Last scattering and a finite visibility interval

A thin surface is a useful approximation

Radiation observed from a distant region can have scattered before reaching us. The probability that a given event was its last scattering depends both on the scattering opportunity there and on the absence of later scattering along the path.

The term last-scattering surface describes a line-of-sight shell associated with a finite range of emission or scattering times. Treating it as infinitely thin is an approximation, rather than a claim about a material wall or identical event time for every photon.

Last scattering also differs from photon creation. A photon can exist earlier and scatter many times before its final scattering in the stated propagation account.

The example below uses an ideal supplied profile with an asymptotically transparent late endpoint. It contains no measured visibility curve, actual redshift-to-time conversion, present observer time or physical cosmic origin.

Less remaining opacity means greater survival

Define conformal time η in seconds by dη = dt/a. Optical depth τ_opt from a candidate scattering time toward a fixed reception endpoint measures accumulated scattering opacity. As the candidate time moves later toward that endpoint, the remaining optical depth decreases.

In this convention −dτ_opt/dη = a n_eσ_T c is positive. The factor c is needed because η is measured in seconds. Other sources may use different symbols or conventions, so the sign and units must be mapped explicitly.

The probability of no subsequent scattering is S = exp(−τ_opt). The visibility density per conformal time is g_η = dS/dη = −τ_opt′ exp(−τ_opt), with inverse-second units. A probability is obtained by integrating that density over an interval.

Large early opacity can provide scattering opportunities while making subsequent unscattered survival very unlikely. Very low late opacity gives high survival but few scattering opportunities. Their combination can peak between those limits.

A finite-width peak follows from a smooth opacity law

Choose a positive width Δη and define u = (η − η_star)/Δη. Supply τ_opt(u) = exp(−u), with an ideal infinitely opaque early limit and transparent asymptotic late limit. The analytic idealization extends over all real u; the displayed window will be only −3 ≤ u ≤ 4.

Then S(u) = exp[−exp(−u)] and g_u = dS/du = exp[−u − exp(−u)]. Here g_u is probability density per unit u, while g_η = g_u/Δη is the density per conformal second.

The density peaks at u = 0, where τ_opt = 1 and g_u = 1/e ≈ 0.367879. It is not zero immediately before or after the peak. The profile describes a distribution across an interval rather than an instantaneous transition.

The integral over the full ideal line is 1 because S tends from zero to one. That normalization relies on the declared ideal endpoints; it should not be assigned automatically to an arbitrary truncated plot or physical history.

Quantiles and omitted tails keep the width honest

Inverting S(u) = p gives u_p = −ln(−ln p). The 10 percent and 90 percent points are about −0.834032 and 2.250367, so the central 80 percent spans about 3.084400 Δη in conformal time.

The median is u_0.5 ≈ 0.366513, which differs from the peak at zero. This asymmetry is part of the supplied profile; peak time, median time and an interval width are distinct summaries.

The displayed window from −3 to 4 contains S(4) − S(−3) ≈ 0.981851071 of the total probability. The omitted early tail is about 1.89218 × 10⁻⁹ and the omitted late tail about 0.018148927.

Those tails are retained in the probability account. Renormalizing the displayed portion silently to one would describe a conditional truncated distribution instead of the stated full profile.

Visibility is one ingredient in an observed sky

The supplied profile demonstrates the relation between opacity, survival and a last-event density. It does not derive atomic populations, solve recombination or include reionization, an actual reception epoch or measured cosmological parameters.

An observed CMB anisotropy calculation combines visibility with temperature, velocity, gravitational and polarization sources along the line of sight. Finite width can affect how those sources are averaged and projected.

Later scattering and gravitational propagation can add further effects. The last-scattering approximation should therefore be identified with the calculation and observable for which it is being used.

When reading a visibility graph, check the time variable, density units, optical-depth sign, endpoint and plotted normalization. These details determine whether an area is a probability and whether a claimed surface is being treated as an approximation.

Worked reasoning

Try the reasoning

1. Is g_η a dimensionless probability?

   Show the answer and reasoning

   No; it is a probability density with inverse-second units.

   Integrating g_η over a conformal-time interval gives a probability.

2. Does last scattering mean that the photon was created at that event?

   Show the answer and reasoning

   No.

   It labels the final scattering in the stated propagation account, not necessarily the photon’s creation.

3. Does the displayed −3 ≤ u ≤ 4 window contain exactly all the profile’s probability?

   Show the answer and reasoning

   No; it contains about 0.981851071.

   The unplotted early and late tails remain part of the declared full distribution.

What to carry forward

• Visibility combines positive scattering opacity with the probability of no later scattering.
• A probability density needs a time convention and integration interval.
• A finite last-scattering shell is neither a creation surface nor an infinitely thin material wall.