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== 11.7#h(0.6em)Evolution of the Early Universe


  In the previous section, we discussed the structure and dynamics of universe. In particular, the universe appears to be expanding and even accelerating. But what was the universe like at the beginning of time? In this section, we discuss what evidence scientists have been able to gather about the early universe and its evolution to present time.


  
    
=== The Early Universe


    Before the short period of cosmic inflation, cosmologists believe that all matter in the universe was squeezed into a space much smaller than an atom. Cosmologists further believe that the universe was extremely dense and hot, and interactions between particles were governed by a single force. In other words, the four fundamental forces (strong nuclear, electromagnetic, weak nuclear, and gravitational) merge into one at these energies . How and why this “unity” breaks down at lower energies is an important unsolved problem in physics.


    #figure(figph[Figure shows a timeline. At 10 to the power minus 43 seconds after big bang, the line splits into two. One branch is gravitational force. The other moves ahead and further splits into two at 10 to the power minus 35 seconds. From here, one branch is strong nuclear force. The other splits into two at 10 to the power minus 12 seconds. The two branches are labeled electromagnetic force and weak nuclear force. The particle energy and the temperature of the universe at the time of the first split are: 10 to the power 19 GeV and 10 to the power 32 K respectively. At the second split, they are: 10 to the power 14 GeV and 10 to the power 27 K respectively. At the third split, they are 100 GeV and 10 to the power 15 K respectively. All four lines continue till they reach the values: 5 into 10 to the power 17 seconds, 10 to the power minus 4 eV and 3 K.], alt: "Figure shows a timeline. At 10 to the power minus 43 seconds after big bang, the line splits into two. One branch is gravitational force. The other moves ahead and further splits into two at 10 to the power minus 35 seconds. From here, one branch is strong nuclear force. The other splits into two at 10 to the power minus 12 seconds. The two branches are labeled electromagnetic force and weak nuclear force. The particle energy and the temperature of the universe at the time of the first split are: 10 to the power 19 GeV and 10 to the power 32 K respectively. At the second split, they are: 10 to the power 14 GeV and 10 to the power 27 K respectively. At the third split, they are 100 GeV and 10 to the power 15 K respectively. All four lines continue till they reach the values: 5 into 10 to the power 17 seconds, 10 to the power minus 4 eV and 3 K.", caption: [The separation of the four fundamental forces in the early universe.])


    Scientific models of the early universe are highly speculative.  shows a sketch of one possible timeline of events.


    #figure(figph[Figure shows a timeline. Inflation starts at 10 to the power minus 43 seconds after big bang, at a temperature of 10 to the power 32 K and an energy of 10 to the power 19 GeV. Inflation ends at 10 to the power minus 35s, 10 to the power 27 K and 10 to the power 15 GeV. This is followed by Age of quarks: quarks, neutrinos, muons, positrons, electrons, and photons. Protons are formed at 10 to the power minus 35 s, 10 to the power 27 K and 10 to the power of 15 GeV. This is followed by the age of hadrons and leptons: leptons, antileptons, baryons, and photons. Nuclear fusion begins at 10 s, 10 to the power 9 K and 10 to the power minus 4 GeV. This is followed by the age of Big Band nucleosynthesis: leptons, H, D, He, Li, and photons. Nuclear fusion ends at 20 minutes years, 3,000 K, and 10 to the power minus 10 GeV. This is followed by the age of ions: leptons, H, D, He, Li, and photons. Cosmic microwave background is at 370,000 years and 10 to the power minus 12 GeV. This is followed by age of atoms. First stars and galaxies are formed at 10 to the power 8 years, 60 K and 10 to the power minus 12 GeV. This is followed by the age of stars and galaxies. Today the temperature is 2.7 K and the energy is 2.3 into 10 to the power minus 13 GeV.], alt: "Figure shows a timeline. Inflation starts at 10 to the power minus 43 seconds after big bang, at a temperature of 10 to the power 32 K and an energy of 10 to the power 19 GeV. Inflation ends at 10 to the power minus 35s, 10 to the power 27 K and 10 to the power 15 GeV. This is followed by Age of quarks: quarks, neutrinos, muons, positrons, electrons, and photons. Protons are formed at 10 to the power minus 35 s, 10 to the power 27 K and 10 to the power of 15 GeV. This is followed by the age of hadrons and leptons: leptons, antileptons, baryons, and photons. Nuclear fusion begins at 10 s, 10 to the power 9 K and 10 to the power minus 4 GeV. This is followed by the age of Big Band nucleosynthesis: leptons, H, D, He, Li, and photons. Nuclear fusion ends at 20 minutes years, 3,000 K, and 10 to the power minus 10 GeV. This is followed by the age of ions: leptons, H, D, He, Li, and photons. Cosmic microwave background is at 370,000 years and 10 to the power minus 12 GeV. This is followed by age of atoms. First stars and galaxies are formed at 10 to the power 8 years, 60 K and 10 to the power minus 12 GeV. This is followed by the age of stars and galaxies. Today the temperature is 2.7 K and the energy is 2.3 into 10 to the power minus 13 GeV.", caption: [An approximate timeline for the evolution of the universe from the Big Bang to the present.])


