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Horizon distance, as i understand is the distance the photon has traveled from its point of emission. But what is the proper distance, as a terminology used in cosmology?

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The proper distance between two nearby observers is the distance (at the time it is being observed) that is measured with rulers. If you're familiar with relativity, the proper distance between two nearby events is the distance between them on a frame in which they happen at the same time.

This is related with another useful definition of distance in cosmology: comoving distance. Which for two nearby observers is the proper distance measured at a given time multiplied by the ratio of the scale factor $a$ then to now. If the observers are separated by an arbitrary distance, then

$$ {d}_{\rm comoving} = \frac{c}{H_0}\int_0^z\frac{{\rm d}z'}{E(z')} $$

with

$$ E^2(z) = \Omega_{m,0}(1 + z)^3 + \Omega_{\gamma,0}(1 +z)^4 + \Omega_{\Lambda} $$

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