What's the advantage of Conformal Time? I'm trying to follow the analysis of CMB Acoustics and several charts are done using conformal time instead of chronological time.  Conformal time corresponds to the quotient of the particle horizon to the speed of light.  What advantage is there to these units?
 A: One situation I've found conformal time very useful in in the context of spacetime diagrams. Since the ratio between proper time $t$ and conformal time $\eta$ is the same as the ratio between proper ("physical") coordinates $d$ and comoving coordinates $\chi$ — namely the scale factor $a$ of the Universe — spacetime diagrams showing conformal time as a function of comoving distance has the aesthetically pleasing property that null geodesics — that is, the paths, or worldlines — of photons are lines that are tilted 45º (provided you've drawn the size of one $\eta$ tick mark in Gyr the same length as one $\chi$ tick mark in Glyr).
Moreover, objects that have no peculiar motion but just follow the expansion of the Universe (comoving objects, such as an average galaxy) are vertical lines.
In contrast, if you show proper time as a function of proper distance — which might at first seem more intuitive — both null geodesics and comoving objects change their slope along the way, so they're difficult to distinguish.
Compare for instance these two spacetime diagrams showing the history of the Universe as $t(x)$ (top) and $\eta(\chi)$ (bottom).

Fig. 1 from Davis & Lineweaver (2004) (excluding the middle panel, and with my own annotations).
A: I think I figured out the answer.  There's not a lot of difference between the conformal time and chronological time in an analysis of the time leading up to recombination.  After all, it's just a coordinate system and there's a simple translation between one coordinate system and the other.  Pah-tay-toe, Pah-tah-toe.
As nearly as I can tell, the primary advantage to using conformal time comes when adding in the effects of the damping trail.  The damping scale is, roughly, the ratio of the mean free path of a photon to the particle horizon, $\sqrt{\frac {\dot\tau}{\eta}}$.  So using conformal time (another way of saying 'particle horizon') seems a more natural axis for integrating the effects of the damping trail.
A: The particle horizon is the maximum distance that light could have traveled to Earth in the age of the universe.
But this is not equal to the speed of light times the age of the universe. It is because space is expanding.
That is why we use the speed of light times the conformal time.


In terms of comoving distance, the particle horizon is equal to the conformal time t  that has passed since the Big Bang, times the speed of light c.
    By convention, a subscript 0 indicates "today" so that the conformal time today  1.48x10^18s. Note that the conformal time is not the age of the universe. Rather, the conformal time is the amount of time it would take a photon to travel from where we are located to the furthest observable distance provided the universe ceased expanding. As such,  it is not a physically meaningful time (this much time has not yet actually passed), though, as we will see, the particle horizon with which it is associated is a conceptually meaningful distance.


https://en.wikipedia.org/wiki/Particle_horizon#Conformal_time_and_the_particle_horizon
The particle horizon recedes constantly, and the conformal time grows. The observed size of the universe always increases. The proper time to the particle horizon is 46.9 billion light years.
