Back to Scientific Theories
Table of Contents
 Core Idea
 Three Pieces of the Big Bang Theory, Colorcoded
 Overview of the Theory
 Development of the Pieces
 Expansion of Space and Recession of Galaxies
 Particle History and CMB Radiation and Light Elements
 Dark Matter
 Dark Energy
 Age and Fate of the Universe
 Friedmann Equation, Rescaled Omega Version
 An Analogy
 Density
 Different Amounts of Stuff
 Almost No Stuff
 Enough Matter and Radiation to Reverse the Expansion
 Minimal Matter and Radiation to Keep the Universe Expanding
 Different Kinds of Stuff
 Matter Dominated Universe
 Radiation Dominated Universe
 Dark Energy Dominated Universe
 Age and Fate of the Universe, Consensus View
 MatterEnergy History of the Universe
 Resources
 Addenda
Core Idea
Space has expanded and the universe cooled from a singularity of infinite density and temperature 12 to 14 billion years ago.
(The age of the solar system is 4.5 billion years.)
Three Pieces of the Big Bang Theory, Colorcoded
General Relativity Applied to the Universe as a Whole
Cosmological Observations of the 20th Century
Particle Physics Applied to the Early Universe
Overview of the Theory
 A scientific theory is:
 designed to explain certain kinds of phenomena
 defined by its postulates
 supported or disproved by its predictions
 For example, Newton’s Theory of Gravitation is:
 designed to explain:
 the motion of falling bodies
 the orbits of heavenly bodies
 defined by its postulates
 Law of Universal Gravitation
 Equation of Motion
 supported by its predictions of
 Kepler’s Laws of Planetary Motion
 Galileo’s Laws of Falling Bodies and Projectile Motion
 designed to explain:
 The Big Bang Theory is
 designed to explain:
 the implications of Einstein’s General Relativity for the universe as a whole
 why galaxies beyond the Milky Way are moving directly away from us, at speeds correlated with their distance.
 what particle interactions took place in the first few seconds and minutes after the Big Bang
 defined by its postulates:
 Expansion of Space
 Based on General Relativity, Einstein’s theory that gravity is the curvature of spacetime.
 Particle History of the Early Universe
 Based on the Standard Model of Particle History, the theory governing the electromagnetic, strong, and weak interactions of elementary particles and fields.
 Expansion of Space
 supported by its predictions of:
 Recession of Galaxies
 Cosmic Microwave Background Radiation
 Abundance of Light Elements
 designed to explain:
Development of the Pieces
 1915 General Relativity
 Einstein set forth General Relativity, his theory that gravity is the curvature of spacetime.
 The theory’s postulates have been famously summarized by John Archibald Wheeler:
 “Matter tells spacetime how to curve, and curved spacetime tells matter how to move.”
 View Space, Time, and Spacetime
 The theory is supported by incredible predictions such as:
 Space expands and contracts.
 Gravitational waves, detected in 2015, are waves of expanding and contracting space.
 Time runs slower in gravitational fields.
 Time runs faster on the atomic clocks on board GPS satellites than it does on atomic clocks on Earth.
 More on General Relativity
 Space expands and contracts.
 1917 Einstein’s Cosmological Model
 Einstein applied General Relativity to the universe as a whole, making two assumptions
 Cosmological Principle
 The universe looks the same on a large scale no matter where in the universe you are (homogeneity) and no matter in what direction you look (isotropy).
 Universe is static, neither expanding nor contracting.
 Cosmological Principle
 To make the universe static, Einstein added a term to the Field Equation of General Relativity: the cosmological constant (Lamda), representing a repulsive energy that balanced the attractive force of gravity.
 Britannica: Cosmological constant
 “The cosmological constant is a term Albert Einstein reluctantly added by to his equations of general relativity to obtain a solution that described a static universe, as he believed it to be at the time. The constant has the effect of a repulsive force acting against the gravitational attraction of matter. When Einstein heard of the evidence that the universe is expanding, he called the cosmological constant the “biggest blunder” of his life. However, recent observations have detected a repulsive force, similar to the cosmological constant, that is called dark energy and is the dominant component of the universe.”
 Einstein applied General Relativity to the universe as a whole, making two assumptions
 1922 Friedmann Equation
 Alexander Friedmann published an equation, based on General Relativity, that is still today the theoretical basis of the expansion of space. Like Einstein, Friedmann assumed the Cosmological Principle. Unlike Einstein, he did not force the universe to be static.
 Alexander Friedmann published an equation, based on General Relativity, that is still today the theoretical basis of the expansion of space. Like Einstein, Friedmann assumed the Cosmological Principle. Unlike Einstein, he did not force the universe to be static.
 19121929 Recession of Galaxies
 American astronomers Vesto Slipher and William Wallace Campbell observed the redshift of distant galaxies, indicating they were receding from us.
 In 1929 Edwin Hubble proposed the Hubble Law:
 The further away a galaxy is, the faster it recedes
 1936present FRW Cosmological Models
 Howard P. Robertson and Arthur G. Walker expanded the Friedmann Equation into a spacetime metric (the RobertsonWalker Metric), a formula that defines the geometry of spacetime on a large scale.
 Since then FriedmannRobertsonWalker (FRW) Cosmological Models have become increasingly sophisticated, incorporating dark matter and dark energy.
