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The Big Bang Theory11 Based on the original document bywjec, which itself was “Freely modifiedfrom various Wikipedia pages”Alby Reid

April 27, 2016

The Big Bang Theory

The Big Bang theory is the prevailing cosmological model for theearly development of the universe. The key idea is that the universeis expanding. Consequently, the universe was denser and hotterin the past. Moreover, the Big Bang model suggests that at somemoment all matter in the universe was contained in a single point,which is considered the beginning of the universe. Modern measure-ments place this moment at approximately 13.8 billion years ago,which is thus considered the age of the universe. After the initialexpansion, the universe cooled sufficiently to allow the formationof subatomic particles, including protons, neutrons, and electrons.Though simple atomic nuclei formed within the first three minutesafter the Big Bang, thousands of years passed before the first electri-cally neutral atoms formed. The majority of atoms produced by theBig Bang were hydrogen, along with helium and traces of lithium.2 2 Big Bang Nucleosynthesis took place

via twelve nuclear interactions thatproduced three isotopes of hydrogen,two of helium, and one isotope eachof lithium and beryllium. Presumablythe Be-7 that was produced is notmentioned here because it has a half-life of only 53 d. Free neutrons werealso produced, but their half-life iseven shorter, at only 10.6 min. Seehttp://MrReid.org/s/ph5bbn for moredetails.

Giant clouds of these primordial elements later coalesced throughgravity to form stars and galaxies, and the heavier elements weresynthesized either within stars or during supernovae.

The Big Bang theory offers a comprehensive explanation for abroad range of observed phenomena, including the abundance oflight elements, the cosmic microwave background radiation (cmbr),large scale structure, and Hubble‘s Law. Today, the distances betweengalaxies is increasing hence, in the past, galaxies were closer together.The known laws of nature can be used to calculate the characteristicsof the universe in detail back to a time when densities and temper-atures were extreme. While large particle accelerators can replicatesuch conditions, resulting in confirmation and refinement of the de-tails of the Big Bang model, these accelerators can only probe so farinto high energy conditions. Consequently, the state of the universein the earliest instants of the Big Bang expansion is poorly under-stood and still an area of open investigation. The Big Bang theorydoes not provide any explanation for the initial conditions of the uni-verse; rather, it describes and explains the general evolution of theuniverse going forward from that point on.3 3 It is tempting to ask what came

“before the Big Bang”, but the Big Bangwas both the creation of space and time,so this is a bit like asking what greensmells like or what colour physics is –the terms do not really apply. It’s bestnot to think about it too much.

Belgian Catholic priest and scientist Georges Lemaître proposedwhat became the Big Bang theory in 1927. Over time, scientists builton his initial idea of cosmic expansion, which, his theory went, couldbe traced back to the origin of the cosmos and which led to the for-

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mation of the modern universe. The framework for the Big Bangmodel relies on Albert Einstein‘s theory of general relativity andon simplifying assumptions such as homogeneity and isotropy4 of 4 Put simply, this means that the uni-

verse has the same composition in alldirections.

space. In 1929, Edwin Hubble discovered that the distances to far-away galaxies were strongly correlated with their red shifts. Hubble‘sobservation was taken to indicate that all distant galaxies and clus-ters have an apparent velocity directly away from our vantage point:that is, the farther away, the higher the apparent velocity, regardlessof direction. The interpretation is that all observable regions of theuniverse are receding from each other.

While the scientific community was once divided between sup-porters of two different expanding universe theories – the Big Bangand the Steady State theory5 – observational confirmation of the Big 5 The term “Big Bang” was coined –

allegedly pejoratively – by Fred Hoyle,a leading proponent of the Steady StateTheory.

Bang scenario came with the discovery of the cmbr in 1964, and laterwhen its spectrum was found to match that of thermal radiation froma black body.

The History of the Universe

Inflation

The earliest phases of the Big Bang are subject to much speculation.In the most common models the universe was filled homogeneouslyand isotropically with an incredibly high energy density and hugetemperatures and pressures and was very rapidly expanding andcooling. Approximately 10−37 seconds into the expansion, a phasetransition caused a cosmic inflation, during which the universe grewexponentially. After inflation stopped, the universe consisted of aquark-gluon plasma, as well as all other elementary particles. Tem-peratures were so high that the random motions of particles were atrelativistic speeds, and particle-antiparticle pairs of all kinds were be-ing continuously created and destroyed in collisions. At some pointbaryogenesis, a reaction that we know little about, violated the con-servation of baryon number, leading to a very small excess of quarksand leptons over antiquarks and antileptons – of the order of onepart in 30 million. This resulted in the predominance of matter overantimatter in the present universe.

