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The universe as an expanding sphere.As the sphere expands, the coordinate system expands, and the radius of curvature changes.The dark circle with the lighter interior represents the part of the universe that we can see, with us at the center of the circle. A photon could have traveled from the circle to the center in the age of the universe. As the universe expands and ages, our horizon expands also. Since our horizon expands at the speed of light, new objects are always coming over the horizon.

The universe as an expanding sphere.As the sphere expands, the coordinate system expands, and the radius of curvature changes.The dark circle with the lighter interior represents the part of the universe that we can see, with us at the center of the circle. A photon could have traveled from the circle to the center in the age of the universe. As the universe expands and ages, our horizon expands also. Since our horizon expands at the speed of light, new objects are always coming over the horizon.

We consider the universe as being confined to the surface of an expanding sphere, as shown in Fig. 20.7. One concept that we can now visualize is that we have a horizon due to the finite age of the universe t (as noted in our discussion of Olbers's paradox). We can only see light emitted toward us within a distance equal to ct. This horizon is growing. We have seen that over small distances even the surface of a sphere appears flat. The curvature becomes apparent as you can survey larger areas. This means that, as our horizon grows, the curvature might become more apparent.

As the universe expands, we would like to keep track of the separation between any two co-moving points. We start by considering nearby points at r and r + dr. The proper distance between those points is

If the points are far apart (like a distant galaxy to here), then we have to integrate that expression.

We can also use our analogy to see that it is meaningless to talk about the radius of the universe. In three dimensions, our sphere has a radius, but in two dimensions we can only talk about the surface. This is one reason why the scale factor R(t) is a better way of keeping track of the expansion. Even though we cannot talk about the radius of the universe in a meaningful way, we can talk about the curvature of our surface. So we can talk about the radius of curvature of the universe. The larger the radius of curvature, the closer the geometry is to being flat.

Our expanding sphere analogy also tells us that it is not very meaningful to talk about the center of the universe. The sphere has a center, but it is not in the universe, which is the surface only. There is nothing special about any of the points on the surface of the sphere. If we go back in time to very small times, our sphere will be very small, and at t = 0 all the points are together, at the center. So the proper way to talk about the center is as a space-time event far in our past.

### 20.4.2 Cosmological redshift

We can also see that the redshift (Hubble's law) fits in as a natural consequence of the expansion (Fig. 20.8). As the universe expands, the wavelengths of all photons expand by the same proportion that cosmic distances expand. That is, they expand in proportion to the scale factor.

Cosmological redshift.As the universe expands, represented by the expanding sphere, the wavelengths of all photons increase in proportion to the scale increase. In this figure the arrow traces the route of a photon, emitted in the first frame, through an expanding universe in the second frame, and absorbed in the third frame.The solid part of the arrow shows where the photon has already been.

where AA

Cosmological redshift.As the universe expands, represented by the expanding sphere, the wavelengths of all photons increase in proportion to the scale increase. In this figure the arrow traces the route of a photon, emitted in the first frame, through an expanding universe in the second frame, and absorbed in the third frame.The solid part of the arrow shows where the photon has already been.

We define the redshift, z, to be AA

where AA

If radiation is emitted at wavelength A1 at epoch t1, and detected at wavelength A2 at epoch t2, then

Remember, since the radiation is emitted before the reference epoch, R(t) < 1, so z > 0.

We can derive an approximate expression for the redshift for radiation emitted some time At in the recent past, where At V t0. Using a Taylor series, we have (see Problem 20.11)

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