Articles: Magic Numbers

By Máire McKay, MAC

Martin Rees describes the deep forces that shape the universe in his book 'Just Six Numbers'. He begins by referring to the way we tune in to the radio station we want to hear, and tells us that our universe is much more fine tuned than that. If only a slight variation were made, our whole world would not exist, nor we, of course.

Mathematical laws underpin the fabric of our universe. The 'magic numbers' have been given letters, for ease of reference.

### The first number is crucially important and is equal to 1 followed by 36 zeros

Rees has given it the letter N. This number measures the strength of the electrical forces that hold atoms together, divided by the force of gravity between them. Gravity weakens in proportion to the square of the distance between two masses. If they are twice as far away, the attraction between them is four times weaker. Gravity is responsible for satellites staying in orbit and also prevents stars etc flying apart. (In a star, there is a balance between centripetal gravity and centrifugal pressure of the hot interior) As for the Earth, the weight of the air above the surface is balanced by the pressure at ground level.

Gravity has more dramatic effects on large objects than on small. The bigger they are, the harder they fall, so mice can be dropped from a great height without harm (please don't experiment!) So being the right size is crucial in the biological world. If an animal's dimensions were doubled their legs wouldn't be able to support them. If a tree were too huge, its branches would fall because the tree wouldn't be able to support their weight. It is different in water, which can support great weight e.g. of whales. In the case of small animals, there are different problems. Here there is a large amount of skin in proportion to their weight so they must eat and metabolise fast to keep warm. Similar restraints would be found on other worlds - in the dense atmosphere of Jupiter, for example, beings would have to be balloon-like creatures. So, what would happen if gravity were stronger than it is (i.e. if N were bigger)? In a universe where gravity was 'only' 10 to the 30th (1 followed by 30 zeros) rather than 10 to the 36th feebler than the electric forces, atoms and molecules would behave as in our universe, but objects would not need to be so large before gravity became competitive with the other forces. The number of atoms needed to make a star would be a billion times less in this imagined universe. The masses of the planets would be scaled down by a billion. The strength of gravity would stunt the evolutionary potential, even insects would need thick legs to support them and no animals could get much larger than insects. Gravity would crush our bodies. As well as this, galaxies would form more quickly and would be miniaturised. Stars would be closer together and collisions frequent so no stable planetary systems could be formed because orbits would be disturbed by passing stars. But also, heat would leak away more quickly from these mini-stars and star life would be a million times shorter.

Now imagine a universe where N was smaller. Here, more elaborate and longer-lived structures could develop but they would resemble long spindly things, blown about by air. Nothing as complex as humankind could have developed.

If gravity were far stronger, as seen in neutron stars (ultra-dense remnants left behind after supernova explosions) where gravity would be million times a million Earth's gravity, a pen dropped from a metre high would impact with the energy of one ton of TNT and the object would be instantly squashed by the intense gravity of the surface of the star. Clocks would run 10 to 20% more slowly near the surface than from further away (gravity can dilate Time too) and light from the surface would be strongly curved, so viewed from afar, you'd see a hemisphere and part of the back side of the world. A body a few times smaller or a few times heavier than a supernova would become a Black Hole.

### 2nd Magic Number - Epsilon

The value of E is 0.007 and it defines how firmly atomic nuclei bind together and how all the atoms on Earth were made. Its value controls the power from the Sun and more sensitively, how stars transmute hydrogen and all the atoms of the periodic table. Carbon and oxygen are common, whereas gold and uranium are rare because of what happens in the stars.

Opposites attract and similars repulse and the electric forces react on the (positive) protons and the (negative) electrons. Gravity is extremely weak at atom size, so what prevents atoms flying apart when 2 positive protons come together? Our number 'E', the 'glue' (non-electric) holding atoms together despite strong electrical repulsion. This is the dominant force in the microworld. It holds the protons in Helium and heavier nuclei together so firmly that fusion is a powerful enough energy source to provide warmth from the Sun. If E were weaker, we could not have emerged, for the Sun would not have given enough heat for long enough.

E is 0.007, i.e. the fuel that powers the Sun converts 0.007 of its mass into energy when the Hydrogen fuses into Helium. This number E determines how long stars can live. The amount of energy released when simple atoms undergo nuclear fusion depends on the strength of the 'gluing 'force. It acts only at short range and is only effective on the atomic scale.

If E were say, 0.006, a proton could not be bonded to a neutron, deuterium could not form and the Universe would be all Hydrogen. Stars could still form but would have no fuel so would deflate and cool and end up as cold remnants. There would be no explosions to scatter debris, so no new stars could form and no elements would exist to form planets like Earth.

