## Abstract

The coincidence problem among the pure numbers of order near 10^{40} is resolved with the Raychaudhuri and Friedmann–Robertson–Lemaitre–Walker equations and a trivial relationship involving the fine structure constant. The fact that the large number coincidence occurs only in the same epoch in which other coincidences among cosmic parameters occur could be considered a distinct coincidence problem suggesting an underlying physical connection. A natural set of scaling laws for the cosmological constant and the critical acceleration is identified that would resolve the coincidence among cosmic coincidences.

## 1. Introduction

Eddington (1931) and Dirac (1937) hypothesized that some unidentified physics was responsible for the ensemble of pure numbers of order near 10^{40} that could be generated from the parameters of the Universe. While a host of explanations have been offered, the problem of the large number coincidence (LNC) has persisted over the decades. Dirac suggested that the LNC would be resolved if the Newtonian gravitation constant were to vary in time (Dirac 1978). Modern discussions of the coincidence have included considerations of anthropic selection mechanisms (Carter 1974) and the holographic *N*-bound conjecture (Mena 2002). However, the coincidence problems among the parameters of nature have only multiplied since the time of Dirac and Eddington with the discovery of other, apparently separate coincidences among cosmological parameters that seem to occur only this epoch.

In §2 of this letter, the LNC is resolved using physical scaling laws from the standard cosmological model that were not known as such in the early twentieth century. However, there is a lingering problem associated with the LNC in that, it occurs only in this epoch in which other coincidences involving cosmological parameters occur. In §3, this new problem is discussed and natural scaling laws for the cosmological constant and the critical acceleration are identified that would resolve the problem. In the rest of this section, the terms constituting the LNC are presented.

Most of the large pure numbers of order near 10^{40} involve cosmological parameters, such as the mass, radius and age of the observable Universe. According to the interpretation of data from the recent Wilkinson Microwave Anisotropy Probe (WMAP) observations, the Big Bang occurred around *T*_{0}≈4.3×10^{17} s ago (Spergel 2003). (Only two significant figures will be employed in this article.) The conformal time, *T*_{0}, corresponds to a cosmic particle horizon *R*_{0}≈1.3×10^{26} m. This distance, *R*_{0}, is the term traditionally used to represent the size of the observable Universe (Lineweaver & Davis 2005). The present average density, *ϵ*_{0}, of energy in the observable Universe is very close to the critical density ≅*c*^{2}9.6×10^{−27} kg m^{−3}, where *H*_{0}≈2.3×10^{−18} s^{−1} is the present value of the Hubble parameter (Spergel 2003). Matter comprises approximately *Ω*_{m}≈27% of the total density and baryonic matter comprises just *Ω*_{b}≈4.4% of the total (Spergel 2003). Some form of vacuum energy, perhaps due to a cosmological constant, *Λ*, comprises *Ω*_{Λ}≈73% of the total cosmic energy density (Spergel 2003). Given that space-time is flat on the largest scales, the total mass, *M*_{0}, of the observable Universe can be calculated as which gives *M*_{0}≅2.4×10^{52} kg.

In the time of Dirac and Eddington, at least four pure numbers of order near 10^{40} were associated with the LNC,(1.1)(1.2)(1.3)(1.4)where *c* is the speed of light in a vacuum, *e* the unit of fundamental electric charge, *G* the constant of gravitation, *m*_{n} and *m*_{e} the masses of the nucleon and electron, respectively, and *r*_{e} the classical radius of the electron. Note that since baryonic matter comprises just *Ω*_{b}/*Ω*_{m}≈16% of all matter equation (1.4) is not exactly the square root of the baryon number of the observable Universe. Since the time of the early investigations, at least three other pure numbers of order near 10^{40} have entered the discourse (Gornitz 1986),(1.5)(1.6)(1.7)where *λ*_{n}≡*h*/(*m*_{n}*c*) is the Compton wavelength of the nucleon, the Planck mass and the reduced Planck constant. The terms in equations (1.1)–(1.7) will be considered to constitute the LNC.

## 2. Resolving the large number coincidence

Three seemingly external equations are found to reduce the gallery of large pure numbers in equations (1.1)–(1.7) to just three that do not constitute a coincidence problem. Two of the reducing equations follow from the Raychaudhuri equation. The Raychaudhuri equation is an important formulation of the motion of cosmological matter that is consistent with the general theory of relativity, but may be derived independently (Raychaudhuri 1955). During the era of matter dominance, the Raychaudhuri equation leads to the scaling law,(2.1)where *H* is the Hubble parameter and *T* the conformal time associated with the co-moving particle horizon *R*. Equation (2.1) is still roughly satisfied in this epoch, since the era of matter dominance ended only recently (cosmologically speaking) and it is one of the equations employed to resolve the LNC.

A second resolving equation is obtained from equation (2.1) and some basic relationships from the standard cosmological model. According to the Friedmann–Robertson–Lemaitre–Walker equations, the Hubble parameter in a Universe with zero curvature is related to the average total energy density *ϵ* by (Bergstrom 1999)(2.2)During the era of matter dominance, the total energy density is *ϵ*≈*c*^{2}*ρ*_{m}, where *ρ*_{m} is the density of matter. Using equation (2.2) to express the density of matter in terms of the Hubble parameter, the total mass *M* contained within the observable Universe would be *M*=*R*^{3}*H*^{2}/(2*G*). With a substitution from equation (2.1), the following scaling law is obtained that is also the second relationship needed to resolve the LNC:(2.3)Like equation (2.1), (2.3) is still roughly satisfied in this epoch even though the energy density of the Universe is no longer matter-dominated. The left side of equation (2.3) is approximately equal to 1.2×10^{16} m^{2} s^{−2} at this time and its proximity to *c*^{2} was regarded as a suggestive coincidence among large numbers before it was known to be a scaling law and before detailed knowledge of cosmological parameters was available (Bondi 1961; Sciama 1969).

