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USAAAO 2024, II кръг — Medium 2. Neutrino decoupling

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Всички материали от USAAAO 2024

2024 National Astronomy Competition · 25 т.

Внимание: Бележка към темата

All eight whole original problems, their native sections, lettered parts, three roman subparts of Long 1(f), full official worked keys and original marking/acceptance notes are retained. Printed scientific discrepancies are preserved in local source notes. The official key adds a Friedmann-variable clarification and a solar apparent magnitude assumption absent from the question booklet, and changes Long1(f)(iii) from a factor to a percentage; these source variants are explicitly attributed. A separate undated common archive constants/reference table is included with its own source identity; its use as the exact 2024 provided attachment is not established, and its constants are not substituted into the printed key. The native cover specifies blank paper rather than a designated answer sheet. April 13th is the end of the examination period, not an established single sitting date.

1 Instructions (Please Read Carefully)

The top 5 eligible scorers on the NAC will be invited to represent USA at the next IOAA. In order to qualify for the national team, you must be a high school student with US citizenship or permanent residency.

This exam consists of 3 parts: Short Questions, Medium Questions and Long Questions.

The maximum number of points is 225 points.

The test must be completed within 2.5 hours (150 minutes).

Please solve each problem on a blank piece of paper and mark the number of the problem at the top of the page. The contestant’s full name in capital letters should appear at the top of each solution page. If the contestant uses scratch papers, those should be labeled with the contestant’s name as well and marked as “scratch paper” at the top of the page. Scratch paper will not be graded. Partial credit will be available given that correct and legible work was displayed in the solution.

This is a written exam. Contestants can only use a scientific or graphing calculator for this exam. A table of physical constants will be provided. Discussing the problems with other people is strictly prohibited in any way until the end of the examination period on April 13th. Receiving any external help during the exam is strictly prohibited. This means that the only allowed items are: a calculator, the provided table of constants, a pencil (or pen), an eraser, blank sheets of papers, and the exam. No books or notes are allowed during the exam. Exam is proctored and recorded. You are expected to have your video on at all times.

We acknowledge the following people for their contributions to this year’s exam:

Abhay Bestrapalli, Orion Foo, Hagan Hensley, David Lee, Sandesh Kalantre, Andrew Liu, Joe McCarty, Leo Yao

Common archive reference — Table of Physical Constants

This separate common USAAAO archive reference is undated in the catalogue. Its use as the exact 2024 exam attachment is not established.

