Deriving Friedmann Equations from EFE and FRW Metric

Hello! Welcome back to our journey into cosmology.

In our last lesson, we established the geometric stage for our cosmic drama. By postulating the Cosmological Principle—the idea that the universe is homogeneous and isotropic on large scales—we derived the Friedmann-Lemaître-Robertson-Walker (FRW) metric. This metric, with its time-dependent scale factor $a (t)$ and spatial curvature $k$ , describes the geometry of such a universe.

However, geometry alone is static. To understand how the universe actually evolves, we need dynamics. Today, we will bring dynamics into the picture by using the Einstein Field Equations to connect the FRW geometry to the matter and energy that fills the universe. Our goal is to apply the EFE to the FRW metric to derive the Friedmann equations, the master equations that govern the expansion of the cosmos.

The process is a direct application of the core principle of General Relativity: "Spacetime tells matter how to move; matter tells spacetime how to curve." We have the spacetime (FRW metric), and we will model the matter (as a perfect fluid). The EFE will provide the link.

Here is a roadmap for our derivation:

Derivation of Friedmann Equations from Einstein Field Equations — This diagram outlines the process we will follow. We will calculate the Einstein tensor (left side of the EFE) from the FRW metric and the stress-energy tensor (right side) from a perfect fluid model. Equating them via the EFE will yield the two Friedmann equations.

1. The Left-Hand Side: Geometry of the FRW Spacetime

The first, and most mathematically intensive, step is to compute the Einstein tensor, $G_{μν} = R_{μν} - \frac{1}{2} R g_{μν}$ , for the FRW metric. As you appreciate mathematical rigor, we won't shy away from the process, which involves:

Calculating the Christoffel symbols ( $Γ_{μν}^{λ}$ ) from the metric $g_{μν}$ .
Using the Christoffel symbols to compute the Ricci tensor ( $R_{μν}$ ).
Contracting the Ricci tensor to find the Ricci scalar ( $R = g^{μν} R_{μν}$ ).
Assembling these parts to construct the Einstein tensor ( $G_{μν}$ ).

The full calculation is lengthy but straightforward. To give you the complete, unabridged derivation you prefer, I will ask you to work through a detailed resource.

An Introduction to FRW Cosmology and dark energy models

The following sections from the paper 'An Introduction to FRW Cosmology and dark energy models' provide a meticulous, step-by-step derivation of all the geometric quantities we need. This is an excellent exercise in applying the formal machinery of GR.

Please study Section 4.3, 'The Friedmann equations', starting from page 42 up to page 54. Begin with the FRW metric components (Eq. 4.6) and their derivatives (Eq. 4.8). Follow the calculation of the Christoffel symbols (pages 44-46). You don't need to check every single one, but trace the logic for a few, like Γ⁰₁₁ and Γ¹₁₀, to understand the procedure. Study the derivation of the Ricci tensor components (pages 46-52). Note the crucial simplification that off-diagonal components are zero. Follow the calculation of the Ricci scalar R (page 53). Finally, see how these are assembled into the Einstein tensor components G⁰₀ and G¹₁ (Eq. 4.19 on page 54). Focus on understanding the methodical process rather than memorizing each intermediate step.

After this detailed calculation, the essential results for the non-zero components of the Einstein tensor (in the $(t, r, θ, ϕ)$ coordinate system, with $c = 1$ for now) are:

Time-Time Component: $G_{00} = 3 [(\frac{a ˙}{a})^{2} + \frac{k}{a ^{2}}]$
Spatial Components: For $i, j \in {1, 2, 3}$ , the spatial part $G_{ij}$ is proportional to the spatial metric $g_{ij}$ . A more convenient form is for the mixed tensor $G_{j}^{i}$ : $G_{j}^{i} = δ_{j}^{i} [2 \frac{a ¨}{a} + (\frac{a ˙}{a})^{2} + \frac{k}{a ^{2}}]$

where $a ˙ = d a / d t$ and $a ¨ = d^{2} a / d t^{2}$ .

