Astrophysics Part IV: Redshift and the Expanding Universe

Nowadays the universe is believed to have began with the big bang. The universe began from a singularity, a spot of infinite density where the four fundamental forces, gravity, the strong and weak nuclear forces, and the electromagnetic force were united. Before this explosion, time and space did not exist.

However, for many years, it was thought that the universe was static, stationary and infinite.  In fact the great physicist and mathematician, Isaac Newton believed this as well, arguing that unless this was correct, the universe would become a spherical mass, as all the stars would have been attracted together, by gravitation.  But it was a German amateur astronomer, Henrich Olber, who disproved Newton’s theory.  The famous Olber’s paradox states that if the universe was infinite and stationary, then the whole sky would be bright.  If one was to look at any direction the night sky would be bright because, there would be an infinite number of stars in any given direction. This can be mathematically shown by a simple proof.

Imagine the stars are distributed in a layer with a density of ρ, thus the number of a stars would seem to be the density, ρ, multiplied by the thickness of the layer, d, multiplied by the surface area of the layer, which is that of a sphere, 4πR². Thus the number of stars that would be visible would be 4πR²ρd.  The volume of visible stars would increase by R², however, luminosity would decrease by 1/R².  Thus luminosity would be equal anywhere. The distance of R² would cancel out, leaving the luminosity to be 4πρd, which would only depend on the thickness d, and the density ρ, not on the location of the observer.

The night sky however, is dark and thus we can conclude that the universe is not infinite or static, but changes with time. The big bang model states that the universe is constantly expanding, and that the distribution of stars is not uniform, resolving Olber’s paradox. However there are many other factors which support the big bang. The first is redshift. This is a phenomena which occurs with radiation emitted from stars that are moving away from us. This is similar to the Doppler Effect.

The Doppler Effect: The changing of frequency which arises from the relative motion between a source and observer

However, the rarefaction or stretching of light waves is actually not due to the Doppler Effect, instead it is due to the fabric of spacetime itself increasing.  Since the big bang, the universe has been increasing in size.  If a photon leaves a nearby galaxy, it spends a considerable time in space. But the space where it is moving is actually being increased in size, thus the longer a particle spends travelling in space, the more the light is rarefacted. As electromagnetic radiation travels at a constant speed, the amount of redshift can be use to determine how far away an object is, and also suggests to us that the universe is expanding. When the particle reaches us, it will have been redshifted, showing us how the universe is expanding, from a central point, suggesting the big bang model is correct.

This relationship can be expressed mathematically by the formula,

Δλ/λ=v/c

Where Δλ = λ’- λ, where λ is the wavelength from a stationary source, and λ’ is the wavelength measured from the source that is moving at speed v , and c is the speed of light

The third factor which supports the big bang is to do with the amount of helium in the universe. In fact 24% of the universe is helium, but the percentage is too high to be solely produced by fusion in stars, suggesting that the helium was made elsewhere. Two physicists, Dicke and Peebles, suggested that this helium was formed in the early universe, when temperatures were large enough to allow hydrogen fusion. In these reactions, high energy photons would have also been produced.  But when later, two physicists Penzias and Wilson positioned an aerial facing space, they encountered constant cosmic background radiation that seemed to have been emitted from a black body of only 2.726K.  (Click this link for information about Wien’s Law) However, the temperatures for hydrogen fusion would have been much larger than this.  The wavelength of this radiation must have increased, supporting the idea of an expanding universe once more, and the big bang.

There are numerous other factors which support the big bang, such as the existence of quasars, and large gas clouds. To summarise we will list the three factors which support the big bang model, which you may have to list in an exam paper.

The three factors which support the Big Bang model

• Olber’s Paradox
• Redshift / The Expanding Universe
• Amount of Helium / Cosmic Background Radiation

Demerit Goods

A demerit good is a good that is overprovided by the market and is deemed to be harmful for society.

These goods or services are considered to be harmful because they have great negative consumption externalities meaning that the consumption of these goods results in spillover effects on a third party and no compensation is paid.

An externality occurs when the actions of consumers or producers give rise to positive or negative side-effects on third parties, and these effects are not paid for or compensated for.

In a demerit good, the benefit to the personal consumer is greater than the benefit received by society, or MPB > MSB. This is a negative externality of consumption. The market is producing at Q_m, but from society’s point of view Q_opt is the preferable level of production. At Q_m, MSB≠MSC, this shows there is market failure, as allocative efficiency has not been achieved.

When allocative efficiency is achieved, the quantity of goods produced and consumed at Q_m, the market equilibrium quantity is not equal to  Q_opt, the quantity deemed most socially desirable. When allocative efficiency is achieved, resources are allocated so well, that if anyone was made better off another party would have to be made worse off.

In the diagram above, the pink triangle has been calculated as the welfare loss, and has been calculated by comparing community surplus at both Q_m and Q_opt and then subtracting the size of the externality up to the quantity of production Q_m. The use of community surplus as a measure of welfare will be discussed in future posts.

A good example of a demerit good is smoking, because the benefit gained by the  consumer is high, but the adverse effects on third parties are great. The externality includes passive smoking and the long term effects which may result from passive smoking, such as development of cancers. Obviously it is difficult to quantify the size of the externality, because when assessing the negative impact, there are elements such as pain or irritation which are difficult to assign a monetary value to.

After government intervention, in this case negative advertising, a decrease in demand occurs and demand shifts left from D to D₁, meaning that the externality has decreased in size. Continuing the smoking example, on most cigarette packets the government enforces mandatory  negative advertising, such as pictures of cancers. After the decrease in demand, D₁ intersects S at a lower quantity, so Q₁ is now closer to Q_opt, meaning that the quantity supplied is now more favourable from society’s point of view. Thus the pink externality has decreased in size, meaning that society is less worse off.

An example of negative advertising, on an Australian cigarette packet.