Quantity	Symbol	Value
Electron rest mass	$m_e$	$9.109\times10^{-31}\ \mathrm{kg}$
Proton rest mass	$M_p$	$1.6726\times10^{-27}\ \mathrm{kg}$
Electronic charge	$e$	$1.6022\times10^{-19}\ \mathrm C$
Speed of light in free space	$c$	$2.9979\times10^8\ \mathrm{m\ s^{-1}}$
Permeability of free space	$\mu_0$	$4\pi\times10^{-7}\ \mathrm{H\ m^{-1}}$
Permittivity of free space	$\epsilon_0$	$8.854\times10^{-12}\ \mathrm{F\ m^{-1}}$
Planck’s constant	$h$	$6.626\times10^{-34}\ \mathrm{J\ s}$
Reduced Planck’s constant	$\hbar=h/2\pi$	$1.0546\times10^{-34}\ \mathrm{J\ s}$
	$\hbar c$	$197.33\ \mathrm{MeV\ fm}$
Boltzmann’s constant	$k_B$	$1.3807\times10^{-23}\ \mathrm{J\ K^{-1}}$
Gas constant	$\mathcal R=k_B/m_H$	$8.250\times10^3\ \mathrm{J\ kg^{-1}\ K^{-1}}$
Molar gas constant	$R$	$8.315\ \mathrm{J\ mol^{-1}\ K^{-1}}$
Avogadro’s number	$N_A$	$6.022\times10^{23}\ \mathrm{mol^{-1}}$
Standard molar volume		$22.414\times10^{-3}\ \mathrm{m^3\ mol^{-1}}$
Unified atomic mass unit ( $^{12}\mathrm C$ scale)	$u$	$931.5\ \mathrm{MeV}/c^2=1.660538\times10^{-27}\ \mathrm{kg}$
Mass of hydrogen atom	$m_H$	$1.0078u=1.6735\times10^{-27}\ \mathrm{kg}$
Bohr magneton	$\mu_B$	$9.274\times10^{-24}\ \mathrm{A\ m^2}$ or $\mathrm{J\ T^{-1}}$
Nuclear magneton	$\mu_N$	$5.051\times10^{-27}\ \mathrm{A\ m^2}$ or $\mathrm{J\ T^{-1}}$
Proton magnetic moment	$\mu_p$	$2.7928\mu_N$
Neutron magnetic moment	$\mu_n$	$-1.9130\mu_N$
Bohr radius	$a_0$	$5.292\times10^{-11}\ \mathrm m$
Fine structure constant	$\alpha=e^2/(4\pi\epsilon_0\hbar c)$	$(137.04)^{-1}$
Compton wavelength of electron	$\lambda_C=h/(m_ec)$	$2.4263\times10^{-12}\ \mathrm m$
Rydberg’s constant	$R_{\infty}$	$1.0974\times10^7\ \mathrm{m^{-1}}$
	$R_{\infty}hc$	$13.606\ \mathrm{eV}$
Stefan-Boltzmann constant	$\sigma$	$5.671\times10^{-8}\ \mathrm{W\ m^{-2}\ K^{-4}}$
Radiation density constant	$a=4\sigma/c$	$7.561\times10^{-16}\ \mathrm{J\ m^{-3}\ K^{-4}}$
Gravitational constant	$G$	$6.673\times10^{-11}\ \mathrm{N\ m^2\ kg^{-2}}$

Rest masses of some leptons and hadrons in $\mathrm{MeV}/c^2$ :

$e^{\pm}$ 0.5110, $\mu^{\pm}$ 105.66, $\tau^{\pm}$ 1777, $\pi^0$ 134.98, $\pi^{\pm}$ 139.57, $\mathrm K^{\pm}$ 493.7, $\mathrm K^0$ 497.7, $\eta$ 547, $\mathrm D^0$ 1865, $\mathrm D^{\pm}$ 1869, p 938.3, n 939.6, $\Lambda^0$ 1115.7, $\Sigma^+$ 1189, $\Sigma^0$ 1193, $\Sigma^-$ 1197, $\Xi^0$ 1315, $\Xi^-$ 1321, $\Omega^-$ 1672, $\mathrm Z^0$ $91.187\times10^3$ , $\mathrm W^{\pm}$ $80.41\times10^3$ .

Quark	Charge	$I_3$	$S$	$C$	$B$	$T$	Mass ( $\mathrm{GeV}/c^2$ )
u	$+\frac23$	$\frac12$	0	0	0	0	$\sim0.003$
d	$-\frac13$	$-\frac12$	0	0	0	0	$\sim0.006$
c	$+\frac23$	0	0	+1	0	0	$\sim1.25$
s	$-\frac13$	0	-1	0	0	0	$\sim0.11$
t	$+\frac23$	0	0	0	0	+1	174.3
b	$-\frac13$	0	0	0	-1	0	4.2

Common archive reference — Astrophysical Data and identities

Astrophysical Data

Quantity	Symbol	Value
1 astronomical unit	AU	$1.496\times10^{11}\ \mathrm m$
1 parsec	pc	$3.086\times10^{16}\ \mathrm m$
Luminosity of Sun	$L_{\odot}$	$3.85\times10^{26}\ \mathrm W$
Mass of Sun	$M_{\odot}$	$1.989\times10^{30}\ \mathrm{kg}$
Radius of Sun	$R_{\odot}$	$6.96\times10^8\ \mathrm m$
Mass of Earth	$M_E$	$5.9742\times10^{24}\ \mathrm{kg}$
Radius of Earth	$R_E$	$6.3781\times10^6\ \mathrm m$