    + #emph[Big Bang] #math.equation(block: false, alt: "open parenthesis t less than 1 0 to the power − 43 s close parenthesis :")[$( t < 1 0^("−" 43) "s" ) :$] The current laws of physics break down. At the end of the initial Big Bang event, the temperature of the universe is approximately #math.equation(block: false, alt: "T equals 1 0 to the power 32 K .")[$T = 1 0^(32) "K" .$]
+ #emph[Inflationary phase] #math.equation(block: false, alt: "open parenthesis t equals 1 0 to the power − 43 to 1 0 to the power − 35 s close parenthesis :")[$( t = 1 0^("−" 43) "to 1" 0^("−" 35) "s" ) :$] The universe expands exponentially, and gravity separates from the other forces. The universe cools to approximately #math.equation(block: false, alt: "T equals 1 0 to the power 27 K .")[$T = 1 0^(27) "K" .$]
+ #emph[Age of quarks] #math.equation(block: false, alt: "open parenthesis t equals 1 0 to the power − 35 to 1 0 to the power − 5 s close parenthesis :")[$( t = 1 0^("−" 35) "to 1" 0^("−" 5) "s" ) :$] As the universe continues to expand, the strong nuclear force separates from the electromagnetic and weak nuclear forces (or electroweak force). Soon after, the weak nuclear force separates from the electromagnetic force. The universe is a hot soup of quarks, leptons, photons, and other particles.
+ #emph[Age of hadrons and leptons] #math.equation(block: false, alt: "open parenthesis t equals 1 0 to the power − 5 to 10 s close parenthesis :")[$( t = 1 0^("−" 5) " to 10 s" ) :$] The universe consists of leptons (electrons, muons, tauons, and neutrinos) and hadrons (such as protons, neutrons, and mesons) in thermal equilibrium. Pair production and pair annihilation occurs with equal ease, so photons remain in thermal equilibrium:
#math.equation(block: true, alt: "γ plus γ ↔ e to the power − plus e to the power plus; γ plus γ ↔ p plus p −; γ plus γ ↔ n plus n − .")[$γ + γ ↔ "e"^("−") + "e"^(+) \ γ + γ ↔ "p" + limits("p")^("−") \ γ + γ ↔ #h(0.2em) "n" + limits("n")^("−") .$]
The number of protons is approximately equal to the number of neutrons until about 1s, through interactions with neutrinos: #math.equation(block: true, alt: "ν sub e plus n ↔ e to the power − plus p; ν − sub e plus p ↔ e to the power plus plus n.")[$ν_(e) + "n" #h(0.2em) ↔ "e"^("−") + "p" \ limits(ν)^("−")_(e) + "p" #h(0.2em) ↔ "e"^(+) + "n."$]
Neutrinos decouple from baryons at around 1s, forming the cosmic neutrino background. #linebreak() 
The temperature of the universe settles to approximately #math.equation(block: false, alt: "1 0 to the power 9 K")[$1 0^(9) "K"$] —much too cool for the continued production of nucleon-antinucleon pairs. The numbers of protons and neutrons begin to dominate over their anti-particles, so proton-antiproton #math.equation(block: false, alt: "open parenthesis p p − close parenthesis")[$( "p" limits("p")^("−") )$] and neutron-antineutron (#math.equation(block: false, alt: "n n −")[$"n" limits("n")^("−")$]) annihilations decline.
+ #emph[Age of Big Bang nucleosynthesis] (10s to 20 minutes): Deuterons (proton-neutron pairs) begin to form. As the universe continues to expand, deuterons react with protons and neutrons to form larger nuclei; these larger nuclei react with protons and neutrons to form still larger stable nuclei, up to #math.equation(block: false, alt: "to the power 7 Li")[$7 "Li"$]. At the end of this period, about 1/4 of the mass of the universe is helium. (This explains the current amount of helium in the universe.) Photons lack the energy to continue electron-positron production, so electrons and positrons annihilate each other to photons only.
+ #emph[Age of ions] (20 minutes to 370,000 years): The universe is hot enough to ionize any atoms formed. The universe consists of electrons, positrons, protons, the light nuclei produced during the Big Bang nucleosynthesis, and photons.
+ #emph[Age of atoms] (370,000 years to about #math.equation(block: false, alt: "10 to the power 8")[$10^(8)$] years): The universe cools below 4000 K and atoms form. Photons do not interact strongly with neutral atoms, so they “decouple” (separate) from atoms. These photons constitute the #strong[cosmic microwave background radiation] to be discussed later. This is also sometimes called the "dark ages" since the universe becomes transparent and the only source of new light is from hydrogen emitting radio waves.
+ #emph[Age of stars and galaxies] (#math.equation(block: false, alt: "10 to the power 8")[$10^(8)$] years to present): When the universe cools to about 60 K, atoms and particles are pulled together by gravity and form large lumps. The atoms and particles undergo nuclear fusion reaction, creating stars and elements up to iron. Eventually, supernovae explode, creating elements past iron and spreading carbon higher throughout the universe.


