### 6.2 Gauge-invariant quantities

Before discussing the detail for the evolution of cosmological perturbations, we construct a number of
gauge-invariant quantities. This is required to avoid the appearance of unphysical modes. Let us consider
the gauge transformation
where and characterize the time slicing and the spatial threading, respectively. Then the scalar
metric perturbations , , and transform as [57, 71, 412]
Matter perturbations such as and obey the following transformation rule

Note that the quantity is also subject to the same transformation: . We express the
scalar part of the 3-momentum energy-momentum tensor as
For the scalar field and the perfect fluid one has and , respectively. This
quantity transforms as
One can construct a number of gauge-invariant quantities unchanged under the transformation (6.20):

Since for single-field inflation with a potential , is identical to [where we
used and ]. In f (R) gravity one can introduce a scalar field as in
Eq. (2.31), so that . From the gauge-invariant quantity (6.31) it is also possible
to construct another gauge-invariant quantity for the matter perturbation of perfect fluids:
where .
We note that the tensor perturbation is invariant under the gauge transformation [412].

We can choose specific gauge conditions to fix the gauge degree of freedom. After fixing a gauge, two
scalar variables and are determined accordingly. The Longitudinal gauge corresponds to the gauge
choice and , under which and . In this gauge one has
and , so that the line element (without vector and tensor perturbations) is given by

where we omitted the hat for perturbed quantities.
The uniform-field gauge corresponds to which fixes . The spatial threading is
fixed by choosing either or (up to an integration constant in the former case). For this
gauge choice one has . Since the spatial curvature on the constant-time hypersurface is
related to via the relation , the quantity is often called the curvature
perturbation on the uniform-field hypersurface. We can also choose the gauge condition or
.