Solving Friedmann Equations for Dominated Universes

1. The Evolving Energy Density

Before we can solve the differential equations for $a(t)$, we must first understand how the energy density $\rho$ changes as the universe expands. In our previous lesson, we noted that the Friedmann equations are consistent with the local conservation of energy, ${\nabla }_{\mu }{T}^{\mu \nu }=0$, which for an FRW universe gives the fluid equation:

This equation links the rate of change of energy density to the expansion rate and pressure. To solve it, we need a relationship between pressure and density, known as the equation of state. For cosmological fluids, this is conveniently parameterized by a dimensionless constant $w$:

Substituting this into the fluid equation gives:

Integrating this simple differential equation yields the crucial relationship for how energy density scales with the expansion of the universe:

where ${\rho }_0$ is the density when the scale factor is $a_0$. Let's see what this means for our three key cosmic components.

The following video and reading summarize this derivation and its physical implications beautifully.

Let's summarize the results:

1. Non-relativistic Matter (Dust): This includes stars, galaxies, and dark matter. These particles are slow-moving, so their pressure is negligible ($p\approx 0$). This corresponds to $w=0$.
   ${\rho }_m\propto a^{-3}$
   Physical reason: The number of particles is conserved, so their density simply dilutes as the volume ($\propto a^3$) increases.

2. Relativistic Particles (Radiation): This includes photons and other highly relativistic particles. For a photon gas, $p=\frac{1}{3}\rho c^2$, which corresponds to $w=1/3$.
   ${\rho }_r\propto a^{-4}$
   Physical reason: The density dilutes with volume ($a^{-3}$), and additionally, the energy of each particle decreases as its wavelength is stretched by the cosmic expansion ($E\propto 1/\lambda \propto a^{-1}$).

3. Vacuum Energy (Cosmological Constant $\Lambda$): This is the intrinsic energy of empty space. As we saw, it's equivalent to a fluid with $p=-{\rho }_{\Lambda }{c}^2$, which corresponds to $w=-1$.
   ${\rho }_{\Lambda }\propto a^0=\text{constant}$
   Physical reason: The energy density per unit volume of space is constant. As space expands, the total vacuum energy increases.

2. Solving the Friedmann Equation for a Flat Universe

Armed with the knowledge of how $\rho$ evolves, we can now solve the first Friedmann equation for the scale factor $a(t)$. We will focus on a spatially flat universe ($k=0$), which is an excellent approximation of our own universe and greatly simplifies the mathematics.

The first Friedmann equation (with $c=1$) becomes:

Substituting $\rho \propto a^{-3(1+w)}$, we get:

where $C$ is a constant of proportionality. This is a separable differential equation which we can write as:

Let's solve this for our three cases, assuming a "Big Bang" start where $a=0$ at $t=0$.

Case 1: Matter-Dominated Universe ($w=0$)

For matter, the exponent is $\frac{1}{2}$. The equation is $\sqrt{a}da=\sqrt{C}dt$.
Integrating both sides:

Solving for $a(t)$ gives the evolution for an Einstein-de Sitter universe:

The expansion is a power law, and it is decelerating ($\ddot{a}<0$).

Case 2: Radiation-Dominated Universe ($w=1/3$)

As you just derived, the solution is:

Case 3: Vacuum-Dominated Universe ($w=-1$)

For vacuum energy, the exponent is $\frac{1+3(-1)}{2}=-1$. The equation becomes:

Note that ${\rho }_{\Lambda }$ is constant, so $\sqrt{C}=\sqrt{\frac{8\pi G{\rho }_{\Lambda }}{3}}$ is just the constant Hubble parameter, $H_0$.
Integrating both sides:

This gives the exponential expansion of a de Sitter universe:

This universe undergoes accelerated expansion.

The "TASI Lectures" reading provides a concise summary of these exact solutions.

3. A Brief History of the Universe

These single-component solutions are powerful because our actual universe has evolved through epochs where one component dominated over the others.

Because ${\rho }_r\propto a^{-4}$, ${\rho }_m\propto a^{-3}$, and ${\rho }_{\Lambda }$ is constant:

1. Early Universe (small $a$): The $a^{-4}$ term for radiation was the largest. The universe was radiation-dominated, and its scale factor grew as $a(t)\propto t^{1/2}$.
2. Intermediate Times: As $a$ grew, the radiation density fell off faster than matter density. Eventually, matter became the dominant component. The universe entered the matter-dominated era, and its expansion law switched to $a(t)\propto t^{2/3}$.
3. Late Times (large $a$): As $a$ continued to grow, the matter density continued to fall. Eventually, the constant vacuum energy density ${\rho }_{\Lambda }$ became the dominant term. Our universe is now in this vacuum-dominated (or dark energy-dominated) era, and its expansion is accelerating exponentially, $a(t)\propto e^{H_{0}t}$.

This sequence provides a compelling, albeit simplified, narrative of cosmic history, all derived from solving our fundamental equations. The following video illustrates this timeline.

4. What if the Universe Isn't Flat?

While our universe is observed to be very nearly flat ($k\approx 0$), it's instructive to see how spatial curvature would change the solutions. Without going into the detailed derivations (which often use a mathematical convenience called conformal time, as seen in Section 2.4 of LINK and Section 4.2 of LINK), we can describe the qualitative behavior for a universe with no cosmological constant ($\Lambda =0$):

• Closed Universe ($k=+1$): The positive curvature term $-k/a^2$ acts like an extra gravitational pull. The universe expands from a Big Bang, reaches a maximum size, and then recollapses in a "Big Crunch".
• Open Universe ($k=-1$): The negative curvature term acts like a form of kinetic energy. The universe expands forever from a Big Bang, though it continuously decelerates.

The presence of a cosmological constant ($\Lambda >0$) dramatically changes this picture. As we saw, it causes exponential acceleration. For any value of $k$ ($+1,0,-1$), if $\Lambda$ is positive and dominant, the universe will ultimately expand exponentially forever, overriding the effect of curvature. This insight broke the old link between "geometry is destiny."

Conclusion

In this lesson, we have moved from the abstract differential equations of cosmology to concrete solutions that describe the expansion history of the universe.

Key Takeaways:

• The evolution of energy density is governed by the equation of state parameter $w$, leading to $\rho \propto a^{-3(1+w)}$.
• For a flat universe, this leads to simple analytical solutions for the scale factor:
  • Radiation-dominated ($w=1/3$): $a(t)\propto t^{1/2}$
  • Matter-dominated ($w=0$): $a(t)\propto t^{2/3}$
  • Vacuum-dominated ($w=-1$): $a(t)\propto e^{Ht}$
• These different solutions correspond to distinct epochs in our universe's history, explaining the transition from a decelerating to an accelerating expansion.

Preview of Next Lesson:

Now that we have derived the functional form of $a(t)$, we have the theoretical basis to make direct contact with one of the most fundamental observations in cosmology. In our next lesson, we will use our understanding of the expanding metric to define cosmological redshift and derive the Hubble-Lemaître law, showing how the observed recession of distant galaxies is a direct consequence of the evolution of $a(t)$.