In their efforts to resolve the large cosmological questions described in the prior pages, astronomers began systematically correlating the brightness and redshift of certain stars in distant galaxies. As it turns out, a particular kind of star has a reliable absolute magnitude (a brightness unrelated to distance). Such a star can be relied on to provide an accurate measure of distance when the inverse-square law is applied to it:

(5)    mr =	ma
	r2

Where:

mr	Relative stellar magnitude as observed.
ma	Absolute stellar magnitude.
r	Distance (radius) between star and observer.

The kind of star chosen for this measurement, a so-called "standard candle" star, is called a "type 1a supernova". Even though supernovae are rather exotic and not yet fully understood, it has been observed about type 1a supernovae that their brightness is consistent from one star to another, and therefore they can be used as a measure of distance. One need only select examples of 1a supernovae in galaxies at varying distances and apply the above equation to convert relative magnitudes to absolute, thereby establishing distances on a cosmological scale.

The next step is to measure redshift. Cosmological redshift is a measure of recession velocity, and the measured amount of redshift is assumed to be linearly related to a galaxy's velocity away from the observer in the present Big Bang cosmological model. By combining the distance measurement made using "standard candle" stars, and the redshift method for establishing velocity, one can correlate distance and speed, thereby putting together a three-dimensional map of the locations and motions of a large number of galaxies.

There is one more factor to consider. Because of the finite speed of light, when we observe a distant galaxy, we are actually seeing it as it was in the past. If we view a galaxy that is located at a distance of five billion¹ light-years² , we are necessarily observing the galaxy as it was five billion years ago — that is, we are collecting the light the galaxy emitted five billion years ago. This turns out to be a critical aspect of the recent discovery.

In the 1990s two teams of observers (the Supernova Cosmology Project at the Lawrence Berkeley National Laboratory and the High-z Supernova Search Team) made a large number of observations of type 1a supernovae and their associated redshifts. Their goal was to improve the observational basis for the cosmological density parameter described earlier, and therefore better understand the large-scale geometry of the universe.

What these teams discovered, however, was that the measured recession velocity was not uniform over time. The most distant galaxies (e.g. those galaxies whose light was emitted in the distant past) showed a smaller recession velocity than galaxies that were closer to us (whose light was emitted more recently).

Because this result contradicted both known physics and common sense, the researchers assumed that this discrepancy resulted from a systematic error in observation, rather than actual physics. Very careful evaluations followed — did type 1a supernovae always have the same absolute magnitude, even in the distant past? Was there some problem with current redshift spectroscopy methods that would skew a result at a great distance? Everyone involved was reluctant to accept this new observation, and many other observers repeated the original work using different equipment and methods ... but with the same result.

This new result has come to be accepted, however reluctantly, and has resulted in the development of two hypotheses to explain it. One, called "quintessence," supposes the existence of a dynamic energy field that assumes different values at different locations and times, according to unknown rules. The other model, favored by myself and many others, is called "dark energy," which, although also quite mysterious and entirely outside current physics, has a consistency over time and space not shared by quintessence.

As it turns out, the present theory of "dark energy" closely resembles Einstein's cosmological constant, and some have even credited Einstein with being far ahead of his time with his original suggestion. The difference is that Einstein's cosmological constant was meant to provide a mathematical basis for a static, unchanging universe, and the new constant is meant to describe a universe that is evolving in an unexpected way, one that confronts common sense.

Here is a formal description of the previously introduced gravitational equation, plus the new "dark energy" factor, using the differential equation notation explained here:

Term	Description
(6)  f''(t) = Λ - G  m1 m2 r2	Second derivative of position, or acceleration, consisting of the previously introduced terms for the gravitational interaction of two masses, plus one new term: f''(t) Acceleration, Newtons, with respect to time. Λ "Dark energy", presently assumed to be a constant term, unaffected by distance or time. G The Universal Gravitational Constant, presently estimated to be equal to 6.6742x10-11 N m2 / kg2. m1,m2 The masses of the two bodies in question. r The distance between m1 and m2.
(7)  f'(0) = v	First derivative of position, or velocity. In Big Bang cosmology, the velocity term v is produced at time = 0 by the Big Bang that starts the universe. Its value is critically associated with the cosmological density parameter discussed earlier.
(8)  f(0) = r	Radius (distance between masses), assumed at time = 0 to also be zero, that is, all the matter in the universe is concentrated at a single point, a singularity.

As shown, this set of differential equation terms contains a glaring problem, one that is present in Big Bang cosmology as well. The problem is that the value of r is equal to zero at time = 0. This means the radius value in term (6) above equals zero at time = 0, which causes the system to break down in both a physical and mathematical sense.

This time-zero problem is unsolved in Big Bang cosmology, and the initial state of the universe has been likened to a black hole, an object of infinite density. Clearly the universe evolved away from that state, but how this process began is not explained by present physical theories, any more than the singularity associated with a black hole is.

To get around the time-zero problem, to create a practical mathematical model of cosmological evolution, one selects an initial time greater than zero, in the same way that one may safely theorize about the region outside the event horizon of a black hole, but not inside.

As it turns out, the differential equation terms listed above represent a reasonable model for cosmological evolution, but there is no known closed-form solution for the system as shown. Instead, various numerical methods are used to produce results in small steps of time, normally using supercomputers to handle the daunting number of required calculations.

Also, to produce useful results, the equations must be applied to many masses simultaneously. In a typical embodiment, an experimental set of masses is created and modeled using a computer numerical algorithm. This article set includes a simulator located here, with which the reader may experiment with the described equation system.

1. My use of "billion" follows the U.S. convention and refers to "thousand million" or 10⁹.
2. "Light-year" is a measure of distance, equaling the distance light travels in a year (about 9.461 x 10¹⁵ meters).