Chapter 21.

Einstein-Friedmann Cosmology

"The evolution of the world can be compared to a display of fireworks that has just ended; some few red wisps, ashes, and smoke. Standing on a cooled cinder, we see the slow fading of the suns, and we try to recall the vanished brilliance of the origin of the worlds." Georges Lemaitre

21.1. What is Cosmology?

• Cosmology is an attempt to reveal underlying unity among seemingly diverse phenomena in the Universe, from subatomic particles to superclusters of galaxies and all in between
• Goal is to develop:
  • From a theory of gravity, such as general relativity
  • Along with some simplifying principles, few as possible
  • A unifying theory of the Universe which accounts over all ages for
    • Existence of observed distribution of galaxies
    • Existence of fundamental elementary particles
    • Existence of physical laws
• What kinds of questions are asked about the Universe?
  • Observation: If galactic recessional velocities mean the Universe is expanding, only conclusion is that in distant past space was considerably smaller than now.
    • Question: But, how much smaller was it?
    • Question: In beginning, could volume have been that of a mathematical point?
    • Question: If larger than a point, how much larger?
  • Observation: If all energy/matter in observable Universe was distributed over smaller volume in the beginning, energy density or temperature should have been immense in the beginning.
    • Question: If immense, what was value time?
    • Question: If temperature was immense, what was matter like, surely not the familiar atoms and molecules?
  • Observation: When expansion began 10-20 x 109 y ago, volume of the Universe increased and consequently temperature must have declined as energy density went down.
    • Question: Can we assume that expansion was uniform in all directions, or was it greater in some directions than others?
    • Question: Did expansion hurry along at some times, slow done at others, maybe even pause, rest awhile, and continue?
    • Question: At that time, was the Universe lumpy or perfectly smooth everywhere as the expansion began?
    • Question: As the temperature declined, what kinds of changes would matter have undergone?
  • Observation: Following the big bang, Universe should have been filled with an overwhelming abundance of very high-energy photons, called the primeval fireball; subsequent expansion cooled this radiation so that today most of its energy should lie in microwave region and in a background sea of neutrinos
    • Question: Does there exist any evidence for this low-temperature radiation, sometimes referred to as "the echo" of the big bang?
    • Question: Does this low-temperature radiation continue to permeate all space, since when the big bang occurred, the volume of the Universe was less but, nevertheless, the Universe began by filling all space, and it continues to fill all space now, except that space is growing ever larger?
  • Observation: Existence of matter means that all matter is exerting a gravitational attraction on every piece of matter.
    • Question: Surely, gravitational attraction must retard expansion, and consequently retard expansion of Universe, since space is what separates pieces of matter?
    • Question: Will expansion continue forever, or will it stop?
    • Question: If expansion stops, what follows? Contraction?
    • Question: If contraction, would the Universe return to whatever small volume it had in the beginning?
    • Question: Would contraction phase be like running movie backward--reversal of everything that happened going forward, or would scenario during contraction be different from that during expansion?
    • Question: When contraction stops, is it followed by another big bang with whole process starting over again?
    • Question: If Universe starts over, does it "remember" anything, physical laws, etc. from previous cycle or does Universe "reinvent" everything, including science?
    • Question: If Universe starts over, how many cycles has it gone through? Finite number or and infinite number?
  • Astronomy and physics attempt to answer the question of how the big bang occurred and what follows afterwards. But to speculate why it occurred is to seek motives which are beyond the realm of science as currently accepted.