 1948 Particle History of the Universe
 George Gamow, Ralph Alpher, and Robert Herman extended the Big Bang Theory to atoms and subatomic particles, setting forth a particle history of the early universe as it cooled and expanded.
 1948 CMB Radiation Predicted
 Ralph Alpher and Robert Herman predicted that, due to the expansion of space, electromagnetic radiation from the early universe would today be in the microwave range, permeating the universe.
 1964 CMB Radiation Detected
 Arno Penzias and Robert Wilson shared the Nobel Prize in Physics for detecting the Cosmic Microwave Background Radiation, the radiation Alpher and Herman had predicted.
 1970s Dark Matter
 Cosmologists recognized that the observed stellar mass of a galaxy does not exert enough gravitational force to keep it together, suggesting the existence of invisible, dark matter.
 1998 Dark Energy
 Saul Perlmutter, Brian P. Schmidt, and Adam G. Riess shared the Nobel Prize in Physics for leading teams that independently found that the expansion of the universe was accelerating, suggesting the existence of antigravitational dark energy.
Expansion of Space
and
Recession of Galaxies
Recession of Galaxies
 American astronomers Vesto Slipher and William Wallace Campbell observed the redshift of distant galaxies, indicating they were receding from the Milky Way Galaxy.
 Galactic redshifts and blueshifts are shifts of lines in the electromagnetic spectrum of a galaxy. Shifts toward red indicate a longer wavelength; those toward blue a shorter wavelength.
 In the diagram, the spectral line under the tip of the arrow head moves to the right from laboratory reference (at the bottom) to very distant galaxy (at the top), resulting in an increase of wavelength from about 440 nm to almost 600 nm.
 A nanometer (nm) is a billionth of a meter.
Image Credit quora.com/Astronomersmeasurethespeedofgalaxiesusingredshiftbutwhatifspaceistintedred
 Spectral shifts commonly result from the Doppler Effect, the change in the length of a wave as it’s emitted or received by a moving object. Thus a galaxy’s redshift indicates it is moving away from us. A galaxy’s blueshift means it’s heading in our direction. The larger the shift, the greater the speed.
Image Credit imagine.gsfc.nasa.gov/features/yba/M31_velocity/spectrum/doppler_more.html
 Spectrographs
 View Spectrograph
 A Passing Train
Hubble Law
 Vesto Slipher and other astronomers had established that distant galaxies are receding, based on their redshift. In 1929 Edwin Hubble correlated their recessional velocities with their distance: the further away the galaxy, the faster it receded — by the same factor.
 The Hubble Law is the mathematical expression of this idea:
 v = H_{0} x D, where
 v is the galaxy’s recessional velocity in kilometers per second
 Based on the galaxy’s redshift
 D is the distance of the galaxy in megaparsecs
 1 megaparsec (Mpc) = 1 million parsecs = 3,262,000 lightyears
 In 1912 Henrietta Swan Leavitt discovered that a certain class of stars, the Cepheid Variables, fluctuate periodically in their brightness and that the periodicity is correlated with their actual luminosity. Hubble used the latter to determine the distance to many Cepheid Variables and thus to the galaxies where they lived.
 H_{0} is the Hubble Constant, about 72 km/s per 1 megaparsec (Mpc)
 72 km/s = 161,059 mph
 v is the galaxy’s recessional velocity in kilometers per second
 v = H_{0} x D, where
 Thus a galaxy 4 Mpc distant moves away at 288 km/s (72 x 4 = 288). A galaxy 2,500 Mpc from us recedes at 180,000 km/s (72 x 2500 = 180,000).
 A graph of recessional velocity (km/s) against distance (Mpc) shows galaxies straddling the Hubble Constant diagonal at 72 km/s per Mpc.
Image Credit: cfa.harvard.edu/~dfabricant/huchra/hubble/
 The significance of the Hubble Law can be explained by analogy.
 Suppose you’re on a hill and see people walking directly away from you in different directions. You detect their velocities and distances with a radar gun:
 Hiker walking north at 2 mph is 8 miles away
 Hiker walking south at 3 mph is 12 miles away
 Hiker walking east at 4 mph is 16 miles away
 Hiker walking west at 5 mph is 20 miles away
 The velocities and distances are not random but fit a pattern: the further away the walker, the faster they walk — by the same factor.
 Velocity = ¼ x Distance
 Graphing speed against miles, the hikers form a line:
 The pattern suggests that the hikers began walking at the same time and place and have been walking for four hours, each at a constant velocity
 Velocity = ¼ x Distance, implying that
 Distance / Velocity = 4
 Distance / Velocity = Elapsed Time
 Therefore, Elapsed Time = 4
 North Hiker: 8 miles / 2 mph = 4 hours
 South Hiker: 12 miles / 3 mph = 4 hours
 East Hiker: 16 miles / 4 mph = 4 hours
 West Hiker: 20 miles / 5 mph = 4 hours
 Suppose you’re on a hill and see people walking directly away from you in different directions. You detect their velocities and distances with a radar gun:
 Likewise the Hubble Law suggests that galaxies began their outward journeys at the same time and place and have been receding for 13.5894 billion years, each at its current velocity.
