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The Cosmological Constant

A universe governed by a universal attractive force cannot remain stable. Matter in it must either fly apart until stopped by the attractive force whereupon it must fall back in on itself, or it must fly apart indefinitely. This is as true of a Newtonian Cosmology as it is of a relativistic one. At the time general relativity was produced, the observable stars showed no evidence of either expansion or collapse. It was widely thought that the universe should endure from everlasting to everlasting. In order to make this possible Einstein made a modification to the field equation, by including a repulsive force to balance gravitational attraction. The only simple modification preserving tensor properties is to add a term proportional to the metric,
The new term must have a value of Λ sufficiently small not to modify gravitational effects observed in the solar system, but plays a role on cosmological scales. Unfortunately it does not lead, as Einstein had hoped, to a static universe, but to an unstable solution, in which the slightest local variation leads to the start of expansion.

In the 1920’s Edwin Hubble made observations on distant galaxies showing that expansion was a fact, formulated as Hubble’s law in 1929. Einstein removed the cosmological term, but in the 1990’s analysis of distant supernova showed a redshift-magnitude relation which cannot be explained by standard general relativity without a cosmological constant. Most cosmologists now accept the reality of the Cosmological constant.

Nonetheless, no theoretical reason for the existence of the cosmological constant has ever been presented. We should be suspicious of a physical law without a physical cause. Fudge factors to make equations fit data are often indicative of a deeper underlying fault in theory. The existence of such a fault in general relativity is known. As seen in the mismatch of Alf and Beth’s Spacetime diagrams, and Einstein himself pointed out, the Levi-Civita connection is not consistent with classical electromagnetism. It seems unreasonable that one would be try to modify the general principle of relativity. We might suspect, as did Einstein, that the Levi-Civita connection is not correct, and that a different redshift law might result from changing it. The teleconnection reconciles general relativity with quantum theory and with quantum electrodynamics, and gives a redshift law consistent with Supernova Redshifts without the requirement of a cosmological constant.

Weyl’s Postulate

The general principle of relativity enables us to define time from clock processes locally, and to claim that a similar definition of time is always possible anywhere in the universe, with the exception of possible singularities. We require a further assumption in order to discuss universal time, or concepts like the age of the universe. This issue was addressed by Herman Weyl in 1923. Weyl argued that in order to discuss the distant we should base our ideas, in so far as is possible, on what we can observe in our own neighbourhood.

An observer can use the radar method to define synchronous surfaces in his neighbourhood with respect to his own proper time.

From the concept of proper time, or time as measured by an observer moving with a clock, we may define Earth time, i.e. proper time for the Earth. Time measured by atomic clocks on satellites is adjusted several times daily to remain synchronised with Earth time.

Although we cannot use radar on the Sun, It remains meaningful to define a synchronous time for the solar system, using adjustments for gravitational and Doppler shifts as required.

Similarly, given the motions of, and distances to, galactic stars, it is meaningful to discuss proper time for the galaxy, or for the local group.

Weyl’s postulate assumes that the overall motion of many galaxies may be likened to a “cosmic fluid”, with each galaxy moving on a geodesic from a point in the finite or infinite past, in such a way that synchronous slices of proper time for each galaxy can be combined to form synchronous slices in the cosmic fluid. These synchronous slices are labelled by cosmic time. Although cosmic time is defined globally, it is not a fundamental global property in the sense of Newtonian absolute time, but rather consists of the synchronisation of proper time for particles in the cosmic fluid.

The Cosmic Fluid

A perfect fluid is characterised by three quantities, defined at each point in the fluid: 4-velocity, u^a, and two scalars, density, ρ, and pressure, p. The most straightforward assumption for the stress-energy tensor is that it is the sum of a part due to inertial motions, and a part due to pressure,
for some symmetric tensor S^ab. The only rank 2 tensors associated with the fluid are u^au^b and the metric, g^ab. The simplest assumption is that, for some constants λ and μ,
The law of local energy-momentum conservation says that the stress-energy tensor should satisfy
We require that it reduces to the continuity equation and to the Navier-Stokes equations in the appropriate limit. From this we find λ =1 and μ = −1. Then the stress-energy tensor is

The Cosmological Principle

The cosmological principle states that in each era as defined by Weyl’s postulate, discounting local variations, the general behaviour of matter is similar in every part of the universe. More precisely, the cosmological principle incorporates assumptions of homogeneity and isotropy.


