The acceleration of the universe

In 1917 Einstein was working out some consequences of the General Theory of Relativity and discovered that the universe should either be expanding or contracting. Astronomers assured him the evidence was for a static universe (neither expanding or contracting). Einstein postulated a cosmological constant, which, if positive, would counteract the attractive force of all the matter.

In 1929 Edwin Hubble discovered that the universe was expanding. Hubble's Law may be stated as follows:


where v is the velocity of recession in km/sec, D is the distance in Megaparsecs, and        is the Hubble constant in km/sec/Mpc. A widely used modern value of the Hubble constant is 72 +/- 8 km/sec/Mpc (Freedman et al. 2001, ApJ, 553, 47).

We note that the Hubble constant has units of 1/time.             is a measure of the age of the universe. With some unit conversion the Hubble time,         , in billions of years is


It turns out that the age of the universe is equal to the Hubble time only if the universe has negligible matter density compared to the critical density (see below), and the cosmological constant is zero. For other situations the age of the universe can be more than        or less than       .

In the past the universe was smaller, and if we can reckon the expansion all the way back, the implication is that the universe was once very small. In the 1940's George Gamow, Ralph Alpher, and Robert Hermann worked out the basics of the Big Bang theory, which predicted that there should be a relic radiation from the Big Bang filling the universe. With the discovery of the Cosmic Microwave Background radiation by Penzias and Wilson (1965, ApJ , 142, 419), the Big Bang theory was strongly confirmed.

If the universe had only matter in it and if the matter density were equal to some critical density, then the expansion of the universe would slow down and slow down. The universe would coast to a stop after an infinite amount of time. If there were ever so slightly more than the critical density, then the universe would eventually halt, then start to contract, leading eventually to a Big Crunch. The quest for the rate of deceleration of the universe engaged astronomers for the second half of the 20th century.

A Type Ia supernova is widely believed to be explosion of a carbon-oxygen white dwarf (WD) whose close companion star has transferred mass to the WD. When the WD mass exceeds 1.4 solar masses, the WD explodes. Since the exploding star always has about the same mass, the explosion gives just about the same amount of energy. In fact, those Type Ia SNe that produce more Nickel-56 are somewhat brighter than those that produce less Ni-56. In any case, at optical wavelengths the brightness of Type Ia SNe at maximum light is related to the rate at which the SNe get fainter (Phillips, 1993, ApJ, 413, L105).

In the late 1990's two groups announced that high-redshift Type Ia supernovae were "too faint" for their redshifts (Riess et al. 1998, AJ, 116, 1009; Perlmutter et al. 1999, ApJ, 517, 565). This has been interpreted as due to the acceleration of the universe. The SNe are "too far away" compared to the (previous) expectation that the universe's expansion should have been slowed by the sum of the gravitational attractions of all the galaxies and other matter in the universe. A personal account of what happened in what order regarding this noteworthy discovery can be found by clicking 

The acceleration of the universe is due to the vacuum energy of the universe being positive. In other words, the vacuum is filled with some kind of Dark Energy which has a repulsive force.

The consequence of a positive cosmological constant is that once the universe got big enough, the repulsive force of the vacuum would cause the universe to accelerate. At what redshift (or look-back time) did this occur?

Let us start with a general version of the Friedman Equation. (For more information click            )



       = Hubble constant (which changes as the universe expands)

         = Hubble constant at the present age of the universe

            = radiation density of the universe

           = matter density

         = Einstein's cosmological constant

        = curvature of the universe

       = velocity of light

and      = the cosmic scale factor.

Space-time expands as the universe grows older. At the time of the Big Bang, R = 0. At the present time R = 1.

It is also the case that the cosmic scale factor is related to the redshift (z) as follows:


At a redshift of 1 the universe was half its present size.

During the early universe it was radiation dominated. Now it is dominated by matter and the Dark Energy, which gives rise to the acceleration. For flat geometry in the present universe, the radiation density is negligible, K = 0, and                              . The Friedman Equation reduces to:


Since                   , the Friedman Equation for a flat universe can be written as:


The purpose of this website is to work out various consequences of this relatively simple-looking equation.

In the plot below we show the rate of change of the cosmic scale factor        divided by the Hubble constant, as a function of the cosmic scale factor R. We have chosen                      and                     , close to the values stipulated by modern data on Type Ia supernovae and the power spectrum of the Cosmic Microwave Background radiation. The function is positive for all R > 0, and it has an obvious minimum. The value of R at the minimum in the curve corresponds to the size of the universe when it underwent a transition from deceleration to acceleration. A simple application of l'Hopital's rule shows that the minimum occurs when


Since                        , it follows that


This is the same as Equation 9 of Turner and Riess (2002, ApJ, 569, 18).                       and                      ,                                  . From Equation 4 or 6 it follows that the transition redshift is 0.6711. The age of the universe for       = 72 km/sec/Mpc,                           and                         is 13.095 billion years (Gyr). The transition between deceleration and acceleration occurred at a look-back time of 5.961 Gyr, or 7.134 Gyr after the Big Bang.

How does one find the function R(t)? Let us consider the simple case of                . From Equation 5,

Simple integration gives:

                                       , or

(7)                                                          .

In other words, the cosmic scale factor increases as the 2/3 power of the time.

Long after the acceleration has taken over, we would have

Simple integration gives:


Raising both sides to the power of e gives

(8)                              .

In other words, the cosmic scale factor eventually increases exponentially with time.

How can one solve the more general equation


If                              , then one can get an explicit solution to the integral. It requires making the rather nasty substitution


It is important to know the following three identities:




For more information on hyperbolic trig functions click 

The reader is directed to Krisciunas (1993, JRAS Canada, 87, 223), where we also find an expression for the age of the universe in Hubble times:


Cosmic time in units of the Hubble time is related to the cosmic scale factor in a geometrically flat universe as follows:



It is not necessary to invert the equation to plot R(t) as a function of t. We can calculate t as a function of R and just reverse the axes in a plot. In the graph below we show a plot of the cosmic scale factor R versus time t for                      ,                     , and a Hubble constant of 72 km/sec/Mpc. As one can see, in the first few billion years after the Big Bang the universe was decelerating, initially proportional to t^2/3. Several billion years ago there occurred the transition between deceleration and acceleration (R = 0.5984, redshift = 0.6711, age = 7.13 Gyr). Since then the dynamics of the universe is more and more driven by the repulsive force of the vacuum energy.
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here. here. here. here. This narrative was put together by Kevin Krisciunas and Sam Gooding.