这是一份UIUC宇宙学课程的作业，学生同时选了好几门物理课，课业压力大，这份作业交给了我们写手完成加答疑。

In this problem put $\sigma=1+1 / \log x$. Show that

$$

\sum_{p>x} \log \left(1-\frac{1}{p^{\sigma}}\right)^{-1}=\sum_{p>x} \frac{1}{p^{\sigma}}+O\left(\frac{1}{x}\right)=\int_{1}^{\infty} \frac{e^{-t}}{t} d t+O\left(\frac{1}{\log x}\right)

$

- An astronomer obtains a spectrum of an elliptical galaxy which shows a strong emission line peaked at a wavelength of $\lambda_{\mathrm{obs}}=7403.2 \AA$ and with a width (i.e., square-root of the variance) of $\sigma_{\lambda}=10.3 \AA$. The astronomer identifies the emission line as $\mathrm{H} \alpha$, which has a laboratory-measured rest-frame wavelength of $\lambda_{\mathrm{lab}}=6563.0 \AA$

(a) What is the redshift of the galaxy?1 (b) Assuming the galaxy has no peculiar motion, what is the distance to the galaxy? (c) What is the velocity dispersion of the galaxy?

Answer:

Using the fact that the photons were all emitted with wavelength $\lambda_{\mathrm{em}}=\lambda_{\text {lab }}$, the galaxy’s redshift is

$$

z=\frac{\lambda_{\mathrm{obs}}-\lambda_{\mathrm{em}}}{\lambda_{\mathrm{em}}}=\frac{7403.2 \AA-6563.0 \AA}{6563.0 \AA}=0.128

$$

Taking Hubble’s constant to be $H_{0}=70 \mathrm{~km} \mathrm{~s}^{-1} \mathrm{Mpc}^{-1}$ (from the lecture notes), the distance to the galaxy would be

$$

d=\frac{c z}{H_{0}}=\frac{2.99 \times 10^{5} \mathrm{~km} \mathrm{~s}^{-1} \times 0.128}{70.0 \mathrm{~km} \mathrm{~s}^{-1} \mathrm{Mpc}^{-1}} \mathrm{Mpc}=548.2 \mathrm{Mpc}

$$

The observed line width can be related to the intrinsic velocity dispersion of the galaxy, $\sigma$, by treating the internal motions and the cosmological term as two separate redshifting processes. This implies that

$$

\begin{aligned}

\frac{\left(\lambda_{\mathrm{obs}} \pm \sigma_{\lambda}\right)-\lambda_{\mathrm{lab}}}{\lambda_{\mathrm{lab}}} &=(1+z)\left(1 \pm \frac{\sigma}{c}\right)-1 \

z \pm \frac{\sigma_{\lambda}}{\lambda_{\text {lab }}} &=z \pm(1+z) \frac{\sigma}{c} \

\sigma &=c \frac{\sigma_{\lambda}}{(1+z) \lambda_{\mathrm{lab}}} \

\sigma &=c \frac{\sigma_{\lambda}}{\lambda_{\mathrm{obs}}} \

\sigma & \simeq 370 \mathrm{~km} \mathrm{~s}^{-1}

\end{aligned}

$$

This implies that the galaxy in question is a very large elliptical galaxy, possibly a cluster dominant (cD) galaxy residing at the centre of a cluster of galaxies.

The observed redshift of the galaxy is the combination of two different effects: the cosmological redshift, $z_{c s m}=H_{0} d / c ;$ and the Doppler shift produced by its (unknown) peculiar velocity, $v$, which is $z_{\mathrm{pec}}=v / c$. (This last expression assumes that $|v| \ll c ;$ it is possible to test whether this approximation is valid at the end of the calculation.) The two redshifts combine according

$$

z=\left(1+z_{\mathrm{csm}}\right)\left(1+z_{\mathrm{pec}}\right)-1=\left(1+\frac{\mathrm{H}_{0} \mathrm{~d}}{c}\right)\left(1+\frac{v}{c}\right)

$$

Given that $z, d, H_{0}$ and $c$ are all known, the above expression can be inverted to give the peculiar velocity as

$$

v=c\left(\frac{1+z}{1+H_{0} d / c}-1\right) \simeq-836 \mathrm{~km} \mathrm{~s}^{-1}

$$

The galaxy is hence moving towards the Sun/Earth, as expected given that the observed redshift is less than would be expected given its distance. The particularly high speed – typical galaxy peculiar motions are a factor of a few smaller than this $-$ is consistent with the above picture of the galaxy as being in a cluster, where such speeds are quite plausible. Still, as high as the galaxy’s speed is, it is still much smaller than the speed of light (i.e., $|v| / c \ll 1)$, hence post-justifying the use of the non-relativistic Doppler formula.

The Milky Way galaxy and the nearby Andromeda galaxy (a comparably-sized spiral galaxy) will “collide”

billions of years from now. Estimate the probability that a given star (e.g., the Sun) in the Milky Way will

actually hit a star in the Andromeda galaxy, and hence estimate the total number of likely star-star collisions

during this encounter. (Given the lack of model specifics, both answers will will rest on making numerous

approximations and assumptions, so try and be explicit about these; it is also important to re-evaluate such

approximations in light of whatever answer they lead to.)

Both the Milky Way and Andromeda are large spiral galaxies, so a reasonable (if very approximate) model of both is a collection of $N \simeq 4 \times 10^{11}$ Sun-like stars (each of which hence has a radius of $R=R_{\odot} \simeq$ $7 \times 10^{8} \mathrm{~m}$ ) distributed uniformly over a galactic disk of radius $a \simeq 50 \mathrm{kpc}$. These simple galaxies could have a “perfect” head-on collision or could only partially overlap; and they could be anywhere between face-on and edge-on when they meet.

