← Back to Contents

1.3: Over-view of the Universe

1.3.1 Expansion of an Energy Continuum

The idea that energy is a substance in its own right rather than a property of matter and electromagnetic radiation is central to the idea of an energy continuum, and it is the energy continuum which is curved and expanding. The space occupied by the continuum is to be viewed as simply the 'carrier' of energy and the properties of curvature and expansion are those of the energy which it contains. Changes in the continuum are mediated by a single, independent, time variable which acts, simultaneously, throughout the whole continuum, as the agent of change.

However, in a large scale analysis of the energy continuum, it is convenient, initially, to assume that the shape of the continuum is a perfect hypersphere. Hence, curvature has the same value at each point, and energy density is uniform throughout. In this sense, it is an ideal universe which is empty since it is assumed that the objects we perceive as being in the universe are regions of higher than average energy density. The virtue of this ideal universe is that it provides a theoretical basis from which deviations from the ideal can be expressed.

A convenient definition of curvature will be given in a later section, here the main concern is with the behavior of total energy density and volume throughout time. These can be found by differentiating the expression for total energy, \(E_u\), with respect to time. We have that \begin{align} E_u = V (t) \rho (t) \nonumber \end{align} where \(V(t)\) is the volume of the universe, and \(\rho (t)\) is the uniform energy density, both of which vary with time.

Assuming that total energy, \(E_u\), is conserved, differentiating with respect to time, yields, \begin{align} V \frac{\partial \rho}{\partial t} + \rho \frac{\partial V}{\partial t} =0 \nonumber \end{align} and hence \begin{align} \frac{1}{\rho} \frac{\partial \rho}{\partial t} = -\ \frac{1}{V} \frac{\partial V}{\partial t} \nonumber \end{align}

If it is assumed that volume, \(V\) and density \(\rho\), are not connected in any other way, then each is related to some function, \(K\), such that, \begin{align} \frac{\partial \rho}{\partial t} = -K \rho \nonumber \end{align}
and
\begin{align} \frac{\partial V}{\partial t} = -K V \nonumber \end{align}

Though theoretically, \(K\) could be made a function of time, doing so would involve complexities which are not found in reality. It is, therefore, convenient to assume that \(K\) is a constant. In the next section it will be shown that \(K\) is related to \(H\), Hubble's constant. Meanwhile, assuming that \(K\) is constant, we can integrate the above expressions with respect to time, which yields \begin{align} \rho(t) =&\ \rho_o\ e^{-Kt} \nonumber \\ V(t)=&\ V_o\ e^{Kt} \nonumber \end{align}

Where \(V_o\) and \(\rho_o\) are volume and energy density respectively at a time designated to be \(t=0\).

This shows that the volume, \(V(t)\), of the ideal universe increases exponentially, while the energy density, \(\rho(t)\), decays exponentially, with the same index, \(K\). It is important to note that \(V\), is curved volume.

The exponential expansion index, \(K\), can be linked to the Hubble constant, \(H\), as follows. Consider a small volume of energy in the form of a cube with sides equal to \(x\) meters. If we assume that at time \(t=0\), \(x = x_o\ e^{Ht}\). According to Hubble's Law \begin{align} \frac{\partial x}{\partial t} = x_o H e^{Ht} \nonumber \end{align}

so that volume will expand according to \begin{align} \frac{\partial (x^3)}{\partial t} = x_o^3\ 3H e^{3Ht} \nonumber \end{align} hence \(K= 3H\).

The above analysis shows that if energy in the universe is conserved, then the energy density and volume of the whole universe changes according to \begin{align} \rho(t) = \rho_o\ e^{-3Ht}\ \ \ and\ \ \ V(t)= V_o e^{3Ht} \nonumber \end{align}
Where \(V_o\) and \(\rho_o\) are volume and energy density respectively at a time designated to be \(t=0\).

This confirms the theory developed in a previous section that, given the existence of a large, closed volume, of conserved energy, exponential expansion of this energy in the shape of a hyper-sphere is a natural process.

The ideal universe, described above, is by its very nature, an empty universe. The presence of mass and radiation in the real universe can be considered as imperfections in the otherwise, smooth and symmetrical, energy continuum. Mass in the universe is almost entirely confined to galaxies, which appear to be both source and sink of radiation.

The total volume of the universe, assuming a radius of \(1.22\ 10^{26}\) meters, is about \(10^{80}\) cubic meters, and is said to contain about one trillion, (\(10^{12}\)), galaxies. Our Milky Way galaxy has a volume of about \(10^{60}\) cubic meters, and assuming that this is a typical galaxy its share of universal volume is about \(10^{68}\) cubic meters. This suggests that, on average, each galaxy is surrounded by one hundred thousand, (\(10^8\)), times more volume of inter-galactic space than it occupies.

← Previous Next →