Protons Forming

The universe continued to decrease in density and fall in tempera-ture, hence the typical energy of each particle was decreasing. Afterabout 10−11 seconds, the picture becomes less speculative, since par-ticle energies drop to values that can be attained in particle physicsexperiments. At about 10−6 seconds, quarks and gluons combined

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Figure 1: The evolution of the Universe.

to form baryons such as protons and neutrons. The small excessof quarks over antiquarks led to a small excess of baryons over an-tibaryons. The temperature was now no longer high enough to createnew proton-antiproton pairs (similarly for neutrons-antineutrons),so a mass annihilation immediately followed, leaving just one in 1010

of the original protons and neutrons, and none of their antiparti-cles. A similar process happened at about 1 second for electrons andpositrons. After these annihilations, the remaining protons, neutronsand electrons were no longer moving relativistically and the energydensity of the universe was dominated by photons (with a minorcontribution from neutrinos).

Nuclear Fusion Begins and Ends

A fraction of a second into the expansion, when the temperaturewas about a hundred billion kelvin (100 GK), neutrons combinedwith protons to form the universe‘s deuterium and helium nucleiin a process called Big Bang nucleosynthesis. However, around 3

minutes after the Big Bang the universe had cooled further so thatfusion was no longer possible. The Big Bang theory itself predictsmass abundances of about 75 % of hydrogen-1, about 25 % helium-4,about 0.01 % of deuterium, trace amounts (in the order of 10−10 of

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lithium and beryllium, and no other heavy elements. That the ob-served abundances in the universe are generally consistent with theseabundance numbers is considered strong evidence for the Big Bangtheory.

The Universe Becomes Transparent

After about 380 000 years the universe cooled to a temperatureof around 3000 K. The electrons and nuclei combined into atoms(mostly hydrogen). This meant that radiation could travel freelywithout forcing free charges to oscillate and continued through spacelargely unimpeded. This relic radiation is known as the cmbr. Itis frequently stated that the cmbr that is detected today started asgamma radiation shortly after the Big Bang. This is not strictly truebecause these photons were scattered and absorbed a long time ago.The cmbr that we can detect now started as mainly infra-red radi-ation 380 000 years after the Big Bang when the universe suddenlybecame transparent. Although the universe previously had been hotenough to emit gamma rays (as a black body radiator), this radiationwas not able to travel very far.

The Modern Universe and The Big Bang Theory

In today‘s universe, the earliest and most direct observational evi-dence of the validity of the theory are the expansion of the universeaccording to Hubble‘s law (as indicated by the red shifts of galaxies),discovery and measurement of the cmbr and the relative abundancesof light elements produced by Big Bang nucleosynthesis. More recentevidence includes observations of galaxy formation and evolution,and the distribution of large-scale cosmic structures. These are some-times called the “four pillars” of the Big Bang theory.

Figure 2: An example of an absorptionspectrum that has been redshifted.

Observations of distant galaxies and quasars show that these ob-jects are red shifted – the light emitted from them has been shifted tolonger wavelengths. This can be seen by taking a frequency spectrumof an object and matching the spectroscopic pattern of emission linesor absorption lines corresponding to atoms of the chemical elementsinteracting with the light. These red shifts are distributed evenlyamong the observed objects in all directions. If the red shift is inter-preted as a Doppler shift, the recessional velocity of the object can becalculated. When the recessional velocities are plotted against thesedistances, a linear relationship known as Hubble‘s law is observed:

v = H0D (1)

where v is the recessional velocity of the galaxy or other distant

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object; D is the distance to the object; H0 is the Hubble constant,measured to be 2.2685× 10−18 s−1.6 6 H0, the Hubble constant, is one of

the least well-known of the physi-cal constants; at various points it hasbeen estimated to be anywhere be-tween 50–90 km s−1 pc−1. The valuewith the lowest error is currentlythat of the Planck Collaboration (seehttp://arxiv.org/abs/1303.5062):(67.2± 1.2) km s−1 pc−1

If Hubble‘s law, v = H0D, is combined with a simple calculationfor the escape velocity from a spherical universe, the so-called criticaldensity of the universe can be calculated:

12

mv2esc −

GMmR

= 0 (2)

where vesc is the escape velocity of an arbitrary mass, m, which is adistance, R, from the ‘centre’ of the universe and M is the mass of theuniverse contained inside the sphere of radius R (upon whose surfacethe arbitrary mass lies). When the volume of the sphere of radius R isalso included, this leads to:

ρc =3H2

08πG

(3)

The critical density, ρc , of the universe can be calculated from thisequation and corresponds to 5 hydrogen atoms per cubic metre. Theobserved mass of the universe (based on counting stars) also leads toa similar value of density.