In the case of a perfect face-on collision, the probability that the Sun collides with a star in Andromeda is he ratio of the area covered by the stars to the area of the whole galaxy

$$

p_{\mathrm{coll}} \simeq \frac{N \pi\left(2 R_{\odot}\right)^{2}}{\pi \mathrm{a}^{2}}=\frac{4 N R_{\odot}^{2}}{a^{2}} \simeq 4 \times 10^{-12}

$$

where the factor of 2 in the numerator comes from the fact that two stars would collide if their centres passed within $2 R_{\odot}$ of each other. The expected total number of stellar collsisions would then be

$$

\bar{N}{\mathrm{coll}} \simeq N p{\mathrm{coll}} \simeq \frac{4 N^{2} R_{\odot}^{2}}{a^{2}} \simeq 0.1

$$

The implication is that there is only $a \sim 10 \%$ chance of any stellar collision, implying that the galaxies would, to some degree, simply pass through each other.

But the main reason for this low probability is the ratio of areas, $\pi R_{\odot}^{2} /\left(\pi a^{2}\right) \simeq$; if the galaxies were completely edge-on then it would a less extreme ratio of length-scales. In this case

$$

p_{\mathrm{coll}} \simeq N \frac{N 2 R_{\odot}}{2 \mathrm{a}}=\frac{N R_{\odot}}{a} \simeq 0.2

$$

The expected total number of stellar collsisions would then be

$$

\bar{N}{\mathrm{coll}} \simeq N p{\mathrm{coll}} \simeq \frac{N^{2} R_{\odot}}{a} \simeq 10^{11}

$$

This is a very different answer, implying that a high chance of catastrophe for any civilisation living on a planet orbiting one of the stars in the two galaxies. But is this model really plausible – if the two galaxies

really were edge-on to each other it is actually more likely that they would just miss either “above” or “below”.

So while it is correct that the edge-on case makes stellar collisions more likely, the above calculation is too much of an approximation – the galaxies need to have some finite height, $h$. A reasonable value for this is $h \simeq 0.1 \mathrm{kpc}$, in which case the collision probability becomes

$$

p_{\mathrm{coll}} \simeq \frac{N \pi\left(2 R_{\odot}^{2}\right)}{a h}=\frac{4 \pi N R_{\odot}^{2}}{a h} \simeq 5 \times 10^{-10}

$$

The expected number of collisions is then

$$

\bar{N}*{\mathrm{coll}} \simeq N p*{\mathrm{coll}} \simeq \frac{\pi N^{2} R_{\odot}^{2}}{a h} \simeq 200

$$

This is much closer to the original low value, implying that the initial result – that a collision between galaxies will not result in many actual collisions between their stars $-$ is a robust result.

The various (other) approximations included:

- The enchanced collision probability caused by “gravitational focusing” (i.e., the fact that two stars in a close encounter are actually pulled towards each other) was ignored.
- The centrally-peaked nature of the distribution of stars (e.g., with an exponential profile) in the stellar disks was ignored.
- It was assumed all stars were Sun-like, when in fact most are much smaller.
- The stars’ positions were assumed to be uncorrelated, whereas in fact many exist in structures like globular clusters, etc..

The list of approximation is almost endless; but the calculated collision probability is so small that only the most extreme model inaccuracy (such as the edge-on disk model above) would change the overall conclusion that the eventual collision of the Milky Way and Andromeda will almost certainly result in just a few stellar collisions.

**PRIMORDIAL GRAVITATIONAL WAVES**

An epoch of rapid expansion in the early universe, known as “inflation”, is thought to have set the stage for the evolution of cosmic structure. This process should also have seeded the universe with a faint hum of primordial gravitational waves, undetectable today but visible as a faint pattern in the polarization of the cosmic microwave background (CMB). Illinois collaborates on instrumentation and data analysis for leading teams seeking to constrain this elusive signature from the South Pole (BICEP) and stratospheric balloons (SPIDER), as well as on future efforts to probe fundamental physics with novel low-temperature detector technology.

Links to research groups and facilities: Jeffrey Filippini, SPIDER

**Primordial Nucleosynthesis and Particle Dark Matter**

The lightest and most abundant elements in the universe were forged from a primordial soup of subatomic particles during the first three minutes of cosmic time. Our group performs state-of-the-art calculations of the primordial abundances of the elements. By combining these theoretical predictions with astronomical observations of light elements and of the cosmic microwave background radiation, we wield the earliest reliable probe of the cosmos. At even earlier times, even higher-energy interactions likely gave rise to exotic particles that gave rise to dark matter today. We use primordial nucleosynthesis and other astrophysical observations to probe dark matter particle physics.

Links to research groups and facilities: Brian Fields

## Faculty working in Cosmology

Peter J. AdsheadAssistant Professor of Physics[email protected]Matias Carrasco KindResearch Assistant Professor[email protected]Patrick I DraperAssistant Professor of Physics[email protected]Brian D. FieldsProfessor[email protected]Jeffrey P FilippiniAssistant Professor of Physics[email protected]Gilbert HolderProfessor of Physics[email protected]Xin LiuAssociate Professor[email protected]Felipe MenanteauResearch Associate Professor[email protected]Gautham NarayanAssistant Professor[email protected]Paul M. RickerProfessor[email protected]Stuart L. ShapiroProfessor of Physics[email protected]Jessie F SheltonAssistant Professor of Physics[email protected]Yue ShenAssociate Professor[email protected]Joaquin VieiraAssociate Professor[email protected]Helvi WitekAssistant Professor[email protected]Dr. Nicolas YunesProfessor of Physics[email protected]