John Bruce Davies, Ph.D.


Dept. of Physics (retd),

University of Colorado, Boulder, CO

Copyright 2012, 2013, 2014 John Bruce Davies All Rights Reserved

Essay awarded Honorable Mention in the Gravity Research Foundation 2014 Awards for Essays on Gravitation

Submitted 28 March 2014


General relativity governs the large-scale gravitational properties of continuum space-time, while the particle and field history of the universe is governed at the microscopic level by quantum mechanics and component phase transitions. We show that these models correlate in their timing and descriptions of important Universe properties and events. The time of decoupling of radiation from matter, resulting in the relic CMBR, coincides with the continuum Universe expanding through its own Event Horizon. Next, the effect of increasing energy density as we go back in time forms a singularity in the continuum and coincides with the cosmic phase change of breaking of electroweak symmetry and we examine whether Dark Matter could be a relic of this age. At the earlier time when the Higg’s Field is postulated to give mass to the quantum system, we investigate the effect of the Uncertainty Principle and whether Dark Energy is the relic from this phase transition. Using our extension of FLRW to the earliest Universe, we show that an exponentially accelerating solution approaches a flat Universe for any initial condition, resulting in an equation of state similar to that of Dark Energy.

Radiation/Matter Decoupling

At an age of about 350,000 years the Universe ended its era where matter and radiation were so tightly coupled that we can see no further back into these dark ages. This quantum prediction implies a slow transition from one dominating component, photons, to the next, matter. In the continuum model of expanding space-time, the matter density decreases inversely with the square of time so that the Universe must originally have developed from a singularity, with an associated Event Horizon later. Intuitively, the concept of an Event Horizon binding photons is remarkably similar to that of a coupled radiation/matter Universe. For a mass like our Sun, the Event Horizon, where all radiation is bound, would be about 3km, so that for a Universe mass of 1023 solar masses, its Event Horizon was about 3*1026 m. Assuming the decoupling and Event Horizon coincided, then expansion of the Universe since the redshift of about 1100 implies a present size of about 3.3*1029 m. The best estimate of the size of the observable Universe is about 9*1026 m, our result implying that the total Universe is nearly 370 times in size. The observed CMBR is thus the relic of the original radiation that decoupled from matter as the Universe expanded through its Event Horizon, a relatively slow cosmic phase transition.

Hadron Epoch

As we go back further in time the energy density of the Universe increases. At about 10 -6 secs, the quantum particle model predicts the start of the hadron era with separation of the weak and electromagnetic forces resulting in the particle/field structure we have today. This occurs at approximately the density of 1017 gm/cc which is the density of neutron stars that have similar constitution. Self-gravitating bodies of higher density than this become singularities and/or black/white holes. Theoretical equations of state of self-gravitating bodies also develop singularities at this density, Davies (1980). Assuming a present Universe energy density of 10-29 gm/cc and age of 4*1017 secs, the inverse square density-time relation yields the time of the singularity at about 10-6 secs, in agreement with the hadron epoch initiation. The rapid cosmic phase transition at this time may also have left a relic of its prior state, whose main interaction properties would be gravitational and non-electromagnetic. Just as in rain condensing out of a more massive cloud, only a fraction of the total prior energy may have condensed into hadrons, the left-over relic energy fitting the major observed properties of Dark Matter.

Higg’s Field

Prior to the separation of strong, electromagnetic and weak forces, the quantum model predicts that the various particles aquire their mass through manifesting of a Higg’s Field at about 10-12 secs. For the large-scale model at these extreme densities, beyond the singularity or inside the hole, we cannot be sure that continuum concepts as space-time, metric and energy density have relevance. However, at these extreme energies and brief times, the quantum model must obey the Uncertainty Principle, ΔE.ΔT > h/2π. Assuming the Higg’s initiates at about 10-12 secs and with a volume of about (ct)3 , we have a density value of about 10 -29 gm/cc. We thus are lead to identify this quantum relic of the mass-giving phase transition with that of Dark Energy, which has equation of state and nearly constant energy density similar to the pre-Higgs original energy in the earliest Universe, as we show below.