21.2. The Cosmological Principle

• Homogeneous universe: three arguments can be advanced to claim that the Universe was homogeneous at least in its earlier days if not now
  • First, radio galaxies, fair fraction of all galaxies, are spread remarkably uniform; they possess from modest to large redshifts
  • Second, diffuse X-ray background radiation is also uniform
    • Irrelevant whether it comes from all-pervading hot gas between galaxies or from discrete objects, such as quasars
    • If radiation comes from high-redshift quasars, matter was apparently distributed in homogeneous fashion at earlier time
  • Third, cosmic background radiation (CBR) is highly isotropic
• 1933, E. A. Milne (1896-1950)
  • Assuming large-scale distribution of matter was homogeneous had considerable philosophical merit and was consistent with Hubble's law of recession and postulates of special relativity
  • Not first to propose that the Universe is homogeneous
  • Assumption called "cosmological principle"
  • Has become a basic postulate of cosmology, i.e., an article of faith
  • Equivalent to Copernican principle from Newtonian cosmology only more precise
  • Cosmological Principle: Since we observe the Universe to be isotropic here, all observers anywhere in space should see the Universe in its essential features in the same way in all directions; that is, the Universe is isotropic, and hence must be homogeneous so that all places are alike at any instant of cosmic time
• Cosmological principle states that our sample of the Universe, except for local or small-scale variations, is no different at present from another sample selected at random at a different place in the Universe
• A homogeneous and isotropic Universe means that any curvature to large-scale space must be uniform; that is, it may not vary from place to place throughout the Universe at one time
• Also Hubble's velocity-distance law is same at any time anywhere in the Universe

21.3. History of Einstein-Friedmann Cosmology

• Static model universes
  • 1917, before discovery of cosmological redshifts, Einstein proposed model universe in which
    • Random galactic motions cancel out, leaving it static
    • Mean density of matter remained constant over time
    • Radius of Universe remained constant over time
    • Closed universe possessing spherical geometry globally
    • Finite in extent, centerless and edgeless
  • Einstein introduced small repulsive force between material particles
    • Force acts over distances separating galaxies
    • Force keeps model from collapsing by its own self-gravitation
    • Known as cosmological constant ()
    • Radius inversely proportional to square root of mean density () of matter, i.e., R
    • Mean density estimated to be 10-29 < < 10-31 g/cm3;
    • Thus radius is 10 x 109 < R < 100 x 109 ly
  • Model predicted no cosmological redshifts for galaxies
  • 1930, Eddington showed
    • Einstein's model actually unstable
    • Model should either expand or contract if perturbed
  • 1917, Willem de Sitter (1872-1934), Dutch astronomer, proposed another apparently static model
    • Model contained no matter in Euclidean space
    • Predicted redshift proportional to distance
    • Not truly static; forerunner of nonstatic or expanding models
    • Predicted expansion would last forever
• Expanding cosmological models
  • 1930, Eddington in England and Georges Lemaitre (1894-1966), Belgian priest, proposed nonstatic models
    • Eddington's model
      • Perturbation of Einstein's static model
      • Begins an expansion that lasts forever
    • Lemaitre's ("father of the big bang") model
      • Begins with big bang
      • Expands for a while
      • Hesitates in a state resembling Einstein's static universe
      • Expands a second time that lasts forever
    • Lemaitre rediscovered cosmological equations of Alexander Friedmann (1888-1925), Russian cosmologist
      • 1922, derived nonstatic models predicting cosmological redshifts
      • Models went unnoticed in scientific community
  • How many different expanding cosmological models does Friedmann's cosmological equations predict?
    • Assume = c,
      • Amount of matter precisely such that mutually gravitational attraction will stop expansion at infinite size
      • Model known as marginally open cosmological model
    • Assume > c,
      • Expansion will stop at a finite size
      • Radius determined by amount of matter
      • Any value of > c, produces one of infinite family of cosmological models
      • Known as closed cosmological models
    • Assume < c,
      • Even when space has become infinite the expansion continues
      • Any value of < c, produces one of infinite family of cosmological models
      • Known as open cosmological models
    • Nonstatic model universes do not require that 0
      • But non-zero value greatly increases number of models
      • Mere proliferation of models used to argue that = 0
  • Proportionality, such as Hubble's law v r, expected in any theory that satisfies cosmological principle