    #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Watch this #link("https://openstax.org/l/21bigbang")[video] to learn more about Big Bang cosmology.
]


    To describe the conditions of the early universe quantitatively, recall the relationship between the average thermal energy of particle (#emph[E]) in a system of interacting particles and equilibrium temperature (#emph[T]) of that system:


    #math.equation(block: true, alt: "E equals k sub B T ,")[$E = k_(B) T ,$]
    where #math.equation(block: false, alt: "k sub B")[$k_("B")$] is Boltzmann’s constant. In the hot conditions of the early universe, particle energies were unimaginably large.


    #examplebox("Example 1")[What Was the Average Thermal Energy of a Particle just after the Big Bang?][
Strategy
The average thermal energy of a particle in a system of interacting particles depends on the equilibrium temperature of that system . We are given this approximate temperature in the above timeline.


      Solution
Cosmologists think the temperature of the universe just after the Big Bang was approximately #math.equation(block: false, alt: "T equals 1 0 to the power 32 K .")[$T = 1 0^(32) "K" .$] Therefore, the average thermal energy of a particle would have been


      #math.equation(block: true, alt: "k sub B T approximately equals open parenthesis 10 to the power −4 eV/K close parenthesis open parenthesis 10 to the power 32 K close parenthesis equals 10 to the power 28 eV = 10 to the power 19 GeV .")[$k_("B") T ≈ ( 10^(−4) #h(0.2em) "eV/K" ) ( 10^(32) "K" ) = 10^(28) attach(#h(0.2em) "eV = 10", t: 19) #h(0.2em) "GeV" "."$]
      Significance
This energy is many orders of magnitude larger than particle energies produced by human-made particle accelerators. Currently, these accelerators operate at energies less than #math.equation(block: false, alt: "1 0 to the power 4 GeV .")[$1 0^(4) #h(0.2em) "GeV" .$]
]


    #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Compare the abundance of helium by mass 10,000 years after the Big Bang and now.


      
      #solutionbox[
about the same
]
]


    Nucleons form at energies approximately equal to the rest mass of a proton, or 1000 MeV. The temperature corresponding to this energy is therefore


    #math.equation(block: true, alt: "T equals the fraction 1000 MeV over 8.62 times 10 to the power 11 MeV times K to the power −1 equals 1.2 times 10 to the power 13 K .")[$T = frac(1000 #h(0.2em) "MeV", 8.62 #h(0.2em) × #h(0.2em) 10^(11) #h(0.2em) "MeV" · "K"^(−1)) = 1.2 #h(0.2em) × #h(0.2em) 10^(13) #h(0.2em) "K" "."$]
    Temperatures of this value or higher existed within the first second of the early universe. A similar analysis can be done for atoms. Atoms form at an energy equal to the ionization energy of ground-state hydrogen (13 eV). The effective temperature for atom formation is therefore


    #math.equation(block: true, alt: "T equals the fraction 13 eV over 8.62 times 10 to the power 5 eV times K to the power −1 equals 1.6 times 10 to the power 5 K .")[$T = frac(13 #h(0.2em) "eV", 8.62 #h(0.2em) × #h(0.2em) 10^(5) #h(0.2em) "eV" · "K"^(−1)) = 1.6 #h(0.2em) × #h(0.2em) 10^(5) #h(0.2em) "K" "."$]
    This occurs well after the four fundamental forces have separated, including forces necessary to bind the protons and neutrons in the nucleus (strong nuclear force), and bind electrons to the nucleus (electromagnetic force).


  
  
    
=== Nucleosynthesis of Light Elements


    The relative abundances of the light elements hydrogen, helium, lithium, and beryllium in the universe provide key evidence for the Big Bang. The data suggest that much of the helium in the universe is primordial. For instance, it turns out that that 25% of the matter in the universe is helium, which is too high an abundance and cannot be explained based on the production of helium in stars.


    How much of the elements in the universe were created in the Big Bang? If you run the clock backward, the universe becomes more and more compressed, and hotter and hotter. Eventually, temperatures are reached that permit #strong[nucleosynthesis], the period of formation of nuclei, similar to what occurs at the core of the Sun. Big Bang nucleosynthesis is believed to have occurred within a few hundred seconds of the Big Bang.


    How did Big Bang nucleosynthesis occur? At first, protons and neutrons combined to form deuterons, #math.equation(block: false, alt: "squared H")[$2 "H"$]. The deuteron captured a neutron to form triton, #math.equation(block: false, alt: "cubed H")[$3 "H"$] —the nucleus of the radioactive hydrogen called tritium. Deuterons also captured protons to make helium #math.equation(block: false, alt: "cubed He")[$3 "He"$]. When #math.equation(block: false, alt: "cubed H")[$3 "H"$] captures a proton or #math.equation(block: false, alt: "cubed He")[$3 "He"$] captures a neutron, helium #math.equation(block: false, alt: "to the power 4 He")[$4 "He"$] results. At this stage in the Big Bang, the ratio of protons to neutrons was about 7:1. Thus, the process of conversion to #math.equation(block: false, alt: "to the power 4 He")[$4 "He"$] used up almost all neutrons. The process lasted about 3 minutes and almost #math.equation(block: false, alt: "25 %")[$25 "%"$] of all the matter turned into #math.equation(block: false, alt: "to the power 4 He")[$4 "He"$], along with small percentages of #math.equation(block: false, alt: "squared H")[$2 "H"$], #math.equation(block: false, alt: "cubed H")[$3 "H"$], and #math.equation(block: false, alt: "cubed He")[$3 "He"$]. Tiny amounts of #math.equation(block: false, alt: "to the power 7 Li")[$7 "Li"$] and #math.equation(block: false, alt: "to the power 7 Be")[$7 "Be"$] were also formed. The expansion during this time cooled the universe enough that the nuclear reactions stopped. The abundances of the light nuclei #math.equation(block: false, alt: "squared H")[$2 "H"$], #math.equation(block: false, alt: "to the power 4 He")[$4 "He"$], and #math.equation(block: false, alt: "to the power 7 Li")[$7 "Li"$] created after the Big Bang are very dependent on the matter density.