 Velocity = 72 x Distance, implying that
 Distance / Velocity = 1/72
 Distance / Velocity = Elapsed Time
 Therefore, Elapsed Time = 1/72
 = 1 Mpc / 72 km/s
 = 3.086 x 10^{19} km / 72 km/s
 = 4.28611 x10^{17} seconds
 = 13.5894 billion years.
 Thus a galaxy 4 Mpc away, receding at 288 km/s, and a galaxy 2,500 Mpc away, receding at 180,000 km/s, both began moving at their current velocities from the same place 13.5894 billion years ago.
 13.5894 billion years is a simple extrapolation from Hubble’s Law.
Expansion of Space
 The Hubble Law has a problem: some galaxies are so far away that it yields velocities beyond the speed of light. For example, the galaxy GNz11, at 9,810 Mpc, should be speeding away at 706,320 km/s, twice the speed of light, violating Einstein’s Special Relativity.
 72 x 9,810 Mpc = 706,320 km/s
 The problem arises because Hubble used the Doppler Effect to calculate velocities from redshifts. But the Doppler Effect is not the only cause of redshifts.
 The Big Bang Theory postulates that galactic redshifts result, not from the Doppler Effect, but from the Expansion of Space
 General Relativity predicts that space expands and contracts. Gravitational waves, detected in 2015, for example, are waves of expanding and contracting space.
 More on Gravitational Waves
 General Relativity predicts that space expands and contracts. Gravitational waves, detected in 2015, for example, are waves of expanding and contracting space.
 According to the Big Bang Theory, space has expanded since the Big Bang singularity, with galaxies spreading apart yet not moving relative to each other Like raisins in rising bread dough,
Image Credit imagine.gsfc.nasa.gov/features/yba/M31_velocity/spectrum/doppler_more.html
 Distances between raisins double without the raisins moving through the dough.
 The expansion of space explains the galactic redshift. As light travels billions of years through expanding space its wavelength increases.
Image Credit imagine.gsfc.nasa.gov/features/yba/M31_velocity/spectrum/doppler_more.html
 The Big Bang Singularity is a point of infinite density and infinite curvature of spacetime where the laws of physics break down.
 More at Big Bang Singularity and Before.
 FAQs about the expansion of space
 If space is expanding, why is the Andromeda Galaxy heading in our direction?
 The Milky Way and Andromeda move toward each other because their gravitational attraction overcomes the expanding space between. For distant galaxies, the gravitational attraction is too small to make a difference.
 If space expands, yardsticks get longer at the same rate distant galaxies separate. Isn’t it theoretically impossible, therefore, to measure the expansion of space?
 Yardsticks don’t get longer: the forces holding the atoms and molecules of the yardstick together overcome the expanding space between them.
 If space is expanding, why is the Andromeda Galaxy heading in our direction?
 As space expands:
 The distance between atomic nuclei and electrons remains the same, since electric forces overcome the expanding space between them.
 The distance between the ends of a yardstick remains the same, since intermolecular forces overcome the expanding space within the yardstick.
 The distance between the Sun and planets remains the same, since gravitational forces overcome the expanding space among them.
 The distance between the Milky Way and the Andromeda Galaxy decreases, since their gravitational attraction overcomes the expanding space between them.
 But the distance from the Milky Way to distant galaxies increases because their gravitational attraction is too weak to overcome the expanding space between them.
Entire line expands but Sun and Earth remain the same distance apart
 Hubble, problematically, stated his law in terms of distance and velocity. The Big Bang Theory, as we’ll see, reformulates Hubble’s Law in terms of the expansion of space.
Friedmann Equation
 In 1922, seven years before the Hubble Law, Russian mathematician Alexander Friedmann introduced a remarkable equation that remains today the theoretical basis for the expansion of the universe.
Understanding the Friedmann Equation without knowing the math.
 The key to understanding the Friedmann Equation is a(t), the scale factor, an index representing the size of space at a time t.
 The equation governs how a(t) changes over time, indicating whether space is expanding or contracting, and how fast.
 For example, if a(t) equaled 1,000 a billion years ago and is 1,200 today, space has expanded 20% since then. Galaxies 50 Mpc apart then are 60 Mpc apart now.
 The equation says, in effect, that the expansion or contraction of space (the left side of the equation) depends on ρ (rho), the density of the universe, the amount and kind of stuff in an average unit volume of space.
 Stuff includes:
 Matter, consisting of
 Ordinary matter
 Dark matter
 Radiation, which includes
 Photons
 Neutrinos
 Dark Energy
 Matter, consisting of
 The equation’s other parameters:
 k is the curvature of spacetime
 k = +1, 0, 1 for closed, flat and open universes
 G is the universal gravitational constant
 c is the speed of light
 k is the curvature of spacetime
 Friedmann derived his equation from Einstein’s General Relativity and the Cosmological Principle:
 The universe looks the same on a large scale no matter where in the universe you are (homogeneity) and no matter in what direction you look (isotropy).
 Thus, just as Earth is not the center of the Solar System, and the Solar System is not the center of the Milky Way, so the Milky Way is not the center of the universe.
 Britannica: Aleksandr Aleksandrovich Friedmann
 “In 1922–24 Friedmann used Einstein’s general theory of relativity to formulate the mathematics of a dynamic (timedependent) universe. In the Friedmann models, the average mass density is constant over all space but may change with time as the universe expands. Friedmann also calculated the time back to the moment when an expanding universe would have been a mere point, obtaining tens of billions of years.”
