Gauge Freedom and Invariant Variables in Perturbation Theory

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

In our last lesson, we took the first crucial step in simplifying the physics of the perturbed universe. We saw how any small perturbation to the geometry ( $g_{μν}$ ) and matter content ( $T_{μν}$ ) of our FRW spacetime can be neatly decomposed into independent scalar, vector, and tensor modes. This SVT decomposition is the foundational organizing principle for all that follows.

However, we ended with a cliffhanger: not all of the perturbation components we defined are physically real. Some are mere artifacts of the coordinate system we use to describe spacetime. You've encountered a similar issue in electromagnetism, where the potentials $A$ and $ϕ$ are not unique; they can be altered by a gauge transformation without changing the physical electric and magnetic fields. Today, we will explore the analogous, but richer, concept of gauge freedom in general relativity. Our goal is to understand this ambiguity and learn how to construct variables that represent true, physical changes in the universe.

This brings us to today's learning outcome: to explain the concept of gauge freedom in perturbation theory and construct gauge-invariant variables (e.g., Bardeen potentials).

1. The Gauge Problem: The Illusion of Perturbation

The core of the gauge problem is that general relativity is "diffeomorphism invariant," meaning the laws of physics are the same regardless of the coordinate system used. When we split the metric $g_{μν}$ into a background part $g ˉ_{μν}$ and a perturbation $δ g_{μν}$ , this separation is not covariant. It depends on our choice of coordinates.

Imagine a perfectly smooth, unperturbed FRW universe. Its metric is:

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

Now, let's make a small, spacetime-dependent change of coordinates (a "gauge transformation"):

τ \to τ ~ = τ + ξ^{0} (τ, x) and x^{i} \to x ~^{i} = x^{i} + ξ^{i} (τ, x)

If you were to calculate the metric tensor in this new, "wiggled" coordinate system, you would find that it now has off-diagonal components and that the diagonal components are no longer functions of time only. It looks like a perturbed metric, with non-zero values for our perturbation functions $A, B_{i}, h_{ij}$ , even though the underlying physical spacetime is completely homogeneous and isotropic.

These are gauge modes: apparent perturbations that are pure artifacts of our coordinate choice. A physical theory of structure formation cannot be built on quantities that might just be coordinate effects. We need a way to distinguish these illusions from reality.

2. How Perturbations Transform

The solution lies in understanding precisely how the perturbation variables change under a gauge transformation. For scalar perturbations, our most general metric is described by four functions: $A, B, C, E$ . The coordinate transformation itself can be generated by two scalar functions: a time shift $T (τ, x)$ and a spatial shift generated by a scalar potential $L (τ, x)$ , such that $ξ^{0} = T$ and $ξ^{i} = \partial^{i} L$ .

The following reading from the Cambridge lecture notes derives how the metric perturbations transform under such a change.

Part III Cosmology

Let's study the mathematical details of gauge transformations in the 'Part III Cosmology' notes. This section shows how an infinitesimal coordinate change induces changes in the scalar metric perturbation functions.

Please read Section 3.2, 'Gauge Problem' (pages 34-35). Focus on: The example of inducing 'fake' perturbations on an unperturbed metric. The general transformation laws for the scalar metric perturbations A, B, C, and E, which are derived from the transformation x^μ → x^μ + ξ^μ. The existence of two arbitrary functions, T and L, which represent our freedom to choose coordinates.

As the notes show, under the scalar gauge transformation generated by $T$ and $L$ , the metric perturbations change as follows (where a prime denotes a derivative with respect to conformal time $τ$ , and $H = a^{'} / a$ ):

A ~ = A - T^{'} - H T B ~ = B + T - L^{'} C ~ = C - H T - \frac{1}{3} \nabla^{2} L E ~ = E - L

The key insight here is that we have two free functions ( $T$ and $L$ ) at our disposal. This implies that of the four scalar metric functions ( $A, B, C, E$ ), only $4 - 2 = 2$ represent true physical degrees of freedom. The other two are redundant gauge modes. Our task is to isolate the two physical modes.

