The age of the Universe is defined as the largest possible value of proper time integrated along a timelike curve from the Earth at the present epoch back to the Big Bang. The time that has elapsed on a hypothetical clock which has existed since the Big Bang and is now here on Earth will depend on the motion of the clock. According to the preceding definition, the age of the universe is just the largest possible value of time having elapsed on such a clock.

It was estimated to be about 13.7 billion (13.7 × 10⁹) years, with an uncertainty of 200 million years, by NASA's Wilkinson Microwave Anisotropy Probe project (WMAP). However this is based on the assumption that the underlying model that was used is correct. Other methods of estimating the age of the universe give different ages.

Some recent studies found the carbon-nitrogen-oxygen cycle to be two times slower than previously believed, leading to the conclusion that the Universe must be at least 14.7 billion years old.

The often quoted age of 13.7+/-0.2 Gyr for the age of the universe comes from the first year WMAP results: This measurement is made by using the location of the first acoustic peak in the microwave background power spectrum to determine the size of the decoupling surface (size of universe at the time of recombination). The light travel time to this surface (depending on the geometry used) yields a pretty good age for the universe. Assuming all the various models used are valid in getting to this number, the accuracy of actual data allows a margin of error around 1%.

However, this age is only accurate if the assumptions built into the various models being used are also accurate. This is referred to as strong priors and essentially involves stripping the potential errors in other parts of the model to render the accuracy of actual observational data directly into the concluded result. Although this is not a totally invalid procedure in certain contexts, it should be noted that the caveat, "based on the fact we have assumed the underlying model we used is correct", then the age given is thus accurate to the specified error (since this error represents the error in the instrument used to gather the raw data input into the model).

The age of the universe based on the "best fit" to WMAP data "only" is 13.4+/-0.3 Gyr (the slightly higher number of 13.7 includes some other data mixed in). This number represents the first accurate "direct" measurement of the age of the universe (other methods typically involve Hubble's law and maximum age of stars, etc). There is a sense of triumphantism in the scientific community surrounding results like this, and therefore a more careful analysis of the methods and assumptions used, tend to be overlooked.

This, of course, is a classic example of how different methods for determining the same parameter (in this case – the age of the universe) can give different answers with no overlap in the "errors". It is quite common to see two sets of uncertainties, one related to the measurement and other the related to the systematic errors of the model. In some cases, this can not be done.

Calculation of the age from the temperature of the universe

The redshift of an object in an expanding or contracting universe is related to the value of the scale factor of the universe at the time of emission of the redshifted photons, by the relation <math>R=R_0/(1+z)<math>. Where <math>R<math> represents the scale factor of the universe as seen at the redshift z, where the current scale is <math>R_0<math>. Contrary to popular belief, or careless language, the scale factor does not represent a measurement of the size of the universe, nor does it represent a radius. However, the shape of the universe is related to the scale factor, and the size in ordinary length units of some region of galaxies (e.g. on a scale of a hundred Mpc or so) does change proportionally to the scale factor. This is described more fully in: shape of the universe. The scale factor is commonly written as <math>a=R/R_0<math>, which is dimensionless.

The temperature of the universe is inversely proportional to its scale; somewhat analogous to a gas that would cool down if expanded, or heat up if compressed, the temperature of the universe is thus related to redshift as <math>T=T_0(1+z)<math>. We can do a quick test by using the current temperature of 2.7K and the redshift of CMB as 1089 to calculate the temperature of the decoupling surface <math>T= 2.7*1090 = 2943\mathrm{K}<math> (this is the temperature of the universe when the CMB was emitted - around the dull red glow of a hot poker.)

One of the most important cosmological models is based on the Friedmann equations.

In a universe like our own, most of the contents is in the form of stuff that does not exert much pressure on its surroundings (clouds of hydrogen gas, stars etc). This is a pressureless, or "dust" model. Here <math>t=t_0(1+z)^{-3/2}<math>, and throwing in the redshift of about 1089 for the cosmic microwave background and a current age of the universe <math>t_0 \approx 13.7<math> Gyr gives us around 380,000 years for the age of the universe when the CMB was emitted. This may not seem so tricky after all, but unfortunately, it is not quite that simple.

Embedded in these models is an assumption that time and space are really just a sort of projection of a single thing: space-time, and therefore should be measured in the same units. However, since the metric distance defined between two points, e.g. each attached to a big cluster of galaxies, in an expanding universe, increases over time, some people interpret this as a fundamental change in the underlying “concept” of distance (and the same situation would also apply to the concept of time), and interpret the phrase space itself expands in this sense. This is erroneous, since general relativity is consistent, in the limit, with special relativity in any relatively small region.

For example, given present observational constraints, the observer could interpret the radial velocities of all observed galaxies in terms of special relativity, apart from the philosophically strange coincidence that she finds that the velocity vectors of other galaxies are all centred on herself. This coincidence, of course, would suggest by the Copernican principle that there must be some theory deeper than special relativity, especially since, according to this interpretation, the most distant galaxies in all directions are travelling very close to the speed of light away from us, but with their velocity vectors centred on us, the observer.

A frequently asked question relating to our CMB calculation above is that if the photons in the CMB went from being hot enough to fry a burger, how come those same photons can't even defrost one today? Where did all that energy go? The answer is that the photons together with the matter, functioning as a system, have increased energy in their combined existence as a system; this increased energy can be called an increase in potential energy.

So this was a very round about way of saying that we can relate the temperature of the universe to the age of the universe. Since we can measure the current temperature and have a model to extrapolate back, all we need to know now is the origin of the graph and read off the age. The earliest valid point in the evolution of the universe where quantum mechanics can be ignored is the Planck time. At this time, the universe had the Planck temperature at a state of essentially zero entropy. The Planck temperature is the maximum attainable temperature in the universe and can be thought of as the Hawking temperature of black hole with a radius of the Planck length.

The Planck temperature Tp comes out to around <math>4.5 \times 10^{30}<math>K, and we can state <math>Tp=To(1+z_{\mbox{max}})<math>, where <math>T_0=2.725<math>K and <math>z_{\mathrm{max}}=1.65\times 10^{30}<math> is the maximum redshift at the Planck time <math>t_p<math>. We know that <math>t_p=t_0(1+z_\mathrm{max})^{-2}<math>, so putting in the Planck time gives us an age of the universe of 11.667 Gyr. This is not the end of the story however: If time was absolute and never changed, then this would be the correct value, but we need to take into consideration the change in time over the age of the universe. This is a fairly simple integration and results in a age one third as much at 15.556 Gyr. The CMB temperature is known to a 2mK accuracy, and with some error in things like the Planck units (mainly from G), the accuracy of this age determination is around 24 Myr.

There is a simplification where if expressed in Planck units, the age (to/tp) is equal to the inverse square of the temperature (To/Tp) of the universe. Dividing To/Tp gives the current temperature expressed in the amount of the Planck temperature <math>6 \times 10^{-31}<math>. Taking the inverse square gives <math>2.72 \times 10^{60}<math> which is the age in Planck units. Multiplying by the Planck time gives the 11.667 Gyr again. There are many other simple relations like this one, including the critical density as the Planck temperature raised to the forth power. In Planck units, the critical density is <math>1.3 \times 10^{-121}<math>, which when multiplied by the Planck density gives <math>3.3 \times 10^{-30}<math> g/cm^3.

See also

• Science 299 (2003) 1532-1533, available here http://arxiv.org/abs/astro-ph/0303180

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