 “Friedmann died of typhoid in 1925 at age 37.”
Particle History
and
CMB Radiation and Light Elements
Particle History of the Early Universe
 From Steven Weinberg’s The First Three Minutes
 “The redshifts of the distant galaxies tell us that the universe is expanding, so its contents must once have been much more compressed than now. The temperature of a fluid will generally rise when the fluid is compressed, so we can also infer that the matter of the universe was much hotter in the past.”
 The early universe was extraordinarily dense and hot. How did atoms and subatomic particles behave under these conditions?
 In the late 1940s particle physicists George Gamow, Ralph Alpher, and Robert Herman tried to reconstruct the early universe using the laws of particle physics. Their idea was that, as the universe cooled and expanded, protons and neutrons would form atomic nuclei, and nuclei and electrons would form hydrogen and helium atoms.
 Examples of laws of particle physics
 A free neutron, with an average lifetime of 14 minutes 40 seconds, decays into a proton, electron, and antineutrino
 A gamma ray of sufficient energy decays into an electron and a positron
 A positron is the antimatter counterpart of the electron.
 The collision of an electron and a positron results in a shortlived atomlike particle called a positronium, which decays in 10^{7} seconds into two gamma rays.
 View Elementary Particles
Overview of the Particle History, from the Britannica
 Primeval Fireball
 Soup of Elementary Particles
 1/10,000 second after Big Bang
 The universe is a condensed soup of elementary particles: quarks, leptons, antiparticles, and photons
 Baryosynthesis (formation of protons and neutrons)
 1/100 second after Big Bang
 The temperature lowers enough to enable quarks to combine to form protons and neutrons.
 A proton consists of two up quarks and a down quark.
 A neutron consists of two down quarks and an up quark.
 View Elementary Particles
 Free protons, neutrons, electrons, and photons zoom around, bouncing off one another.
 Nucleosynthesis (formation of atomic nuclei)
 3.5 minutes after Big Bang
 The temperature lowers enough to enable protons and neutrons to form nuclei of hydrogen and helium.
 Antiparticles have been annihilated.
 Recombination (formation of light atoms)
 380,000 years after the Big Bang
 The temperature lowers to 3,000 K, low enough to enable nuclei and electrons to form hydrogen and helium atoms.
 Photons now travel uninterrupted, since they no longer collide with free electrons.
Image Credit britannica.com/science/bigbangmodel/imagesvideos
 Three eras in the densitytemperature history of the universe:
 Radiation Era
 From the Big Bang to 50,000 years after the Big Bang
 Radiation density is greater than the density of either matter or dark energy.
 Matter Era
 From 50,000 years to 10 billion years after the Big Bang
 Matter density is greater than the density of either radiation or dark energy.
 Dark Energy Era
 From 10 billion years after the Big Bang.
 Dark energy is greater than the density of either matter or radiation.
 Radiation Era
 Eras are divided into epochs.
 The closer you get to the Big Bang, the more speculative the physics becomes.
Prediction of CMB Radiation
 In 1948 Ralph Alpher and Robert Herman calculated that, due to the expansion of space, electromagnetic radiation from the early universe would today be in the microwave range, detectable by radio telescopes.
 380,000 years after the primordial fireball, during Recombination, the temperature of the universe lowered to 3,000 Kelvin (4,940 Fahrenheit), low enough to enable nuclei and electrons to form hydrogen and helium atoms. Photons, which had been colliding with free electrons, were able to travel uninterrupted. These photons today permeate the universe. Due to the expansion of space, their wavelength increased from 0.0001 centimeters (redinfrared) to 0.19 centimeters (microwaves).
 More at Distribution of Cosmic Radiation
 Their prediction was ignored and forgotten until 1964.
Electromagnetic Spectrum
Image Credit imagine.gsfc.nasa.gov/science/toolbox/emspectrum1.html
Detection of CMB Radiation
 In 1964 Arno Penzias and Robert Wilson of Bell Labs built a horn antenna to detect radio waves from space.
 But they kept getting lowlevel noise in the microwave range, no matter where they pointed the antenna.
 From astronomers at MIT and Princeton they learned that the lowlevel noise was the Cosmic Microwave Background Radiation predicted by Alpher and Herman in 1948.
 They received the Nobel Prize in 1978
Light Elements
 The Big Bang Theory makes three predictions:
 Recession of Galaxies
 Cosmic Microwave Background Radiation
 Abundance of Light Elements
 It used to be thought that all naturallyoccurring elements formed in stars. But there was a problem with helium. Helium is indeed formed in stars by nuclear fusion from hydrogen, but not enough to account for its abundance in the universe, about 25% by mass.
 The Big Bang Theory explains the abundance of helium:
 3.5 minutes after the Big Bang, during Nucleosynthesis, protons and neutrons combined to form deuterium nuclei (1 proton + 1 neutron) and helium nuclei (2 protons + 2 neutrons), resulting in approximately 25% helium nuclei and 75% hydrogen nuclei (protons). The proportion is the same today.
Dark Matter
 In 1933 Fritz Zwicky found that the Coma cluster of galaxies did not contain enough mass in its stars to keep the cluster together gravitationally.