Other data and conversion factors

Quantity	Symbol	Value
1 ångstrom	Å	$10^{-10}\ \mathrm m$
1 fermi	fm	$10^{-15}\ \mathrm m$
1 barn	b	$10^{-28}\ \mathrm{m^2}$
1 pascal	Pa	$1\ \mathrm{Nm^{-2}}$
1 standard atmosphere		$1.0132\times10^5\ \mathrm{Pa}$
Standard acceleration due to gravity	$g$	$9.807\ \mathrm{m\ s^{-2}}$
1 electron volt	eV	$1.6022\times10^{-19}\ \mathrm J$
	$\mathrm{eV}/hc$	$8.065\times10^5\ \mathrm{m^{-1}}$
	$\mathrm{eV}/k_B$	$1.1604\times10^4\ \mathrm K$
Wavelength of 1 eV photon		$1.2399\times10^{-6}\ \mathrm m$

Trigonometrical identities

\sin(\theta+\phi)=\sin(\theta)\cos(\phi)+\cos(\theta)\sin(\phi)

\cos(\theta+\phi)=\cos(\theta)\cos(\phi)-\sin(\theta)\sin(\phi)

\sin\alpha+\sin\beta=2\sin\frac12(\alpha+\beta)\cos\frac12(\alpha-\beta)

\cos\alpha+\cos\beta=2\cos\frac12(\alpha+\beta)\cos\frac12(\alpha-\beta)

\cos\alpha-\cos\beta=2\sin\frac12(\alpha+\beta)\sin\frac12(\beta-\alpha)

In a triangle ABC, $a/\sin A=b/\sin B=c/\sin C$

and $a^2=b^2+c^2-2bc\cos A$

Prefixes

Prefix	Name	Factor
T	tera	$10^{12}$
G	giga	$10^9$
M	mega	$10^6$
k	kilo	$10^3$
c	centi	$10^{-2}$
m	milli	$10^{-3}$
$\mu$	micro	$10^{-6}$
n	nano	$10^{-9}$
p	pico	$10^{-12}$
f	femto	$10^{-15}$

Source note: This undated common table prints $G=6.673\times10^{-11}\ \mathrm{N\ m^2\ kg^{-2}}$ , $M_E=5.9742\times10^{24}\ \mathrm{kg}$ and $R_E=6.3781\times10^6\ \mathrm m$ . The 2024 eclipse key instead uses $G=6.674\times10^{-11}$ , Earth mass $5.976\times10^{24}\ \mathrm{kg}$ and Earth radius $6.371\times10^6\ \mathrm m$ . Both original sets of values are preserved separately; this reference is not an established exact dated 2024 attachment.

Условие

At the beginning of the Universe, immediately following inflation, neutrinos were in thermal equilibrium with other radiation and matter, maintained through weak interactions. As the Universe expanded and the temperature of the Universe dropped, the rate of these weak interactions decreased until thermal equilibrium was no longer maintained, and neutrinos began to propagate freely through the Universe. This event is called neutrino decoupling, and occurred at about $t_n=1\mathrm{sec}$ after the Big Bang. Initially, the energy density of radiation was greater than that of matter, but as energy of radiation decreased with cosmological expansion, eventually the energy density of radiation fell below that of matter. This earlier period is called the radiation-dominated era, while the later period is called the matter-dominated era, with a transition time of about $t_r=50000\mathrm{yr}$ .

(a) (4 points) Write proportionalities for the energy density of matter $\rho_m(a)$ and the energy density of radiation $\rho_r(a)$ as a function of scale factor. Your answer should look like $\rho_m(a)\propto f(a)$ and $\rho_r(a)\propto g(a)$ for some single variable functions $f,g$ .

Hint: radiation is redshifted with cosmological expansion, while matter is not.

