The Equations of Cosmology

It is going to be initially suggested that the weight of authority underpinning modern cosmology rests on just 3 equations and 1 metric of expanding space. The formal derivation of these equations is said to be grounded in the general theory of relativity, which is based on a set of nonlinear partial differential equations. Unfortunately, these equations are complex and not easy to solve or understand, although it is worth highlighting that in weak gravitational fields and at relatively low speed, general relativity is required to converge to the same basic results of Newton's law of universal gravitation .


In some ways, this leaves the door open to a more ‘classical’ discussion of the equations of cosmology, which might be of benefit in terms of understanding some of the basic assumptions that now underpin much of accepted cosmology. Of course, those already well versed in the mathematics of general relativity may question the validity of some of the following discussions, especially where some of the accepted conclusions of modern cosmology might appear to be challenged, e.g.

To what extent is the mathematics of general relativity like a computer program, i.e. garbage in, garbage out?

While it is admitted that any authoritative discussion of the mathematics associated with general relativity is still beyond the ability of this website, an introduction to this subject is provided under the heading of ‘Overview of GR Mathematics. However, the question above is not actually challenging the mathematics, as it is only attempting to highlight that logic driven by mathematics is still dependent on the assumptions made about the physical laws at work within the universe, not only in the present era, but across all of time. For example,

Did the universe expand in accordance with the conservation of energy?
Did the universe expand in accordance with the accepted laws of thermodynamics?
Did the universe expand into existing space or did space itself expand?
Could the universe have a centre of gravity?

How these questions are addressed could profoundly affect the outcome of any cosmological model and the answers may not necessarily be found within general relativity alone. For example, it appears that the theory of general relativity makes no explicit statement about the expansion of the universe, only that it cannot be a static system. Therefore, the assumption that cosmology is predicated on general relativity, and defined by its mathematics, may be as premature as some of the other assumptions of cosmology. However, while issues will be raised that appear to question the current weight of authority, the primary goal of this section is simply to present the accepted position of cosmology and defer the discussion of some of the more contentious issues to the final section: The Limits of Inference. So, as indicated above, it might be said that there are 3 equations and 1 metric that underpin most of the basic assumptions of what is sometimes described as the `Hot Big Bangmodel. Collectively these equations are said to define, in mathematical terms, the expansion, density and acceleration of the universe. However, these equations do not account for the process called inflation, which is thought to explain the expansion of the very early universe and therefore this concept will be introduced, separately, in a subsequent section.

  • Friedmann:         as derived from the conservation of energy
  • Fluid:                  as derived from the 1st law of thermodynamics
  • Acceleration:      as derived from the Friedmann & Fluid equations
  • FRW metric:       as related to the Minkowski metric

By way of introduction, the following discussion will now try to outline some issues for consideration prior to addressing the actual derivations based on more classical principles.  From a historical context, Alexander Friedmann was a Russian mathematician who, in 1922, suggested there was not one unique solution to Einstein's equations of general relativity; but rather there was a set of possible solutions. This set of solutions could then be used to underpin a range of different cosmological models of the universe based on a set of given assumptions. However, even today, Friedmann’s equation is still described as one of the most important equations in cosmology and while the original derivation was rooted in the mathematics of general relativity, by changing mass-density to energy-density, these equations can still be derived from Newtonian physics. It is appropriate to first introduce Friedmann's equation, in its accepted form and then outline the basic inference between the Hubble parameter [H], the density [ρ] of space and the assumption of an expanding universe.

[1]      1

We can see that it is the definition of the Hubble parameter [H=v/d] that contains the inference to the universe not being a static system, as there is an implicit suggestion of a velocity [v]. However, the equation itself makes no actual statement as to whether the direction of this velocity is associated with expansion or contraction. Only later were the redshift findings of Slipher and Hubble taken to infer that all galaxies were generally receding and that the universe had to be expanding. In addition to the Hubble parameter [H], the density of space [ρ] and the gravitational constant [G]; there are a number of other parameters that require some initial clarification. The parameter [a] is often described as a scaling parameter, while [k] is referred to as the spatial curvature linked to the geometry of the universe. It was Einstein who originally added the term known as the cosmological constant [Λ] in order to retain a static universe, which is now often excluded from the common form. While the meaning of [k] is explained below, it is assumed to be a numeric number that has no units, which then helps define the units of [a]. Examination of [1] required that each term resolves to units of [1/s2], which then requires [a] to have the units of metres and, in the context of [1], might initially be thought to be reflective of the radius of the universe. So while the value of [a] is a function of time in an expanding universe, i.e. a=r(t), the basic scope of [1] still reflects a snapshot result at some point in the time evolution of the universe.

