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There is one remaining interesting problem with the isotropy of the background radiation. It is actually too isotropic. For the background to be isotropic, the early universe, apart from the fluctuations that would become galaxies, must have been quite uniform. We even incorporate this uniformity into our cosmological models. However, as illustrated in Fig. 21.13, conditions can only be identical at different locations if x0

Decoupling

Decoupling   Region Reached by Signal Before Decoupling

Region Reached by Signal Before Decoupling

Fig 21.13.

Causality problems in the early universe. In this sequence, we look at the section of the universe that can be affected by light signals starting at some point just after the expansion started, traveling at the speed of light until decoupling. (a) The light signals start at the center of the grid. (b) As the universe expands (indicated by the larger grid), the light signals spread out in all directions.The green shaded area shows the region through which these signals have passed. (c) The expansion continues and the light signals spread farther. Eventually decoupling is reached.The green shaded area shows that region containing material that could have absorbed the radiation before decoupling.This is the only part of the universe that could have been affected before decoupling by the conditions at the center of the grid at the beginning. (d) After decoupling, no more material can be affected by the spreading light signals.The region that has been affected (shaded in purple) now simply expands with the universe.This is indicated by the fact that the shaded area in (d) covers the same number of grid sections as in (c).

Fig 21.14. Relationship between linear and angular scales that could have communicated in the lifetime of the universe.

they have some way of communicating with each other. Something must have regularized the structure in the early universe. However, such communication can travel no faster than the speed of light. Two objects separated by a distance greater than that which light can travel in the age of the universe cannot affect, or cause events to happen at, each other. This is called the causality problem.

We can use Fig. 21.14 to show the scales on which this is a problem. We must relate the current angular separation between two points with their linear separation at the time of decoupling. We now let the two points be separated by an angle 90 (where 90 V1 rad). If the points are a distance d from us, their current separation is

If we are just seeing the light from these points, it must have been traveling for t0, the current age of the universe. (We ignore the small difference between the current age of the universe and the time since decoupling, since decoupling occurred very early in the history of the universe.) This means that d = ct0

XQ 0octQ

If we take our reference epoch to be now, so that R(t0) = 1, and we let R be the scale factor at the time of decoupling, then the separation between the two points at the time of decoupling is R times their current separation, or x = 90ct0R

The time for light to travel the distance x is x/c, or

For these two points to communicate, At(x) must be less than or equal to t, the age at decoupling. The farthest the two points can be is when At(x) = t, so t = W

We now solve for 00, the maximum current angular separation between the two points that could have been causally connected before decoupling:

Example 21.1 Causality in the early universe If the age of the universe at decoupling was 105 yr, and the current age is 1.5 X 1010 yr, find the maximum current angular separation between two points that can be causally connected.

solution

To use equation (21.8), we have to find R, the scale factor at the time of decoupling. We know that the temperature of the background radiation at any time t, T(t), is proportional to 1/R(t). The background temperature is 3 K now and was 3000 K at decoupling, so R at decoupling must have been 10~3. This means that

0 (10~3)(1.5 X 1010 yr) = 7 X 10~3rad This is approximately 0.4°.

There is another source of small-scale anisotropy in the background radiation that is not actually cosmological. It has to do with the interaction of the radiation with clusters of galaxies as it passes

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