Linearized Einstein Equations for Scalar Perturbations

Hello! Welcome to your third lesson in the "Cosmological Perturbation Theory" module.

In the previous two lessons, we established the fundamental framework for our analysis. First, we performed the Scalar-Vector-Tensor (SVT) decomposition, which showed that at linear order, different types of perturbations evolve independently. Then, we tackled the subtle issue of gauge freedom, learning how to distinguish physical perturbations from coordinate artifacts by either fixing a gauge (like the intuitive Newtonian gauge) or constructing fully gauge-invariant variables (like the Bardeen potentials).

We now have all the necessary conceptual machinery to tackle the central task of this module. Our goal today is to derive the dynamical equations that govern the evolution of these physical perturbations. We will do this by taking the full Einstein Field Equations, $G_{μν} = 8 π G T_{μν}$ , and linearizing them—expanding both the geometry ( $G_{μν}$ ) and the matter content ( $T_{μν}$ ) to first order around the homogeneous and isotropic FRW background.

This brings us to today's learning outcome: to derive the linearized Einstein equations for scalar perturbations in an expanding background.

To make the physics as transparent as possible, we will work in the Longitudinal (or Newtonian) gauge, which you'll recall has a simple, diagonal metric.

1. Setting the Stage: The Perturbed Universe

Let's begin by writing down the spacetime metric in the Newtonian gauge. The line element is given by:

d s^{2} = g_{μν} d x^{μ} d x^{ν} = a^{2} (τ) [- (1 + 2Ψ) d τ^{2} + (1 - 2Φ) δ_{ij} d x^{i} d x^{j}]

Here, $a (τ)$ is the scale factor as a function of conformal time $τ$ . The functions $Ψ (τ, x)$ and $Φ (τ, x)$ are the scalar metric perturbations. As you proved in the last lesson, in this specific gauge, these two functions are precisely equal to the gauge-invariant Bardeen potentials. $Ψ$ acts as the gravitational potential, while $Φ$ describes the perturbation to the spatial curvature.

Our strategy is to compute the Einstein tensor $G_{μν}$ for this metric and the energy-momentum tensor $T_{μν}$ for a perturbed fluid, keeping only terms up to the first order in $Ψ$ , $Φ$ , and the fluid perturbations.

2. The Left-Hand Side: Perturbing Geometry ( $δ G_{μν}$ )

The first, and most labor-intensive, part of our task is to compute the perturbed Einstein tensor, $δ G_{μν}$ . The process is a direct application of the definitions you learned in the first two modules of this course, but the algebra is lengthy. The logical sequence is as follows:

Metric $g_{μν}$ : We start with the Newtonian gauge metric above. Its inverse, $g^{μν}$ , is found by inverting the components, keeping terms to first order: $g^{00} = - a^{- 2} (1 - 2Ψ)$ and $g^{ij} = a^{- 2} (1 + 2Φ) δ^{ij}$ .
Christoffel Symbols $Γ_{μν}^{α}$ : We compute the connection coefficients using their definition in terms of metric derivatives. Each symbol will have a background part (related to $H = a^{'} / a$ ) and a perturbed part ( $δ Γ$ ) involving derivatives of $Ψ$ and $Φ$ .
Ricci Tensor $R_{μν}$ : From the Christoffel symbols, we compute the components of the Ricci tensor. This will contain background terms ( $R ˉ_{μν}$ ) and first-order perturbations ( $δ R_{μν}$ ).
Ricci Scalar $R$ : We contract the Ricci tensor, $R = g^{μν} R_{μν}$ , to find the Ricci scalar, again separating it into background and perturbed parts, $R ˉ + δ R$ .
Einstein Tensor $G_{μν}$ : Finally, we assemble the Einstein tensor $G_{μν} = R_{μν} - \frac{1}{2} g_{μν} R$ , and isolate the first-order perturbation, $δ G_{μν}$ .