Earliest Universe

One of the most important and least explained of the properties of our present Universe is why it is so flat for so long. Also, an early burst of exponential inflation is postulated by quantum models and has recently received observational proof. We extend the FL­­RW equations to show that these previously-unexplained properties can be related through a self-consistent causal continuum Universe. The fundamental equations governing the expansion of the scale-factor R(t) of the Universe is:

(1) Rt2 = 8πG/3.ρR2 + Λ/3.R2 - k

where G is the Gravitational Constant, ρ is the total energy density of the Universe, Λ is a cosmological constant term and k=0,1 or -1 so we do not impose either a flat, closed or open Universe. The second fundamental equation for this standard model governs the acceleration of the scale-factor:

(2) Rtt = -4πG/3.(ρ+3P)R + Λ/3.R

where P is the pressure. Einstein obtained his equation relating Scale Factor acceleration with energy density based on the assumption of conservation of energy, only approximately valid in the matter-dominated era. This assumption forced Einstein to invent a hypothetical cosmological constant. As we specifically do not assume conservation of energy, we thus set Λ=0 and investigate only the effect on acceleration of the pressure and density. Differentiating (1), which removes any effect of the k term, dividing by Rt and equating with acceleration in (2), yields the equation relating pressure to density, its gradient and Universe expansion:

(3) P = -ρ – ρt.(R/3Rt)

Thus, in this FLRW Universe, the pressure depends not only on density but also on the rate of Universe expansion and the rate of energy density lost or gained, ρt. As ρt => 0, P => -ρ, which is also the observed equation of state of Dark Energy. This is the equation of state of a nonlinear material and is similar to the viscous equation of fluid flow when (R/Rt) is constant. For such materials the solutions are usually exponential decays or growths when impulse sources are initial conditions.

The acceleration is obtained by substituting pressure P from equation (3) into (2), i.e.

(4) Rtt = 4πG/3.(2ρ + Rρt /Rt).R

From this relation (4) we can obtain the conditions for acceleration or deceleration of the universe expansion .We see that any energy density input or generation that is faster than the energy density decrease due to expansion, ρt ≥ 0, contributes toward acceleration. The point at which acceleration changes to deceleration is when (2ρ + Rρt /Rt) = 0, i.e. acceleration if ρt ≥ -2ρRt/R and deceleration otherwise. This implies that when ρ α R-3, we get deceleration, as expected and observed during the matter-dominated expansion.

Inflation Solutions

As an accelerated expansion is postulated by quantum models to be of cosmological importance in early times, we look for solutions where we take R = Ro exp(λ t), with λ constant in the simplest case, and where Ro is the initial Universe scale factor at t=0. Note that starting at a finite time is essentially just changing by scale the size of the initial Scale Factor. Substituting for R and its time derivatives in (4), we get:

(5) λ2 = 4πG/3 (2ρ + ρt/λ)

which gives a first order differential equation for density:

(6) ρt + λ2ρ - λ3.3/4πG = 0

This is solved by inverting and integrating:

(7) λ ∫ dt = ∫ dρ/( λ2.3/4πG - 2ρ)

So that we get:

(8) λt = -1/2 ln( λ2.3/4πG - 2ρ) + constant

which gives:

(9) ρ = λ2.3/8πG + C exp(-2λt) ; ρt = -2λ C exp(-2λt) ; R = Ro exp(λt)

where the constant C is chosen to satisfy boundary conditions. Independent of initial conditions, the energy density approaches a constant for large time, t >> 1/λ , this value corresponding to that of a flat Universe.

We examine three main possible starting situations; the Big-Bang, with initial super-high density decaying rapidly; in (9), at t=0, C is large and positive implying that the energy density gradient starts out large and negative;

(10) t=0 : ρ = λ2.3/8πG + C ; ρt = -2λ C ; R = Ro

The initial zero-density solution: the Big-Bubble, at t=0, C is negative implying that the energy density gradient starts out large and positive

(11) t=0 : ρ = 0 => C = -λ2.3/8πG ; ρt = -2λ C ; R = Ro

The constant-density solution: C=0, and the energy density gradient is zero:

(12) ρ = λ2.3/8πG ; ρt = 0 for all R

In all cases, the energy density at sufficiently long time approaches a constant, equal to that of a flat Universe, and ending with negligible energy density change. From (3), we see that the equation of state in this flat limit is P => - ρ , which is similar to that observed for Dark Energy. That the effective end of this initial intrinsic inflation is always close to a flat Universe, as observed since the time of decoupling, gives confidence in this simple solution for the earliest energetic continuum FLRW Universe.


These correlations between the continuum and quantum models allow a deeper understanding of the cosmic phase transitions, their properties and long-term effects on our present Universe. As a retired but active research physicist, I have empathy for these Relics of the Dark Ages; the CMBR, Dark Matter and Dark Energy.


Davies, J.B. (1980) : Honorable Mention, Gravity Research Foundation Essay Competition.