(b) (6 points) Using the Friedmann equation:

H^2=\left(\frac{\dot a}{a}\right)^2=\frac{8\pi G}{3}\rho-\frac{kc^2}{a^2}

Derive proportionalities for the evolution of a matter-dominated and a radiation-dominated Universe, $a_m(t)$ and $a_r(t)$ . Take $t$ to be the time since the start of the Universe in each case, assuming the Universe was always made of only matter or only radiation. Assume no curvature or cosmological constant. Derivation required for credit.

Hint: guess a form $a(t)=bt^c$ with $b,c$ constants, which has derivative $\dot a(t)=c\cdot bt^{c-1}$ , and equate exponents.

From this point forwards, take the simple (and unrealistic) assumption that in the radiation-dominated era, our Universe was entirely made up of radiation from the start up to some time $t_r$ , then was entirely made up by matter afterwards in the matter-dominated era.

(c) (10 points) Using the conditions that the scale factor $a(t)$ and expansion rate $H(t)=\frac{\dot a(t)}{a(t)}$ of our Universe must be continuous, write expressions for the scale factor $a(t)$ of our Universe for $t<t_r$ and $t>t_r$ , up to a single scaling constant. Do not substitute in the value of $t_r$ yet.

Hint: naively trying to equate the expressions of the previous part will not allow both $a(t)$ and $H(t)$ to be continuous. What assumption holds in those expressions that no longer holds here, giving us an extra degree of freedom?

(d) (2 points) Solve for the scaling constant in the previous part in terms of $t_r$ and $t_0$ , the current age of our Universe.

Hint: what is the current value of $a_0=a(t_0)$

(e) (3 points) For a boundary between matter and radiation-dominated eras of $t_r=50000\mathrm{yr}$ , estimate the temperature $T_n$ of the Universe at the time of neutrino decoupling. The current age of the Universe is $t_0=13.8$ billion years, and the current temperature of the Universe is $T_0=2.73\mathrm K$ .

Отговори

Решение

Key-only statement addition (solution PDF page 8, part (b)): After the Friedmann equation, the key adds “where $a$ is the scale factor and $k$ is the curvature.” This sentence is absent from the question booklet.

(a) Consider matter in the Universe as made up of numerous tiny dust particles. As the Universe expands, the dust particles get further away from each other, but the total number of dust particles remains unchanged. Expansion also does not cause dust particles to be created or to disappear, or their masses to change.

Therefore, the total energy density of matter in the Universe remains unchanged with time, so matter density is inversely proportional to volume, which goes as the cube of the scale factor:

\rho_m(a)\propto a^{-3}

Expansion also does not create or destroy radiation. However, the energy of photons is affected due to cosmological redshift: a photon of wavelength $a_1\lambda$ is redshifted to a wavelength $a_2\lambda$ , leading to an energy per particle inversely proportional to $a$ . Then, energy density of radiation goes as the fourth power of the scale factor (3 for volume, 1 for redshift):

\rho_r(a)\propto a^{-4}

(b) As we are assuming a flat Universe, we have no curvature, $k=0$ , and can discard the second term. We guess a solution of the form $a(t)=bt^c$ , for $b,c$ constants, and can drop non-exponential constants as we are solving for a proportionality.

For matter density:

\begin{aligned} \left(\frac{\dot a}{a}\right)^2&\propto a^{-3}\\ \frac{t^{2(c-1)}}{t^{2c}}&\propto t^{-3c}\\ t^{-2}&\propto t^{-3c}\\ c&=\frac23\quad\Longrightarrow\quad a_m(t)\propto t^{2/3} \end{aligned}

For radiation density:

\begin{aligned} \left(\frac{\dot a}{a}\right)^2&\propto a^{-4}\\ \frac{t^{2(c-1)}}{t^{2c}}&\propto t^{-4c}\\ t^{-2}&\propto t^{-4c}\\ c&=\frac12\quad\Longrightarrow\quad a_m(t)\propto t^{1/2} \end{aligned}

a_r(t)=b_rt^{1/2}

a_m(t)=b_mt^{2/3}

And the corresponding Hubble constants:

H_r(t)=\frac{\frac12b_rt^{-1/2}}{b_rt^{1/2}}=\frac1{2t}

H_m(t)=\frac{\frac23b_mt^{-1/3}}{b_mt^{2/3}}=\frac2{3t}

Where $t$ is defined to be the time since the start of the Universe in the case where the Universe has always been made of only radiation or only matter.