With the left-hand side of the EFE in hand, we now turn to the right-hand side.

2. The Right-Hand Side: Matter Content of the Universe

The Cosmological Principle implies that the matter and energy filling the universe must also be homogeneous and isotropic on large scales. The perfect physical model for such a substance is a perfect fluid.

The stress-energy tensor for a perfect fluid is given by:

T^{μν} = (ρ c^{2} + p) U^{μ} U^{ν} + p g^{μν}

where $ρ$ is the total energy density, $p$ is the pressure, and $U^{μ}$ is the 4-velocity of the fluid.

In the FRW framework, we use comoving coordinates, which are coordinates that are "carried along" with the cosmic expansion. In this frame, the fluid is, by definition, at rest. This leads to a simple form for the 4-velocity:

U^{μ} = (c, 0, 0, 0) and U_{μ} = g_{μν} U^{ν} = (- c, 0, 0, 0)

Plugging this into the stress-energy tensor, we find the components:

$T_{00} = (ρ c^{2} + p) U_{0} U_{0} / c^{2} + p g_{00} = (ρ c^{2} + p) (- c) (- c) / c^{2} + p (- c^{2}) = (ρ c^{2} + p) - p = ρ c^{2}$
$T_{ij} = p g_{ij}$ (since the spatial parts of $U^{μ}$ are zero).

So, the stress-energy tensor is a simple diagonal matrix:

T_{μν} = ρ c^{2} 000 0 0 p g_{ij} 0

3. Assembling the Friedmann Equations

We are now ready to assemble the full Einstein Field Equations, including the cosmological constant $Λ$ , which plays a vital role in modern cosmology:

G_{μν} + Λ g_{μν} = \frac{8 π G}{c ^{4}} T_{μν}

Because both $G_{μν}$ and $T_{μν}$ are diagonal, we only get two independent equations: the 00 component (time) and the ii component (space).

The following video provides a very clear walkthrough of this final assembly process.

Relativity 110f: Cosmology - Friedmann Equations Derivation + Universe Evolution Models (FINALE)

This video from eigenchris masterfully combines the geometric and matter components to derive both Friedmann equations. It's an excellent summary of the final algebraic steps.

Watch the segment from 03:10 to 06:58. The first part (to 05:10) covers the derivation of the First Friedmann Equation from the 00-component of the EFE. Follow the substitution and algebraic rearrangement. The second part (to 06:58) derives the Second Friedmann Equation (or the acceleration equation) by taking the trace of the EFE. This is an elegant alternative to using the spatial ii-components directly.

Let's summarize the results, re-inserting the factors of $c$ where the videos might have omitted them.

The First Friedmann Equation

From the 00-component: $G_{00} + Λ g_{00} = \frac{8 π G}{c ^{4}} T_{00}$ .

Substituting our derived expressions (and using $g_{00} = - c^{2}$ from the FRW metric in some conventions, or $g_{00} = - 1$ if setting c=1, be careful with conventions):

3 [(\frac{a ˙}{a})^{2} + \frac{k c ^{2}}{a ^{2}}] - Λ c^{2} = \frac{8 π G}{c ^{4}} (ρ c^{2})

Rearranging this gives the first Friedmann equation:

(\frac{a ˙}{a})^{2} = \frac{8 π G}{3} ρ - \frac{k c ^{2}}{a ^{2}} + \frac{Λ c ^{2}}{3}

This equation relates the expansion rate of the universe ( $a ˙ / a$ , the Hubble parameter squared) to its energy density, spatial curvature, and the cosmological constant.

The Second Friedmann Equation (The Acceleration Equation)

From the spatial components, after some combination with the first equation (as shown in the eigenchris video using the trace method, or in the resource LINK using the ii-component), we arrive at the second Friedmann equation, often called the acceleration equation:

\frac{a ¨}{a} = - \frac{4 π G}{3} (ρ + \frac{3 p}{c ^{2}}) + \frac{Λ c ^{2}}{3}

This equation governs the acceleration of the cosmic expansion. It shows that both energy density $ρ$ and pressure $p$ contribute to the gravitational "braking" of the universe, while a positive cosmological constant $Λ$ provides a repulsive force, causing acceleration.

