The Expanding Universe

Show explicit units, and run a sanity check on every result. Worked solutions are released after the homework due date.

Useful constants:

Constant	Value
$H_0$	$70~\mathrm{km\,s^{-1}\,Mpc^{-1}}$
$c$	$3.0 \times 10^{5}~\mathrm{km\,s^{-1}}$
$1~\mathrm{Mpc}$	$3.086 \times 10^{19}~\mathrm{km}$
$1~\mathrm{yr}$	$3.16 \times 10^{7}~\mathrm{s}$
Parallax distance	$d[\mathrm{pc}] = 1/p[\mathrm{arcsec}]$

Useful relations: Hubble’s law $v = H_0 d$ ; the Hubble distance $d_H = c/H_0$ ; the Hubble time $1/H_0$ ; cosmological redshift $1+z = \lambda_\text{obs}/\lambda_\text{emit} = a_0/a_\text{emit}$ , with $a_\text{emit} = 1/(1+z)$ for $a_0 = 1$ . The forms $v = H_0 d$ and $v \approx cz$ are low-redshift approximations.

Conceptual

Problem

⭐⭐ One rung of the ladder. A Cepheid in a distant galaxy has a pulsation period that implies a luminosity $L$ , and you measure its flux $F$ .

(a) State the observable, the model, and the inference that turn this Cepheid into a distance.
(b) Why does this rung need a lower rung (parallax) to work at all?
(c) Why reach a galaxy $500~\mathrm{Mpc}$ away with a Type Ia supernova rather than a Cepheid?

Problem

⭐⭐ Not an explosion into space. A student pictures the Big Bang as a grenade: galaxies flung outward from one point, with empty space beyond the blast.

(a) Identify two things wrong with the grenade picture.
(b) What is actually expanding, and why is there no center?
(c) If every observer sees the same Hubble law, where is the “edge” of the expansion?

Problem

⭐ Two kinds of redshift. In one or two sentences each:

(a) How does a cosmological redshift differ physically from an ordinary Doppler shift?
(b) Why can’t you read a $z = 6$ galaxy’s redshift as a recession “velocity” of $6c$ ?

Calculation

Problem

⭐⭐ How fast, and how far is too far? Use $H_0 = 70~\mathrm{km\,s^{-1}\,Mpc^{-1}}$ .

(a) Find the recession speed of a galaxy at $d = 250~\mathrm{Mpc}$ .
(b) At what distance does $v = H_0 d$ give exactly the speed of light? (This is the Hubble distance.)
(c) A galaxy is observed at $z = 6$ . Explain why you should not compute its speed as $v = cz = 6c$ , and what you would report instead.

Problem

⭐⭐ Reading a redshift as a scale factor. A quasar has redshift $z = 6$ (take $a_0 = 1$ ).

(a) Compute the wavelength stretch factor $1+z$ .
(b) What was the scale factor $a_\text{emit}$ when its light was emitted?
(c) By what linear factor has the universe expanded since that light left?

Problem

⭐⭐ A first age of the universe. Use $H_0 = 70~\mathrm{km\,s^{-1}\,Mpc^{-1}}$ .

(a) Convert $H_0$ to $\mathrm{s^{-1}}$ and compute the Hubble time $1/H_0$ in years.
(b) The measured age of the universe is about $13.8~\mathrm{Gyr}$ . Compare it to your $1/H_0$ .
(c) Why is $1/H_0$ only an estimate of the age — what would have to be true for it to be exact?

Synthesis

Problem

⭐⭐ A photograph that is also a clock. A deep field shows thousands of galaxies at many distances.

(a) Why is a more distant galaxy in the frame also seen at an earlier cosmic time?
(b) A faint red smudge could be a nearby dim galaxy or a distant young one. What single measurement settles which?
(c) Explain what it means to call a deep field a “time archive.”

Problem

⭐⭐ One measurement, two limits. A galaxy has $z = 1$ (take $a_0 = 1$ ).

(a) What was the scale factor when its light was emitted?
(b) Does $z = 1$ by itself tell you the exact lookback time (how many Gyr ago the light left)? Why or why not?
(c) What additional information would convert $z$ into a precise time?

Problem

⭐⭐⭐ The whole ladder, end to end. Carry one chain of measurements from a parallax angle to the Hubble flow. Use $H_0 = 70~\mathrm{km\,s^{-1}\,Mpc^{-1}}$ .

(a) Rung 1 — parallax: a nearby Cepheid has parallax $p = 0.002''$ . Find its distance in parsecs.
(b) Rung 2 — standard candle: explain in one sentence how the parallax distance to that Cepheid calibrates the Cepheid period–luminosity relation, and how Cepheids then calibrate Type Ia supernovae.
(c) Rung 3 — Hubble flow: a Type Ia supernova in a distant galaxy gives $d = 300~\mathrm{Mpc}$ . Find its recession speed from $v = H_0 d$ , and state what the single point $(d, v)$ contributes.
(d) At each rung, name the observable and the inference.