ASTRONOMY 9: HISTORY OF COSMOLOGY

Handout #22

J. E. Baker
UC Berkeley, Spring 2000

The Expanding Universe

Our Cosmic Location
- The Copernican Principle
  - Our location in the cosmos is not special
  - Old idea: Pythagoreans, Copernicus, ...
- The Cosmological Principle (C.P.)
  - No location in the cosmos is special
  - Also, no direction is special
  - That is, the universe is homogeneous and isotropic
  - If you average over very large volumes of space (10⁸ light-years or more)
  - Cosmology can't proceed without this idea! But it may not be testable: there are parts of the universe which are unobservable (maybe even in principle)
  - First stated explicitly by E. A. Milne (1933); implicit in earlier work of Einstein (1917)
- The Perfect Cosmological Principle (P.C.P.)
  - The universe looks the same (in large-scale average), not only at all points in space (C.P.), but also at all points in time
  - Universe is eternal, no single creation event
  - Dominant in cosmological thinking until 1930, contrast with evolutionary world-view or Christian creation mythology
  - MacMillan (1918) and Millikan hypothesize individual stars are born and die, but new matter is created to replace lost energy, and universe overall doesn't change
  - Steady-state theory (1948-1965, Hoyle and others)
Models of the Universe
- Solve Einstein's G.R. equations to predict the structure and evolution of the whole universe!
1.
The Einstein Model (1917)
- Einstein suggests universe can be closed: finite but no edge (like surface of sphere)
- But gravity would cause the universe to collapse!
- G.R. equations allow some freedom: a ``constant of integration'', the cosmological constant (called )
  - $\Lambda$ acts like an energy which fills even ``empty'' space: ``vacuum energy''
  - If $\Lambda > 0$ , acts like material with a negative pressure! Or a kind of cosmic anti-gravity: galaxies are effectively repulsed with a force that increases with distance: $F_\Lambda \propto r$ !
  - Effect over smaller distances (eg, size of the solar system) is tiny and undetectable
  - $\Lambda$ might also be <0, in which case it acts as a cosmic attraction like gravity (but stronger at larger distances)
- Einstein realizes if $\Lambda$ is just right, can balance the attractive force of gravity
- Allows a static, eternal, closed universe
- Later realized that this model is unstable: any perturbation will cause it to collapse or expand exponentially
2.
The de Sitter Model (1917, Dutch)
- Universe with no matter (density $\rho=0$ ) but some positive $\Lambda$
- Expands exponentially in time, $R\propto e^{\lambda t}$ where $\lambda$ is a constant related to $\Lambda$ , R is the distance between points in the universe
- Note expansion does not have a center: all points are expanding away from all other points
- Expansion also does not have a beginning: R does not reach zero unless $t=-\infty$
- de Sitter model originally interpreted as static (using different space/time coordinates)
3.
Alexander Friedmann (1922, Russia)
- Explores wide variety of models: the dynamic universe
- Interest in mathematical calculations, no effort to relate to the real physical universe
- Vary average density $\rho$ of matter-energy and $\Lambda$
- Density may change with time as universe expands or contracts
- Even models with $\Lambda=0$ can be expanding
- Geometry (open, flat, or closed) depends on total density of matter-energy and $\Lambda$
- Critical density divides open from closed:
  - $\rho > \rho_c$ : k=+1 (positive curvature, ``spherical''), closed (spatially finite)
  - $\rho = \rho_c$ : k=0, flat (spatially infinite)
  - $\rho < \rho_c$ : k=-1 (negative curvature, ``hyperbolic''), open (spatially infinite)
- $\rho_c$ is about 1 H atom per cubic meter!
- Friedmann equation allows you to compute R(t) given a cosmic recipe
- Notes that some models have R=0 a finite time in the past!
- Such universes have an origin and are not eternal (later identified with ``Big Bang'')
4.
Lemaitre-Eddington Model (1927)
- Starts off as infinitely old Einstein model (unstable)
- Gets perturbed at some point and starts expanding
- Turns into de Sitter model (density of matter decreases, gravity has less importance, $\Lambda$ takes over--effectively empty universe)
- Need new data to decide between the models!
Advances in Astronomy to the 1920s
- Large photographic plates developed after 1878
- No longer dependent on capricious nature of observers' eyes
- Astronomy increasingly professionalized
  - Big money, big telescopes, largely American
  - Funded by private philanthropy (Carnegie, Rockefeller)
  - New Mt Wilson observatory (1917), 100'' telescope
- Recall Doppler shift can be used to measure radial velocities
- How to measure distances to astronomical objects?
  - Parallax: only useful out to distances of tens of parsecs
  - Inverse-square law: apparent brightness b fades as d²:
    