 In the 1970s American astronomers Vera Rubin and W. Kent Ford confirmed that the mass of the stars visible within a typical galaxy is only about 10 percent of that required to keep the stars orbiting the galaxy’s center. Most of a galaxy’s matter must be dark.
 Conjectures regarding the nature of Dark Matter:
 Hidden Baryonic (Atomic) Matter
 MACHOs (Massive Compact Halo Objects)
 Black holes, neutron stars, white dwarfs, Jupitersized planets, brown dwarfs
 MACHOs (Massive Compact Halo Objects)
 NonBaryonic Matter
 Hot Dark Matter
 Neutrinos
 Cold Dark Matter
 WIMPs (Weakly interacting massive particles)
 Hot Dark Matter
 Hidden Baryonic (Atomic) Matter
Dark Energy
 In 1998 two international teams of astronomers discovered dark energy. In investigating very distant Type Ia supernovae, they discovered that billions of years ago the universe was expanding more slowly than it is today. So the universe today is expanding faster than in the past, which doesn’t make sense because the matter in the universe should be slowing the expansion.
 A supernova is a cataclysmic explosion of a star. The star brightens suddenly to many millions times it normal luminosity. Then it gradually fades over the subsequent weeks and months.
 Analogy
 Like throwing a ball straight up in the air except that, after slowing down, the ball speeds up.
 Dark Energy: No Answers but More Questions, by Adam G. Riess and Mario Livio, Scientific American, 2016 (Reiss received the Nobel Prize in 1998 for leading one of two teams that discovered dark energy.)
 Scientists have a number of hypotheses for what might be driving the acceleration of the universe.
 First Hypothesis: Empty Space, teeming with virtual particles, is inherently antigravitational.
 The leading candidate arises from the nature of empty space. In quantum physics a vacuum is not “nothing”—rather it is teeming with pairs of “virtual” particles and antiparticles that spontaneously appear and annihilate one another within a tiny fraction of a second. As strange as it may sound, this sea of ephemeral particle pairs carries energy, and energy, just like mass, can produce gravity. Unlike mass, however, energy can create either an attractive or a repulsive gravity, depending on whether its pressure is positive or negative. The vacuum energy in empty space, according to theory, should have a negative pressure and thus may be the source of the repulsive gravity driving the accelerated expansion of the universe.
 This idea is equivalent to the “cosmological constant,” a term Albert Einstein added to his general relativity equations that represents a constant energy density throughout space. As the name implies, this hypothesis holds that the density of dark energy is constant—that is, unvarying—over space and time. So far the astrophysical evidence we have best fits with the cosmological constant explanation, with some discrepancies
 Second Hypothesis: Dark energy is a changing antigravitational energy field permeating the universe called quintessence.
 Alternatively, dark energy may be an energy field dubbed “quintessence” that pervades the universe, imbuing every point in space with a property that counteracts the pull of gravity. Physicists are familiar with fields—the everyday forces of electromagnetism and gravity act via fields (although these usually arise from localized sources and do not pervade all of space).
 If dark energy is a field, it would not be a constant and so might change over time. In that case, dark energy might once have been stronger or weaker than it is now and could have affected the universe differently at different times. Likewise, its strength and impact on the evolution of the universe might alter in the future. In the socalled freezingfield version of this idea, dark energy evolves more and more slowly as time progresses; in the thawing variant, the field changes slowly at first and faster later.
 Third Hypothesis: General Relativity is incomplete.
 A third option may account for cosmic acceleration: there is no dark energy, and the quickening expansion of the universe results from physics not explained by Einstein’s theory of gravity (general relativity), which is incomplete. It is possible that in truly extreme regimes, such as the breadth of galaxy clusters or the entire observable universe, the laws of gravity work differently than the theory predicts, and gravity misbehaves. Physicists have put forth a few interesting theoretical suggestions along these lines, but no selfconsistent theory that agrees with all the observations currently exists, so dark energy seems to have the upper hand over this option for now. (Previous ideas, such as the notion that cosmic acceleration is a manifestation of an uneven distribution of matter throughout the universe or the result of a network of geometric defects in the structure of space, have by now largely proved to be inconsistent with observational data.)
Age and Fate of the Universe
 Will space expand and cool forever, ending in the Big Chill? Or is there enough stuff in the universe so that gravity slows and reverses the expansion, resulting in the Big Crunch?
 How old is the universe?
 The Rescaled Omega version of the Friedmann Equation provides a framework for answering these questions.
Friedmann Equation, Rescaled Omega Version
 The Rescaled Omega version of the Friedmann Equation says in effect that:
 The rate of expansion/contraction of space is a function of:
 The Hubble Constant
 The average density and kind of stuff in the universe, in particular:
 the average density of matter
 the average density of radiation
 the average density of dark energy.
 The rate of expansion/contraction of space is a function of:
 The equation is solved for given values for the input variables: H_{0}, Ω_{m}, Ω_{r}, Ω_{ν}
 is then converted to , using
 is the scale factor, representing the relative size of space at a time t.
 The scale factor indicates whether the universe is expanding, contracting, or staying the same.
 Thus, if a(t) equaled 1,000 a billion years ago and is 1,200 today, space expanded 20%. Galaxies 50 Mpc apart are now 60 Mpc distant.
 is the rate of expansion or contraction over time.