    The predicted abundances of the elements in the universe provide a stringent test of the Big Bang and the Big Bang nucleosynthesis. Recent experimental estimates of the matter density from the #strong[Wilkinson Microwave Anisotropy Probe (WMAP)] agree with model predictions. This agreement provides convincing evidence of the Big Bang model.


  
  
    
=== Cosmic Microwave Background Radiation


    According to cosmological models, the Big Bang event should have left behind thermal radiation called the cosmic microwave background radiation (CMBR). The intensity of this radiation should follow the blackbody radiation curve (Photons and Matter Waves). Wien’s law states that the wavelength of the radiation at peak intensity is


    #math.equation(block: true, alt: "λ sub max equals the fraction 2 . 898 times 1 0 to the power − 3 m-K over T ,")[$λ_("max") = frac(2 . 898 #h(0.2em) × #h(0.2em) 1 0^("−" 3) #h(0.2em) "m-K", T) ,$]
    where #emph[T] is temperature in kelvins. Scientists expected the expansion of the universe to “stretch the light,” and the temperature to be very low, so cosmic background radiation should be long-wavelength and low energy.


    In the 1960s, Arno #strong[Penzias] and Robert #strong[Wilson] of Bell Laboratories noticed that no matter what they did, they could not get rid of a faint background noise in their satellite communication system. The noise was due to radiation with wavelengths in the centimeter range (the microwave region). Later, this noise was associated with the cosmic background radiation. An intensity map of the cosmic background radiation appears in . The thermal spectrum is modeled well by a blackbody curve that corresponds to a temperature #math.equation(block: false, alt: "T equals 2.7 K")[$T = 2.7 "K"$] .


    #figure(figph[An oval shape showing patterns of blue and yellow. Some red areas are also visible.], alt: "An oval shape showing patterns of blue and yellow. Some red areas are also visible.", caption: [This map of the sky uses color to show fluctuations, or wrinkles, in the cosmic microwave background observed with the WMAP spacecraft. The Milky Way has been removed for clarity. Red represents higher temperature and higher density, whereas blue indicates lower temperature and density. This map does not contradict the earlier claim of smoothness because the largest fluctuations are only one part in one million.])


    #figure(figph[Graph of I subscript v in W per m squared s per r per Hertz versus Frequency in GHz and Wavelength in cm. The curve rises gradually, peaks and falls sharply. The curve is for 2.73 K blackbody. There are various types of dots marked along the curve. On the rising slope of the curve are dots labeled LBL Italy, White Mt and South Pole. Above these is a dot labeled Princeton, ground and balloon. Above this are three dots labeled DMR COBE satellite. Near the peak, on its either side are two dots labeled Cyanogen, optical. On the peak and the falling curve are several dots labeled UBC, sounding rocket as well as dots labeled FIRAS COBE satellite.], alt: "Graph of I subscript v in W per m squared s per r per Hertz versus Frequency in GHz and Wavelength in cm. The curve rises gradually, peaks and falls sharply. The curve is for 2.73 K blackbody. There are various types of dots marked along the curve. On the rising slope of the curve are dots labeled LBL Italy, White Mt and South Pole. Above these is a dot labeled Princeton, ground and balloon. Above this are three dots labeled DMR COBE satellite. Near the peak, on its either side are two dots labeled Cyanogen, optical. On the peak and the falling curve are several dots labeled UBC, sounding rocket as well as dots labeled FIRAS COBE satellite.", caption: [Intensity distribution of cosmic microwave background radiation. The model predictions (the line) agree extremely well with the experimental results (the dots). Frequency and brightness values are shown on a log axis.])


    The formation of atoms in the early universe makes these atoms less likely to interact with light. Therefore, photons that belong to the cosmic background radiation must have separated from matter at a temperature #emph[T] associated with 1 eV (the approximate ionization energy of an atom) . The temperature of the universe at this point was


    #math.equation(block: true, alt: "k sub B T ∼ 1 eV ⇒ T equals the fraction 1 eV over 8 . 617 times 1 0 to the power 5 eV / K ∼ 10 to the power 4 K .")[$"k"_("B") "T" ∼ "1 eV" ⇒ T = frac("1 eV", 8 . 617 #h(0.2em) × #h(0.2em) 1 0^(5) #h(0.2em) "eV" "/" "K") ∼ 10^(4) #h(0.2em) "K" .$]
    According to cosmological models, the time when photons last scattered off charged particles was approximately 380,000 years after the Big Bang. Before that time, matter in the universe was in the plasma form and the photons were “thermalized.”


