# 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?

• You may find this classic article by Davis and Lineweaver illuminating. Expanding Confusion: common misconceptions of cosmological horizons and the superluminal expansion of the Universe – PM 2Ring Jul 21 at 2:16
• Nice article, but I'm not confused about what Conformal Time is. I want to know what's the advantage to using it in CMB acoustic analysis over chronological time. – Quarkly Jul 22 at 13:21
• Ah, ok. I assume by chronological time you refer to comoving time (the time according to the clock of an observer in the comoving frame of the CMB). The conformal time is to the proper distance as the comoving time is to the comoving distance. I don't know much about CMB acoustics (so I won't attempt to write an answer), but I suppose it makes sense to analyze it in terms of conformal time & proper distance, rather than in comoving time & distance, since the latter subtracts out the predominant motion of the CMB. – PM 2Ring Jul 22 at 18:58
• I'm not sure I've come across the concept of 'comoving time'. It seems like an oxymoron. Can you tell me a little bit more about what comoving time is? – Quarkly Aug 12 at 14:17
• Comoving time is the proper time of an observer who moves along with the Hubble flow, thus they see the CMB as isotropic. See en.wikipedia.org/wiki/… – PM 2Ring Aug 12 at 14:47

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.

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).

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.

• Thanks but I know what Conformal Time is. What I asked is what's the advantage to using it for CMB analysis. – Quarkly Jul 21 at 23:34
• @DonaldAirey "For perturbations to be meaningful, we have to define what is unperturbed spacetime. A natural choice is a spatially homogeneous and isotropic expanding FLRW spacetime. In this case, there are prefered coordinates. Threading corresponds to the worldlines of observers who see zero momentum density at their position. To them, expansion appears isotropic and their worldline are geodesics. Slicing is orthogonal to the threading, and on each slice, space is homogeneous. – Árpád Szendrei Jul 23 at 9:27
• @DonaldAirey "In presence of perturbations, no coordinates exist in which all these properties are verified. The minimal requirement is that they reduce to those of RWFL spacetime when perturbations are absent. There are many reasonable choices which satisfy these properties. Each choice defines the perturbations and perturbations can look very different between two coordinate choices, very much like the gauge potential does in electromagnetism. Hence the nomenclature. – Árpád Szendrei Jul 23 at 9:27
• @DonaldAirey "The gauge adopted here is called the Conformal Newtonian Gauge, although for the sake of illustration we also consider so-called comoving gauges." Please see here: www2.ulb.ac.be/sciences/physth/SF.pdf – Árpád Szendrei Jul 23 at 9:29