### A Thirty-Year Search for just Two Numbers

Allan Sandage "the Father of Modern Astronomy" at the Carnegie Institution of Science, 2010

In 1926 the astronomer Victor Slipher measured the spectra of light from distant galaxies, in his soon-to-be, revolutionizing work he found that they were almost all shifted to longer wavelength by some amount, $\lambda$. What Slipher had in fact discovered would send tremors across the scientific community, what he noticed was the red shift of galaxies $\Delta \lambda /\lambda =v/c$ but he didn't appreciate yet, what this meant.

Eighteen years earlier, the Harvard astronomer Henrietta Swan Leavitt discovered a relationship between the period of Cephid variable stars and their luminosity which allowed astronomers (like Hubble) to calculate their distance from earth, provided one first knew what period of their evolution they were in.

Edwin Hubble was able to use the relation discovered by Leavitt to calculate the distances of various spiral nebulae (galaxies) which contained Cephid variable stars; the two-year study Hubble published in 1925 provided not only the first evidence of objects outside our galaxy but something even more profound came about when he compared his work with Slipher.

Hubble in a very famous 1929 article 'A Relation between Distnce and Radial Velocity among Extra-galactic Nebulae' compared the luminosity of distant galaxies, the apparent magnitude of their recession, to their distance from us. What became known as the "Hubble diagram", the data showed a positive correlation between the two. Something predicted by George Lemaitre in 1927. In affect this was the first evidence of the expanding universe, the first empirical support for the Big Bang.

$v = H_{0}x$

This equation is the mathematical expression of Hubble's law. Definitive confirmation of the Big Bang theory wasn't reached until the discovery of the Cosmic Microwave Background Radiation in 1965 (predicted in 1946 based off of the Big Bang model by George Gamow).

This much of the story is probably widely known by anybody reading this but there's an epilogue. Hubble's constant has units of $s^{-1}$ so its inverse should, in principle give us the age of the universe provided its expansion has been perfectly well behaved over time. In reality however, its not that simple. Calculating the age of the universe depends not just on $H_{0}$ but also the strength of gravity $F_{G}$ slowing down the recessional velocity of galaxies.

This pioneering search for just two numbers was spearheaded by the astronomer Allan Sandage and its something he dedicated almost thirty years of his life to. While Sandage was the first astronomer to give a reliable date for the age of the universe, he wasn't the first to try. Edwin Hubble tried to get a measurement of the Hubble constant using measurements of Cephlid variable stars, to calculate the distance to more distant stars and then ultimately to distant galaxies. This method in observational cosmology is called a "cosmic distance ladder" and Hubble was out by a factor of ten, his results told him the universe was less than 2 billion years old.

A more refined measurement for $H_{0}$ is needed. This was achieved in a series of three different measurements, first the distances were measured to ionized hydrogen in interstellar space, then to isolated galaxies and then to remote galaxies that are sufficiently distant enough to have noticeable red shifts.

Observations leading to estimates of $H_{0}$ were obtained first from the Virgo cluster at $57.0 \pm 6 _{kms^{-1}Mpc^{-1}}$ then, when Sandage along with G. A. Tammann also looked at the red shifts of individual galaxies they obtained a Hubble constant of virtually the same. These determined the local Hubble constant; a global value of the Hubble constant required observations of more distant galaxies, which again turned out to be very close to the local value, $55.0 \pm 5 _{kms^{-1}Mpc^{-1}}$. If one considered this data alone, one would expect the universe to be 18 billion years old and this number is called simply the "Hubble time".

Only if the universe's expansion is uniform would its age be equal to one Hubble time but in the standard Big Bang model we need to also take into account the deceleration parameter $q_{0}$.

Today we also know that one should consider the effects of dark energy (which would make the universe older) and dark matter (which would make the universe slightly younger) but neither of these were known about at the time, when one takes all of this into account, the Hubble Space Telescope gives us the most accurate, modern reading of $H_{0}$ to be $72 \pm 8 _{kms^{-1}Mpc^{-1}}$ but lets consider only the consequences of a pre-Lambda FRW model.

Now that we know the Hubble constant, we now figure out the density parameter, the ratio of the critical density to the actual density of the universe. The critical density, $\rho_{0}=3H_{0}^{2}/{8\pi G}$ is the density required the produce a flat universe, above this value and the universe is closed, below this value and it is open. In Sandage's era, before the discovery of dark matter and dark energy, most physical cosmologists thought the universe was open (some theoreticians e.g., Leonard Susskind still do).

When describing this equation some pop-level descriptions of cosmology equate the local geometry of the universe (i.e., open, closed or flat) with its global topology. When cosmologists say "the universe is flat" they don't mean its like a two-dimensional sheet, they mean simply that its described the axioms of euclidean geometry on large scales. If you were to construct a large enough triangle, using the CMB you'd find that the internal angles add up to 180 degrees.