21.4. Friedmann Universes

• Einstein-Friedmann formulation of field equations of general relativity
  • Assuming = 0, three types of cosmological models, known as Friedmann universes, possible
    • Mean density () of matter distinguishes the three types
    • Cosmological expansion is slowed by mutual gravitational attraction of matter
  • Hubble's constant decreases with cosmic time at different rates for different Friedmann models;
    • For example, since c H2, and if H = 15 km/s/Mly
    • Hence c 5 x 10-30 g/cm3 or 3 atom/m3
• Open Friedmann Universe, Not Enough Matter
  • Defined by < c, insufficient matter to stop expansion
    • Universe will expand forever
    • Size (R) increases as R t
  • Universe is open with negative curvature
  • Universe is "infinite" in extent
  • Geometry of space is hyperbolic
  • One expansion
    • Begins with big bang from state of extremely high energy density
    • Proceeds toward an infinite value for separation between galaxies
    • Mean density of matter decreases to zero
  • Light arriving from most distant objects
    • Just able to reach us
    • Light is redshifted almost beyond perception
    • This outermost limit of observation is our cosmic horizon; but not an edge to space
    • Only observe events that were close enough that light has had enough time to reach us since the big bang
    • Cosmic horizon distance (h) is roughly velocity of light (c) multiplied by age (t) of Universe: h ct
    • Size (R) of Universe is proportional to t1/2 or t2/3
    • During expansion, h increases faster than R
    • Conversely, going back in time, h decreases faster than R
    • Consequently, cosmic horizon encloses smaller and smaller portion of Universe as we approach beginning
    • This means that for objects at large redshifts, horizon is proportionally less of the Universe
    • Or, objects exist whose separation from our position in beginning is such that they are not yet inside cosmic horizon
• Marginally Open Friedmann Universe (Einstein-de Sitter universe), Just Enough Matter
  • Defined by = c, sufficient mass to halt expansion but not before Universe becomes infinite in extent
  • Universe is a marginally open with zero curvature
  • Geometry of universe is Euclidean
  • Starts with big bang
    • Expansion goes as R t2/3
    • Expands more slowly than an open universe
  • Self-gravitation causes expansion to cease
    • Occurs at some time in infinite future
    • Universe will have infinite extent
    • Mean density falls from an infinite density when time began to zero at end
• Closed Friedmann Universe, More Than Enough Matter
  • Defined by < c, gravitational attraction by matter is sufficient to halt expansion
    • Universe stops expanding at a finite size
    • Followed by to collapse
  • Universe is closed
  • Universe possesses spherical geometry with positive curvature
  • Finite volume of space
    • No center and no boundary, since that would violate cosmological principle
    • Finite time for expansion and collapse
  • Expansion slowed by self-gravitation
    • Universe reaches a maximum size, depending on how much larger the mean density is than the critical value
  • Contraction starts slowly and accelerates toward big crunch
    • Matter collapses toward superheated, superdense state
    • The greater the mean density, the less time needed to reach big crunch
  • If Universe is closed, it should rebound into a new cycle
    • Recycled Universe need not have same physical details as previous one
    • Universe could have different physical and chemical properties
      • For example, only fundamental constants--velocity of light, gravitational constant, and Planck's radiation constant--might remain unchanged
    • Universe might begin anew with no remembrance of past cycles of expansion and contraction
      • For example, new cycle defines differently such things as fundamental constants and physical laws
  • Friedmann Universea, Beginning, Curvature of space, Geometry, Mean density (g/cm3), Age of universeb (y), Fate, Extent
  • Open, Big bang, Negative, Hyperbolic, <5 x 10-30, >13 x 109, Expands forever, Infinite
  • Marginally open, Big bang, Zero, Euclidean, 5 x 10-30, 13 x 109, Expands forever, Infinite
  • Closed, Big bang, Positive, Spherical, >5 x 10-30, <13 x 109, Expands stops and contracts, Finite
    • a Friedmann model with Einstein's cosmological constant equal to zero
    • b For H = 15 km/s/Mly or 50 km/s/Mpc

