are thought to be very slightly cooler.


COBE stopped taking readings on 23rd December 1993.  Its task was passed over to WMAP.


WMAP made a number of important discoveries, including:


Planck is studying the Cosmic Microwave Background in unprecedented detail.  It has found that the Universe is expanding at a slightly slower rate than thought previously, making the Universe slightly older (13.82 thousand million years compared with 13.7 thousand million years).  The proportions of dark matter and dark energy have changed slightly.  Dark matter is thought to be 22 %, while dark energy is about 74 % of the Universe.  Planck has also shed more light on the notion of inflation.

Image from Wikimedia Commons


The picture above compares the resolution available from each of the satellites.


Cosmological Origin of Redshift

We know that red-shift results from the Doppler effect.  The equations involved are:



In the earliest days the red-shift was regarded as just a random phenomenon.  Some galaxies were moving away (red-shift), while others were moving towards us (blue-shift).  However, as the nature of the expanding universe was understood more fully, the idea of cosmological red-shift as well as Doppler red-shift.


Consider a binary star system:


B is moving towards the observer, while A is moving away, hence blue and red-shift respectively.  The wavelength of a photon depends on the motion of the star when the photon was emitted.


In cosmological red-shift, the wavelength is not due to the movement of objects.  It results from the expansion of space itself.  The further away the source of the photons is, the greater the red-shift and the greater the rate of expansion.   Consider the two galaxies in this diagram:



The two galaxies emit light at exactly the same frequency.  The light from the closer galaxy is slightly red-shifted.  The light from the far distant galaxy is more red-shifted.  In this case, this is due to the expansion of space rather than the random movement of the galaxies themselves.


The recession velocity can be worked out using the cosmological red-shift.  The mathematical models used are complex, but an approximation can be obtained from this simpler model:



z - cosmological constant;

H - Hubble Constant;

D - co-moving distance;

v - recession speed.