7.3.1 How do peculiar velocities build up?

When some part of the Universe contains more matter than average, its increased gravity brakes the expansion more strongly. Suppose that the average density is (t), so that the ratio (t) = / crit, and the average expansion is described by the scale factor (t), and the Hubble parameter (t). Then within the volume we are studying, we write

 (7.25)

If our region is approximately spherical, we can follow the same steps that led us to Equation 7.15: the matter outside will not exert any gravitational force within it. If the region is large enough that we can ignore gas pressure (see below), then it will behave just like part of a denser, more slowly expanding, cosmos.

Where the local density exceeds the critical value, with 1 + (t) > -1 (t), expansion will be halted, allowing bound groups and clusters of galaxies to form. Matter will fall in on itself, growing ever denser until pressure intervenes to arrest the collapse, or random motions become important. Where there is less matter than average, the expansion is faster; the region becomes even more diffuse relative to its surroundings. Once the Universe is matter dominated, we have (see the following problem)

 (7.26)

so (t) always approaches unity as z -> , or t -> 0. Even if 0 < 1 and the average density is now below critical, at early times only a tiny fractional change would be required to give > 1.

 Problem 7.14: Use Equation 7.19 to show that a2 H2 (t) [ 1 - (t) ] = H02 [ 1 - 0 ]. Substitute for H (t) from Equation 7.21 to show that Equation 7.26 holds when (t) a-3.

The problem becomes much simpler if we are in the linear regime, where and of Equation 7.25 are much less than unity. We noted in Section 7.1 (in the discussion following Equation 7.4) that fluctuations on scales larger than about 8 h-1 Mpc are linear at present; earlier, even smaller structures were in this regime. We can substitute into Equation 7.20, ignoring terms in 2, , 2, and higher powers of these variables. Remembering that terms involving only barred average quantities will cancel, we find

 (7.27)

the last term represents the change in the present density and expansion rate within our denser region. When the Universe is matter dominated, a3 is constant, so = 3 . Using Equation 7.22 for (t), we have

 (7.28)

Early on, z is large and the contrast in density grows proportionally to (t). At late times, when the average motion is given by t, matter coasts outward with constant speed. Its gravity is too weak to have any effect on the expansion, so remains fixed. If we now live in a low-density Universe with 0 ~ 0.1, large structures ceased to become denser around redshift z ~ 8. If 0 1, clusters continue to become denser, and voids in the galaxy distribution to expand, up to the present day.

 Problem 7.15: Substitute t2/3 into Equation 7.27 to show that where > 0, the expansion is less than average - which is why these regions become denser.

 Problem 7.16: Show that (t) t-1 is also a solution to Equation 7.27; some irregularities will smooth themselves out as the Universe expands. Show that now expansion is faster than average in regions where > 0.

Any denser-than-average region pulls the surrounding galaxies more strongly toward it. While the fractional deviations (x, t) from uniform density remain small, Equation 7.28 tells us that over a given time, (x) increases by an equal factor everywhere. Because the pull on a galaxy from each overdense region increases in the same proportion, its acceleration, and hence its peculiar velocity, is always parallel to the local gravitational force. So by measuring peculiar motions, we can reconstruct the force vector, and hence the distribution of mass.

To see how this works, we can write the velocity u(x, t) of matter at point x as the sum of the average cosmic expansion directly away from the origin, and a peculiar velocity v:

 (7.29)

The equation of mass conservation relates the velocity field u(x, t) to the density, which we write as (x, t) = (t) [ 1 + (x, t)]:

 (7.30)

Remembering that terms involving only the barred average quantities will cancel, and dropping terms in 2, v and v2, we have

 (7.31)

Setting x = (t) r, we switch to the coordinate r comoving with the average expansion. The time derivative following a point at fixed r is

 (7.32)

and since (t) r = x, Equation 7.31 simplifies to

 (7.33)

Defining a velocity potential v such that v = x v, we rewrite this as

 (7.34)

For a small enough volume, if we assume that the Universe beyond is homogenous and isotropic, we can use Newton's laws to calculate the gravitational force Fg corresponding to local deviations from the average density , and the potential g such that Fg = - g. Equation 3.9, Poisson's equation, tells us that

 (7.35)

- which looks suspiciously like the equation for v. Equation 7.28 assures us that all perturbations grow at the same rate, so (x, t) ð (x, t) / ð t. Then as long as both v (x, t) and Fg diminish to zero as | x | increases, they must also be proportional: the peculiar velocity is in the same direction as the force resulting from local concentrations of matter. Dividing the right-hand sides of the last two equations, we find

 (7.36)

From Equation 7.28, in a matter-dominated Universe we have f = 1 for 1, and f -> 0 as -> 0; in general, f () 0.6 is a good approximation. Using Equation 3.5 for the force, we can write the peculiar velocity as

 (7.37)

 Problem 7.17: Show that if the density is uniform apart from a single overdense lump at x = 0, then distant galaxies move toward the origin with v (x, t) 1/ x2.

Problem 7.18: In the expanding coordinate r, show that

 (7.38)

Use Equation 7.28 for (r) to show that the peculiar velocity v t1/3 at early times, while 1. In an open
matter-dominated Universe, show that at late times when (t) t, v f(), decreasing roughly as t-0.6.

So if we can measure the overdensity (x) of the nearby rich galaxy clusters, and the peculiar velocities of the galaxies around them, we should be able to test Equation 7.37 and solve for the density parameter 0. First, we determine the average peculiar motion v (x) of our galaxies. We must assume that the Universe is homogeneous and isotropic on even larger scales, so that forces from galaxies outside our survey volume will average to zero. Inverting Equation 7.37 should then yield the product f (0) . (x), from which we can find 0.

But the mass distributions predicted from measured peculiar velocities do not match the observed clustering of galaxies very well. Alternatively, we could say that the forces calculated from the galaxies at their observed positions do not yield the measured peculiar motions. The pull of matter outside the volume of our present surveys appears to be significant. In particular, we have still not identified the concentration of matter responsible for most of the Local Group's peculiar motion of ~ 600 km s-1. Work is under way on this problem, and galaxy surveys are being extended as techniques for finding distances improve.

Locally, we can use the crude model of Figure 7.8 for the Virgocentric infall to estimate the density parameter 0. Let dV 16 Mpc be the distance of the Local Group from the center of the Virgo cluster. Within a sphere of radius dV about the cluster center, the density of luminous galaxies is roughly 2.4 times the mean; if the mass density is increased by the same factor, then the overdensity 1.4. Although Equation 7.36 was derived for << 1, we can use it to make a rough calculation of f().

Assuming that the Virgo cluster is roughly spherical, the additional gravitational pull on the Local Group is Fg 4 G dV / 3, just as if all the cluster's mass had been concentrated at its center. So our peculiar motion toward Virgo is

 (7.39)

Cosmic expansion is pulling the cluster away from us at a speed H0 dV 1400 km s-1, so this yields 0 0.2. Either the mean density of the Universe is well below the critical value = 1, or the Virgo cluster contains more than its fair share of luminous galaxies, so that the excess mass density < 1.4. Some astronomers favor the suggestion that the distribution of luminous matter is lumpier than that of the total. For example, perhaps galaxies can form and make their stars only in the densest regions of the Universe. If the overdensity of galaxies is a factor b greater than that of the matter, then = 1.4 / b. If 0 = 1, we require b ~ 4 to explain the Virgocentric inflow. This idea is known as biassed galaxy formation; we will see in the next section why it has been popular.

Further reading: On a graduate level, see T. Padmanabhan, 1993, Structure Formation in the Universe (Cambridge University Press, Cambridge, UK).
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