For a flat matter-dominated universe this gives us:
13
Expression 14: "z" left parenthesis, chi , "T" , right parenthesis equals StartFraction, "T" Superscript, 2 thirds , Baseline Over left parenthesis, "T" Superscript, 1 third , Baseline minus chi Superscript, 1 third , Baseline , right parenthesis squared , EndFraction minus 1 Superscript, , Baseline left parenthesis, chi Superscript, 1 third , Baseline less than "T" Superscript, 1 third , Baseline , right parenthesiszχ,T=T23T13−χ132−1χ13<T13
14
The redshift drift is:
15
Expression 16: "z" Subscript, "T" , Baseline prime left parenthesis, chi , "T" , right parenthesis equals StartFraction, ∂ "z" Over ∂ "T" , EndFraction left parenthesis, chi , "T" , right parenthesiszT′χ,T=∂z∂Tχ,T
16
Which gives us for a flat matter-dominated universe:
17
Expression 18: "z" Subscript, "T" , Baseline prime left parenthesis, chi , "T" , right parenthesis equals StartFraction, 2 left parenthesis, StartNestedFraction, chi NestedOver "T" , EndNestedFraction , right parenthesis Superscript, 1 third , Baseline Over 3 left parenthesis, chi Superscript, 1 third , Baseline minus "T" Superscript, 1 third , Baseline , right parenthesis cubed , EndFraction left parenthesis, chi Superscript, 1 third , Baseline less than "T" Superscript, 1 third , Baseline , right parenthesiszT′χ,T=2χT133χ13−T133χ13<T13
18
As:
19
Expression 20: chi Superscript, 1 third , Baseline less than "T" Superscript, StartFraction, 1 Superscript, , Baseline Over 3 , EndFraction , Baselineχ13<T13
20
and
21
Expression 22: chi greater than 0 Superscript, , Baselineχ>0
22
Expression 23: "T" Superscript, , Baseline greater than 0T>0
23
The redshift drift is always negative for a flat matter-dominate universe, i.e.:
24
Expression 25: "z" Subscript, "T" , Baseline prime left parenthesis, chi , "T" , right parenthesis less than 0zT′χ,T<0
