Alternative to the Epoch Time Solution Patching Structure of the Cosmological Standard Model in the Friedman Dust Universe with Einstein's Lambda
| Scientific Paper | |
|---|---|
| Title | Alternative to the Epoch Time Solution Patching Structure of the Cosmological Standard Model in the Friedman Dust Universe with Einstein's Lambda |
| Read in full | Link to paper |
| Author(s) | James G Gilson |
| Keywords | Einstein, universe, time, cosmological constant, General Relativity |
| Published | 2010 |
| No. of pages | 12 |
Read the full paper here
Abstract
The need for the cosmological constant, Lambda, in Einstein's field equations to be an absolute mathematical constant over all the time that they are used to describe some astrophysics process is demonstrated. Only if that condition holds will the conservation laws of mass and momentum hold as in classical physics. The Friedman equations that can be deduced rigorously from general relativity are consequently equally restricted to a constant valued Lambda and for the same reasons. However, the standard cosmological model, is not constructed from one solution of the Friedman equations but rather from at least three different but rigorous solutions patched together at times where they are physically thought to join. This is because the known solutions are thought to represent different conditions of mass movement, highly erratic or thermal at time near the big bang or more particle like and organized into systems at time near now, just to mention two types of activity when there obviously could be a continuous range of activities of mass types. Clearly this idea of how things have evolved after the big bang is very plausible, if the big bang idea is accepted as fact. The apparent need to patch solutions together over time creates great mathematical difficulties for cosmology because the three functions selected have to join smoothly which is the same as saying that they have to be differentiable not once but twice if accelerations are taken into account as they must be if the Friedman equations are to hold through the join. It is not clear whether or not this patching process can be rigorously achieved. However it is clear that the big bang concept does violate Einstein's field equations at t = 0 because this concept implies that mass and momentum comes from nowhere. It is shown that all of these problems can be removed by introducing a continuously variable over time structure into the definition of temperature for the dust universe model. This only affects the value of the temperature that is associated with a given time and make no difference to the validity of the dust universe model with regard to it being a rigorous solution to the Einstein Field equations for all time from minus infinity to plus infinity.
Overview
This 2010 paper by James G Gilson of Queen Mary University of London is a short methodological argument about how cosmological models are built rather than a new solution of the field equations. Gilson makes two claims. The first is that the cosmological constant Λ appearing in Einstein's modified field equations must be an absolute constant — not merely slowly varying, and not a function of time — because only then does the Λ term satisfy the same covariant conservation condition that the Einstein tensor and the stress-energy tensor satisfy identically. The second is that the standard cosmological model violates this requirement, and violates ordinary conservation of mass and momentum as well, because it is not a single solution of the Friedman equations at all but three physically distinct solutions — an inflationary phase, a radiation-dominated phase and a matter-dominated phase — "time wise patched together" at epochs chosen on physical rather than mathematical grounds.
Against this Gilson sets his own dust universe model, introduced in A Dust Universe Solution to the Dark Energy Problem and developed in later papers, which is a single closed-form solution of the field equations with constant Λ valid, in his words, "for all time from minus infinity to plus infinity". The obstacle to using it as a full cosmology had been an assumption he calls the strong assumption: that the thermal mass of the cosmic microwave background and the remaining mass of the universe are each separately conserved, which freezes the universe into a single matter character. The paper's contribution is to drop that assumption. Gilson shows that the radiation mass can be made an arbitrary function of time without disturbing the solution at all, because it enters only the definition of temperature and nothing else. The universe can then change its "matter character type" continuously as it evolves, and no patching is needed.
The argument
Λ must be an absolute constant
Gilson begins with Einstein's 1917 modified equations,
Rμν − ½gμνR + gμνΛ = −κTμν,
together with the two identities Gμν;μ = 0 and Tμν;μ = 0. Both are first-order tensor (vector) equations obtained by contracting one index of a second-order tensor through differentiation, and both express conservation of energy and momentum. Since the geometry side and the matter side each conserve separately, consistency requires the added term to conserve as well: (gμνΛ);μ = 0. Expanding, and using the fact that the covariant derivatives of the metric vanish, this reduces to gμν(Λ);μ = 0. Λ is never taken to depend on the space coordinates, so only the time derivative survives, and because the covariant derivative of a scalar is the ordinary derivative the condition becomes simply
∂Λ/∂t = 0.
Gilson's conclusion is blunt: "If a space or time variable Λ is used in Einstein's field equations their physical mathematical validity is totally compromised. Lambda is rightly referred to as The Cosmological Constant."
The patching problem in the standard model
He then applies this test to the standard picture. Inflation supposedly involves "a massive value for the cosmological constant which clearly cannot be matched with the small values following at later times", so the constancy condition fails outright at the earliest epoch. At the later joins — radiation to matter, deceleration to acceleration at some tc — the requirement is more subtle but still severe: the pieced-together scale factor must be not merely continuous but twice differentiable, since the Friedman equations involve accelerations and must hold through the join. Gilson does not claim to prove this is impossible, only that "it seems to me that this smooth connection scenario is not mathematically demonstrably achieved", and that the Big Bang itself violates the field equations at t = 0 by having mass and momentum "come from nowhere".
