# 11.6 The big bang  (Page 3/8)

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The fate of this expanding and smooth universe is an open question. According to the general theory of relativity, an important way to characterize the state of the universe is through the space-time metric:

$d{s}^{2}={c}^{2}d{t}^{2}-a{\left(t\right)}^{2}d{\text{Σ}}^{2},$

where c is the speed of light, a is a scale factor (a function of time), and $d\text{Σ}$ is the length element of the space. In spherical coordinates ( $r,\theta ,\varphi \right)$ , this length element can be written

$d{\text{Σ}}^{2}=\frac{{dr}^{2}}{1-k{r}^{2}}+{r}^{2}\left(d{\theta }^{2}+{\text{sin}}^{2}\theta d{\phi }^{2}\right),$

where k is a constant with units of inverse area that describes the curvature of space. This constant distinguishes between open, closed, and flat universes:

• $k=0$ (flat universe)
• $k>0$ (closed universe, such as a sphere)
• $k<0$ (open universe, such as a hyperbola)

In terms of the scale factor a , this metric also distinguishes between static, expanding, and shrinking universes:

• $a=1$ (static universe)
• $da\text{/}dt>0$ (expanding universe)
• $da\text{/}dt<0$ (shrinking universe)

The scale factor a and the curvature k are determined from Einstein’s general theory of relativity. If we treat the universe as a gas of galaxies of density $\rho$ and pressure p , and assume $k=0$ (a flat universe), than the scale factor a is given by

$\frac{{d}^{2}a}{d{t}^{2}}=-\frac{4\pi G}{3}\left(\rho +3p\right)a,$

where G is the universal gravitational constant. (For ordinary matter, we expect the quantity $\rho +3p$ to be greater than zero.) If the scale factor is positive ( $a>0$ ), the value of the scale factor “decelerates” ( ${d}^{2}a\text{/}d{t}^{2}<0$ ), and the expansion of the universe slows down over time. If the numerator is less than zero (somehow, the pressure of the universe is negative), the value of the scale factor “accelerates,” and the expansion of the universe speeds up over time. According to recent cosmological data, the universe appears to be expanding. Many scientists explain the current state of the universe in terms of a very rapid expansion in the early universe. This expansion is called inflation.

## Summary

• The universe is expanding like a balloon—every point is receding from every other point.
• Distant galaxies move away from us at a velocity proportional to its distance. This rate is measured to be approximately 70 km/s/Mpc. Thus, the farther galaxies are from us, the greater their speeds. These “recessional velocities” can be measure using the Doppler shift of light.
• According to current cosmological models, the universe began with the Big Bang approximately 13.7 billion years ago.

## Conceptual questions

What is meant by cosmological expansion? Express your answer in terms of a Hubble graph and the red shift of distant starlight.

Cosmological expansion is an expansion of space. This expansion is different than the explosion of a bomb where particles pass rapidly through space. A plot of the recessional speed of a galaxy is proportional to its distance. This speed is measured using the red shift of distant starlight.

Describe the balloon analogy for cosmological expansion. Explain why it only appears that we are at the center of expansion of the universe.

Distances to local galaxies are determined by measuring the brightness of stars, called Cepheid variables, that can be observed individually and that have absolute brightnesses at a standard distance that are well known. Explain how the measured brightness would vary with distance, as compared with the absolute brightness.

With distance, the absolute brightness is the same, but the apparent brightness is inversely proportional to the square of its distance (or by Hubble’s law recessional velocity).

## Problems

If the speed of a distant galaxy is 0.99 c , what is the distance of the galaxy from an Earth-bound observer?

$\left(0.99\right)\left(299792\phantom{\rule{0.2em}{0ex}}\text{km}\text{/}\text{s}\right)=\left(\left(70\frac{\text{km}}{\text{s}}\right)\text{/}\phantom{\rule{0.2em}{0ex}}\text{Mpc}\right)\left(d\right),\phantom{\rule{0.2em}{0ex}}d=4240\phantom{\rule{0.2em}{0ex}}\text{Mpc}$

The distance of a galaxy from our solar system is 10 Mpc. (a) What is the recessional velocity of the galaxy? (b) By what fraction is the starlight from this galaxy redshifted (that is, what is its z value)?

If a galaxy is 153 Mpc away from us, how fast do we expect it to be moving and in what direction?

$1.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{4}\phantom{\rule{0.2em}{0ex}}\text{km/s away from us}\text{.}$

On average, how far away are galaxies that are moving away from us at $2.0\text{%}$ of the speed of light?

Our solar system orbits the center of the Milky Way Galaxy. Assuming a circular orbit 30,000 ly in radius and an orbital speed of 250 km/s, how many years does it take for one revolution? Note that this is approximate, assuming constant speed and circular orbit, but it is representative of the time for our system and local stars to make one revolution around the galaxy.

$2.26\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{8}\phantom{\rule{0.2em}{0ex}}\text{y}$

(a) What is the approximate velocity relative to us of a galaxy near the edge of the known universe, some 10 Gly away? (b) What fraction of the speed of light is this? Note that we have observed galaxies moving away from us at greater than 0.9 c .

(a) Calculate the approximate age of the universe from the average value of the Hubble constant, ${H}_{0}=20\phantom{\rule{0.2em}{0ex}}\text{km/s}·\text{Mly}$ . To do this, calculate the time it would take to travel 0.307 Mpc at a constant expansion rate of 20 km/s. (b) If somehow acceleration occurs, would the actual age of the universe be greater or less than that found here? Explain.

a. $1.5\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{10}\phantom{\rule{0.2em}{0ex}}\text{y}=15\phantom{\rule{0.2em}{0ex}}\text{billion years}$ ; b. Greater, since if it was moving slower in the past it would take less more to travel the distance.

The Andromeda Galaxy is the closest large galaxy and is visible to the naked eye. Estimate its brightness relative to the Sun, assuming it has luminosity ${10}^{12}$ times that of the Sun and lies 0.613 Mpc away.

Show that the velocity of a star orbiting its galaxy in a circular orbit is inversely proportional to the square root of its orbital radius, assuming the mass of the stars inside its orbit acts like a single mass at the center of the galaxy. You may use an equation from a previous chapter to support your conclusion, but you must justify its use and define all terms used.

$v=\sqrt{\frac{GM}{r}}$

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