The cosmological principle is not open to direct empirical test, since we can only observe a small part of the universe. Cosmological models have been developed which do not obey it. However, no effects contravening it have been observed and it gains some support from the degree of homogeneity and isotropy in the microwave background. Clearly the universe is not homogeneous on the scale of the Solar system, or of the Milky Way, and nor is it homogeneous on the scale of the local cluster or supercluster of galaxies. On much larger scales the distribution of matter does appear uniform. The scale on which the matter distribution becomes uniform may be seen in the rotating slice movie, courtesy of the 2dFGRS team. On the scale of the slice, galaxies appear as specks of dust, bearing in mind that the region plotted, up to redshifts, z ≈ 0.2, is small compared to the universe as a whole.

Spaces of Constant Curvature

It follows from homogeneity and isotropy, together with Einstein’s field equation that, on the large scale, the universe has constant curvature. This means that any plane in tangent space can be mapped onto a space of constant curvature in such a way that scaling distortions are removed. If the universe has positive curvature in the space coordinates, then any synchronous plane in tangent space can be mapped onto a sphere. This does not mean that space is really wrapped around a sphere, but that the metric field, is an identical function on tangent space to that of a spherical geometry. The point of contact, or “North pole” is completely arbitrary; tangent space can be drawn at any point and the map to a sphere is essentially unchanged.

Using polar coordinates (r, φ) to describe any plane, with an origin at the observer, where is determined from lightspeed, the metric locally is
This simply states that the distance associated with a small coordinate change dr is dr, and the distance associated with a coordinate change dφ; is rdφ. A flat space in two dimensions has this metric globally, multiplied by a constant scale factor.

The extension of this metric on a space of uniform positive curvature is identical to the metric of an imagined sphere. Radial coordinate distance is mapped to an arc of angle ρ on a sphere of radius a, so that r = aρ. A small arc of angle dρ corresponds to radial distance adρ, and a coordinate change dφ corresponds to distance asinrdφ. Thus, in ρ-φ coordinates, the metric is
a is called the scale factor, and gives a measure of the scale on which curvature is apparent. In some old accounts a was called the radius of the universe, but this terminology becomes meaningless for spaces of zero or negative curvature. Uniform negative curvature brings in √−1. Instead of sinρ we have sinhρ = isin(iρ).
This can be checked by an explicit calculation of the Riemann tensor. Let f(ρ) = sinρ, ρ, or sinhρ for positive, zero and negative curvature, respectively. Then the three possibilities can be written in one,
Now consider three dimensions, using spherical coordinates (ρ, θ, φ). At a zenith angle of θ = 90°, small changes, dρ, dθ and dφ, in ρ, θ and φ lead to distance changes, dρ, af(ρ)dθ and af(ρ)dφ. So, the metric at θ = 90° is,
Rotate the plane through angle 90 − θ° about φ = 0 (blue to red; the position of φ = 0 is arbitrary, so there is no loss of generality). After rotation, there is no change to distances in the ρ or θ directions, but the distance associated with a change dφ is reduced by a factor sinθ. The metric is
This is usually written in the form,
where ds is the proper distance due to small changes in coordinate position, (dρ, dθ, dφ). This form of the metric is known as the line element, since it gives the length of a small displacement in any direction.

Cosmological Expansion

The introduction of a time coordinate, t, allows a to vary in time; a : t → a(t). With a (+, −, −, −) signature, the metric is
After the substitution, adτ = dt, the metric in τ−ρ coordinates is
In these coordinates the radial speed of light is 1. Homogeneity and isotropy require that the particles of the cosmic fluid (galaxies) remain on lines of constant ρ (a change in ρ would define a prefered direction). A Penrose diagram plots the Universe in τ−ρ coordinates.


In the Penrose diagram we can see the meaning of the scale factor more clearly. At each point the diagram scales by the factor, a(t), relative to locally Minkowski coordinates. Cosmological expansion is seen as the “shrinking” of matter in inverse proportion to a(τ), which, in turn, is due to the changing rate of cosmic time with respect to τ.

In conventional distance units, r = a(t)ρ. The metric in t-r coordinates is
which reduces to Minkowski metric for
Given the scale factor as a function of time, the universe may be plotted relative to locally defined distances (i.e. such that the size of galaxies does not change).