  
  
    
=== Antimatter and Matter


    We know from direct observation that antimatter is rare. Earth and the solar system are nearly pure matter, and most of the universe also seems dominated by matter. This is proven by the lack of annihilation radiation coming to us from space, particularly the relative absence of 0.511-MeV #math.equation(block: false, alt: "γ")[$γ$] rays created by the mutual annihilation of electrons and positrons. (Antimatter in nature is created in particle collisions and in #math.equation(block: false, alt: "β to the power plus")[$β^(+)$] decays, but only in small amounts that quickly annihilate, leaving almost pure matter surviving.)


    Despite the observed dominance of matter over antimatter in the universe, the Standard Model of particle interactions and experimental measurement suggests only small differences in the ways that matter and antimatter interact. For example, neutral kaon decays produce only slightly more matter than antimatter. Yet, if through such decay, slightly more matter than antimatter was produced in the early universe, the rest could annihilate pair by pair, leaving mostly ordinary matter to form the stars and galaxies. In this way, the vast number of stars we observe may be only a tiny remnant of the original matter created in the Big Bang.


  
  
    
=== Dark Matter and Dark Energy


    In the 1970s, accomplished astronomer Vera Rubin began investigating the movement of spiral galaxies. She observed that their outermost reaches were rotating as quickly as their centers, which was not the predicted phenomenon. She calculated that the rotational velocity of galaxies should have been enough to cause them to fly apart, unless there was a significant discrepancy between their observable matter and their actual matter. This became known as the galaxy rotation problem. Rubin proposed that massive amounts of unseen, or dark, matter must be present in and around the galaxies for the rotation speeds to occur.


    In the last two decades, new and more powerful techniques have revealed that the universe is filled with #strong[dark matter]. This type of matter is interesting and important because, currently, scientists do not know what it is. However, we can infer its existence by the deflection of distant starlight. For example, if light from a distant galaxy is bent by the gravitational field of a clump of dark matter between us and the galaxy, it is possible that two images of the same galaxy can be produced . The bending of light by the gravitational field of matter is called gravitational lensing. In some cases, the starlight travels to an observer by multiple paths around the galaxy, producing a ring .


    Based on current research, scientist know only that dark matter is cold, slow moving, and interacts weakly with ordinary matter. Dark matter candidates include neutralinos (partners of Z bosons, photons, and Higgs bosons in “supersymmetry theory”) and particles that circulate in tiny rings set up by extra spatial dimensions.


    #figure(figph[Figure shows a star on the left and earth on the right. There is a galaxy in the center. Two rays originate from the star and bend around the galaxy to reach the earth. The back extensions of the bent rays connect to two objects, both labeled star image, one at the top and the other at the bottom of the star.], alt: "Figure shows a star on the left and earth on the right. There is a galaxy in the center. Two rays originate from the star and bend around the galaxy to reach the earth. The back extensions of the bent rays connect to two objects, both labeled star image, one at the top and the other at the bottom of the star.", caption: [Light from a distant star is bent around a galaxy. Under the right conditions, two duplicate images of the same star can be seen.])


    #figure(figph[Photograph of a black sky. A yellow light is seen at the center. A circle of white light surrounds it.], alt: "Photograph of a black sky. A yellow light is seen at the center. A circle of white light surrounds it.", caption: [Light from a distant star is bent around a galaxy. Under the right conditions, we can see a ring of light instead of a single star.])


    Increasingly precise astronomical measurements of the expanding universe also reveal the presence of a new form of energy called #strong[dark energy]. This energy is thought to explain larger-than-expected values for the observed galactic redshifts for distant galaxies. These redshifts suggest that the universe is not only expanding, but expanding at an increasing rate. Virtually nothing is known about the nature and properties of dark energy. Together, dark energy and dark matter represent two of the most interesting and unsolved puzzles of modern physics. Scientists attribute #math.equation(block: false, alt: "68.3 %")[$68.3 "%"$] of the energy of the universe to dark energy, #math.equation(block: false, alt: "26.8 %")[$26.8 "%"$] to dark matter, and just #math.equation(block: false, alt: "4.9 %")[$4.9 "%"$] to the mass-energy of ordinary particles . Given the current great mystery over the nature of dark matter and dark energy, Isaac Newton’s humble words are as true now as they were centuries ago:


    “I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.”


    #figure(figph[A pie chart shows 26.8 percent dark matter, 4.9 percent atoms and 68.3 percent dark energy.], alt: "A pie chart shows 26.8 percent dark matter, 4.9 percent atoms and 68.3 percent dark energy.", caption: [Estimated distribution of matter and energy in the universe.])


  
  
    
=== Summary


    - The early universe was hot and dense.
- The universe is isotropic and expanding.
- Cosmic background radiation is evidence for the Big Bang.
- The vast portion of the mass and energy of the universe is not well understood.


