SVT Decomposition of Metric and Matter Perturbations

Hello! Welcome to the first lesson of our ninth module, Cosmological Perturbation Theory.

In the last module, we delved into the quantum realm of curved spacetime, culminating in our discussion of the black hole information paradox. We were probing the ultimate limits of physics where the end-state of gravitational collapse challenges the foundations of quantum mechanics. Now, we pivot from the end of spacetime to its very beginning. Our goal in this module is to understand how the universe evolved from a nearly uniform primordial state into the rich, structured cosmos of galaxies, clusters, and voids we see today. The key to this lies in understanding the evolution of tiny initial inhomogeneities—cosmological perturbations.

Today's lesson provides the foundational mathematical tool for this entire field. Our learning outcome is to decompose metric and matter perturbations into scalar, vector, and tensor modes (SVT decomposition). This powerful classification scheme will allow us to break down the complex dynamics of the early universe into simpler, independent parts. We will see how this decomposition applies to both the geometry of spacetime (the metric) and the matter-energy content within it (the stress-energy tensor).

1. Perturbing the Universe

We begin with our standard model of cosmology: the flat Friedmann-Robertson-Walker (FRW) universe. In conformal time $τ$ (where $d τ = d t / a (t)$ ), the metric is remarkably simple:

g ˉ_{μν} d x^{μ} d x^{ν} = a^{2} (τ) (- d τ^{2} + δ_{ij} d x^{i} d x^{j})

This describes a perfectly homogeneous and isotropic spacetime. To account for the structures we observe, we must introduce small perturbations, $δ g_{μν}$ , to this background metric, so that $g_{μν} = g ˉ_{μν} + δ g_{μν}$ . The most general, symmetric perturbation $δ g_{μν}$ has 10 independent components. We can write the full perturbed line element as:

d s^{2} = a^{2} (τ) [- (1 + 2 A) d τ^{2} + 2 B_{i} d x^{i} d τ + (δ_{ij} + h_{ij}) d x^{i} d x^{j}]

Here, $A$ , $B_{i}$ , and $h_{ij}$ are small, arbitrary functions of spacetime $(τ, x)$ .

$A$ is a scalar function (1 component).
$B_{i}$ is a 3-vector field (3 components).
$h_{ij}$ is a symmetric 3x3 tensor field (6 components).

This gives us a total of $1 + 3 + 6 = 10$ functions, representing the 10 degrees of freedom of the metric perturbation. The challenge is that these components are all coupled together in the Einstein Field Equations. The SVT decomposition provides the crucial simplifying step.

2. The Scalar-Vector-Tensor Decomposition

The central idea is to decompose the geometric objects $B_{i}$ and $h_{ij}$ into pieces that transform irreducibly under rotations of the spatial coordinates $x^{i}$ . This is a generalization of the Helmholtz decomposition you know from electromagnetism, where a vector field is split into a curl-free part (gradient of a scalar) and a divergence-free part.

A thorough explanation of this decomposition is provided in the lecture notes from the University of Cambridge.

Part III Cosmology

Let's study the core mathematical procedure of the SVT decomposition from the 'Part III Cosmology' lecture notes by George Efstathiou and Will Handley (notes transcribed by Hiranya Peiris and edited by Thomas Laurence). This resource provides a clear, systematic breakdown.

Please read Section 3.1, 'Perturbed metric' (pages 33-35). Focus on: How the vector field B_i is split into the gradient of a scalar B and a divergenceless vector B_i^V. How the symmetric tensor h_ij is split into a trace part (related to scalar C), a trace-free scalar part (from scalar E), a vector part (from divergenceless vector E_i^V), and a transverse-traceless tensor part E_{ij}^T. Follow the degree-of-freedom counting at the end of the section to confirm that we have correctly accounted for all 10 initial components.