  • k=0, if the density of the universe is equal to a critical value, the universe is assumed to expand forever, but at an ever-decreasing rate. 

  • k>0, if the density is high enough, the gravitational attraction is assumed to eventually stop the expansion, causing the universe to collapse back on itself in what is sometimes called the `big crunch`.

  • k<0, if the density is too low, the universe is assumed to simply expand forever, as there will not be enough gravitational attraction to stop the expansion.

If we run a little ahead of ourselves, observations in the present era appear to support the assumption that the universe can be generally described as spatially flat, i.e. k=0. If we also ignore the requirement of the cosmological constant [Λ], introduced to maintain a static universe, the Friedmann equation would reduce to:

[2]      2

So what does this equation tell us about the universe?

Well, in this form, the main variable that will affect the velocity [v] is the energy-density [ρ] of space. However, the ability of [ρ] to expand or contract the universe, as a whole, very much depends on the assumptions we might wish to make about the homogeneous energy-density and the nature of space it occupies. As already outlined, the Hubble parameter [H] can be defined in terms of a velocity [v/d] divided by distance, i.e. metres/second per metre. However, this definition can also be transposed into a description of the expansion of space itself with time, i.e. metres/metre per second. Within this difference lies a key issue of debate that will be touched on throughout many of the remaining discussions:

Did the material universe expand into pre-existing space or did it create space?

If we assume the universe was finite within the concept of some form of singularity, where nothing can exist outside; then space must have been created or expanded. The only other reasonable alternative being that the expansion of our material universe, as defined by the physics of the energy-density [ρ], took place within pre-existing space. At this stage, we shall simply table this debate for further discussion and say that the standard model generally assumes the velocity [v] to be a recessional velocity based on measured redshift [z] of objects, e.g. a galaxy, from the relationship [v=zc], where [c] is the speed of light. It is highlighted that this basic relationship needs to be modified as velocity [v] approaches [c], but see ‘Redshift and Hubble's Lawfor more details, while the issue of determining the  distance [d] is discussed under the heading ‘Cepheid Variables & Hubble’s Constant . While Cepheid stars do provide a relative distance measure, an absolute reference is still required for calibration and this has proved to be very difficult. As such, the distance required by the Hubble Constant [H] can only be given within certain limits of accuracy. Even so, an approximate estimate of [H] allows a number of other important estimates to be made:

[3]      3

The first of the terms in [3] above shows [H] being described as a recessional velocity [v] with distance [d]. However, the units of distance can be cancelled out leaving an inference to some measure of time, i.e. [1/t], which might suggest some correspondence to the age of an expanding universe. Equally, having determined an estimate of [H] from observations, the critical density [ρC] of a homogeneous universe can be calculated from [2]:

[4]      4

The inclusion of the units in [4] clarifies that both [4] and [2] are still presenting the density [ρ] in terms of a mass-density, which actually needs to be multiplied by [c2] to become an energy-density, i.e. E=mc2:

[5]      5

Such issues are typical of much of the apparent complexity surrounding cosmology, which stems from the use of various systems of units, including some that normalise constants, such as [c, G, h], to unity. While there may be good reasons for the use of some systems, it can make the process of learning more difficult as it leads to the same equation appearing in multiple forms. So while some equations might be considering density in terms of matter-density, the early universe is assumed to have been dominated by radiation, which aligns more readily to the concept of an energy-density. Therefore, irrespective of the units used, the present discussion will always be inferring an energy-density, which can be resolved by the presence or absence of [c2]. At this point, we need to introduce a second equation that helps define the wider scope of energy-mass density. This equation is called the `Fluid Equation`.

[6]      6                        

The equations in [6] are again indicative of the complexity caused by presenting the equation in a number of different, but equivalent forms. The first terms of both [3a] and [3b] represent the rate of change of density with time using the differential dot notation. While [3c] highlights that the units of the Hubble constant [H] reduces to the reciprocal of time [1/t] that then suggests density [ρ] must falls with time, which appears consistent with the idea of an expanding universe. Actually, this is a very important assumption, which seems to be supported by the laws of thermodynamics and the general idea of entropy, which will also be touched upon within the derivation sub-sections. The final term contains both density [ρ] and pressure [P] and so, at this point, it might be useful to add a note about the relationships between these 2 quantities.