 Input Variables:
 H_{0} is the Hubble Constant, the recessional velocity of distant galaxies divided by their distance from us. It appears inside , which is defined as .
 The Density Parameters are a measure of the relative amounts of matterenergy in an average unit volume of space.
 Ω_{m} is for ordinary and dark matter
 Ω_{r} is for radiation, mostly photons
 Ω_{ν} is for dark energy
 Ω_{c}, which equals 1 – Ω_{m} – Ω_{r} – Ω_{ν}, represents the curvature of spacetime.
An Analogy
 A railgun is a cannon that fires a projectile using electromagnetic force rather than gunpowder. Its muzzle velocity is over twice that of a highpowered rifle.
Image Credit wikipedia.org/wiki/Railgun
 Say we transport a railgun to the Moon, where it’s fired straight up into space.
 Two factors determine whether the projectile falls back to the Moon or keeps going:
 The projectile’s muzzle velocity.
 The Moon’s gravitational force.
 Analogs with the Big Bang Theory:
 The Big Bang is like the firing of the railgun.
 The expansion and contraction of space are like the projectile’s motion away from and toward the Moon.
 The density of the universe is like the Moon’s mass, exerting a gravitational pull on the projectile.
 The Hubble Constant is like the velocity of the projectile divided by its distance from the Moon.
 Ω is like the ratio of the projectile’s muzzle velocity to the Moon’s escape velocity.
Density
 Density
 Density is the quantity of something per unit volume
 Density of the Universe
 The density ρ (rho) of the universe is the average mass and energy per unit volume, usually given in units of mass per volume:
 Grams per cubic centimeter, g/cm^{3}
 Kilograms per cubic meter, kg/m^{3}
 ⍴ is the sum of;
 ⍴_{m}, the average matter density, consisting of:
 Ordinary matter
 Dark matter
 ⍴_{r}, the average radiation energy density, consisting of:
 Photons
 Neutrinos
 ⍴_{ν}, the average dark energy density
 ⍴_{m}, the average matter density, consisting of:
 The density ρ (rho) of the universe is the average mass and energy per unit volume, usually given in units of mass per volume:
 Critical Density
 The critical density of the universe, ρ_{crit}, is defined as that density such that
 Any density less than or equal to it results in an open universe that expands forever
 Any density greater than it results in a closed universe that eventually stops expanding and contracts.
 The (reverse) analog of critical density in the railgun scenario is the Moon’s escape velocity:
 Any lesser velocity results in the projectile returning to the Moon.
 Any greater velocity results in the projectile traveling into space.
 The formula for critical density is derived from the Friedmann Equation:
 ρ_{crit} = 3H_{0}^{2}/8πG
 If H_{0} is 69 km/s per Mpc, the value of ρ_{crit} is:
 8.94 x 10^{27 }kg/m^{3}
 This amounts to about 10 atoms per cubic meter.
 The critical density of the universe, ρ_{crit}, is defined as that density such that
 Density Parameters
 Ω is the ratio of the present density of the universe to the critical density
 Ω = ρ(t_{0})/ρ_{crit}
 If Ω ≤ 1, the universe is open and expands forever
 If Ω > 1, the universe is closed and eventually contracts
 Ω is the ratio of the present density of the universe to the critical density
Different Amounts of Stuff
 In the following graphs
 the horizontal axis is the age of the universe in billions of years (Gyr)
 the vertical axis is the scale factor, signifying the relative size of space.
 In the first graph, for example, the distance between remote galaxies at 30 Gyr, where a(t) = 8, is twice that between galaxies at 15 Gyr, where a(t) = 4.
 the dotted, vertical line indicates the present time.
Almost No Stuff
The expansion rate stays the same since there’s virtually no matter, radiation, or dark energy to affect it, the density being zero.
Enough Matter and Radiation to Reverse the Expansion
The density of matter and radiation is large enough to reverse the expansion, resulting in the Big Crunch around 90 billion years after the Big Bang.
Minimal Matter and Radiation to Keep the Universe Expanding
The density of matter and radiation slows the expansion just enough to keep the universe expanding forever. Any greater density would result in the Big Crunch.
Different Kinds of Stuff
 The Density Parameters for matter, radiation, and dark energy:
 Ω_{m} is the ratio of the matter density to the critical density
 Ω_{m} = ρ_{m}(t_{0})/ρ_{crit}
 Ω_{r} is the ratio of the radiation density to the critical density
 Ω_{r} = ρ_{r}(t_{0})/ρ_{crit}
 Ω_{ν} is the ratio of the dark energy density to the critical density
 Ω_{ν} = ρ_{ν}(t_{0})/ρ_{crit}
 Ω_{m} is the ratio of the matter density to the critical density
 The following graphs show the expansion of space for universes where matter, radiation, and dark energy dominate.
 The graphs show that:
 Radiation slows the expansion slightly more than matter.
 Dark energy, being repulsive, speeds up the expansion.
Matter Dominated Universe
Radiation Dominated Universe
Dark Energy Dominated Universe
Age and Fate of the Universe, Consensus View
 The Omega version of the Friedmann Equation predicts the age and fate of the universe, based on the Hubble Constant and density parameters.
 Measurements of the Hubble Constant range from 67 to 77 km/s per Mpc.
 Measurements of the density of matter yield values of Ω_{m} around 0.3, mostly dark matter.
