A-level Physics (Advancing Physics)/Big Bang Theory

Big Bang theory states that space-time began as a single point, and that, as time passed, space itself expanded.

Hubble's Law
Hubble's Law describes the expansion of the universe mathematically:

$$v = H_0d$$,

where v is the velocity of recession of a celestial object, and d is the distance to the object. H0 is the Hubble constant, where H0 = 70 km s−1 Mpc−1. The '0' signifies that this is the Hubble constant now, not in the past or the future. This allows for the fact that the Hubble constant might be changing, but very slowly.

The Age of the Universe
Imagine a galaxy which flies out from the big bang at the speed of light (c). The distance it has travelled d is given by:

$$d = vt$$,

where t is the age of the universe, since the galaxy has been travelling since the beginning. If we substitute in Hubble's Law for v, we get:

$$d = H_0dt$$

$$1 = H_0t$$

$$t = \frac{1}{H_0}$$

So, the reciprocal of the Hubble constant is the age of the universe - but be careful with the units.

More Doppler Effect
We have already seen that red-shift z is given by:

$$z = \frac{\Delta\lambda}{\lambda_s} = \frac{v_s}{c}$$,

where &Delta;&lambda; is the amount by which radiation is red-shifted from a celestial object, &lambda;s is the wavelength of the radiation relative to the celestial object, vs is the velocity of recession of the object, c is the speed of light, and v is much less than c. If &lambda; is the wavelength of the radiation relative to us:

$$z = \frac{\lambda - \lambda_s}{\lambda_s} = \frac{\lambda}{\lambda_s} - 1$$

$$z + 1 = \frac{\lambda}{\lambda_s}$$

However, if it is actually space that is being stretched, then this is actually the ratio of the distances between us and the celestial object at two times: the time at which the radiation was emitted, and the time at which the radiation was received. We can apply this to any distance between any two stars:

$$\frac{R_{now}}{R_{then}} = z + 1$$

Red Shift
If we measure the red shift of celestial objects, we see that most of them are moving away from us - the light from them is red-shifted. This is not true of all celestial objects - the Andromeda galaxy, for example, is blue-shifted; it is moving towards us due to the gravitational attraction of the Milky Way. Some galaxies are partly red-shifted, and partly blue-shifted. This is due to their rotation - some parts of the galaxy are rotating towards us, while others are rotating away from us. However, the majority of celestial objects are moving away from us. If we extrapolate backwards, we find that the universe must have started at a single point. However, we are assuming that the universe has always expanded and that movement alone can cause redshift. Red shift provides evidence for a Big Bang, but does not prove it.

Cosmic Microwave Background Radiation
Models of the Big Bang show that, at the beginning of the universe, radiation of a relatively short wavelength would have been produced. Now, this radiation, due to the expansion of space, has been stretched - it has become microwave radiation. Cosmic microwave background radiation fits in extremely well with Big Bang theory, and so is strong evidence for it.

Questions
1. What is the Hubble Constant in s−1?

2. How old is the universe?

3. What effect might gravity have had on this figure?

4. Polaris is 132pc away. What is its velocity of recession, according to Hubble's Law?

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