In the following the latin scripts range from 1 thru 3, for spacial coordinates. Greek scripts range from 0 thru 3 and include the time dimension. The qualification “for each i” means to suppress summation on the qualified expressions.

Several popular description of Guth’s inflation suggests a metric like
g_αβ = 0 except for each i g_ii = −e^2It and g₀₀ = 1.
where I is the inflation rate, a large positive real, and t = x⁰ or time. (x¹, x², x³ are spatial coordinates.) Note that the distance between two ‘points’, such as (t, 0, 0, 0) and (t, 1, 0, 0) increases exponentially. Lets see what Einstein’s equation says about such a place and what sort of stuff is there.

∂g_αβ/∂x^γ = 0 except ∂g_ii/∂x⁰ = −2Ie^2It.
Γ^γ_αβ = ½ g^γδ (−∂g_αβ/∂x^δ + ∂g_δβ/∂x^α + ∂g_δα/∂x^β)
g^αβ = 0 except gⁱⁱ = −e^−2It and g⁰⁰ = 1
Γ^γ_αβ = 0 except for each i (Γ⁰_ii = −Ie^2It and Γⁱ_i0 = Γⁱ_0i = I).
Wikipedia gives the following formula of the Ricci tensor:
R_αβ = ∂_ρΓ^ρ_βα − ∂_βΓ^ρ_ρα + Γ^ρ_ρλΓ^λ_βα − Γ^ρ_βλΓ^λ_ρα.
∂_ρΓ^ρ_βα = 0 except for each i (∂₀Γ⁰_ii = −2I²e^2It).
Γ^ρ_ρλ = 0 except Γ^ρ_ρ0 = 3I
Γ^ρ_ρλΓ^λ_βα = 0 except Γ^ρ_ρ0Γ⁰_βα = 3IΓ⁰_ii = 3I(−Ie^2It)
i.e. Γ^ρ_ρλΓ^λ_βα = 0 except Γ^ρ_ρλΓ^λ_ii = −3I²e^2It.
Γ^ρ_βλΓ^λ_ρα = 0 except ...

I have currently abandoned this. Perhaps this note is a reparameterization of the space I seek here. Its metric is:
g_ij =
−t⁻² 0
0 t⁻²

A stab at automating this error prone stuff.
crumbs