  In the same way as the energy density associated with the tensor modes of the
geometry modifies the evolution of the curvature perturbations, the scalar
modes may also indirectly affect the cosmic backgrounds of relic gravitons by
inducing higher-order corrections that are only superficially gauge-invariant.
This spurious gauge-invariance gets manifest when the effective anisotropic
stresses, computed in different coordinate systems, are preliminarily expressed
in a form that only depends on the curvature inhomogeneities defined on
comoving orthogonal hypersurfaces and on their corresponding time derivatives.
Using this observation we demonstrate in general terms that the higher-order
contributions derived in diverse coordinate systems coincide when the
wavelengths are smaller than the sound horizon defining the evolution of the
curvature inhomogeneities but they lead to sharply different results in the
opposite limit. A similar drawback arises when the energy density of the relic
gravitons is derived from competing energy-momentum pseudo-tensors and should
be consistently taken into account in the related phenomenological discussions.
