Question

In Example 2.32, \(a(t)\) was determined for a curved universe with nonrelativistic matter for \(\Omega_{0}>1\). Derive the parametric equations for \(\Omega_{0}<1\), $$ \begin{aligned} &a=\frac{\Omega_{0}}{2\left(1-\Omega_{0}\right)}(\cosh \eta-1) \\ &t=\frac{\Omega_{0}}{2 H_{0}\left(1-\Omega_{0}\right)^{3 / 2}}(\sinh \eta-\eta) \end{aligned} $$ for \(\eta \geq 0\)

Short answer

The required parametric equations are \( a = \frac{\Omega_{0}}{2(1-\Omega_{0})}(\cosh \eta - 1) \) and \( t =\frac{\Omega_{0}}{2 H_{0}(1-\Omega_{0})^{3/2}}(\sinh \eta - \eta) \).

Step-by-step solution

Recall the Friedmann Equations

The Friedmann equations, which are derived from the Einstein Field Equations, describe the evolution of the universe. For a universe filled with nonrelativistic matter, the scale factor \(a\) evolves according to the equation: \( \frac{da}{dt} = \sqrt{\frac{8\pi G \rho}{3} a^2 - k} \), where the universe has a curvature \(k\), and \(\rho\) is the density of the universe.

Introduce the Density Parameter \(\Omega_{0}\)

The density parameter \(\Omega_{0}\), is defined as the ratio of the actual density to the critical density, which is the density required to make the universe flat. The equation becomes: \( \frac{da}{dt} = Ha\sqrt{\Omega_{0} a^{-3} - (1-\Omega_{0})} \), where \(H= \sqrt{\frac{8\pi G \rho_{c}}{3}}\) is the Hubble constant.

Redefine Time and Scale Factor

To solve the above equation, redefine time and scale factor in terms of a new variable \(\eta\), choose \(a = \frac{\Omega_{0}}{2(1-\Omega_{0})}(\cosh \eta - 1)\), and \(t =\frac{\Omega_{0}}{2 H_{0}(1-\Omega_{0})^{3/2}}(\sinh \eta - \eta) \). This makes the differential equation easier to solve.

Solve for \(\eta\)

By differentiating these expressions with respect to \(\eta\), you can eliminate \(t\), leaving an equation for \(a(\eta)\), which can be solved to obtain \( \eta(a)\).

Key concepts

Friedmann Equations

The Friedmann equations are the cornerstone of understanding the evolution of universes in cosmology. They originate from the Einstein Field Equations, which are the mathematical manifestation of the theory of General Relativity describing the fabric of spacetime. In a cosmological context, the Friedmann equations particularly help us understand how the universe expands over time.

For a universe filled with nonrelativistic, or 'normal' matter, the equations relate the growth rate of the universe's scale factor, denoted by \(a\), to the total energy content of the universe. One critical element is the density parameter \(\Omega_0\), which compares the actual density of the universe with the critical density necessary for a flat universe. When \(\Omega_0<1\), as in the presented exercise, we expect a universe that is open, meaning it will continue to expand forever.

In simple terms, the Friedmann equations allow us to calculate how the universe's dimensions change with time. They have profound implications, not just for the future of the universe, but also for its past, including the conditions at its birth in the Big Bang.

Scale Factor Evolution

The scale factor, \(a(t)\), is a critical concept in cosmology, representing the size of the universe relative to its size at a specific time. Its evolution determines how the distances between distant galaxies change over time, indicative of the universe's expansion or contraction.

In the case where \(\Omega_0<1\), indicative of an open universe, the scale factor evolves according to the parametric equation involving the hyperbolic cosine (\cosh), reflecting a universe without recollapse, in perpetual expansion. The variable \(\eta\) acts as a convenient mathematical tool to express the relationship between the scale factor and cosmic time \(t\), facilitating the computation and understanding of how the universe evolves.

Understanding how \(a(t)\) changes are central to predicting cosmic phenomena like redshifts in the light from distant galaxies, which is directly related to the rate at which the universe expands.

Cosmological Density Parameter

The cosmological density parameter \(\Omega_0\) serves as a dimensionless measure, assessing the density of the universe relative to the critical density \(\rho_c\). This parameter remains crucial for classifying the geometry of the universe: whether it is flat, open, or closed. It is defined mathematically as \(\Omega_0 = \rho/\rho_c\), where \(\rho\) is the present-day density of the universe.

When this parameter is less than one (\(\Omega_0<1\)), it suggests that the universe has insufficient mass density to halt its expansion, resulting in an open universe. The parametric equations provided in the step-by-step solution represent precisely this scenario. The equations also describe that the scale factor smoothly increases, echoing an eternally expanding universe, without the possibility of a future 'Big Crunch'.

By studying \(\Omega_0\), cosmologists can infer important conclusions about the fate of our universe and the types of matter-energy contents that prevail within it.

Nonrelativistic Matter Universe

In cosmological models, a nonrelativistic matter universe, also known as a matter-dominated universe, is characterized by a predominance of matter whose energy density dilutes with the expansion of the universe according to \(\rho \propto a^{-3}\). In such a universe, the dominant components are forms of matter that do not travel at relativistic speeds, like most forms of dark matter and baryonic matter, which includes stars and galaxies.

The parametric equations for \(\Omega_0<1\) reflect the dynamics of a nonrelativistic matter universe that is open. As the equations show, these universes are not bound to collapse back on themselves, rather, they are fated to expand forever, with the scale factor \(a(t)\) increasing over time. The hyperbolic functions in the equations also hint towards an asymmetry in time, owing to the open nature of these universes.

This understanding of a nonrelativistic matter universe is fundamental to the big picture of cosmology, as it has implications for the universe's past, namely the era of matter domination following the radiation-dominated era of the early universe.