I've seen in some lecture notes that the proper distance d(adsbygoogle = window.adsbygoogle || []).push({}); _{p}(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 t_{e}transforming to z and t_{0}to z= 0 - there is a minus sign which creeps in when you find dz/dt because da = - a^{2}dz 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 = t_{e}the red shift should be 0 . Why am I wrong here?

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# Proper distance integral limits seem wrong

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