Not only can we calculate a mean density for the modern uni-verse, we can also calculate a mean temperature. From the cmbr,if the universe is assumed to be a black body then a temperature of(2.725± 0.001)K is obtained. Moreover, the microwave spectrumfollows a perfect black body spectrum shape.7 7 See Figure 3.

Since the early 1980s more and more evidence for larger scale or-der of matter in the universe has been discovered. Stars are organisedinto galaxies, which in turn form galaxy groups, galaxy clusters,superclusters, sheets, walls and filaments, which are separated by im-mense voids, creating a vast foam-like structure sometimes called the“cosmic web”.8 All these enormous scale structures have been simu- 8 See Figure 4.

lated by computer and all seem to agree with the Big Bang theory.

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Figure 3: Comparison between thepredicted spectrum of the cmbr (line)and the spectrum as measured by cobe

(diamonds).

Figure 4: The large-scale structure ofthe Universe.

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Questions

Where necessary, use the values for constants provided in the wjec formula booklet.

1. Explain the following terms in your own words:

(a) cosmic microwave background radiation

(b) redshift

(c) homogeneity (as it applies in this context)

(d) black body

(e) inflation

(f) annihilation

(g) relativistic speeds

(h) baryogenesis

(i) quark

(j) lepton

(k) gluon

(l) quasar

(m) Hubble constant

(n) Doppler shift

(o) recessional velocity

(p) escape velocity

(q) critical density

(r) [galactic] supercluster

2. What limits our ability to simulate conditions in the earliest moments of the Big Bang?

3. In Big Bang nucleosynthesis there are two processes by which neutrons interact to create protons:

n + e+ ←→ νe + p

n + νe ←→ p + e−

Show that both of these processes conserve charge, baryon number and lepton number.

4. The cmbr has a black body temperature of (2.725± 0.001)K.

(a) What is the peak frequency of the radiation emitted by the cosmic microwave background?

(b) What is the uncertainty in this frequency?

(c) What would the peak wavelength be at 380 000 yr after the Big Bang, and in which part of the elec-tromagnetic spectrum would this lie?

5. Describe the structure of baryons, and give three examples.

6. Why did the Universe become transparent?

7. The bright sodium line at 589.995 nm is observed from the smaller star of a binary system to be oscillat-ing between 678.789 nm and 678.200 nm.

(a) At what speed is the binary system moving away from Earth?

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(b) At what speed is the smaller star orbiting the larger star?

8. Edwin Hubble‘s first value for the Hubble constant was 500 km s−1 Mpc−1.

(a) Show that the standard unit for the Hubble constant (km s−1 Mpc−1) has the same units as the unitquoted in the text (s−1).

(b) Convert Hubble’s first value of the Hubble constant to s−1.

9. Using the value of H0 provided, calculate the recessional velocity (as a fraction of the speed of light) ofthe quasar 3C 273 which lies 2.443 gigalightyears from Earth.

10. Using your existing knowledge of gravitational fields:

(a) Derive Equation 2.

(b) Derive Equation 3.

11. Calculate the critical density of the Universe in kg m−3 and show that this is approximately “5 hydro-gen atoms per cubic metre”.

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Answers

1. No answers provided

2. Our ability to simulate the conditions in the earliest moments of the Big Bang is limited by the energiesachievable in particle accelerators.

3. (a)

n e+ νe p

Q 0 +1 0 +1B +1 0 0 +1L 0 −1 −1 0

(b)

n νe p e−

Q 0 0 +1 −1B +1 0 +1 0L 0 +1 0 +1

4. (a) Using Wein‘s Law:

λpeak =bT

cf=

2.90× 10−3 m K2.725 K

3.00× 108 m s−1

f= 1.06× 10−3 m

f =3.00× 108 m s−1

1.06E− 3f = 282× 109 s−1

f = 282 GHz

(b) The percentage error in the peak frequency will be the same as the percentage error in the tem-perature of the cosmic microwave background (we assume that there is no error in the values of theconstants provided).

