In the 1920s, British astronomer Sir Arthur Eddington became fixated on a curious coincidence regarding a huge ratio number, 10^40 2
The ratio of the electric force to gravitational force (presumably a constant), is a large number (about 10^40), while the ratio of the observable size of the universe (when reaching an equilibrium radius, see later in this article) to the size of an elementary particle is also a large number, surprisingly close to the first number (also about 10^40) 3 It is hard to imagine that two very large and unrelated numbers would turn out to be so close to each other. Why are they?
1937, Dirac wrote a 650-word letter to the journal Nature. The letter considered the number 10^40, the ratio of the strength of the electromagnetic force to the gravitational force. But Dirac compared this number to the ratio of the radius of the universe to the radius of a proton. The results were very similar, certainly similar enough to convince Dirac that there was a connection. It is certainly unusual to find such a huge number in science, and even more surprising to find approximately the same number arising from two different calculations. As John Barrow said: "There must exist some undiscovered mathematical formula linking the quantities involved. They must be consequences rather than coincidences." This is called the Dirac large numbers hypothesis.
The Dirac large numbers hypothesis (LNH) is an observation made by Paul Dirac relating ratios of size scales in the Universe to that of force scales. The ratios constitute very large, dimensionless numbers: some 40 orders of magnitude in the present cosmological epoch. According to Dirac's hypothesis, the apparent similarity of these ratios might not be a mere coincidence but instead could imply a cosmology with these unusual features. 1
The ratio of the strength of the electromagnetic force to the gravitational force is 10^40. The modified gravity hypothesis (MGH) suggests that the universe has a certain equilibrium radius, and the universe will expand until it reaches that radius.
So the strength of the electromagnetic force controls the size of the atom, and the strength of gravity determines the size of the universe. We are dealing with objects which are in an equilibrium state, meaning the forces
holding them together are precisely equal to the forces pulling them apart. This results in an equilibrium radius for the object, and it is this equilibrium distance which interests us. The MGH suggests the universe has an equilibrium radius in much the same way that an atom has an equilibrium radius.
Gravity dominates the universe and determines its size in much the same way as the electromagnetic force determines the overall size of atoms. It is as though the universe is a scaled-up atom!
Neil Turok considered this simplicity in his 2015 talk at the Perimeter Institute called The Amazing Simplicity of Everything
"The astonishing thing about recent discoveries in physics is that they tell us the universe is surprisingly simple and regular, on the tiniest scale and on the hugest scale. It's only complicated in the middle. To first approximation, the universe is absolutely uniform in all directions. The whole universe is as simple as the simplest atom. If you think about a hydrogen atom, how many numbers do you need to describe an atom? An atom is a pretty simple thing: you have a nucleus, you have an electron going around it, you have the force of electrical attraction between the nucleus and the electron. Well, it turns out to describe the universe you need just one number. That number describes the universe — fewer numbers than you need to describe a single atom. So the universe turns out to be the simplest thing we know."
At these two extremes of scale — the atom and the universe — we find similar situations. We find objects in stable, equilibrium situations dominated by a single particular force. This results in simplicity at the two extremes of scale.
The equilibrium radius which reveals deep truths about the universe. The ratio of this radius of the universe, AND the radius of an atom ARE BOTH 10^40. This is a prediction of a well-founded hypothesis, constructed from
first principles, a hypothesis which makes predictions and agrees with known measurements.
So far, we have calculated the value for the ratio of the size of the universe to the size of an atom, and found the value equal to 10^40. Crucially, this value does not necessarily have to alter with time. For the next step, Dirac considered this 10^40 value and found it was the same as the ratio of the strength of the electromagnetic force to the gravitational force. Is this a just a crazy coincidence? Or could the radiuses of the universe and an atom be related to the strengths of the electromagnetic and gravitational forces in some way?
Firstly, considering the atom. What forces control the size of the atom? Well, the strong force is short range, confined to the nucleus, and so does not play a role. And gravity is too weak at these scales. So it is the electromagnetic force which controls the overall shape of the atom, holding electrons in orbit around the nucleus.
Secondly, considering the universe, the electromagnetic force tends to cancel in atoms, positive charge equalling negative charge, so large objects become electrically neutral. Hence, the electromagnetic force does not play a role in shaping the universe. It is gravity which is the dominant force in the overall size of the universe, even though it is by far the weakest of the four forces. Gravity dominates because all mass has the same gravitational charge (there is no such thing as negative mass). Hence, for very large objects, the force of gravity steadily accumulates until it becomes the dominant force.
The problem of fine tuning is one of the biggest embarrassments facing modern physical and biological science. These “coincidences” may be indicating the existence of some deep, underlying unity involving the fundamental constants, linking the microcosm to the macrocosm just as the ancients saw without mathematics. 3 The anthropic principle just states the obvious: “We are here because we are here.” It has little explanatory power.