Scalar Perturbations

Scalar perturbations in a flat universe are represented by plane waves $Q^{(0)}= \exp( i \vec{k} \cdot \vec{x} )$,which are the eigenfunctions of the Laplacian operator  

$$\nabla^2 Q^{(0)}= -k^2 Q^{(0)},$$ (18)

and their spatial derivatives. For example, vector and symmetric tensor quantities such as velocities and stresses based on scalar perturbations can be constructed as

$$Q_i^{(0)}= - k^{-1} \nabla_i Q^{(0)}, \qquad Q_{ij}^{(0)}= [k^{-2} \nabla_i \nabla_j + {1 \over 3} \delta_{ij}] Q^{(0)}.$$ (19)

Since $\vec{\nabla} \times \vec{Q}^{(0)}= 0$, velocity fields based on scalar perturbations are irrotational. Notice that Q⁽⁰⁾= G₀⁰, nⁱ Q_i⁽⁰⁾= G₁⁰ and ${n}^i {n}^j Q_{ij}^{(0)}\propto G_{2}^0$,where the coordinate system is defined by $\hat{e}_3 = \hat{k}$.From the orthogonality of the spherical harmonics, it follows that scalars generate only m=0 fluctuations in the radiation.