- #1
deneve
- 37
- 0
I've seen in some lecture notes that the proper distance dp(t) can be written as
##\int_{t_e}^{t_0} c dt/a = \int_0^z c dz /H(z)##
I can perform this integral ok using
##H =\dot a/a## and the fact that ##1 + z = 1/a(t_e)## but it requires associating the limits of the integration as te transforming to z and t0 to z= 0 - there is a minus sign which creeps in when you find dz/dt because da = - a2dz so the limits have to be switched. Thus they don't match as it appears when you read the integral.
I don't see how to interpret this because I feel it should be the other way round. That is t 0 should be associated with red shift z (that's what we measure today) and at time t = te the red shift should be 0 . Why am I wrong here?
##\int_{t_e}^{t_0} c dt/a = \int_0^z c dz /H(z)##
I can perform this integral ok using
##H =\dot a/a## and the fact that ##1 + z = 1/a(t_e)## but it requires associating the limits of the integration as te transforming to z and t0 to z= 0 - there is a minus sign which creeps in when you find dz/dt because da = - a2dz so the limits have to be switched. Thus they don't match as it appears when you read the integral.
I don't see how to interpret this because I feel it should be the other way round. That is t 0 should be associated with red shift z (that's what we measure today) and at time t = te the red shift should be 0 . Why am I wrong here?