21.5. Age of the Universe

• Reciprocal of Hubble's constant estimate an age for Universe
  • For non-accelerated motion, relation between distance, velocity, and time is r = vt, or t = r/v
  • By analogy, assuming uniform expansion, from Hubble's law of recession v = Hr or 1/H = r/v
  • Reciprocal of Hubble's constant has units of time, and is known as Hubble time
    • Hubble's Constant (km/s/Mly), Hubble Time (y)
    • 15, 20 x 109
    • 20, 15 x 109
    • 25, 12 x 109
    • 30, 10 x 109
  • Such values compatible with ages of oldest globular clusters in our Galaxy
    • For example, order of 10-20 x 109 years
  • Universe must be younger than values in table if expansion has slowed with time
• Age of marginally open universe given by
  • tEuclidean = 2/3(1/H)
  • For H equal to 15 km/s/Mly, then tEuclidean = 13 x 109 y
• Comparison of ages of Friedmann universes
  • Age of open universe lies between Hubble time and age of marginally open universe
  • Age of closed universe less than that of marginally open universe

21.6. Geometry of the Universe

• Geometries of interest in cosmology
  • Are those which are uniform everywhere
  • Hence satisfy cosmological principle
• Locally, geometries must resemble Euclidean, but can be quite different globally
  • Universe may possess zero curvature globally
    • Described by Euclidean geometry
    • Will become infinite in extent
  • Universe may possess positive curvature globally
    • Described by a spherical geometry, like the two-dimensional surface of a sphere
    • Will remain finite in extent but unbounded
  • Universe may possess negative curvature globally
    • Described by a hyperbolic geometry, like the two-dimensional geometry of a saddle-shaped surface
    • Will become infinite in extent

21.7. Alternatives to a Big-Bang Cosmology

• Observations underlying Einstein-Friedmann cosmology
  • Hubble's interpretation of recessional velocities as expansion of space; supported by velocity-distance relationship
  • Gamow's prediction that spectrum of cosmic background radiation is blackbody spectrum; found by Penzias and Wilson
• Is some other interpretation of Hubble's velocity-distance law and Gamow's CBR acceptable other than big bang occurred long ago?
  • Number of competent scientists have tried alternative interpretations
  • None has succeeded in convincing science that their interpretations are more consistent with observations than Hubble's and Gamow's interpretations
• Steady-state cosmology
  • 1948, Herman Bondi (b. 1919), Thomas Gold (b. 1920), and Fred Hoyle (b. 1915) sought an alternative to the big-bang cosmology not based on Einstein's general relativity
  • Nonstatic model universe
    • General appearance remains unaltered forever; i.e., there is no beginning and no ending
    • Model is not static but allows for expansion as suggested by cosmological redshifts
  • Bondi and Gold extended cosmological principle to even more stringent conditions
    • Known as perfect cosmological principle
    • Not only does Universe appear the same to all observers everywhere, but it looks the same to them at all times
  • Space expands exponentially with time toward infinity
  • Hubble's constant does not vary with time as in evolving models
  • Galaxies form, evolve, and disappear, while mean density of matter in space remains constant
  • To keep mean density constant, theory assumes that new matter--hydrogen--is continuously being created, which compensates for expansion that thins out matter
  • Example: mean rate of creation in large classroom is about 1 hydrogen atom in 50 x 106 years (2.8 x 10-40 g/cm3/s), a rate well beyond detection
  • Because galaxies form at steady rate to keep pace with expansion, average separation between galaxies remains unchanged
  • Model is inconsistent with observed properties of Universe
    • Radio sources and quasars at great distances were more numerous in distant past (i.e., numbers increase with distance)
    • CBR is consequence of big bang, and does not occur in steady-state universe, since universe never in superdense condition
• Non expansion interpretations of redshift
  • Proposed cosmologies for which galactic redshifts mean other than expansion of Universe; known as non expansion cosmologies
  • "Tired-light" hypothesis
    • Photons lose energy traveling from distant galaxies by amount proportional to length of path through Universe
    • Wavelengths lengthened proportionate to distance of source, giving impression of cosmological expansion
    • No known physical processes can produce tired-light effect across entire electromagnetic spectrum
    • Argument for a "new law of physics" operating on a cosmic scale; difficult to overcome
    • Only convincing counter argument is to raise enough legitimate objections that "new law" becomes self-contradictory and dies from lack of adherents
  • Apparent association of high-redshift quasars with low-redshift galaxies suggests redshifts mean something other than expansion of space
    • Compelling point not just close proximity of quasar to low-redshift galaxies, but that most associated galaxy or galaxies are peculiar in some regard
    • Number of cases in which it appears as if several quasars and small peculiar galaxies ejected from giant peculiar galaxy, with all having different redshifts
    • New proposal is that only portion of redshift is due to cosmological expansion
    • most of redshift is intrinsic property of peculiar galaxies
    • possibly associated with its compact structure
    • Majority of astronomers do not accepted this contention
    • Reasonable body of observational material that is suggestive if not convincing