3. Strategy 1: Gauge Fixing

The most direct way to deal with the two redundant degrees of freedom is to use our freedom to choose $T$ and $L$ to eliminate two of the perturbation functions. This is called gauge fixing. By setting two functions to zero, we are essentially choosing a specific coordinate system that makes the equations simpler. Popular choices include:

Synchronous Gauge: Choose $T$ and $L$ such that $A = 0$ and $B = 0$ . This gauge is computationally convenient and is used in numerical codes like CMBFAST. It corresponds to a family of observers for whom proper time is the same as the background cosmic time.
Spatially Flat Gauge: Choose $T$ and $L$ such that $C = 0$ and $E = 0$ . This means the spatial hypersurfaces of constant time have no intrinsic curvature perturbation.
Longitudinal (or Newtonian) Gauge: Choose $T$ and $L$ such that $B = 0$ and $E = 0$ . This choice is physically very intuitive because the metric becomes diagonal.

In the longitudinal gauge, the line element simplifies immensely:

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

It is conventional to define two new potentials, $Ψ \equiv A$ and $Φ \equiv - C$ , so the metric becomes:

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

Here, $Ψ$ can be interpreted as the generalization of the Newtonian gravitational potential, and $Φ$ represents the perturbation to the intrinsic spatial curvature. As we will see, for a perfect fluid, the Einstein equations demand $Ψ = Φ$ .

4. Strategy 2: Constructing Gauge-Invariant Variables

While gauge fixing is practical, a more powerful and conceptually elegant approach is to construct combinations of the original perturbation variables that, by their very structure, do not change under any gauge transformation. These are gauge-invariant variables.

The most famous examples were constructed by James Bardeen and are directly related to the potentials in the longitudinal gauge.

Part III Cosmology

Let's return to the same section of the Cambridge notes, this time focusing on the construction of the Bardeen potentials.

Please read the part of Section 3.2 subtitled 'Work with Gauge invariant quantities' (page 35-36). Pay close attention to the definitions of the Bardeen potentials, Ψ_B and Φ_B.

The Bardeen potentials are defined as:

Ψ_{B} \equiv A + H (B - E^{'}) + (B - E^{'})^{'}

Φ_{B} \equiv - C - H (B - E^{'}) + \frac{1}{3} \nabla^{2} E

The remarkable property of these combinations is that they are invariant under gauge transformations.

Test your understanding!

Let's prove the gauge invariance of $Φ_{B}$ . Using the transformation rules for $C, B, E$ that we saw earlier:
$C ~ = C - H T - \frac{1}{3} \nabla^{2} L$
$B ~ = B + T - L^{'}$
$E ~ = E - L$
Show that $Φ ~_{B} = Φ_{B}$ .

Show answer

Let's compute the transformed potential $Φ ~_{B}$ :

Φ ~_{B} = - C ~ - H (B ~ - E ~^{'}) + \frac{1}{3} \nabla^{2} E ~

First, let's find the transformed combination $B ~ - E ~^{'}$ :

B ~ - E ~^{'} = (B + T - L^{'}) - (E - L)^{'} = (B - E^{'}) + T - L^{'} + L^{'} = (B - E^{'}) + T

Now substitute this and the transformations for $C ~$ and $E ~$ into the definition of $Φ ~_{B}$ :

Φ ~_{B} = - (C - H T - \frac{1}{3} \nabla^{2} L) - H ((B - E^{'}) + T) + \frac{1}{3} \nabla^{2} (E - L)

Let's expand and collect terms:

Φ ~_{B} = - C + H T + \frac{1}{3} \nabla^{2} L - H (B - E^{'}) - H T + \frac{1}{3} \nabla^{2} E - \frac{1}{3} \nabla^{2} L

The terms involving the gauge parameters $T$ and $L$ cancel out:

(H T - H T) = 0 and (\frac{1}{3} \nabla^{2} L - \frac{1}{3} \nabla^{2} L) = 0

We are left with:

Φ ~_{B} = - C - H (B - E^{'}) + \frac{1}{3} \nabla^{2} E

This is precisely the definition of $Φ_{B}$ . Thus, $Φ ~_{B} = Φ_{B}$ , and the Bardeen potential $Φ_{B}$ is gauge-invariant. A similar calculation shows the same for $Ψ_{B}$ .

The Bardeen potentials $Ψ_{B}$ and $Φ_{B}$ are the true physical degrees of freedom of the scalar metric perturbation. They represent the gravitational potential and spatial curvature perturbation in a way that is independent of our coordinate choice. Notice that in the longitudinal gauge ( $B = E = 0$ ), we have $Ψ_{B} = A = Ψ$ and $Φ_{B} = - C = Φ$ . This shows that the Bardeen potentials are precisely the metric potentials as measured in that specific, physically intuitive coordinate system.