  
  
    
=== Key Equations


    #figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Momentum of a charged particle in a cyclotron], [#math.equation(block: false, alt: "p equals 0.3 B r")[$p = 0.3 B r$]]),
  [Center-of-mass energy of a colliding beam machine], [#math.equation(block: false, alt: "W squared equals 2 [ E sub 1 E sub 2 plus open parenthesis p sub 1 c close parenthesis open parenthesis p sub 2 c close parenthesis ] plus open parenthesis m sub 1 c squared close parenthesis squared plus open parenthesis m sub 2 c squared close parenthesis squared")[$W^(2) = 2 [ E_(1) E_(2) + ( p_(1) c ) ( p_(2) c ) ] + attach(( m_(1) c^(2) ), t: 2) + attach(( m_(2) c^(2) ), t: 2)$]],
  [Approximate time for exchange of a virtual particle between two other particles], [#math.equation(block: false, alt: "Δ t equals the fraction h over E")[$"Δ" t = frac(h, E)$]],
  [Hubble’s law], [#math.equation(block: false, alt: "v equals H sub 0 d")[$v = H_(0) d$]],
  [Cosmological space-time metric], [#math.equation(block: false, alt: "d s squared equals c squared d t squared minus a open parenthesis t close parenthesis squared d Σ squared")[$d s^(2) = c^(2) d t^(2) − a attach(( t ), t: 2) d "Σ"^(2)$]],
))


  
  
    
=== Conceptual Questions


    
      What is meant by a “cosmological model of the early universe?” Briefly describe this model in terms of the four fundamental forces.


    
    
      Describe two pieces of evidence that support the Big Bang model.


    
    #solutionbox[
The observed expansion of the universe and the cosmic background radiation spectrum.
]


    
      In what sense are we, as Newton once said, “a boy playing on the sea-shore”? Express your answer in terms of the concepts of dark matter and dark energy.


    
    
      If some unknown cause of redshift—such as light becoming “tired” from traveling long distances through empty space—is discovered, what effect would that have on cosmology?


    
    #solutionbox[
If light slow down, it takes long to reach Earth than expected. We conclude that the object is much closer than it really is. Thus, for every recessional velocity (based on the frequency of light, which we assume is not disturbed by the slowing), the distance is smaller than the “true” value, Hubble’s constant is larger than the “true” value, and the age of the universe is smaller than the “true” value.
]


    
      In the past, many scientists believed the universe to be infinite. However, if the universe is infinite, then any line of sight should eventually fall on a star’s surface and the night sky should be very bright. How is this paradox resolved in modern cosmology?


    
  
  
    
=== Additional Problems


    
      Experimental results suggest that a muon decays to an electron and photon. How is this possible?


    
    
      Each of the following reactions is missing a single particle. Identify the missing particle for each reaction.


      #math.equation(block: true, alt: "(a) p plus p − → n plus ?; (b) p plus p → p plus Λ to the power 0 plus ?; (c) π to the power minus plus p → Σ to the power − plus ?; (d) K to the power − plus n → Λ to the power 0 plus ?; (e) τ to the power plus → e to the power plus plus ν sub e plus ?; (f) ν − sub e plus p → n plus ?")[$"(a)" #h(0.2em) "p" + limits("p")^("−") → "n" + ? \ "(b)" #h(0.2em) "p" + "p" → "p" + "Λ"^(0) + ? \ "(c)" #h(0.2em) π^(−) + "p" → "Σ"^("−") + ? \ "(d)" #h(0.2em) "K"^("−") + "n" → "Λ"^(0) + ? \ "(e)" #h(0.2em) τ^(+) → "e"^(+) + ν_("e") + ? \ "(f)" #h(0.2em) limits(ν)^("−")_("e") + "p" → "n" + ?$]


    
    #solutionbox[
a. #math.equation(block: false, alt: "n −")[$limits(n)^("−")$]; b. #math.equation(block: false, alt: "K to the power plus")[$"K"^(+)$]; c. #math.equation(block: false, alt: "K to the power plus")[$"K"^(+)$]; d. #math.equation(block: false, alt: "π to the power −")[$π^("−")$]; e. #math.equation(block: false, alt: "ν − sub τ ;")[$limits(ν)^("−")_(τ) ;$] f. #math.equation(block: false, alt: "e to the power plus")[$e^(+)$]
]


    
      Because of energy loss due to synchrotron radiation in the LHC at CERN, only 5.00 MeV is added to the energy of each proton during each revolution around the main ring. How many revolutions are needed to produce 7.00-TeV (7000 GeV) protons, if they are injected with an initial energy of 8.00 GeV?


    
    
      A proton and an antiproton collide head-on, with each having a kinetic energy of 7.00 TeV (such as in the LHC at CERN). How much collision energy is available, taking into account the annihilation of the two masses? (Note that this is not significantly greater than the extremely relativistic kinetic energy.)


    
    #solutionbox[
#math.equation(block: true, alt: "14.002 TeV approximately equals 14.0 TeV")[$"14.002 TeV" ≈ 14.0 #h(0.2em) "TeV"$]
]


    
      When an electron and positron collide at the SLAC facility, they each have 50.0-GeV kinetic energies. What is the total collision energy available, taking into account the annihilation energy? Note that the annihilation energy is insignificant, because the electrons are highly relativistic.