 Measurements of dark energy yield values of Ω_{ν} around 0.7.
 Measurements of radiation, mostly the CMB, show Ω_{r} near zero.
 Age and fate of the universe based on a reasonable estimate of the parameters:
 Thus
 The universe is about 13.6 billion years old
 Space expands forever.
 The rate of the expansion is increasing, thanks to dark energy.
MatterEnergy History of the Universe
 The Friedmann Equation can be solved for past and future values of Ωr, Ωm, and Ωv, establishing periods when each was dominant:
 Radiationdominated Era
 From the early universe to 50,000 years after the Big Bang, Ωr was greater than either Ωm or Ωv
 Matterdominated Era
 From 50,000 to 10 billion years after the Big Bang, Ωm was greater than either Ωr or Ωv
 Dark Energydominated Era
 Ωv has been greater than either Ωm or Ωr, since 10 billion years after the Big Bang.
 Radiationdominated Era
 Matter and Radiation densities decrease with time, at different rates. The former overtakes the latter at the Matterradiation crossover point.
 Dark energy density remains the same, overtaking matter and radiation density 10 billion years after the Big Bang
 Right Pie Chart
 Dark energy dominates today’s universe.
 Dark and atomic matter make up 30%.
 Left Pie Chart
 Dark and atomic matter dominated the 380,000 yearold universe,
 Radiation, consisting of photons and neutrinos, made up 25%.
Image Credit: britannica.com/science/darkmatter
Resources
Websites
 Australia Telescope National Facility
 Britannica
 Cosmos (Swinburne University)
 Hawking Center of Theoretical Cosmology
 NASA Lambda
 NASA Universe 101
 Preposterous Universe (Sean Carroll)
 Wikipedia
Books
 First Three Minutes, Steven Weinberg
 Steven Weinberg received the Nobel Prize in Physics in 1979 for developing the electroweak theory, unifying electromagnetism with the weak nuclear force.
 Gravity: An Introduction to Einstein’s General Relativity, James Hartle
 A graduatelevel textbook on General Relativity and Cosmology.
 James B. Hartle is Research Professor and Professor of Physics Emeritus at the University of California, Santa Barbara
 wikipedia.org/wiki/HartleHawking_state
 In theoretical physics, the Hartle–Hawking state (named after James Hartle and Stephen Hawking) is a proposal concerning the state of the Universe prior to the Planck epoch
Addenda
Contents
 Big Bang Singularity and Before
 Calculation of Inverse Scale Factor from Omega Version of Friedmann Equation
 Distribution of Cosmic Radiation
 Existence of Galaxies
 Friedmann Equations
 Friedmann Equation, Rescaled Omega Version
 Redshift and its Causes
 Scale Factor and Hubble Concepts
 Spectrograph
Big Bang Singularity and Before
 Stephen Hawking, A Brief History of Time, page 133
 If the classical theory of general relativity was correct, the singularity theorems that Roger Penrose and I proved show that the beginning of time would have been a point of infinite density and infinite curvature of spacetime. All the known laws of science would break down at such a point.
 James Hartle, What Came Before the Big Bang, his textbook page 381
 The big bang is a singular moment of infinite density and curvature.
 Spacetime breaks down at a singularity. General Relativity has no way of determining what happened before the big bang from events after it.
 Time began at the big bang.
 Ned Wright
 The standard Big Bang model is singular at the time of the Big Bang, t = 0. This means that one cannot even define time, since spacetime is singular.
 Stephen Hawking from The Origin of the Universe
 The problem of what happens at the beginning of time is a bit like the question of what happened at the edge of the world, when people thought the world was flat. … However, when one combines General Relativity with Quantum Theory, Jim Hartle and I realized that time can behave like another direction in space under extreme conditions. This means one can get rid of the problem of time having a beginning, in a similar way in which we got rid of the edge of the world. Suppose the beginning of the universe was like the South Pole of the earth, with degrees of latitude playing the role of time. The universe would start as a point at the South Pole. As one moves north, the circles of constant latitude, representing the size of the universe, would expand. To ask what happened before the beginning of the universe would become a meaningless question, because there is nothing south of the South Pole.
 wikipedia.org/wiki/Gravitational_singularity
 A gravitational singularity, spacetime singularity or simply singularity is a location in spacetime where the density and gravitational field of a celestial body is predicted to become infinite by general relativity in a way that does not depend on the coordinate system. The quantities used to measure gravitational field strength are the scalar invariant curvatures of spacetime, which includes a measure of the density of matter. Since such quantities become infinite at the singularity point, the laws of normal spacetime break down
 wikipedia.org/wiki/Singularity_(mathematics)
 In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be wellbehaved in some particular way, such as by lacking differentiability or analyticity.
 In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be wellbehaved in some particular way, such as by lacking differentiability or analyticity.
Friedmann Equations
 First and Second Friedmann Equations
 The equations are derived from
 the Field Equation of Einstein’s General Relativity
 the RobertsonWalker Metric:
 The Second Friedmann Equation can also be derived from:
 the First Friedmann Equation
 First law of Thermodynamics for homogenous, isotropic universes , where
 where
 ρ is the density of the universe
 p is the pressure exerted by matter
Friedmann Equation, Rescaled Omega Version
Calculation of Inverse Scale Factor from Omega Version of Friedmann Equation
Distribution of Cosmic Radiation
 A system in thermal equilibrium at a given temperature emits a pattern of electromagnetic radiation equivalent to that emitted by a black body at that temperature.