    $\begin{displaymath}b = \frac{L}{4\pi d^2} \end{displaymath}$
    
    where L (luminosity) is total energy emitted per second
  - If L is known, can measure b and compute d!
  - Henrietta Leavitt (1908-1912) studies stars called Cepheids which vary in brightness
  - Discovers luminosity L is related to period P (easy to measure)!
  - Harlow Shapley (1917) calibrates relationship using stars at same distance (in clusters)
  - Measure P and b, get L, then can compute d!
The ``Great Debate'': The Nature of the Spiral Nebulae
- Debate takes place in April 1920, NAS, Washington DC
- Harlow Shapley (Mt. Wilson) vs. Heber Curtis (Allegheny Obs.)
- Shapley's position
  - Uses Cepheids to measure distribution of globular clusters around the Milky Way
  - A globular cluster is a swarm of up to about a million stars
  - Concludes Milky Way is much bigger than anyone thought: diameter of 300,000 l.y.!
  - Also, Sun is not at the center of the galaxy, but 2/3 of the way out towards the edge
  - If M.W. is so big, believed spiral nebulae could not possibly be external galaxies, must be local to M.W.
  - Only one galaxy in the universe
  - Also cites other data
    - 1885: nova (actually a supernova) in Andromeda galaxy
    - van Maanen uses Doppler shifts to conclude nebulae are rotating too fast (wrong)
- Curtis' position
  - Finds novae in more spiral nebulae using old photos
  - Assumes they have same luminosity, then can estimate distance
  - Rejects Andromeda nova as anomalous (correct), concludes spiral nebulae are actually millions of l.y. away!
  - ``Island universe'' hypothesis, M.W. is not special
  - Argues for smaller M.W., diameter only 20,000 l.y.
- Results
  - Curtis judged ``winner'' of debate
  - Shapley's position more influential for several years
  - Shapley was correct in arguing for large M.W. (though he overestimated size by factor of 2 or 3)
  - Curtis was correct about the nature of the spiral nebulae: external galaxies!
Hubble's Discovery
- Edwin Hubble (1859-1953): studied law, Rhodes scholar (1912), but turned to his true passion: astronomy!
- For Ph.D. (1917), classified galaxies: ``tuning fork'' or Hubble classification
  - Continuum of galaxy types: elliptical, spiral, barred spiral
- Hubble resolves the Great Debate
  - Discovers Cepheids in Andromeda galaxy (M31)
  - Concludes distance must be about 10⁶ light-years!
  - Shapley: ``here is the letter that destroyed my universe''
  - Spiral nebulae (as well as elliptical galaxies) really are external galaxies at great distances
  - M.W. really is just an average galaxy!
  - Curtis knew this from novae, but Cepheids were more convincing
- Hubble discovers the expansion of the universe
  - Slipher had earlier measured redshifts for galaxies: some are redshifted, few are blueshifted
  - Hubble extends Slipher's work, measuring more redshifts
  - Also measures distances by finding Cepheids in the galaxies
  - Discovers simple relationship between redshift z and distance d: more distant galaxies have bigger z!
  - If interpret redshifts z as due to a radial velocity v, can express in km/sec using (relativistic) Doppler formula
  - Hubble finds that more distant galaxies seem to be receding faster!
  - Hubble's law: velocity is proportional to distance:
    
    v = Hd
  - Consistent with C.P.: there is no center nor edge to the expansion--all galaxies will see all other galaxies receding according to Hubble's law
- Notes on the Expansion
  - Don't really observe velocity, but rather redshift z, linear redshift-distance relation is only an approximation, good for small $z\ll 1$
  - Redshift is $z = \Delta\lambda / \lambda$
  - Observed cosmological redshift is not really a Doppler shift!
  - Recall, galaxies are not expanding through space, rather, space is expanding between them
  - If define v as the rate at which physical distances between galaxies gets bigger, then v=Hd is exactly true
  - But we observe distant galaxies in the past, when universe may have been expanding differently
  - As universe expands, light waves get stretched:
    
    $\begin{displaymath}1+z = \frac{R_0}{R} \end{displaymath}$
The Einstein-de Sitter Model
- In many ways, the simplest cosmological model
- Model was generally predominant from 1930s through 1990!
- With discovery of expanding universe, Einstein realizes no need for $\Lambda$ to hold up the universe, calls its introduction his ``greatest blunder'' (but it reappears in the 1960s and again in the late 1990s!)
- Einstein-de Sitter model has no $\Lambda$ , only matter with exactly the ``critical'' density (note this is not the same as a cross between an Einstein and a de Sitter model!)
- Expands forever (just barely), $R\propto t^{2/3}$
- Geometry is flat, no curvature, and infinite
- Note: geometry is destiny (unless there's !)
  - If density had been larger, would be spherical (finite) and re-collapse
  - If smaller, would be hyperbolic (infinite) and also expand forever
The Hubble ``constant'' and age controversies
- Hubble constant H: units are 1/time, represents how rapidly the universe is expanding
- H is constant in space but not time: expansion can speed up (e.g., de-Sitter model) or slow down
- 1/H is called the Hubble time
- If expansion has not sped up nor slowed down, 1/H is the same as the age of the universe!
- The value of the Hubble constant today is called H₀
- Hubble's measurement: $H_0\approx 550$ km/sec/Mpc!
- Nearly $10\times$ too high for a number of reasons (e.g., two types of Cepheid)
- Gives $1/H\approx 2$ billion yrs, and without $\Lambda$ (universe is decelerating), age of universe is <1/H!
- Way too short: universe must obviously be older than everything in it!
- Measurements of H₀ come down, but age problems crop up through the mid-1990s
- Errors were always (and are still) thought to be about 10%, even with Hubble's very high value!
- Best estimate today: 65-75 km/s/Mpc (HST and other projects), age about 14 billion years (with $\Lambda$ )
- Does seem to be older than everything in it: no current age controversy
Changes in cosmology
- Until 1920s, cosmology largely European, theoretical (Einstein, de Sitter, Friedmann, Lemaitre, Eddington)
- Focus shifts to large American observatories
- Conception of the cosmos changes rapidly with Hubble around 1930
- Single Milky Way system in a static universe gives way to vast multitude of galaxies in an evolving universe
- By 1940s, another largely American pursuit has a dramatic influence: nuclear physics

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