In cosmology it's frequently said that photons from the early universe carry information from that time. However, wouldn't they also carry data from later interactions? How do we differentiate between the data from various time periods? As an analogy, consider photons that strike your face and reach my eyes. We say that that photon carries information about your face which then helps me to identify you, but don't these photons collide midway with air molecules, gaining information from the air? By the same logic that I can see your face through the information gained with collisions with your face, why can't I see the photon's collisions with the air, or even something the photons hit before they collided with you, like fans, bridges, etc?
As an analogy, consider the photon that strike your face and reach my eyes, we say that that photon carries information about your face which then helps me to identify you,
You are confusing the individual photons with the electromagnetic wave that is light, which is composed out of a zillion photons. It is the superposition of the wavefunctions of individual photons that can carry information.
but don't these photons collide midway with air molecules, and if they still somehow retain the information,
An individual photon can be absorbed or scattered out of the beam it is composing. A few photons scattered out of the beam that is carrying information make no difference as the number of photons is enormous.
If a photon is scattered coherently with the other zillion photons the beam itself carries information of the way all these photons are building it up, in the total wave function, which is a superposition. That is how images can be refracted and reflected.
At some point in opaque and non reflecting surfaces the coherence is lost and any information the beam has been carrying is lost.
then going by the same logic why don't I see the image of things that photon collided before coming to your face like fans, bridge etc
The individual photon is distinguished by its energy and direction, +1 or-1 spin projection on its momentum. When you see an image, you see the collective superposed interaction of the beam of photons, light. When light strikes a "fan, a bridge etc." it disperses and carries information from the object. The part of the light reflected from the bridge when it hits a second object loses the previous coherence by striking and reflecting back, and coheres in a new image pattern.
Generally coherence is lost as electromagnetic beams interact with matter.
In cosmology it's frequently said the photon from the early universe carry information of that time.
The experiments measuring cosmic microwave background radiation are very careful to look at clear regions of the sky. Some are on satellites for this reason so as not to have interference from backgrounds. The photons they are measuring have not interacted before interacting with the detectors of the experiment. If they had been absorbed they would not have arrived at the detector.
The individual photons carry information of their directions, which way the were coming, and their energy, which was absorbed to give the count. That is how the CMB maps are made.
The ensemble of CMB photons arriving at the detectors may have interacted with cosmic dust, for example, or been deflected by strong gravitational fields, and this will affect the polarization displayed by the "beam" of arriving photons. This polarization has been measured in the Planck experiment and with BICEP2 and it is a research question whether it is an original polarization from the inflationary period due to gravitational waves, or due to scatterings from dust. This information is a collective information carried by the beam the CMB photons build up.
But don't they also carry the data afterwards and how do we then differentiate between the data's.
So some information can be carried by the ensemble of photons, as described above.