    
    
      The core of a star collapses during a supernova, forming a neutron star. Angular momentum of the core is conserved, so the neutron star spins rapidly. If the initial core radius is #math.equation(block: false, alt: "5.0 times 10 to the power 5 km")[$5.0 #h(0.2em) × #h(0.2em) 10^(5) #h(0.2em) "km"$] and it collapses to 10.0 km, find the neutron star’s angular velocity in revolutions per second, given the core’s angular velocity was originally 1 revolution per 30.0 days.


    
    #solutionbox[
#math.equation(block: true, alt: "964 rev/s")[$964 #h(0.2em) "rev/s"$]
]


    
      Using the solution from the previous problem, find the increase in rotational kinetic energy, given the core’s mass is 1.3 times that of our Sun. Where does this increase in kinetic energy come from?


    
    
      (a) What Hubble constant corresponds to an approximate age of the universe of #math.equation(block: false, alt: "10 to the power 10")[$10^(10)$] y? To get an approximate value, assume the expansion rate is constant and calculate the speed at which two galaxies must move apart to be separated by 1 Mly (present average galactic separation) in a time of #math.equation(block: false, alt: "10 to the power 10")[$10^(10)$] y. (b) Similarly, what Hubble constant corresponds to a universe approximately #math.equation(block: false, alt: "2 times 10 to the power 10")[$2 #h(0.2em) × #h(0.2em) 10^(10)$] years old?


    
    #solutionbox[
a. #math.equation(block: false, alt: "H sub 0 equals the fraction 30 km/s over 1 Mly equals 30 km/s times Mly;")[$H_(0) = frac("30 km/s", "1 Mly") = 30 "km/s" · "Mly;"$] b. #math.equation(block: false, alt: "H sub 0 equals the fraction 15 km/s over 1 Mly equals 15 km/s times Mly")[$H_(0) = frac(15 #h(0.2em) "km/s", 1 #h(0.2em) "Mly") = 15 "km/s" · "Mly"$]
]


  
  
    
=== Challenge Problems


    
      Electrons and positrons are collided in a circular accelerator. Derive the expression for the center-of-mass energy of the particle.


    
    
      The intensity of cosmic ray radiation decreases rapidly with increasing energy, but there are occasionally extremely energetic cosmic rays that create a shower of radiation from all the particles they create by striking a nucleus in the atmosphere. Suppose a cosmic ray particle having an energy of #math.equation(block: false, alt: "10 to the power 10 GeV")[$10^(10) #h(0.2em) "GeV"$] converts its energy into particles with masses averaging #math.equation(block: false, alt: "200 MeV/ c squared")[$200 "MeV/" c^(2)$].


      (a) How many particles are created? (b) If the particles rain down on a #math.equation(block: false, alt: "1 .00-km squared")[$1 ".00-km"^(2)$] area, how many particles are there per square meter?


    
    #solutionbox[
a. #math.equation(block: false, alt: "5 times 10 to the power 10")[$5 #h(0.2em) × #h(0.2em) 10^(10)$]; b. divide the number of particles by the area they hit: #math.equation(block: false, alt: "5 times 10 to the power 4 particles / m squared")[$5 #h(0.2em) × #h(0.2em) 10^(4) "particles" "/" "m"^(2)$]
]


    
      (a) Calculate the relativistic quantity #math.equation(block: false, alt: "γ equals the fraction 1 over the square root of 1 minus v squared / c squared")[$γ = frac(1, sqrt(1 − v^(2) "/" c^(2)))$] for 1.00-TeV protons produced at Fermilab. (b) If such a proton created a #math.equation(block: false, alt: "π to the power plus")[$π^(+)$] having the same speed, how long would its life be in the laboratory? (c) How far could it travel in this time?


    
    
      Plans for an accelerator that produces a secondary beam of K mesons to scatter from nuclei, for the purpose of studying the strong force, call for them to have a kinetic energy of 500 MeV. (a) What would the relativistic quantity #math.equation(block: false, alt: "γ equals the fraction 1 over the square root of 1 minus v squared / c squared")[$γ = frac(1, sqrt(1 − v^(2) "/" c^(2)))$] be for these particles? (b) How long would their average lifetime be in the laboratory? (c) How far could they travel in this time?


    
    #solutionbox[
a. 2.01; b. #math.equation(block: false, alt: "2.50 times 10 to the power −8 s")[$2.50 #h(0.2em) × #h(0.2em) 10^(−8) #h(0.2em) "s"$]; c. 6.50 m
]


    
      In supernovae, neutrinos are produced in huge amounts. They were detected from the 1987A supernova in the Magellanic Cloud, which is about 120,000 light-years away from Earth (relatively close to our Milky Way Galaxy). If neutrinos have a mass, they cannot travel at the speed of light, but if their mass is small, their velocity would be almost that of light. (a) Suppose a neutrino with a #math.equation(block: false, alt: "7 -eV/ c squared")[$7 "-eV/" c^(2)$] mass has a kinetic energy of 700 keV. Find the relativistic quantity #math.equation(block: false, alt: "γ equals the fraction 1 over the square root of 1 minus v squared / c squared")[$γ = frac(1, sqrt(1 − v^(2) "/" c^(2)))$] for it. (b) If the neutrino leaves the 1987A supernova at the same time as a photon and both travel to Earth, how much sooner does the photon arrive? This is not a large time difference, given that it is impossible to know which neutrino left with which photon and the poor efficiency of the neutrino detectors. Thus, the fact that neutrinos were observed within hours of the brightening of the supernova only places an upper limit on the neutrino’s mass. (#emph[Hint:] You may need to use a series expansion to find #emph[v] for the neutrino, since its #math.equation(block: false, alt: "γ")[$γ$] is so large.)