The dust universe and the strong assumption
In the dust model the total non-dark-energy mass is split as MU = MΔ + MΓ, where MΓ is the thermal mass of the cosmic microwave background and MΔ is everything else. MU is taken as strictly conserved. The original strong assumption held MΔ and MΓ separately constant too — a restriction Gilson candidly says he imposed "in the mistaken belief that it made progress with the theory possible", and whose removal was urged on him by Professor C. W. Kilmister. The relevant solution quantities are
ρ(t) = (3/(8πG))(c/RΛ)2 sinh−2(3ct/(2RΛ)),
T(t) = ±(MΓ3c4/(8πa(RΛ)2MUG sinh2(3ct/(2RΛ))))1/4,
T4(t) = ±MΓc2ρ(t)/(a MU),
where a here is the radiation constant and RΛ the length scale set by Λ. The third relation follows from the first two, and with MΓ constant it gives the simple result that the ratio of temperatures at two epochs is the fourth root of the ratio of densities: (T(t1)/T(t2))4 = ρ(t1)/ρ(t2).
Releasing the radiation mass
If instead MΓ is an arbitrary function MΓ(t), that relation acquires a factor
Γ(t1,t2) = MΓ(t1)/MΓ(t2), with Γ(t1,t2) = Γ−1(t2,t1),
and the whole modification of the earlier theory reduces to the single substitution rule (T(t1)/T(t2))4 → Γ(t2,t1)(T(t1)/T(t2))4. The key observation is that MΓ appears nowhere except in the temperature, so making it time-dependent "can have no effect on ρ(t)" and the solution's status as a rigorous solution of the field equations is untouched. The only constraints are MΓ(t) ≤ MU for all t and MΔ(t) = MU − MΓ(t); otherwise the function is free, to be chosen by the user of the theory to fit "any theoretical or measured continuous time sequence of mass character". Gilson concludes that a single solution with infinite time range and continuous adaptability is "obviously a great advance" over a patching scheme confined to three finite time ranges.
Assessment
The paper is unusually clear about what it is doing, and the mathematics it does present is correct. The conservation argument for constant Λ is the standard textbook one and is stated cleanly. The dust solution itself is genuine general relativity: ρ ∝ sinh−2(3ct/(2RΛ)) is exactly the matter density of a flat Friedman universe containing dust and a constant Λ, with RΛ2 = 3/Λ, and the prefactor 3c2/(8πGRΛ2) is precisely the dark-energy density Λc2/(8πG). Substituting ρ into T4 = MΓc2ρ/(aMU) reproduces equation (4.3) term for term, so the three displayed formulae are mutually consistent. Gilson is also refreshingly honest in retracting his own earlier "strong assumption" and naming the colleague who pushed him to it.
The difficulties lie in what the freedom actually buys. The paper's own argument for why MΓ(t) can be released — that it "occurs only in the temperature" and affects nothing else — is simultaneously the reason the release is empty. In this model the CMB makes no contribution to the dynamics at all; it is a label attached to the solution, not a source in the field equations. Allowing that label to vary with time therefore does not produce a radiation-dominated era, it only relabels a dust era. The standard model's radiation phase is not a bookkeeping convenience: radiation energy density gravitates, and its a−4 scaling is what sets the expansion rate during primordial nucleosynthesis and fixes the acoustic peak structure of the CMB. A model in which radiation is dynamically inert has removed the phenomenon rather than explained it.
The characterisation of the standard model as three solutions "patched together" is also not accurate as a description of modern practice. The Friedman equation with ρr + ρm + ρΛ on the right-hand side is one ordinary differential equation integrated continuously; the radiation-dominated and matter-dominated power laws are asymptotic approximations to that single solution, not separate solutions requiring joins. Gilson's twice-differentiability objection applies to the approximations, not to what is actually computed. Inflation is a fair target — it does involve an effective Λ far larger than today's — but inflation is normally driven by a scalar field in Tμν, not by the geometric Λ term, and Gilson's own conservation argument does not reach it. The same escape applies to variable-Λ cosmologies generally: move Λ to the matter side as a vacuum fluid and ∇μTμν = 0 permits time dependence provided the vacuum exchanges energy with matter. That option is not discussed.
A more specific difficulty follows from the paper's own equations. With MΓ constant, T4 ∝ ρ ∝ a−3, so T ∝ a−3/4 and the temperature would scale as (1+z)3/4 — in conflict with direct measurements of the CMB temperature at redshift, from fine-structure excitation in quasar absorption systems and from the Sunyaev–Zel'dovich effect in clusters, which follow (1+z) to a few per cent out to z ≈ 3. The new freedom can repair this, but only by choosing MΓ(t) ∝ 1/a, so that T4 ∝ a−4. That choice makes MΓ grow without limit toward early times, and it collides with the paper's own constraint MΓ(t) ≤ MU once a/a0 falls below the present radiation-to-matter mass ratio — that is, at roughly the redshift of matter–radiation equality, the very epoch the extension was meant to cover. The free function is thus not as free as stated: fixing the one observable it governs exhausts it, and the bound (4.13) then fails exactly where a radiation era is required.
Finally, the claim that the model is valid "from minus infinity to plus infinity" should be read carefully. ρ(t) diverges as sinh−2 → ∞ at t = 0, so the density singularity is not removed by extending the time axis; it is mirrored into negative t. What the paper legitimately claims is that no mass is created — MU is constant throughout — which is a real conceptual difference from a Big Bang origin, but it is a statement about the mass content, not about the curvature or density singularity. Readers should note also a bibliographic slip: references [33] and [35] are different papers given the same arXiv identifier.