5. The Zoo of Gauge-Invariant Variables

The Bardeen potentials are not the only useful gauge-invariant quantities. Different physical situations call for different variables. For example, when connecting inflation to the later universe, it is useful to measure the curvature on surfaces that have a special physical meaning.

This leads to other important gauge-invariant variables, such as:

The Comoving Curvature Perturbation, $R$ : This measures the spatial curvature on hypersurfaces where the total cosmic fluid is at rest (i.e., the fluid has no peculiar velocity).
The Curvature Perturbation on Uniform Density Hypersurfaces, $ζ$ : This measures the spatial curvature on hypersurfaces where the total energy density is constant.

The following advanced review provides an excellent catalogue of these variables and the physical reasoning behind their construction.

Cosmological perturbations

Now that we've mastered the Bardeen potentials, let's survey other physically motivated gauge-invariant variables. Different variables are better suited for different physical regimes. The review paper 'Cosmological perturbations' by Malik and Wands provides a comprehensive catalogue. We will not read it in its entirety but will focus on the definitions of a few key variables.

Please skim Section 7, 'Gauge-invariant variables' (starts on page 28). You don't need to follow every derivation, but focus on understanding the concept behind each gauge choice and the resulting gauge-invariant variable: Section 7.1 (Longitudinal gauge): Quickly review this to see the definitions of Φ and Ψ, which we've already covered (note the notation φ₁ and ψ₁ in the paper correspond to A and -C in our other resource, and Φ and Ψ are the Bardeen potentials). Section 7.4 (Comoving orthogonal gauge): Read this to understand the definition of the comoving curvature perturbation R. The key idea is to define a coordinate system tied to the fluid's rest frame. Section 7.6 (Uniform density gauge): Read this to understand the definition of the curvature perturbation on uniform density hypersurfaces ζ. Here, the coordinate system is tied to surfaces of constant energy density. Pay attention to the physical idea: we define our coordinates based on a property of the matter (e.g., fluid velocity is zero or density is constant) and then calculate the curvature in that frame.

The crucial takeaway from this survey is that we have a toolbox of gauge-invariant variables.

$Ψ$ and $Φ$ (Bardeen Potentials) are best for studying the evolution of structures on scales smaller than the cosmological horizon, where they behave like Newtonian potentials.
$R$ and $ζ$ are invaluable for studying the super-horizon evolution of perturbations. A key result, which we will prove later in the course, is that for adiabatic perturbations (where all fluid components are perturbed in the same way), $R$ and $ζ$ are conserved on super-horizon scales. This conservation law is the magic bridge that allows us to take the predictions from the theory of inflation and use them to calculate the temperature anisotropies in the Cosmic Microwave Background.

Conclusion

Today, we've navigated the subtle but critical issue of gauge freedom in cosmology. We've seen that the apparent perturbations to spacetime are not always physical, but we have developed robust strategies to isolate the true degrees of freedom.

Key Takeaways:

The decomposition of the metric into a background and a perturbation is not unique and depends on the choice of coordinates, giving rise to gauge freedom.
Spurious gauge modes can appear as perturbations when they are merely artifacts of the coordinate system.
We can handle this freedom by fixing a gauge (e.g., the intuitive Longitudinal/Newtonian gauge) or by constructing gauge-invariant variables.
The Bardeen potentials, $Ψ$ and $Φ$ , are the fundamental gauge-invariant scalar metric perturbations, corresponding to the gravitational potential and spatial curvature perturbation.
Other important gauge-invariant quantities, like the comoving curvature perturbation $R$ and the uniform-density curvature perturbation $ζ$ , are defined by measuring curvature on physically meaningful hypersurfaces. These are essential for studying super-horizon physics and are conserved in simple inflationary models.

Preview of the Next Lesson:

We now have all the conceptual tools in place. We have decomposed perturbations (SVT) and learned how to describe them physically using gauge-invariant variables. In the next lesson, we will put everything together to derive the linearized Einstein equations for scalar perturbations. We will do this in the Newtonian gauge, which will yield a set of dynamical equations for the Bardeen potentials $Ψ$ and $Φ$ , sourced by the physical perturbations in density, pressure, and velocity of the matter in the universe.

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