We need to enforce the conditions that $a(t)$ and $H(t)$ are continuous at the transition point $t=t_r$ . If we naively try to equate the proportionalities, for the $a(t)$ condition we get:

b_rt_r^{1/2}=b_mt_r^{2/3}

b_r=b_mt_r^{1/6}

However, there is no way to get the Hubble constants $H_r(t)$ and $H_m(t)$ to be equal! This suggests that there is an assumption we have made in the expressions for $a(t)$ that do not hold in this case.

The assumptions in our scale factor expressions are that $t=0$ is the start of the Universe, and the Universe has always been made up of the specific substance since $t=0$ . This is true for our radiation-dominated phase, but is not the case for our matter-dominated phase, as it comes after the radiation-dominated phase.

If we evolve our matter-dominated Universe back in time, the time at which $a(t)=0$ does not necessarily match our coordinate choice $t=0$ , as it is defined to be the time for the radiation-dominated phase. Therefore, we must introduce an offset $t^*$ to our matter-dominated Universe expression:

a_r(t)=b_rt^{1/2}

a_m(t)=b_m(t-t^*)^{2/3}

We can now equate Hubble constants:

\frac1{2t_r}=\frac2{3(t_r-t^*)}

t^*=-\frac13t_r

Our matter-dominated phase therefore is the same as a Universe that was always made up of matter, but started at time $t=-\frac13t_r$ instead of $t=0$ .

Substituting back in:

a_r(t)=b_rt^{1/2}

a_m(t)=b_m\left(t+\frac{t_r}{3}\right)^{2/3}

Equating scale factor $a(t)$ at $t=t_r$ :

b_rt_r^{1/2}=b_m\left(\frac43t_r\right)^{2/3}

b_r=b_m\left(\frac43\right)^{2/3}t_r^{1/6}

Then expressions for scale factor $a(t)$ before and after turning point:

a(t)=\begin{cases}b_m\left(\frac43\right)^{2/3}t_r^{1/6}t^{1/2}&t<t_r\\b_m\left(t+\frac{t_r}{3}\right)^{2/3}&t>t_r\end{cases}

(d) The current value of the scale factor is $a_0=1$ . Enforcing $a(t_0)=1$ :

b_m\left(t_0+\frac{t_r}{3}\right)^{2/3}=1

b_m=\left(t_0+\frac{t_r}{3}\right)^{-2/3}

(e)

a(1\mathrm{sec})=\left(t_0+\frac{t_r}{3}\right)^{-2/3}\left(\frac43\right)^{2/3}t_r^{1/6}t^{1/2}

Substituting $t_0=1.38\times10^{10}\cdot3.15\times10^7$ , $t_r=5\times10^4\cdot3.15\times10^7$ , and $t=1$ gives scale factor:

a(1\mathrm{sec})=2.276\times10^{-10}

And as temperature is inversely proportional to scale factor:

T_n=\frac{a_0}{a(1\mathrm{sec})}T_0=1.2\times10^{10}\mathrm K

Adapted and extended from a problem posed by Alan Guth.

Source note: The matter explanation says “total energy density ... remains unchanged” while deriving density proportional to a^-3. Its radiation proportionality prints a_m(t), although the following derivation uses a_r(t). Both native expressions are retained. The key-only Friedmann variable clarification is attributed separately.

Теми и трудност

Оценена трудност: 4/5.

Галактики и космология · Космология · Разширение на Вселената

Предпоставки: Гимназиална подготовка, Разширена олимпиадна подготовка

Оригинал в Архива: second_exam_2024.pdf · официални решения: second_exam_2024-solutions.pdf · допълнителни данни: physical-constants.pdf