matter redshift

For a flat matter-dominated universe this gives us:

Espressione 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 parenthesis

The redshift drift is:

Espressione 16: "z" Subscript, "T" , Baseline prime left parenthesis, chi , "T" , right parenthesis equals StartFraction, ∂ "z" Over ∂ "T" , EndFraction left parenthesis, chi , "T" , right parenthesis

Which gives us for a flat matter-dominated universe:

Espressione 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 parenthesis

As:

Espressione 20: chi Superscript, 1 third , Baseline less than "T" Superscript, StartFraction, 1 Superscript, , Baseline Over 3 , EndFraction , Baseline

and

Espressione 22: chi greater than 0 Superscript, , Baseline

Espressione 23: "T" Superscript, , Baseline greater than 0

The redshift drift is always negative for a flat matter-dominate universe, i.e.:

Espressione 25: "z" Subscript, "T" , Baseline prime left parenthesis, chi , "T" , right parenthesis less than 0

Actual maths/graph

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