Is the Rate of Expansion of the Universe Monotonically Increasing?

The rate of expansion of the universe is a fascinating topic, and understanding how this rate changes over time is crucial for exploring the nature of the universe. Contrary to some popular beliefs, the expansion rate of the universe is not monotonically increasing. Instead, it has a nuanced and dynamic relationship that involves complex mathematical equations and fundamental constants. Through the lens of general relativity, particularly Einstein's field equations, we can delve into why this phenomenon occurs.

The Role of Einstein's Field Equations

At the heart of cosmology lies Einstein's field equations, a set of ten interdependent equations that describe the gravitational effects of matter and energy on the geometry of spacetime. The equation

R_{munu} - frac{1}{2}R g_{munu} - Lambda g_{munu} frac{8pi G}{c^4} T_{munu}

defines the relationship between the curvature of spacetime and the distribution of energy and momentum.

Understanding the Metrics and Equations

To model the expansion of a homogeneous and isotropic universe, we often turn to the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric. This metric is a simplified model that captures the large-scale behavior of the universe, even though the universe itself is not perfectly homogeneous and isotropic.

The FLRW Metric

The FLRW metric in spherical coordinates is given by:

g_{munu} begin{pmatrix} 1 - frac{10,000}{a^2 t^2} 0 0 0 0 -a^2 t^2 0 0 0 0 -a^2 t^2 r^2 0 0 0 0 -a^2 t^2 r^2 sin^2 theta end{pmatrix}

For a spatially flat universe (k0), the metric simplifies to:

g_{munu} begin{pmatrix} 1 - frac{10,000}{a^2 t^2} 0 0 0 0 -a^2 t^2 0 0 0 0 -a^2 t^2 r^2 0 0 0 0 -a^2 t^2 r^2 sin^2 theta end{pmatrix}

This metric effectively combines the flat Minkowski space-time metric with a scale factor that accounts for the expansion of the universe.

Deriving the Friedmann Equations

From the FLRW metric, we can derive the Friedmann equations, which describe the evolution of the scale factor (a(t)) and the Hubble parameter (H(t) frac{dot{a}}{a}).

Friedmann Equations

frac{dot{a}^2}{a^2} frac{8pi G rho}{3} frac{Lambda c^2}{3}

frac{ddot{a}}{a} -frac{4pi G}{3} (rho 3p c^2) frac{Lambda c^2}{3}

Here, (dot{a}) and (ddot{a}) represent the first and second time derivatives of the scale factor (a(t)), (H frac{dot{a}}{a}) is the Hubble parameter, (rho) is the energy density, and (p) is the pressure. The term (Lambda) represents the cosmological constant, a form of dark energy that causes the expansion of the universe to accelerate.

The Dynamics of Expansion

The relationship between the density (rho) and the scale factor (a(t)) is given by (rho propto a^{-3}), reflecting the inverse relationship between density and time. As the universe expands, the density decreases, leading to a decrease in the expansion rate.

The Hubble parameter (H) can be expressed as (H sqrt{frac{8pi G rho}{3} frac{Lambda c^2}{3}}). In the current stage of the universe, known as the Lambda-dominated era, both the expansion rate and the acceleration approach a constant value determined by (Lambda).

Although (H) is almost constant, the scale factor still increases exponentially as:

a propto exp(H_0 t)

where (H_0 approx 2.27 times 10^{-18} text{s}^{-1}), meaning the universe expands at a rate of about 7.2 percent per billion years.

As time approaches infinity, the Hubble parameter (H) and the acceleration term (ddot{a}/a) both approach constant values set by (Lambda). Specifically, (H_{infty} approx 1.82 times 10^{-18} text{s}^{-1}) (or 5.7 percent per billion years) and (A_{infty} frac{Lambda c^2}{3} approx 3.3 times 10^{-36} text{s}^{-2}).

Thus, the expansion rate will never fall below this constant value, and the acceleration will never rise above it, marking a fundamental limit in the dynamics of the universe.