If on the other hand, E were 0.008, no Hydrogen would have survived from the Big Bang, two protons would be able to glue together directly and none of the Hydrogen would remain to fuel ordinary stars and no water would have existed. A change in E would also alter the periodic table, a weaker nuclear force would lead to impoverished chemistry and a stronger, to more stable heavy atoms. If E were other than 0.007, no carbon-based Universe could exist.

### The Cosmic Number Omega

This measures the amount of material in our universe- galaxies, diffuse gas and 'dark matter'. It tells us the relative importance of gravity and expansion energy in the universe Whether the universe goes on expanding forever, reaches a certain stage then continues indefinitely, or eventually begins to slow down and contract depends on the 'competition' between gravity and the expansion energy. Will gravity halt the expansion? This depends on how much stuff is exerting a gravitational pull. The denser the material, the stronger the gravity. The ratio of the actual density (calculated as 5 atoms to each cubic metre) is crucial to the fate of our universe. This ratio is denoted by the letter Omega. If omega = one, the expansion energy and the gravitational energy are in exact balance.

Mathematicians can calculate the density of visible matter (galaxies, stars etc.) but not of Dark matter, which seems to make up a great deal of the total matter. There have been many suggestions as to just what Dark Matter is but the question has not yet been decided.

However, Omega must have been tuned amazingly close to One in the early universe. If expansion had been too fast, gravity could never have pulled regions together to form stars or galaxies, whereas if the initial impetus had been insufficient, a premature Big Crunch would have quenched evolution when it had hardly begun.

### The 4th Number: Lambda

In 1998 astronomers discovered that the rate of expansion of the universe seemed to be accelerating. This was surprising as it was thought gravity would slow it down. This acceleration implies that there must be an extra force that causes a 'cosmic repulsion' even in a vacuum. This would be indiscernible in the Solar System and our galaxy , but it could overwhelm gravity in the more rarefied environment of intergalactic space. We have to add another 'magic number' to our list to describe the strength of this 'anti-gravity'.

As early as 1917 Einstein postulated this force to keep gravity in check, but then lost interest in it and even called it his 'biggest blunder', but now astronomers are beginning to consider it again. The cosmic number Lambda describes this weakest force in nature which seems to control the universe's expansion and its eventual fate. Fortunately for us, Lambda is very small. If it were much higher than it is, it would have overwhelmed gravity early on and would have prevented galaxies forming. Our existence requires that Lambda should not have been too large.

### The 5th Number: Q

The fabric of our universe depends on this. If our universe had started off completely smooth and uniform it would have remained so throughout its expansion. It would be cold and dull after 10,000 million years, containing only thinly spread Dark Matter and hydrogen and helium so rarefied that there would be less than one atom in each cubic metre, so no galaxies, no stars, no periodic table, no complexity no people.

The presence of very slight irregularities in the early phases were extremely important and made a crucial difference because density contrasts amplify during expansion. Gravity amplified slight ripples in the almost featureless fireball, enhancing density contrasts till over-dense regions stopped expanding and condensed into structures held together by gravity. Stars, galaxies and clusters of galaxies are all held together by gravity How tightly they are bound together ( or how much energy would be required to break them asunder) can be expressed as a proportion of their total 'rest-mass energy' (mc squared) For the biggest structures in our universe the answer is about one part in a 100,000. This ratio of two fundamental energies is called Q and is in the order of ten to the minus five or one over 100,000.

If Q were even smaller, the universe would have been inert and without structures If Q were much larger, the universe would be a very violent place in which no stars or solar system could survive, dominated by vast Black Holes. Any surviving gas would get so hot it would emit intense X-rays and gamma rays.

### The 6th Number: D

D refers to the number of spatial dimensions that exist in our universe, so D = 3 If D = 2 ('flatland') the world would be as a picture on a paper. An object could not have a channel through it (e.g. a digestive tract) without being divided in two. There would be even more limited scope in one dimension ('pointland') Time is our '4th dimension ' but is different in that we seem to be dragged only one way( forward) whereas in spatial dimensions we can move E or W; N or S; up or down Einstein taught us that Space-Time is united but his ideas retain a distinction between Time and Space, between what's in Space and what lies in the past or future. Notice also that these 3 spatial dimensions refer to our 'common-sense world'. Astronomers have postulated many more dimensions e.g. in Super String theory, however these are not yet reconcilable with past 3+1 dimension theories, so we'll leave them aside for now.

As far as we know, life couldn't exist if D=2 or 4.

### Conclusion- Is there a connection between these numbers?

At the moment we can't be sure, we just know that if they were different even by a small amount, we wouldn't be here. Of course, there may be more universes beside our own, multiverses even, where these numbers are different, but we are not there to see them!