Modern History of Cosmology
Spring 2001

Instructor: Helge Kragh, History of Science Department, University of Aarhus

(page numbers refer to Pedersen & Kragh, Fra Kaos til Kosmos)

#1 Newton, Kant, Herschel (156-57, 177-93, 231-35)
#2 Early astrophysics and chemistry (195-210, 235-39)
#3 "The Great Debate" (239-42, 216-19, 254-61)
#4 Relativistic cosmology, I (252-54, 271-79)
#5 The expanding universe (264-69, 279-84)
#6 Earliest big band models (284-91)
#7 Heterodox models (Dirac; Milne) (294-95)
#8 Gamow's big bang (303-16)
#9 The age of the world (292-95)
#10 Steady state theories (322-33)
#11 The cosmological controversy (325-41)
#12 Cosmic microwaves (316-19, 341-44)
#13 Developments since 1970 (355-74)

> Read and try to make sense of the sources

Presentations (about 20 min.)

(1) Who was Wm. Crookes? What's he--a chemist--doing here, in a cosmological context? Give a brief account of Crookes' life and career, and explain the context of his presidential address and the problem area he is addressing.
See, e.g., R. DeKosky, "Spectroscopy and the elements in the late nineteenth century," British Journal for the History of Science 6 (1973), 400-423 [IVH].

(2) Explain the background for Boltzmann's cosmic scenario about multiple worlds, including "pockets" with decreasing entropy. What was the context of Boltzmann's speculation? Account in non-technical terms for the background in Boltzmann's probabilistic view of the 2nd law of thermodynamics and the current discussion about this issue.
See, e.g., S. Brush, The Kind of Motion We Call Heat, vol. 2 (Amsterdam, 1976), pp. 598-627.

Textbook reading assignment: Pedersen & Kragh, pp. 235-45, 254-64.

Sources and more background: Find material on

(or search for 'great debate'). This site includes a wealth of information, including the texts by Shapley and Curtis. There is much more than you can consummate, but browse through the material and pay particular attention to the first item ("What the Great Debate was...") and the lecture in the lesson plan ("The universe and the Curtis-Shapley debate").

After having understood the nature of the debate, answer the review questions (end of the lesson plan) #1, 3, 4, 5, 6, 7, 8 and 9. I assume that you all answer these simple questions.

Individual student assignment (20 min. presentation): Give an account of the main points in Virginia Trimble's nice paper on "The 1920 Shapley-Curtis discussion."

PS: www.alltheweb.com is an excellent and very quick search machine. Use it!

Copied sources include Friedmann's 1922 paper in English translation, and a general introduction by Einstein of 1920, taken from Munitz, Theories of the Universe. Friedmann's paper is highly technical, but much of it can be understood without great difficulties. I have marked which sections should be read.

Problems to be solved by all. They refer to Friedmann's work, 3-5 also to my note.

[1] According to Friedmann (end of paper), if it is assumed that the mass of the universe is 5 x 1021 sun masses, the "world period" of a cyclical model will be approx. 1010 years. Check numerically this statement.
[2] From Friedmann's note 8 it appears that he uses K = 8piG/c2 for Einstein's gravitational constant (based on mass density rather than energy density; the convention K = 8piG/c4, which I use in the note, is based on the energy density). Show that the dimension and numerical value for K, as given by Friedmann, are correct.
[3] The Friedmann eqs., with p = 0, include the static Einstein universe. Show that the universe can only be static if l = .5PKc2. Is this result stated in Friedmann's paper? What will happen if l is smaller/larger than this value?
[4] In de Sitter's model, k = 0 and P = p = 0. In modern interpretation, it is an ever-expanding universe according to R(t) ~ exp(ct{sqrt(l/3)}). Show that the de Sitter model, too, is among the solutions to the Friedmann equations.
[5] We define H = R'/R and O = P/Pc, where Pc is the critical density 3H2/8piG. Assuming that l = 0, show that kc2 = H2R2(O - 1). What does this result imply for how the curvature of space depends on the average mass density?
[6] In Einstein's model, the "radius" is given by R2 = 2/KP (see note 4 in Einstein source). In 1917, he believed that P was approx. 10-22 g/cm3, a value much too large. With this value, what will the radius of the universe, expressed in light-years, be? (Present estimates of visible matter give P approx. 10-31 g/cm3).

Friedmann (or Friedmann-Lemaître) models

The most general formulations of the equations, as given by Lemaître in 1927, are

3(R'/R)2 + 3kc2/R2 = (l + KPc2)c2 (1)

(R'/R)2 + 2R"/R + kc2/R2 = (l-PK)c2 (2)

where R' = dR/dt, R" = d2R/dt2, and R is the scale factor (measure of distance)

k is the space curvature parameter (flat space: k = 0; closed space: k = +1; open space: k = -1).
K is Einstein's constant of gravitation, ( = 8(G/c4 (with G Newton's constant).
l is the cosmological constant.
P is the average density of matter.
p is the pressure.
c is the velocity of light.