[7]      7      

We can see that in terms of units there is a similarity between pressure [P] and energy-density [ρE]. However, as previously highlighted, the Friedmann and Fluid equations are still based on the definition of mass-density [ρM], which is resolved by the conversion factor [c2]:

[8]      8         

However, while pressure and density can be related to the same units, the definition of pressure [P] involves a weighted number [ω], which is referred to as the ‘equation of state’. Exactly how [ω] is determined is discussed in more details under the heading ‘Pressure & Equations of State, but for now we will simply introduce some examples:

Matter    w =0
Radiation    w =1/3
Dark Energy    w <0

In the context of the whole homogeneous universe, the density of matter is very low, in the order 10-20 particles per cubic metre. To get a better visual perspective of this scale, if these particles were scaled up to the size of a beach ball, then a neighbouring particle would not even be in our solar system. Therefore, cosmologist often see matter in homogeneous space as acting like `dust` in that it exerts no pressure, as the probability of collision is virtually zero, hence w=0. In theory, matter and radiation are both just forms of energy, but in practice, matter moves at very slow speeds compared to the speed of light. Therefore, most of the energy content of matter is its mass-energy, i.e. E = m0c2, and the associated kinetic energy of motion is negligible, i.e. v<<c. In contrast, photons move at the speed of light and have zero rest mass, but still have non-zero momentum. In general, the total energy of a particle with non-zero mass can be approximated by the equation:

[9]      9

However, as [v] approaches [c], the second term begins to dominate due to the effects of special relativity. Particles with relativistic velocity, e.g. photons, exert pressure given by the equation: P=ρc2/3. If we assume that the total mass of dust in a homogeneous universe is directly proportional to its volume, this makes the density [ρ] inversely proportional to the cube of its radius [1/a3]. By the same argument, radiation is also inversely proportional to volume and therefore its energy density must also reflect the [1/a3] factor, although its overall energy density must take account of another factor. As the universe expands, the energy associated with radiation, i.e. E=hf, is reduced as the expansion is also assumed to increase the wavelength of the radiation and so reduces its energy by a further [1/a] factor. Therefore, the radiation energy density becomes inversely proportional to [1/a4] by this argument. At this point, we are only introducing the `Acceleration Equation` because it is thought to generally define the accelerating expansion of the universe:

[10]    10

The term on the left is equivalent to acceleration over distance [a/d] using derivative dot notation, where [ρ] is the density and [P] is pressure. We might get an initial understanding of [10] in the context of a dust-dominated universe, as [P] goes to zero:

[11]    11

As such, we have reduced [10] to the form in [11], which is possibly reflective of the Newtonian equation for gravity, although [11] suggests that the acceleration is, by default, negative. However, a negative acceleration would oppose the velocity of expansion, i.e. it suggests that expansion must be slowing down. While this appears to be a very logical assumption, let us just stop for a moment to consider some of the parallels with Newtonian gravity:

[12]    12

Basically, the form of the first equation in [12] can be transposed into [2] by simply substituting for [M=ρV], where [V] is the volume of the homogeneous density [ρ]. However, there is nothing in either [2] or [12] that can explain why the universe expands rather than contracts under gravity. Of course, there could be some fundamental differences in the geometry of an expanding homogeneous universe, as suggested by Newton’s Shells, but ultimately any normal mass-density must either contract under gravity towards its centre of mass or experience no net gravity due to an absence of any centre of gravity. So let us rephrase the earlier question:

Does the universe have a centre of gravity?

Well, according to the assumptions of the cosmological principle, the answer would appear to be no. Today, many models attempt to explain the evolution of structure within the universe based on local perturbation in the uniformity of the energy-density, which then causes localised gravitational centres, e.g. solar systems and galaxies. If so, it is possible that the apparent slow-down in the expansion rate of the universe, as a whole, might have little to do with gravitation and more to do with the conservation of energy as implied in the thermodynamics of the Fluid equation. In fact, if we simply re-arrange the Fluid equation, as in [13], we might realise that the recession velocity [v], implied by [H], has to be ultimately explained as a function of density and pressure, which we have not yet really addressed in any detail.

[13]    13                           

We might also see that the acceleration equation in [10] reflects a similar dependency on both the density and pressure, which we might reasonably assume changes as a function of time within an expanding universe. However, before pursuing the questions being raised within this preliminary introduction, we should possibly first present some form of derivation for each equation in turn.