Cosmological Coincidences.

Seen from Earth, the Universe is a great Sphere, centred on the Earth, with a radius stretching across the vast distance of inter-galactic space, all the way to the Big Bang, which is the Sphere’s circumference.

This Sphere is expanding as the Universe ages, so the circumference is becoming more distant in space the further back in time it becomes.

How old is the Universe? According to measurements carried out by the NASA Goddard Space Flight Center’s Wilkinson Microwave Anisotropy Probe (WMAP), the Universe is 13.73 billion years old.

The mass of the Universe, it turns out, can be calculated from an equation,

M = m_PL⁴/m_pm_nm_e = 8.79674 × 10⁵² kg ,

where m_PL is the Planck mass, (ħc/G)^½ = 2.1767 × 10^-8 kg, and m_p, etc., are the masses of the proton, neutron and electron, respectively.

It is a simple matter to check this. The Schwarzschild radius of the Universe (the radius the Universe would have, if it were an enormous black hole) is the same as its actual radius, on the condition that the value of something called the total density parameter, Ω_TOT, is exactly equal to 1.

This radius can be expressed as a time, so on that basis the age of the Universe should be equal to 2GM/c³ (given that the Schwarzschild radius, R = 2GM/c² = ct, with t being the age of the Universe).

A quick calculation suffices to show that, if the above equation for the mass is correct, then t = 4.357 × 10¹⁷ s, which is 13.8 billion years.

That’s pretty close to the WMAP figure, which has a degree of uncertainty of ±0.876%, or ±120,000,000 years (see: http://map.gsfc.nasa.gov/).

The question is, why is this equation right? What connection is there between the mass and age of the Universe and the masses of the sub-atomic particles, and the values of the Dirac constant, the speed of light in vacuo and the Newtonian gravitational constant?

One would expect to see some connection, at least between those last two and the the Universe’s mass. The ‘total density parameter’, which determines the large-scale geometry of space-time, is given by the Friedmann Equation:

H₀² = 8πGρ/3 – kc²/a² + Λc²/3 .

Here, H₀ is the Hubble parameter at the present epoch, G is the gravitational constant, ρ is the baryonic + dark matter density of the Universe, Λ is the cosmological constant, k is the space-time curvature parameter (–1 for Bolyai-Lobachevskian, or hyperbolic, space-time; 0 for Euclidean, or ‘flat’, space-time; and +1 for Riemannian, or spherical, space-time), and a is the scale factor, which is taken to be 1. The above is re-arranged, thus:

Ω_TOT = 8πGρ₀/3H₀² = Ω_B + Ω_D + Ω_Λ – Ω_k ,

where Ω_B is the ‘baryon’ (i.e., ordinary) matter density parameter of the Universe, Ω_D and Ω_Λ are the dark matter and ‘dark energy’ density parameters, respectively, and Ω_k is the space-time curvature density parameter.

The value of Ω_Λ is currently estimated to have a value of 0.72 – in other words, 72% of the mass of the Universe is thought to be comprised of dark energy, generated by the action of the cosmological constant. Some 23.3% of the remaining mass is dark matter, and 4.6% ordinary atoms (‘baryonic’ matter + electrons).

The critical density, ρ₀ (ordinary + dark matter + dark energy, less [negative] curvature density), is ~9.4 × 10^-27 kg m^-3, and the Hubble parameter is ~70.8 km s^-1 per megaparsec (a parsec is 3.2616 light years, or 3.0857 × 10¹³ km, so 1 megaparsec is 3.0857 × 10¹⁹ km, and H₀ = 2.2944 × 10^-18 s^-1; the reciprocal of H₀ is the Hubble time, τ₀, which is 13.82 billion years, roughly the age of the Universe; see: http://wmap.gsfc.nasa.gov/universe/uni_expansion.html).

Putting the numbers in above, we have:

Ω_TOT = 0.99817 = 0.72 + 0.233 + 0.046 – 0.00083 ,

which gives us the value of the space-time curvature density parameter, Ω_k = (-) 0.00083. If this is true, and if Ω_TOT < 1, then the curvature parameter k = –1, and space-time is Bolyai-Lobachevskian, i.e., open. With a positive non-zero value for Λ, its expansion is also accelerating over time, so that the past – the Big Bang – will retreat to infinity at a steadily increasing rate.

Long before then, the Sun will have died, as eventually all stars will die, and all black holes will evaporate, via Bekenstein-Hawking radiation.

It is a bleak prospect, but there are literally trillions of years to go before then, so there is really no point in worrying about it!

Of much more immediate concern is the fate of the Sun, which will become too hot for us, even without the greenhouse effect, in a few hundred million years from now. Of course, not only we not be here, but the human race might not be here – the dinosaurs, all the many different classes, genuses and species of them, lasted for hundreds of millions of years, and genus Homo has only been around for 2 million thus far. As for sub-species H. sapiens sapiens, we’re johnny-come-latelys, who have only been around for ~200,000 years, 0.0044% of the age of the Earth, or 0.00145% of the age of the Universe.

It is a sobering thought. However, as Blaise Pascal said, in the Pensées (1670, No.347), L’homme n’est qu’un roseau, le plus faible de la nature; mais c’est un roseau pensant. (‘Man is a reed, the weakest in nature; but he is a thinking reed.’)