This derivation is a rite of passage in cosmology. The following resource provides an exceptionally detailed, step-by-step calculation, which is perfectly suited to your "no compromise" preference for mathematical rigor.

Scalar Cosmological Perturbations

The paper 'Scalar Cosmological Perturbations' by Ureña-López & Reyes-Ibarra provides a complete, explicit derivation of the perturbed Einstein tensor. We will use it as our guide for the geometric part of the calculation.

Please study sections 3.1 through 3.5 (from 'The metric' to 'The Einstein tensor', pages 5-11). Your goal is not to memorize the derivation, but to follow its logic and verify the structure of the result. Focus on: Section 3.1 & 3.2: How the perturbed metric (Eq. 14) is used to calculate the Christoffel symbols (Eqs. 20-24). Try re-deriving one or two simple components yourself, like Γ⁰₀₀ or Γⁱ₀₀, to get a feel for the process. Section 3.3 & 3.4: How these Christoffel symbols are substituted into the definition of the Ricci tensor and Ricci scalar to find their perturbed components (Eqs. for δR₀₀, δR₀ᵢ, Rᵢⱼ, and δR). Section 3.5: The final assembly of the perturbed Einstein tensor components δG⁰₀, δG⁰ᵢ, and δGᵢⱼ (Eqs. 33, 34, and 35). Note that the paper calculates mixed-index components like δG⁰₀, but the principle is the same. This is a dense calculation. Take your time to trace how each piece is built from the last. The beauty of this paper is that almost no steps are skipped.

After this detailed work, we have the left-hand side of our linearized equation. The Cambridge lecture notes by Fergusson (resource LINK), which we will use next, provide a concise summary of the results for the mixed-tensor components $δ G_{ν}^{μ} = g ˉ^{μα} δ G_{α ν} + δ g^{μα} G ˉ_{α ν}$ (after converting from cosmic time $t$ to conformal time $τ$ , which the resource does implicitly):
$δ G_{0}^{0} = \frac{1}{a ^{2}} (- 2 \nabla^{2} Φ + 6 H Φ^{'} + 6 H^{2} Ψ)$
$δ G_{i}^{0} = \frac{2}{a ^{2}} \partial_{i} (Φ^{'} + H Ψ)$
$δ G_{j}^{i} = \frac{1}{a ^{2}} [(\nabla^{2} (Φ - Ψ)) δ_{j}^{i} - \partial^{i} \partial_{j} (Φ - Ψ) + δ_{j}^{i} (2 Φ^{''} + (4 H^{'} + 2 H^{2}) Ψ + (2 H^{'} + 4 H^{2}) Φ + 2 H Ψ^{'} + 4 H Φ^{'})]$
These expressions look formidable, but their structure contains the key physics we are after.

3. The Right-Hand Side: Perturbing Matter ( $δ T_{μν}$ )

Now we turn to the source term. We model the cosmic medium as a perfect fluid, whose energy-momentum tensor is:

T^{μν} = (ρ + P) U^{μ} U^{ν} + P g^{μν}

We introduce first-order perturbations to the density $ρ = ρ ˉ + δ ρ$ , pressure $P = P ˉ + δ P$ , and 4-velocity $U^{μ} = U ˉ^{μ} + δ U^{μ}$ . The background is at rest, so $U ˉ^{μ} = (a^{- 1}, 0)$ . For scalar perturbations, the peculiar velocity $δ U^{i}$ can be written as the gradient of a scalar potential, $v$ , such that $δ U^{i} = a^{- 1} \partial^{i} v$ .

The following reading explains how to construct the perturbed components of $T_{ν}^{μ}$ .

Part III Cosmology

Let's turn to the 'Part III Cosmology' lecture notes by Fergusson, which models the matter as a general perfect fluid. This section details the components of the perturbed energy-momentum tensor.