Test your understanding!

The two Friedmann equations are not independent. One can be derived from the other using the law of local energy conservation, $\nabla_{μ} T^{μν} = 0$ . For a perfect fluid in an FRW background, this conservation law simplifies to the continuity equation:

ρ ˙ + 3 \frac{a ˙}{a} (ρ + \frac{p}{c ^{2}}) = 0

Show that you can derive the second Friedmann (acceleration) equation by taking the time derivative of the first Friedmann equation and substituting in the continuity equation. (This is a great check of your understanding and algebra!)

Show answer

Start with the first Friedmann equation (rearranged slightly for convenience): $a ˙^{2} = \frac{8 π G}{3} ρ a^{2} - k c^{2} + \frac{Λ c ^{2}}{3} a^{2}$
Take the time derivative of both sides: $2 a ˙ a ¨ = \frac{8 π G}{3} (ρ ˙ a^{2} + 2 ρ a a ˙) + \frac{Λ c ^{2}}{3} (2 a a ˙)$
Divide by $2 a a ˙$ (assuming $a ˙ \neq = 0$ ): $\frac{a ¨}{a} = \frac{4 π G}{3} (\frac{ρ ˙ a}{a ˙} + 2 ρ) + \frac{Λ c ^{2}}{3}$
Use the continuity equation to substitute for $ρ ˙$ . First, rearrange it: $ρ ˙ = - 3 \frac{a ˙}{a} (ρ + p / c^{2})$ .
Substitute $ρ ˙$ into the equation from step 3: $\frac{a ¨}{a} = \frac{4 π G}{3} (\frac{- 3 \frac{a ˙}{a} ( ρ + p / c ^{2} ) a}{a ˙} + 2 ρ) + \frac{Λ c ^{2}}{3}$
Simplify: $\frac{a ¨}{a} = \frac{4 π G}{3} (- 3 (ρ + p / c^{2}) + 2 ρ) + \frac{Λ c ^{2}}{3}$ $\frac{a ¨}{a} = \frac{4 π G}{3} (- 3 ρ - 3 p / c^{2} + 2 ρ) + \frac{Λ c ^{2}}{3}$ $\frac{a ¨}{a} = \frac{4 π G}{3} (- ρ - 3 p / c^{2}) + \frac{Λ c ^{2}}{3}$
Final Result: $\frac{a ¨}{a} = - \frac{4 π G}{3} (ρ + \frac{3 p}{c ^{2}}) + \frac{Λ c ^{2}}{3}$

This matches the acceleration equation exactly, demonstrating the consistency between the geometric evolution (EFE) and local energy conservation.

Conclusion

Today, we have accomplished one of the central tasks in theoretical cosmology. By applying the Einstein Field Equations to the symmetrically constrained FRW metric, we have derived the dynamical laws governing the universe's evolution.

Key Takeaways:

The dynamics of a homogeneous and isotropic universe are governed by the two Friedmann equations.
The first Friedmann equation relates the expansion rate $H (t) = a ˙ / a$ to the universe's content ( $ρ$ ), geometry ( $k$ ), and the cosmological constant ( $Λ$ ).
The second Friedmann (acceleration) equation describes how the expansion accelerates or decelerates, revealing that pressure $p$ acts as a source of gravity, similar to energy density $ρ$ .
These equations form a system of differential equations for the scale factor $a (t)$ , whose solution describes the history and future of the universe.

Preview of Next Lesson:

We now possess the fundamental equations of cosmology. The next logical step is to solve them. In the next lesson, we will solve the Friedmann equations for universes dominated by different components—namely matter, radiation, and vacuum energy (the cosmological constant). By analyzing these simplified but powerful solutions, we will build a picture of the different epochs in our universe's history.