[Web page editor's note: I have been unable to find a standard format among browsers for displaying mathematical formula. Thus some of the constants above are non-standard (e.g. english letters for Greek, such as l for lambda), and the red numerals denote superscript numerals. I apologize for the inelegance of my solution.]

(2) Lemaître's 1927 paper (copied in its English translation of 1930).

Student presentation (20-25 min.): About Edwin Hubble, based on Allan Sandage's centennial article in J. RAS Canada 1989 and reproduced in
www.antwrp.gsfc.nasa.gov/diamond_jubilee/d_1996/sandage_hubble.html

Questions relating to Lemaître's paper, to be solved by all students:
(1) In which ways do Lemaître's eqs. (2) and (3) differ from the corresponding equations given in Friedmann's 1922 paper?
(2) Compare Lemaître's eq. (11) with eq. (21) in Friedmann's paper. For which value of B do the equations coincide?
(3) Lemaître's last eq. can be written as v = (c/Rsqrt{3})r. This is the Hubble relation, with H = c/Rsqrt{3}. Show that this result follows as an approximation to Lemaître's eq. (25). Also, what is Lemaître's value of H, expressed in the standard unit km·sec-1·Mpc-1?

Student presentation: Give an account (20-25 min.) of Lemaître's ideas of the relationship between cosmology and theology. Use sources supplied by the teacher.

Questions relating to the Einstein-De Sitter paper (all students):
[1] Show that eq. (2) follows from the Friedmann-equations.
[2] Eq. (3) corresponds to the critical density, usually written as P = 3H2/8piG (or O = 1) with H being the Hubble constant. Show this.
[3] What is the numerical value of P based on the best modern value of H? (You may find H on the internet).
[4] The Einstein-De Sitter model is a typical big bang model, yet this feature is conspicuously "suppressed" in the paper. Show that R varies with time according to R(t) = at2/3 + B, where a and B are two constants.
[5] Show that the age of the Einstein-De Sitter universe is given by t = 2/3H = 2T/3 (where T is the Hubble time, T = H-1, or the age corresponding to a uniform expansion). Give in a R(t) diagram a geometrical interpretation of the result.
[6] A uniformly expanding model (with t = T) is sometimes called a Milne model. Can such a model be described by the Friedmann-equations with l = 0 ?

"The mysterious Eddington-Dirac number" is nicely explained in http://www.geoastro.de/astro/stars/diracnumber.htm

Problems:
[1] Read carefully Dirac's note and understand his argument for G(t) ~ t-1 and N ~ t2 (where N is the number of nucleons in the universe). Is Dirac's choice of a "natural" or atomic time unit, namely e2/mc3, the only one that can be constructed from fundamental constants?
[2] In 1938 Dirac argued that T = H-1 = ca. 5 x 1016 sec and P-1 (the inverse of the mean density of matter) = ca. 2 x 1030 cm3/g are roughly equal numbers of magnitude 1040 if expressed in atomic units. Show from Dirac's large number hypothesis (as stated in his 1937 paper) that we then are led to the result P-1 = kR/R' (where k approx. 1).
[3] If matter conservation is assumed, P approx. R-3. Combine this with the result P-1 = kR/R' and find how R varies with time, i.e. the R(t) function. (You should get R ~ t1/3).
[4] Now find (from T = R/R') the relation between T and t, and hence the age of the universe in terms of T (you should get to = T/3).
[5] Dirac's G(t) prediction was quantitative, namely, that the rate of change is given by abs. val. {G'/G} = 3H. How is that? With Dirac's (or Hubble's) value of H, what is the change per year?
[6] Read carefully Teller's 1948 criticism of Dirac's hypothesis and understand his reasoning. How will the modern value of the Hubble time (say 15 billion years) affect Teller's argument?

Student assignment: Give an account of how R. Alpher and R. Herman in 1948 predicted the existence of a cold (ca. 5 K) cosmic background radiation. You may use their papers (Nature 162 [1948], 774 and Phys. Rev. 75 [1949], 1089) and/or Kragh, Cosmology and Controversy (1996), 119-20.

Gamow was a colourful and versatile scientist. Read about his life and career, and see what he looked like, in the nice article: www.media.gwu.edu/~physics/gwmageh.htm

Problems:
[1] Is a blackbody radiation of T approx. 5 K a microwave radiation?
[2] Read Gamow's paper and try to understand it in a general way.
[3] Show that Gamow's eq. (1b) follows from the Friedmann-equations.
[4] In 1953, Gamow obtained a much too large value for the cosmic background temperature by simply inserting the age of the universe (then about 3 billion years) into his eq. (14). What was his result? Where did he go wrong?

> In reading Tolman's paper, you may ignore the equations; but pay attention to his general line of reasoning. (The clip from Nature is just to indicate philosophers' interest in the matter.)