Let's summarize the results of this decomposition. The original 10 metric perturbation components are regrouped into:

Scalar (Spin-0) Modes: These are constructed from scalars or by taking derivatives of scalars. They describe compressional fluctuations.
- $A$ (from $g_{00}$ )
- $B$ (from the curl-free part of $B_{i}$ )
- $C$ (from the trace of $h_{ij}$ )
- $E$ (from the trace-free scalar part of $h_{ij}$ )
- Total: 4 scalar degrees of freedom.
Vector (Spin-1) Modes: These are divergenceless vector fields. They describe rotational, or vorticity, perturbations.
- $B_{i}^{V}$ (from the divergence-free part of $B_{i}$ , has 2 d.o.f since $\partial^{i} B_{i}^{V} = 0$ )
- $E_{i}^{V}$ (from the vector part of $h_{ij}$ , has 2 d.o.f since $\partial^{i} E_{i}^{V} = 0$ )
- Total: 4 vector degrees of freedom.
Tensor (Spin-2) Modes: This is the remaining part of $h_{ij}$ , which is both transverse (divergenceless) and traceless.
- $E_{ij}^{T}$ (satisfies $\partial^{i} E_{ij}^{T} = 0$ and $δ^{ij} E_{ij}^{T} = 0$ ). This corresponds to gravitational waves and has 2 independent components (the '+' and 'x' polarizations).
- Total: 2 tensor degrees of freedom.

The total number of degrees of freedom is $4 (S) + 4 (V) + 2 (T) = 10$ , as expected.

The profound importance of this decomposition, as noted in the text, is that at linear order, the scalar, vector, and tensor modes evolve completely independently of each other when plugged into the Einstein equations. Scalar sources (like density changes) only generate scalar metric perturbations, vector sources (like vorticity) only generate vector perturbations, and tensor sources only generate tensor perturbations.

Test your understanding!

Consider a simple metric perturbation in a flat FRW background where the only non-zero component is $h_{11} = - h_{22} = f (τ, z)$ , with all other $h_{ij}$ components being zero. What type of perturbation is this?

Show answer

This is a tensor perturbation. Let's check the conditions.

Traceless: The trace is $δ^{ij} h_{ij} = h_{11} + h_{22} + h_{33} = f - f + 0 = 0$ . The condition is satisfied.
Transverse (Divergenceless): We need to check $\partial^{i} h_{ij} = 0$ .
- For $j = 1$ : $\partial^{i} h_{i 1} = \partial_{1} h_{11} + \partial_{2} h_{21} + \partial_{3} h_{31} = \frac{\partial}{\partial x} (f) + 0 + \frac{\partial}{\partial z} (0) = 0$ .
- For $j = 2$ : $\partial^{i} h_{i 2} = \partial_{1} h_{12} + \partial_{2} h_{22} + \partial_{3} h_{32} = 0 + \frac{\partial}{\partial y} (- f) + 0 = 0$ .
- For $j = 3$ : $\partial^{i} h_{i 3} = \partial_{1} h_{13} + \partial_{2} h_{23} + \partial_{3} h_{33} = \frac{\partial}{\partial x} (0) + \frac{\partial}{\partial y} (0) + \frac{\partial}{\partial z} (f) = \frac{\partial f}{\partial z}$ .
  The transverse condition is only met if the wave is not propagating in the $z$ direction (i.e., if $f$ is not a function of $z$ ). The question states $f = f (τ, z)$ , indicating a wave propagating along the z-axis. This corresponds to the $h_{+}$ polarization of a gravitational wave. The condition $\partial^{i} h_{ij} = 0$ assumes we are taking derivatives with respect to the spatial coordinates of the perturbation's variation, not the propagation direction itself. For a plane wave $h_{ij} \propto e^{i k_{l} x^{l}}$ , the condition is $k^{i} h_{ij} = 0$ , meaning the wave is transverse to the direction of propagation. In our case, the propagation is in the z-direction ( $k = (0, 0, k_{z})$ ), and the perturbation is in the x-y plane, so it is indeed a transverse tensor mode.

3. Decomposing Matter Perturbations

Just as we decomposed the metric, we must also decompose the perturbations in the stress-energy tensor, $δ T_{ν}^{μ}$ , which act as the source for the metric perturbations via the Einstein equations, $δ G_{ν}^{μ} = 8 π G δ T_{ν}^{μ}$ .

For a perfect fluid, $T_{ν}^{μ} = (ρ + p) u^{μ} u_{ν} + p δ_{ν}^{μ}$ . We perturb the density $ρ \to ρ ˉ + δ ρ$ , pressure $p \to p ˉ + δ p$ , and four-velocity $u^{μ} \to u ˉ^{μ} + δ u^{μ}$ . The spatial part of the velocity perturbation, $δ u^{i}$ , is the peculiar velocity field $v^{i}$ .