21.8. Tests of Cosmological Models

• Mean density of matter in Universe
  • Since for Friedmann models, c H2
    • If 15 H 30 km/s/Mly, then 5 x 10-31 14 x 10-30 g/cm3
    • Most widely accepted value closer to 5 x 10-30 g/cm3, or 3 atom/m3
  • To measure directly, volumes of space must be selected that are comparable to clusters of galaxies or larger
    • Estimates yield 0.05c to 0.1c, or 0.06 to 0.2 atom/m3 (i.e., 2 x 10-31 to 5 x 10-31 g/cm3)
    • Conclusion is Universe should be an open, or hyperbolic, and will expand forever
  • Problem of missing mass or dark matter in clusters of galaxies
    • Mass estimates based primarily on luminous matter and do not account for dark or non luminous matter
    • Apparent stability of clusters of galaxies suggests 10 times more matter present than is visible
    • Still short of mass needed to close Universe
    • More perplexing problem is form of dark matter, since it represents about 90% of all matter in Universe
    • X-ray observations of clusters reveal hot gas lying outside constituent galaxies
    • amount may be as much as is visible in galaxies
    • far short of required amount to close Universe
    • Missing mass supplied by neutrinos
    • masses of neutrinos 100,000 times or so smaller than electrons
    • but neutrinos fill Universe in incredible numbers
    • Prize to be gained in this search may be as much as 10 to 100 times more mass than current estimates, or enough to close it
• Measuring geometry of space
  • Over distances comparable to sizes of clusters of galaxies, curvature of space is negligible, except for local deformations caused by black holes
  • On cosmological scale of distances, effect of curvature of space is appreciable
  • As more distant galaxies observed, spatial curvature has greater effect on distance measurements
  • Equation for direct measure of spatial curvature derived from Einstein's field equations connecting apparent brightness of galaxies with their redshift
    • Which assumes galaxies maintain a constant brightness throughout life span--questionable assumption
  • For positively curved space (a closed Universe)
    • Since distance increases less rapidly the farther into space, galaxies at a given redshift, or recessional velocity, would have smaller apparent distances than in a flat space
    • This can be seen in two dimensions for surface of a sphere
  • For negatively curved space (an open Universe)
    • Since distance increases more rapidly the farther into space, galaxies at great distances, or a given redshift, should have larger apparent distances than they do in flat space
  • Problem is that galaxies may become intrinsically fainter or brighter as they age; thus it may be galactic evolution, not the curvature of space, that is being measured
• Number counts of radio sources
  • If radio galaxies distributed at random, then in flat space numbers of galaxies appears evenly distributed over space
  • If Universe is positively curved, there will appear to be more nearby galaxies than distant ones
  • If Universe is negatively curved, there will appear to be more distant than nearby galaxies
  • Number counts for radio galaxies, point to more of radio galaxies at great distances than even open universe predicts
• Angular-size distances
  • Another measure of curvature of space is the variation of angular size with redshift
  • In flat Euclidean space angular diameter declines with increasing distance
  • For positive curvature of spherical universe, angular diameter is larger at large redshifts than that for Euclidean space with zero curvature
  • For negative curvature of hyperbolic universe, angular diameter is smaller at large redshifts than that for Euclidean space
  • Angular diameters estimated for core regions of rich clusters of galaxies possessing different redshifts suggest Universe is open
  • Problem: must be cautious in assuming no evolutionary effects for clusters of galaxies when many reasons to believe that clusters do change with age
• Summary
  • Four classical tests suggest that Universe is an open Friedmann universe
  • But because of great uncertainty of effects of evolutionary changes on galaxies and clusters of galaxies, this is a very tentative conclusion