The third equation employed to resolve the LNC is most probably a simple coincidence,(2.4)where *α*≈7.3×10^{−3} is the fine structure constant. In the context of large pure numbers of order near 10^{40}, the factor of 10 may be ignored and it can be said that the fine structure constant is roughly of order near the ratio of the electron mass to the nucleon mass.

Using equations (2.1), (2.3) and (2.4), the gallery of large pure numbers in equations (1.1)–(1.7) may be reduced as follows. First, equation (2.4) causes the Compton length of the nucleon to have the same order of magnitude as the classical radius of the electron, and the term in equation (1.7) thus follows from equation (1.1). Equation (1.2) also follows immediately from equation (1.1) due to the equation (2.1). Equation (1.3) follows from equations (1.1), (1.4) and (2.3). Finally, equation (1.1) follows from equations (1.4), (1.5), (2.3) and (2.4). The only remaining irreducible large pure numbers of order near 10^{40} are therefore those in equations (1.4)–(1.6),(2.5)The group of terms in equation (2.5) does not constitute a coincidence problem, and the LNC can be said to have been resolved with equations (2.1), (2.3) and (2.4). Dirac and Eddington were therefore correct to hypothesize that some implicit physics was involved in generating the coincidence problem. Note that the terms in equation (2.5) may be connected by some yet-unspecified physics. For instance, the Eddington–Weinberg relation,(2.6)follows from equations (2.1), (2.3) and (2.5), and the holographic *N*-bound conjecture may provide some insight towards explaining that relationship (Mena 2002). Also, anthropic selection mechanisms could perhaps be identified that would influence the relationships in equation (2.5) or others among cosmological parameters.

## 3. The coincidence of cosmic coincidences

It is important to note that, while the LNC is resolved with equations (2.1), (2.3) and (2.4), the numerical coincidence problem associated with it could occur only in this cosmological epoch. When considered in conjunction with other coincidence problems that are unique to this time, that fact may create a distinct coincidence problem. It is considered remarkable that, only in this epoch, the energy density of matter in the Universe is of the order of the vacuum energy density attributed to the cosmological constant *Λ*. This coincidence is known as the cosmic coincidence and may be expressed as(3.1)where the term on the left is approximately 2.3×10^{−10} J m^{−3}. The cosmological constant is approximately *Λ*≈3.9×10^{−36} s^{−2} and corresponds to a vacuum energy density *ϵ*_{vac}=3*Λc*^{2}/(8*πG*)≅6.2×10^{−10} J m^{−3}. A fundamental scaling that is responsible for the value of *Λ* has not been derived.

So, this cosmological epoch could be special for at least two presumably distinct reasons: the LNC and the cosmic coincidence occur only in this epoch. It may be more reasonable to hypothesize that an underlying physical connection is implicit in the simultaneous occurrence of two, presumably distinct coincidence problems among similar terms rather than to stipulate that only the chance is responsible. A scaling law that could explain the coincidence of cosmic coincidences is(3.2)That is to say that the energy density associated with the cosmological constant may be scaled to the gravitational energy density of the nucleon mass confined to a sphere, whose radius is the Compton wavelength of the nucleon. The right side of equation (3.2) is approximately equal to 3.7×10^{−31} s^{−2}, but replacing *λ*_{n} with *bλ*_{n}, where *b*∼10, could reasonably account for the difference between the right and left sides. The relationships in equation (2.5) would follow from equation (3.2) and the cosmic coincidence, and the LNC would thus naturally result from the cosmic coincidence. Also note that equation (3.2) follows immediately from equation (2.6) and the fact that *H*_{0}∼*Λ*^{1/2}, which results from equation (2.2) and vacuum-dominance.

There may be yet another curious coincidence among cosmological parameters, which is unique to this epoch. The observed motions of clusters of galaxies and material within the galaxies may be interpreted to indicate that the laws of dynamics deviate from Newtonian models at accelerations smaller than some critical acceleration *a*_{0}∼10^{−10} m s^{−2} (Milgrom 1983). It so happens that the Hubble acceleration, *H*_{0}*c*, is of the order 10^{−10} m s^{−2} only in this epoch. With the substitution *H*_{0}∼*Λ*^{1/2}, the coincidence of the critical acceleration may be expressed in the more suggestive form,(3.3)The coincidence, *a*_{0}∼*H*_{0}*c*, is well known (Milgrom 1983). The point of this present analysis is that, insofar as it represents a problem, the coincidence of the critical acceleration generates a distinct coincidence problem in that it occurs only in the same epoch in which the cosmic coincidence and the LNC occur. However, if the critical acceleration were scaled to the characteristic gravitational acceleration of the nucleon mass at its Compton length,(3.4)which follows from equations (3.2) and (3.3), then the coincidence of the critical acceleration would follow from the cosmic coincidence. The right side of equation (3.4) is approximately 6.3×10^{−8} m s^{−2}, but as with equation (3.2), replacing *λ*_{n} with *bλ*_{n}, where *b*∼10 could reasonably account for the difference between the right and left sides. The proposed scaling laws in equations (3.2) and (3.4) may also be linked to the holographic *N*-bound conjecture (Mena 2002).

## Acknowledgements

This work benefited from discussions with Lloyd Knox, Moti Milgrom, Tom Mongan and Joe Tenn.

## Footnotes

- Received January 30, 2006.
- Accepted May 19, 2006.

- © 2006 The Royal Society