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Opinion Mathematics 29 September 2016
4 minute read 

The big baffling number at the heart of a cosmic coincidence

One huge ratio helps explain two very different phenomena. Paul Davies explains.

The number 1040 has fascinated scientists throughout history
Jeffrey Phillips

Scientists are used to dealing with very large and very small numbers. Take the age of the universe, for example. Dated at 13.8 billion years old, it has existed for a hundred thousand times longer than Homo sapiens. At the opposite end of the number spectrum, the rapid speed of atomic and subatomic processes are measured in tiny slivers of time. It takes light a mere trillion-trillionth of a second to cross an atomic nucleus.

The ratio of these two time scales, macro and micro, is itself a very big number – about 1040. In the 1920s, British astronomer Sir Arthur Eddington became fixated on a curious coincidence regarding this huge number.

An atom of hydrogen – the simplest atom of all – consists of one electron, negatively charged, bound by electric forces to a positively charged proton. But there is a tiny gravitational attraction between these two particles as well. The ratio of these two forces is also about 1040.

Why should the two numbers – one concerning the age of the universe, the other the strength of two fundamental forces of nature – be so nearly the same? Is there something that connects them?

Years later, physicist Paul Dirac came up with a possible explanation. He pointed out that the age of the universe is not a fixed number, but grows over time. At one second after the Big Bang, for example, the time scale ratio was not 1040 but 1024. Dirac thought it was too much of a coincidence that humans just happened to live when the two ratios were about the same.

He was convinced there was an unseen link between the two. As the universe changes over time, Dirac proposed, the ratio of gravitational and electric forces must change along with it. Each year, he suggested, gravity weakens by about one part in 10 billion.


When Dirac published his theory in 1937, it became the subject of much discussion. If the force of gravity really did decline with time, it would have a profound effect on the structure of stars and galaxies. It would also mean that the orbits of the planets grow slowly larger; Earth, for example, would gradually move away from the Sun. But at the time, there was no accurate way to test this prediction.

A few decades later, in the 1960s, Princeton astrophysicist Robert Dicke offered a very different explanation for the big number coincidence. The starting point for his theory was the connection between biological evolution and the evolution of the universe.

The current age of the universe now, he reasoned, is not some random moment; it’s the moment when life has evolved enough to produce beings able to measure it. Dicke wondered what the pre-requisites were for intelligent life in the universe, and how that might relate to the mystery of 1040.

All known life is based on the element carbon. But carbon did not exist in the early universe; it was formed later by nuclear processes inside stars. When large stars grow old and die, they explode, and their life-encouraging carbon can end up in the next generation of stars and their planets. That meant, Dicke reasoned, that life in general, and intelligent beings in particular, could not exist until at least one generation of stars had lived and died.

Stars are made mainly of hydrogen – a form of nuclear fuel – held together by gravity. They burn steadily through this fuel until it is spent and the energy released has radiated away.

If gravity were stronger, the stars would consume their nuclear fuel faster, because they would be squeezed harder and burn brighter. But a star only dies once all the heat produced from burning its fuel has radiated into space, and its escape depends on electromagnetism: photons formed deep inside the stars have to plough through a thick soup of electrons and protons, scattering this way and that.

Dicke showed that the lifetime of a typical star hinges precisely on the ratio of electric to gravitational forces in the hydrogen atom.

The upshot of Dicke’s calculation is that the current age of the universe is about the same as the ratio of electric to gravitational forces. If gravity were weaker, stars would live longer and carbon-based life would have emerged later. And if gravity were stronger, life would have arisen sooner.

So there is a simple explanation for the otherwise baffling big-number coincidence – without the need for Dirac’s new theories of gravitation. A 1976 NASA mission to Mars helped prove it.

While the Viking mission’s primary aim was to look for life, it produced a useful by-product. By accurately timing the radio signals between the landers and Earth over several years, astronomers could determine whether the orbit of Mars was slightly changing.

By 1984, it was clear that it wasn’t. Dirac died the same year. The theory that he loved was laid to rest at the same time.

While Dicke’s argument tells us why the two ratios should be roughly the same, it says nothing about why those ratios are so big – that is, where the number 1040 comes from. Scientists may be adept at handling huge numbers, but this one continues to baffle even the biggest brains.

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Contrib pauldavies.jpg?ixlib=rails 2.1
Paul Davies is a theoretical physicist, cosmologist, astrobiologist and best-selling author.

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