    
    
      Assuming a circular orbit for the Sun about the center of the Milky Way Galaxy, calculate its orbital speed using the following information: The mass of the galaxy is equivalent to a single mass #math.equation(block: false, alt: "1.5 times 10 to the power 11")[$1.5 #h(0.2em) × #h(0.2em) 10^(11)$] times that of the Sun (or #math.equation(block: false, alt: "3 times 10 to the power 41 kg")[$3 #h(0.2em) × #h(0.2em) 10^(41) #h(0.2em) "kg"$]), located 30,000 ly away.


    
    #solutionbox[
#math.equation(block: true, alt: "the fraction m v squared over r, equals, the fraction G M m over r squared ⇒; v, equals, open parenthesis the fraction G M over r close parenthesis to the power 1 / 2 equals [ the fraction open parenthesis 6.67 times 10 to the power −11 N times m squared / kg squared close parenthesis open parenthesis 3 times 10 to the power 41 kg close parenthesis over open parenthesis 30,000 ly close parenthesis open parenthesis 9.46 times 10 to the power 15 m/ly close parenthesis ] equals 2.7 times 10 to the power 5 m/s")[$frac(m v^(2), r) & = & frac(G M m, r^(2)) ⇒ \  \  \ v & = & attach(( frac(G M, r) ), t: 1 "/" 2) = [ frac(( 6.67 #h(0.2em) × #h(0.2em) 10^(−11) #h(0.2em) "N" · "m"^(2) "/" "kg"^(2) ) ( 3 #h(0.2em) × #h(0.2em) 10^(41) #h(0.2em) "kg" ), ( "30,000 ly" ) ( 9.46 #h(0.2em) × #h(0.2em) 10^(15) #h(0.2em) "m/ly" )) ] = 2.7 #h(0.2em) × #h(0.2em) 10^(5) #h(0.2em) "m/s"$]
]


    
      (a) What is the approximate force of gravity on a 70-kg person due to the Andromeda Galaxy, assuming its total mass is #math.equation(block: false, alt: "10 to the power 13")[$10^(13)$] that of our Sun and acts like a single mass 0.613 Mpc away? (b) What is the ratio of this force to the person’s weight? Note that Andromeda is the closest large galaxy.


    
    
      (a) A particle and its antiparticle are at rest relative to an observer and annihilate (completely destroying both masses), creating two #math.equation(block: false, alt: "γ")[$γ$] rays of equal energy. What is the characteristic #math.equation(block: false, alt: "γ")[$γ$]-ray energy you would look for if searching for evidence of proton-antiproton annihilation? (The fact that such radiation is rarely observed is evidence that there is very little antimatter in the universe.) (b) How does this compare with the 0.511-MeV energy associated with electron-positron annihilation?


    
    #solutionbox[
a. 938.27 MeV; b. #math.equation(block: false, alt: "1.84 times 10 cubed")[$1.84 #h(0.2em) × #h(0.2em) 10^(3)$]
]


    
      The peak intensity of the CMBR occurs at a wavelength of 1.1 mm. (a) What is the energy in eV of a 1.1-mm photon? (b) There are approximately #math.equation(block: false, alt: "10 to the power 9")[$10^(9)$] photons for each massive particle in deep space. Calculate the energy of #math.equation(block: false, alt: "10 to the power 9")[$10^(9)$] such photons. (c) If the average massive particle in space has a mass half that of a proton, what energy would be created by converting its mass to energy? (d) Does this imply that space is “matter dominated”? Explain briefly.


    
    
      (a) Use the Heisenberg uncertainty principle to calculate the uncertainty in energy for a corresponding time interval of #math.equation(block: false, alt: "10 to the power −43 s")[$10^(−43) "s"$]. (b) Compare this energy with the #math.equation(block: false, alt: "10 to the power 19 GeV")[$10^(19) #h(0.2em) "GeV"$] unification-of-forces energy and discuss why they are similar.


    
    #solutionbox[
a. #math.equation(block: false, alt: "3.29 times 10 to the power 18 GeV approximately equals 3 times 10 to the power 18 GeV")[$3.29 #h(0.2em) × #h(0.2em) 10^(18) #h(0.2em) "GeV" ≈ 3 #h(0.2em) × #h(0.2em) 10^(18) #h(0.2em) "GeV"$]; b. 0.3; Unification of the three forces breaks down shortly after the separation of gravity from the unification force (near the Planck time interval). The uncertainty in time then becomes greater. Hence the energy available becomes less than the needed unification energy.
]