In their article 'Steps Toward a Hubble Constant' Sandage and Tammann review several different ways of determining the deceleration parameter. Some of these are fairly obvious (1) Try to determine upper and lower limits of the age $T_{0}$ of the universe, from halo globular clusters, the oldest stars and certain heavy elements. Compare it to the Hubble time and calculate the difference to work out the deceleration parameter. (2) Count the number of galaxies in the observable universe, multiply it by the mass of an average galaxy and divide it by the volume of the universe, $nM/V$. (3) Direct observations of how nearby galaxies red shifts' are being decelerated.

Some methods however are far more technical, like (4) which involves the in-fall of matter into clusters and taking observations of fluxes in electromagnetic radiation and (5) compare the observed abundance of deuterium in the universe, with the amount of production expected based on calculations, in the Big Bang model to determine how the density has dissipated over time.

Ultimately, utilizing the first method to obtain an upper limit on the density parameter Sandage and Tammann used galaxies local velocities and summed up their densities, altogether this gave them a deceleration parameter of $q_{0} < 0.28 \pm 0.09$ the deceleration due to gravity is essentially, negligible. Since deceleration is a function of density, $\Omega$ and the deceleration parameter is less than $1/2$ Sandage and Tammann concluded that the universe had to be open and approximately 15 billion years old.

Independent studies by Gott, Gunn, Schramm and Tinsley confirmed a density $\Omega < 0.09$ and also suggested an open universe aged between 13 and 20 billion years old.

Today we in fact know that the universe's expansion is accelerating, which could still support either an open universe or still a closed universe with a positive cosmological constant like the early Lemaitre model. Sandage and Tammann believed the universe was open and therefore its expansion would never stop and collapse in on itself, the "creation event" in their words "was unique".

1. Even this conclusion that the universe will expand forever is open to question, an open universe can begin to contract. If we look at this from today's standpoint we think the universe has a positive cosmological constant but it could decay into a more stable negative vacuum state (if such a state could exist) and then the universe's expansion would halt and contract, forming a Big Crunch. These conclusions are tentative but still, incredibly interesting.

2. This comment has been removed by a blog administrator.

1. Removed, because it's spam.
- Andrew

### Set Theory

This post is a very brief introduction to some of the basic concepts of set theory. Set theory is a branch of mathematical-logic, that has wide applications across disciplines. Its not just used in the obvious way of studying the foundations of mathematics by mathematicians but also in physics, social science, and even by philosophers as a theory of semantics for predicate logic (although you can do propositional logic without set theory).

A set is a collection of elements, or members; the notation for a set is specified by listing its components. So the set of even numbers can be represented a
$E: \left \{ 2,4,6,8 ... \right \}$$E: \left \{ x: x > 0 \wedge even\right \}$ Either of these notations is valid. Further, elements of a set can only be in that set, once. So   $E: \left \{ 2,2,2,4,4,6,8 ... \right \} = E: \left \{ 2,4,6,8 ... \right \}$ The notation used to indicate that something is an element of a set, is using the Greek symbol "epsilon". That is: $4 \epsilon S$…

### William Lane Craig and the Hartle-Hawking No Boundary Proposal

Classical standard hot Big Bang cosmology represents the universe as beginning from a singular dense point, with no prior description or explanation of classical spacetime. Quantum cosmology is different in that it replaces the initial singularity with a description in accord with some law the "quantum mechanical wave function of the universe", different approaches to quantum cosmology differ in their appeal either to describe the origin of the material content of the universe e.g., Tyron 1973, Linde 1983a, Krauss 2012 or the origin of spacetime itself e.g., Vilenkin 1982, Linde 1983b, Hartle-Hawking 1983, Vilenkin 1984.

These last few proposals by Vilenkin, Hartle-Hawking and others are solutions to the Wheeler-DeWitt equation and exist in a category of proposals called "quantum gravity cosmologies" which make cosmic applications of an approach to quantum gravity called "closed dynamic triangulation" or CDT (also known as Euclidean quantum gravity). I&#…

### Can inflation be eternal into the past?

Back in 2003 a paper appeared on the arXiv titled "Inflationary spacetimes are not past complete" that was published by Arvind Borde, Alan Guth and Alexander Vilenkin which has had considerable amounts of attention online. The theorem is rather uninteresting but simple and doesn't require a very complicated understanding of math. So I thought I'd explain the result here.

It's purpose is to demonstrate that inflationary models are geodesically incomplete into the past which they take as "synonymous to a beginning" but Vilenkin stresses that the theorem can be extended to non inflationary models so long as the condition of the theorem that the average rate of expansion is never below zero is met. These models too then are incomplete into the past. Consider the metric for an FRW universe with an exponential expansion

Where the scale factor is

Since the eternal inflation model is a "steady state cosmology" the mass density and the Hubble paramet…