> The age of the universe (and the earth) plays some role in the present (mainly American) creationist debate. You may get an impression by clicking on "The age of the earth and the universe" at the Creation Science homepage: emporium.turnpike.net/C/cs/

> A good site for cosmology (mainly non-historical) is: www.yourphysicslink.com/relativity/cosmology/

Problems:
[1] Go carefully through McCrea's argument on p. 9 that in steady state (SS) theory the average age of galaxies in any large region of space is T/3. (You may also consult Kragh, Cosmology and Controversy, p. 187). Prepare to give a clear and concise presentation of the argument.
[2] Although SS theory does not comply with general relativity, it shares with FL-models that it satisfies the Robertson-Walker (RW) metric. What is the RW metric? (You may look it up in any textbook, or on the internet).
[3] In 1948, Bondi and Gold showed from the perfect cosmological principle that the SS universe must be flat (k = 0) and expand like R(t) ~ exp(Ht), that is, like what is known as a de Sitter model. Show this. (Hint: According to SS, the rate of particle creation is affected by the curvature k/R2. Both this quantity, as well as the Hubble parameter, must be constant.)
[4] It is possible to get a kind of SS model from the FL-equations (with l = k = 0 and P > 0) if a negative pressure (p = - Pc2) is allowed as a kind of cosmic tension. Show this.
[5] In the late 1980s, Hoyle and a few other cosmologists developed a QSSC model (Quasi Steady State Cosmology). What is the essence of this model? How does it differ from the old SS model? (See, e.g., www.astro.ucla.edu/~wright/stdystat.htm and/or www.ias.ac.in/pramana/dec1999/c3.html)
[6] Hoyle argued that the big bang model violates the principle of energy conservation even more grossly than SS theory. For according to BB all the material content of the universe was suddenly created at t = 0, which is a huge violation of energy conservation. Consider the soundness of Hoyle's argument.

Problems (for all to consider):
[1] (Radio astronomical counts, cp. Pedersen & Kragh, p. 330). Let S be the flux density of a celestial object, say a radio source (flux density = the power emitted in a certain frequency range through a unit area; dimension Wm-2s). Assume that the sources are distributed spherically in a static Euclidean space with a constant density. Then show that the number of sources N with a flux density larger than or equal to S varies as N ~ S-3/2.
[2] According to the Einstein-de Sitter model, the apparent angular diameter of an object (say a cluster of galaxies) will vary with the redshift z as
delta theta = (DH/2c)(1 + z)3/2 [(1 + z)1/2 - 1]-1
where D = absolute diameter and H = Hubble constant. What is the minimum (apparent) diameter? For what value of z will it appear? What will happen with the diameter in the limit z approaching infinity?

Presentation:
In the Soviet Union between ca. 1947 and 1960, cosmology was considered "politically incorrect". Give a brief account of how the two major world systems (big bang and steady state) were conceived by Soviet ideologues and how this politization of the area affected Soviet astronomy and cosmology. The presentation may be based on Kragh, Cosmology and Controversy, pp. 259-68.

Problems (paper by Dicke et al.)
[1] On p. 418 there is a reference to Mach's principle. What is this principle? (You may look it up on the internet).
[2] On p. 416, lower part, the authors refer to Peebles 1965 ("Phys. Rev. (in press)"). Did this paper ever appear?
[3] On same page it is suggested that Pm may be as small or smaller than 3 x 10-32 gcm-3 (Pm = matter density). Is this suggestion convincing?
[4] On same page it is stated that "the present radiation ... varies as the cube root of the assumed present mean density of matter." As early as 1948 Alpher and Herman showed that the product PrPm-4/3 remains constant in time (Pr = radiation density).
4.a. Show that this follows from equations given in Gamow (1949), p. 370.
4.b. Show that the Alpher-Herman relation leads to the cited statement.

Presentation: The discovery of the 3K-radiation in 1965, the way in which earlier work was ignored, and the subsequent Nobel prize for the discovery caused a great deal of frustration among G. Gamow, R. Alpher and R. Herman (who had predicted the radiation as early as 1948). The discovery and the merit that followed is interesting not only from a scientific point of view, but also from one of sociology. Read selected parts of Kragh, Cosmology and Controversy (1996) and give an account of these sociological features. Were Alpher and Herman treated "fairly"? Were they justified in their complaints? Was the 1978 Nobel prize "just"? etc. (Relevant sections include pp. 139-41, 350-51, 353-54, 375-76).

Sources:
(1) V. Rubin, "Dark matter in the universe," Scientific American, March 1988, to be found on the internet as: www.sciam.com/specialissues/0398cosmos/0398rubin.html
(2) N. Straumann, "The mystery of the cosmic vacuum energy density and the accelerated expansion of the universe," Eur. J. Phys. 20 (1999), 419-27 to be found in the electronic library of Aarhus State Library (Statsbiblioteket).

This final session in the course will be a quick survey of some recent developments in big bang cosmology, including: particle physics and the very early universe; inflation theories; dark matter; the revival of the cosmological constant; new observations; attempts at quantum cosmologies; many-universe theories; ...