 380,000 years after the primordial fireball, the universe was in thermal equilibrium at 3,000 degrees Kelvin.
 Therefore, 380,000 years after the primordial fireball, the universe emitted a pattern of electromagnetic radiation equivalent to that emitted by a black body at that 3,000 degrees of Kelvin.
 The typical wavelength of a black body at 3,000 K is 1000 nanometers, in the redinfrared range.
 1000 nanometers = 0.0001 centimeters
Distribution of Cosmic Radiation at 3,000 degrees Kelvin
Typical wavelength is 1000 nanometers = 0.0001 centimeters
 Due to 13.8 billion years of expanding space, the typical wavelength of cosmic radiation increased from 0.0001 centimeters (redinfrared) to its current value of 0.19 centimeters (microwaves).
Distribution of Cosmic Radiation at 2.7245 degrees Kelvin
Typical wavelength is 1.9 millimeter = 0.19 centimeters
Image Credit wmap.gsfc.nasa.gov/universe/bb_tests_cmb.html
Redshift and its Causes
 Redshift is the fractional increase in wavelength from emission to reception (observation).
 Equivalent expressions for 𝒛:
 A negative redshift is a blueshift. Thus, light from the Andromeda Galaxy, which is heading for us, has a redshift of z = −0.001001.
 Causes of Redshift
 Doppler Redshift
 Light is redshifted as it travels from a source moving away from an observer.
 If velocity v is small compared to the speed of light c:
 Otherwise:
 Cosmological Redshift
 The cosmological redshift is due to the expansion of space as a wave travels from source to observer.
 Gravitational Redshift
 Light traveling from a stronger gravitational field to a weaker one is redshifted because time runs faster as the wave travels. Therefore, the wave’s period decreases, its frequency increases, and its wavelength decreases.
 More on Gravitational Redshift.
 Doppler Redshift
Scale Factor and Hubble Concepts
 The scale factor, a(t), is an index representing the size of space at a time t
 Scale Factor Principle

 where d_{0} = d(t_{0})
 and t_{0} is the present time
 That is, the distance between two points at a time t equals the distance today multiplied by the scale factor at time t

 a(t_{0}) = 1
 Proof:
 Applying the Scale Factor Principle to present time t_{0} yields:
 d(t_{0}) = a(t_{0}) d(t_{0})
 From which it follows that:
 a(t_{0}) = d(t_{0}) / d(t_{0}) = 1
 Applying the Scale Factor Principle to present time t_{0} yields:
 Proof:
 Example:
 If d_{0} = 100 meters and a(1 billion years ago) = 0.5, d(1 billion years ago) = 50 meters.
 Hubble Constant
 Hubble formulated the Hubble Constant in terms of velocity and distance: 72 km/s per megaparsec.
 The Big Bang Theory redefines the Hubble Constant in terms of the scale factor:
 That is, the Hubble Constant equals the current rate of change of the scale factor divided by today’s scale factor.
 The redefinition is based on the cosmological redshift
 Hubble Parameter
 The Hubble Parameter generalizes the Hubble Constant, applying to every time t rather than just the present moment t_{0}.
 The Hubble Parameter varies over time.
 The Hubble Parameter generalizes the Hubble Constant, applying to every time t rather than just the present moment t_{0}.
 Hubble Law
 Hubble stated his law in terms of velocity and distance:
 v = H_{0} x D
 The problem with his formulation is that some galaxies are so far away that their velocity would be faster than light.
 The Big Bang Theory reformulates the law in terms of distance and the scale factor.
 That is, the rate of change of distance equals the Hubble Parameter times distance.
 The reformulated Hubble Law follows from the Scale Factor Principle and the definition of the Hubble Parameter, derived below.
 Hubble stated his law in terms of velocity and distance:
 Hubble Time
 The Hubble Time is the inverse of Hubble’s Constant.
 The dimension of t_{H} is time, since the dimension of H_{0} is velocity / distance, meaning that the dimension of t_{H} is distance / velocity = time.
 t_{H} = (1 Mpc / 72 km/s) = (3.086 x 10^{19} km / 72 km/s) = (4.28611 x10^{17} seconds) = (13.5894 billion years).
 The Hubble Time in the Big Bang Theory is today’s scale factor divided by its rate of change.
 The Hubble Time is not the exact age of the universe, since the scale factor changes over time.
 The Hubble Time is the inverse of Hubble’s Constant.
 Derivation of the Hubble Law from Scale Factor Principle and Hubble Parameter

 Scale Factor Principle

 Differentiating both sides of #1 with respect t

 From #1

 From #2 and #3

 Hubble Parameter

 From #4 and #5

Existence of Galaxies
 Before the 1920s astronomers disagreed about the nature of spiral nebulae: were they nearby stars or faraway galaxies?
 In 1923 Edwin Hubble settled the matter, establishing that the spiral nebula Andromeda was much further away than the stars.
 The Andromeda Galaxy is 2,480,000 lightyears away; the star Alpheratz, for example, only 97 lightyears.
Andromeda Galaxy
Spectrograph
A spectrograph splits light into its component wavelengths.