σT =uncertaintyactual value

σT =0.001 K2.725 K

σT = 3.67× 10−4 = 0.0367 %

σf = 0.0367 %

σf = 0.0367 %× 282 GHz

σf = ±103 MHz

Therefore f = (281.897± 0.103)GHz.

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(c) Again using Wein‘s Law, and the value of 3000 K from the text

λpeak =bT

λpeak =2.90× 10−3 m K

3000 Kλpeak = 9.67× 10−7 m

λpeak = 967 nm

Given that the visible spectrum runs from approximately 390–700 nm this would be in the infraredpart of the spectrum.

5. Baryons are particles composed of three quarks. Examples of baryons include protons, neutrons, thedelta baryons (∆++, ∆+, ∆0 and ∆−), the lambda baryons (Λ0, Λ0

b, Λ+c and Λ+), etc.

6. The Universe became transparent because electrons, which had previously scattered photons (viaThompson scattering), became bound to protons to form neutral hydrogen.

7. (a) To calculate the recessional velocity of the binary system we find the mid-point (i.e. the average) ofthe two wavelengths – this is the wavelength that would be measured if the stars were not orbitingeach other.

λrecession =678.789 nm + 678.200 nm

2λrecession = 678.495 nm

Using this wavelength in the equation for Doppler shift yields the recessional velocity:

∆λ

λ=

vc

678.495 nm− 589.995 nm589.995 nm

=v

3.00× 108 m s−1

0.150001 =v

3.00× 108 m s−1

v = 0.150001× 3.00× 108 m s−1

v = 45.0× 106 m s−1

(a) To calculate the orbital velocity, we use the difference between either of the received wavelengthsand lab wavelength.

∆λ

λ=

vc

678.789 nm− 589.995 nm678.495 nm

=v

3.00× 108 m s−1

88.7940 nm589.995 nm

=v

3.00× 108 m s−1

0.150 500 =v

3.00× 108 m s−1

v = 0.150 500× 3.00× 108 m s−1

v = 45.150× 106 m s−1

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When we remove the recessional component from this we are left with the orbital velocity: vorbital =

0.150× 106 m s−1 or 150 km s−1.

8. (a) The kilometre and parsec are both units of distance.

km s−1 Mpc−1 = s−1

m s−1 m−1 = s−1

s−1 = s−1

(b) One parsec is 3.09× 1016 m.

H0 =500 km s−1

1 Mpc

H0 =500× 103 m s−1

3.09× 1016 mH0 = 1.62× 10−11 s−1

9. 2.443× 109 lightyears is 2.443× 109 × 3.00× 108 m s−1 × 365× 24× 60× 60 = 2.31× 1025 m.

v = H0D

v = 2.2685× 10−18 s−1 × 2.31× 1025 m

v = 52.4 Mm s−1

v = 0.174c

10. (a) The gravitational potential (VG) of an object with mass m at radius R from a body with mass Mis: VG = −GM/R. The gravitational potential energy of this object is then U = GMm/R. In order toescape the gravitational field of mass M, the object must have kinetic energy equal to U, which yieldsEquation 2.

12

mv2esc =

GMmR

12

mv2esc −

GMmR

= 0

(b) The density of the Universe is ρ = M/V. Beginning with Equation 2:

GMmR

=12

mv2

M =Rv2

2G

We now substitute in our equation for v in terms of Hubble‘s constant and the radius of the Universev = H0R, and divide by the volume of the Universe V = 4/3πR3.

M =RH2

0 R2

2GMV

=H2

0 R3/2G

4/3πR3

ρ =3H2

08πG

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11. Using the equation provided:

ρc =3H2

08πG

ρc =3×

(2.2685× 10−18 s−1)2

8π × 6.67× 10−11 N m2 kg−2

ρc =3× 5.15× 10−36 s−2

1.68× 10−9 N m2 kg−2

ρc =3× 1.54× 10−35 s−2

1.68× 10−9 kg−1 m3 s−2

ρc = 9.21× 10−27 kg m−3

The mass of a hydrogen atom is equal to the mass of a proton plus the mass of an electron:

mH = mp + me

mH = 1.66× 10−27 kg + 9.11× 10−31 kg

mH = 1.66× 10−27 kg

Therefore, in terms of hydrogen masses, the critical density is:

ρc =9.21× 10−27 kg m−3

1.66× 10−27 kg

ρc = 5.55 kg−1