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## A Perturbation equations with G ( t ) & Λ( t ) 

The perturbed metric can be written as the sum of the background metric ¯ g$_{αβ}$ , and the pertur- bation h$_{αβ}$ , g$_{αβ}$ = ¯ g$_{αβ}$ + h$_{αβ}$ . We choose the background to be flat homogeneous and isotropic FLRW metric: 

<img src='content_image/326649.jpg'>

with scale factor a ( t ), and the perturbation decomposed into (spatial) scalar, vector, and tensor modes. Hence, the line element becomes 

<img src='content_image/326650.jpg'>

where A, B, E and F are scalars, C$_{i}$ and G$_{i}$ are divergenceless vectors and D$_{ij}$ is a symmetric divergenceless–traceless tensor. 

Analogous to the metric decomposition, the perfect fluid energy–momentum tensor, ¯ T$_{αβ}$ = ¯ p g$_{αβ}$ + (¯ p + ¯ ρ )¯ u$_{α}$ ¯ u$_{β}$ is perturbed as: 

<img src='content_image/326651.jpg'>

<img src='content_image/326653.jpg'>

<img src='content_image/326652.jpg'>

where a bar over any quantity represents its non–perturbed value. δp and δρ are pressure and density perturbations. The velocity perturbation is decomposed as δu$_{i}$ = ∂$_{i}$δu + δu V i to a scalar velocity potential δu and a divergenceless vector δu$_{i}$ . The other parameters in δT$_{ij}$ , i.e. scalar π S i , divergenceless vector π V i and symmetric divergenceless-traceless tensor π T $_{ij}$, characterize departure from energy-momentum tensor of perfect fluid. 

Since the CMB inhomogeneity comes from scalar perturbation, in the following we omit the vector and tensor perturbations and just consider the scalar mode. Separating these terms on both sides of the Einstein perturbed equation, δR$_{αβ}$ = − 8 πG ( t )( δT$_{αβ}$ − 1 $_{2}$¯ g$_{αβ}$δT λ λ − 1 $_{2}$h$_{αβ}$ ¯ T λ $_{λ}$), 