Please read Section 3.3, 'Perturbed Matter' (page 36). Focus on: The definition of the perturbed energy-momentum tensor (Eq. 3.23). The introduction of anisotropic stress Π, which is the traceless part of the spatial stress. For a perfect fluid, this is zero. The final expressions for the perturbed components δT⁰₀, δT⁰ᵢ, and δTᵢⱼ (Eqs. 3.27-3.30).

From this, we find the key components of the mixed tensor $δ T_{ν}^{μ}$ :

Energy Density Perturbation: $δ T_{0}^{0} = - δ ρ$
Momentum Density Perturbation: $δ T_{i}^{0} = (ρ ˉ + P ˉ) \partial_{i} v$
Pressure & Stress Perturbation: $δ T_{j}^{i} = δ P δ_{j}^{i} + (\partial^{i} \partial_{j} - \frac{1}{3} δ_{j}^{i} \nabla^{2}) Π$

4. Assembling the Linearized Einstein Equations

We are finally ready to equate the two sides, $δ G_{ν}^{μ} = 8 π G δ T_{ν}^{μ}$ . This will give us a set of coupled differential equations for our perturbation variables. We will again use the Cambridge notes, which present the final equations clearly.

Part III Cosmology

This is the moment of truth. We will now equate our expressions for δG^μ_ν and δT^μ_ν. The 'Part III Cosmology' notes present the resulting equations and their physical interpretation.

Please read Section 3.4, 'Linearised Equations' (pages 37-40). This is the most important section for this lesson. Focus on: How the trace-free part of the ij equation leads to the conclusion that Ψ = Φ in the absence of anisotropic stress (Π=0). This is a crucial simplification. The 00 component equation, which yields the cosmological Poisson equation (labeled G1). The 0i component equation, which relates the potentials to the fluid momentum (labeled G2). The trace part of the ij equation, which provides a second-order evolution equation for the potential (labeled G3). The derivation of the Continuity and Euler equations from the conservation law ∇_μT^μν=0.

Let's summarize the results, which form the core of cosmological perturbation theory.

The Anisotropic Stress Equation

From the traceless part of the $δ G_{j}^{i} = 8 π G δ T_{j}^{i}$ equation, we find:

(\partial^{i} \partial_{j} - \frac{1}{3} δ_{j}^{i} \nabla^{2}) (Φ - Ψ) = 8 π G a^{2} (\partial^{i} \partial_{j} - \frac{1}{3} δ_{j}^{i} \nabla^{2}) Π

This magnificent equation tells us that any difference between the two gravitational potentials is sourced exclusively by anisotropic stress in the matter. For many important cosmological fluids (including perfect fluids and canonical scalar fields), the anisotropic stress $Π$ is zero. Therefore, for the majority of our work, we can make the vital simplification:

Ψ = Φ

The Gravitational Field Equations

Assuming $Ψ = Φ$ , the remaining Einstein equations become:

The (0,0) Equation (Cosmological Poisson Equation):
$\nabla^{2} Φ - 3 H Φ^{'} - 3 H^{2} Φ = 4 π G a^{2} δ ρ$
This is the GR analogue of the Poisson equation $\nabla^{2} ϕ_{N e w t o n} = 4 π Gρ$ . It relates the spatial curvature perturbation $Φ$ to the energy density perturbation $δ ρ$ , but with extra terms due to the cosmic expansion.
The (0,i) Equation (Momentum Constraint):
$\partial_{i} (Φ^{'} + H Φ) = - 4 π G a^{2} (ρ ˉ + P ˉ) \partial_{i} v$
This equation links the time evolution of the gravitational potential to the peculiar velocity of the cosmic fluid.
The (i,i) Equation (Evolution Equation):
$Φ^{''} + 3 H Φ^{'} + (2 H^{'} + H^{2}) Φ = 4 π G a^{2} δ P$
This is a second-order differential equation describing the time evolution of $Φ$ , sourced by pressure perturbations $δ P$ .