More generally, for an imperfect fluid, the spatial part of the stress-energy tensor can also have a traceless part, $Π_{ij}$ , known as the anisotropic stress. This term represents internal friction or viscosity in the cosmic fluid.

Just like the metric perturbations, these matter perturbations can be decomposed.

Part III Cosmology

Let's return to the 'Part III Cosmology' notes to see how the stress-energy tensor is perturbed and decomposed.

Please read Section 3.3, 'Perturbed Matter' (page 36). Focus on: The components of the perturbed energy-momentum tensor, δT^μ_ν. The decomposition of the peculiar velocity v_i into a scalar part v. The introduction of the anisotropic stress Π_{ij} and its scalar part Π.

To summarize, the matter perturbations are classified as follows:

Scalar: Density perturbation $δ ρ$ , pressure perturbation $δ p$ , the scalar potential for the peculiar velocity $v$ , and the scalar part of the anisotropic stress $Π$ .
Vector: The divergenceless part of the peculiar velocity $v_{i}^{V}$ , describing vorticity in the fluid.
Tensor: The transverse-traceless part of the anisotropic stress $Π_{ij}^{T}$ .

4. The Decoupled Einstein Equations

The power of the SVT decomposition is fully realized when we write down the linearized Einstein equations. The equations naturally separate into three independent systems. This is a profound simplification that makes cosmological perturbation theory tractable.

To see this structure explicitly, let's examine a different set of notes that presents the final, separated equations.

Week 6: Perturbations to the FRW metric

The notes 'Week 6: Perturbations to the FRW metric' by Marc Kamionkowski, while using a slightly different notational convention, provide an excellent demonstration of the final, decoupled equations. This makes the abstract power of the decomposition concrete.

Please read the section labeled 'The next task will be to separate the perturbed Einstein equations...' which is part of Section 4, 'Synchronous gauge'. Don't worry about the specifics of 'synchronous gauge' for now; that is a topic for our next lesson. Focus on the list of 7 equations for G_μν. Notice how they are grouped: there are equations for scalar parts (involving h and ξ, which are scalar potentials), vector parts (involving h_i), and a final, separate equation for the tensor part (h_{ij,T}). This explicitly shows the decoupling.

As you saw, the system of 10 coupled partial differential equations has fractured into:

A set of equations coupling only the scalar metric perturbations to the scalar matter perturbations.
A set of equations coupling only the vector metric perturbations to the vector matter perturbations.
A single wave equation for the tensor modes, sourced by the tensor part of the anisotropic stress.

In most simple cosmological models (like single-field inflation), vector perturbations are not generated and, even if present, they decay away with the expansion of the universe. For this reason, they are often ignored. This leaves us with two primary, and decoupled, problems to solve: the evolution of scalar perturbations (which lead to galaxy formation) and the evolution of tensor perturbations (primordial gravitational waves).

Conclusion

In this lesson, we have developed the primary organizational principle of cosmological perturbation theory. By classifying perturbations according to their behavior under spatial rotations, we have untangled the intricate dynamics of the perturbed universe.

Key Takeaways:

Any general, linear perturbation to the FRW metric and the stress-energy tensor can be uniquely decomposed into scalar, vector, and tensor (SVT) parts.
Scalar modes represent compressional perturbations like fluctuations in density and pressure.
Vector modes represent rotational perturbations, or vorticity, which decay with cosmic expansion and are typically not sourced by inflation.
Tensor modes are transverse-traceless tensor fields that represent gravitational waves.
The crucial result is that at linear order, the Einstein Field Equations decouple for these three types of modes. This allows us to study the physics of structure formation (scalars) and primordial gravitational waves (tensors) as separate problems.

Preview of the Next Lesson:

While we have successfully grouped our 10 initial metric perturbations into S, V, and T modes, not all of them represent true physical degrees of freedom. Some simply reflect our choice of coordinates. This is the problem of gauge freedom. In our next lesson, we will explore this concept and learn how to construct gauge-invariant variables—combinations of perturbations that correspond to real, observable physics. This is the next essential step toward deriving the equations that govern the growth of structure in our universe.

Recommended resources

1. Perturbing the Universe

2. The Scalar-Vector-Tensor Decomposition

3. Decomposing Matter Perturbations

4. The Decoupled Einstein Equations

Conclusion