In a positive curvature universe, each plane with the observer at the origin is the geometrical equivalent of a sphere of radius a(t). If a is increasing with t, the universe expands relative to locally defined coordinates based on cosmic time. Note that the central galaxy in the figure is quite arbitrary. Any galaxy could be chosen as central, and the overall picture would be unchanged. Expansion takes place for the universe as a whole, relative to local distances in which the size of galaxies remains fixed. It does not make sense to talk of expansion locally, because locally we can only talk about scale relative to local reference matter. Empty space cannot be measured. We can talk of the expansion of the universe, but the expansion of space is an empirically meaningless concept. It does not make sense to talk of expansion locally.

Within general relativity, one coordinate system is as good as another. The Penrose diagram and the expanding universe diagram have equal status as descriptions of reality. It is as valid to say that the universe expands relative to the size of galaxies as it is to say that local distances shrink with respect to the universe. However, the Penrose diagram has the important feature of retaining the constancy of the speed of light, according to which local coordinate systems are defined in practice, and shows the paths of matter as straight lines (ignoring local gravity), respecting Newton’s first law. It will gain an increased significance when the teleconnection is defined in relational quantum gravity. We may be encouraged to think that the Penrose diagram, with its picture of a universe of constant size is in some sense the more fundamental view of the universe. In this view curvature and universal attraction are understood as effects of the changing speed of clocks with the scale factor.

Cosmological Redshift

Inertial radial motions are locally constant with respect to the inertial motions of galaxies, hence they are constant everywhere and are straight lines on a Penrose diagram. Because radial paths of inertial objects and of light are straight lines, parallel transport is given by translation. A vector representing the momentum of a signal from a distant galaxy to the origin is shown in red on the diagram. A local displacement vector, showing the size of a galaxy, is shown in cyan. The relative length of these vectors is changes proportional to the scale factor, a(t). Thus a signal from a galaxy sent at cosmic time t, and detected on Earth at t₀ (now) is subject to a cosmological redshift proportional to expansion,
At time t, the distance, r(t), to a galaxy at current distance coordinate r₀ = r(t₀)


The Hubble parameter is not a constant of Nature, but varies over cosmological timescales. It may be taken as constant in our era.


We have
For nearby galaxies, the distance coordinate may be identified with distance (this is not a useful, or even very meaningful, measure of distance for a galaxy far enough away that the universe expands appreciably in the time taken for light to travel from that galaxy). Differentiating gives Hubble’s law, for the speed, v, of recession of nearby galaxies,

The Friedmann Equation

In 1922 Alexander Friedmann solved Einstein’s field equation for a homogeneous isotropic cosmology (calculation).


All but two of the solutions of the Friedman equation contain an initial singularity, known as the Big Bang, and which gives its name to big bang cosmology. Density, ρ includes both matter density and energy density, including radiation. Observation shows that the universe is matter dominated, and has been since it was about 1⁄10 000 of its current size, 350 000 or so years after the big bang. In this case mass conservation requires that ρa³ is constant:
It is convenient to define the cosmological parameters
Then Friedmann’s equation is
Setting t = t₀ gives the identity
Thus, for no-Λ models, Ω = 1, or ρ_c = 3H² ⁄ 8πG, is critical density, above which the universe has positive curvature and is finite, and below which it has negative curvature and is infinite. Experiments in observational cosmology are much concerned with determining the values of the cosmological parameters, H₀, Ω, Ω_k and Ω_Λ.

Friedmann Models

Cosmological models in which expansion is governed by the Friedmann equation are called Friedmann models, or FRW models in honour of Robertson and Walker who worked on the problem some ten years after Friedmann. The Friedmann equation may be solved for different values of the cosmological parameters. (Solution). Near the big bang, all models are approximated by the Einstein-de Sitter, flat space, no-Λ model.


Using parallel transport of photon momentum under the Levi-Civita connection, there is strong evidence from supernova data and WMAP for the “concordance model” (magenta), a flat space model with Ω ≈ 0.27 and Ω_Λ ≈ 0.73. Using the teleconnection, supernova data favours an “Einstein preferred” model (cyan) with positive curvature, Ω_Λ ≈ 0 and Ω ≈ 2.

Timeline for Processes from the Big Bang

High energy processes in elementary particle physics can be investigated empirically on earth using particle accelerators. Based on what is found we can calculate the processes taking place in the high energy densities near the Big Bang. Big bang models based on general relativity get a great deal of empirical support, not only from the the observation of the cosmic microwave background (CMB), but also from the observed ratio of hydrogen to helium and other light elements. This is a prediction of Big Bang nucleosynthesis (BBN), and is critically dependent on the age of the universe, which determines the rate of expansion near the big bang and the length time during which free neutrons were able to decay to protons.



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