The Fluid Equations from Covariant Conservation

As you know, the Bianchi identity $\nabla_{μ} G^{μν} = 0$ implies that the energy-momentum tensor is covariantly conserved: $\nabla_{μ} T^{μν} = 0$ . Expanding this conservation law to first order gives two equations governing the fluid perturbations themselves:

The Continuity Equation:
$δ ρ^{'} + 3 H (δ ρ + δ P) = - (ρ ˉ + P ˉ) \nabla^{2} v + 3 (ρ ˉ + P ˉ) Φ^{'}$
This describes how the energy density perturbation changes over time due to cosmic dilution, fluid flowing in or out of a region ( $\nabla^{2} v$ ), and the compression/rarefaction of space itself ( $Φ^{'}$ ).
The Euler Equation:
$((ρ ˉ + P ˉ) v)^{'} + 3 H (ρ ˉ + P ˉ) v = - δ P - (ρ ˉ + P ˉ) Φ$
This is Newton's second law for the fluid. It states that the change in momentum is driven by forces from pressure gradients (- $δ P$ ) and gravity (- $Φ$ ).

Test your understanding!

Consider a static, non-expanding universe ( $a = 1, H = 0$ ) filled with non-relativistic matter ( $P ≪ ρ$ , so we can set $P ˉ = δ P = 0$ ). How do the linearized Einstein equations simplify in this limit? What familiar equations do you recover?

Show answer

In this limit ( $a = 1, H = 0, P = 0$ ):

The (0,0) equation becomes: $\nabla^{2} Φ = 4 π G δ ρ$ . This is exactly the Newtonian Poisson equation, identifying $Φ$ with the Newtonian gravitational potential.
The (0,i) equation becomes: $\partial_{i} Φ^{'} = - 4 π G ρ ˉ \partial_{i} v$ .
The Euler equation becomes: $(ρ ˉ v)^{'} = - ρ ˉ Φ$ . Taking a spatial gradient gives $\nabla (ρ ˉ v)^{'} = - ρ ˉ \nablaΦ$ . This shows that fluid acceleration is driven by the gradient of the gravitational potential, which is the gravitational force.
The Continuity equation becomes: $δ ρ^{'} = - ρ ˉ \nabla^{2} v$ . This is the standard continuity equation $ρ ˙ + \nabla \cdot (ρ v) \approx 0$ when linearized.

The system reduces to the equations of Newtonian gravity and fluid dynamics, confirming that our relativistic framework has the correct classical limit.

Conclusion

You have just completed one of the most important derivations in modern cosmology. By systematically linearizing Einstein's theory, we have transformed it into a set of tractable differential equations that precisely describe how small inhomogeneities in the early universe evolve under the influence of gravity and cosmic expansion.

Key Takeaways:

The process of deriving the linearized equations involves a direct, albeit lengthy, calculation of the perturbed Christoffel symbols, Ricci tensor, and Einstein tensor.
Working in the Newtonian gauge provides significant physical intuition, with the potentials $Φ$ and $Ψ$ corresponding to the gauge-invariant Bardeen potentials.
A crucial result is that for perfect fluids (or scalar fields), there is no anisotropic stress, leading to the simplification $Ψ = Φ$ .
The final system of equations consists of two types:
- Gravitational equations (Cosmological Poisson, etc.) that show how matter perturbations ( $δ ρ, δ P, v$ ) source the metric perturbations ( $Φ$ ).
- Fluid equations (Continuity and Euler) that show how the fluid perturbations evolve under pressure, expansion, and the gravitational fields they themselves create.

Preview of the Next Lesson:

We now have the equations of motion, but where do the initial perturbations come from? An engine that can run needs a spark to start it. In the next lesson, we will discover that spark. We will model the quantum fluctuations of the inflaton field during the epoch of cosmic inflation as the ultimate origin of these primordial perturbations, providing the initial conditions for the equations we have derived today.

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