The age of the universe and the Big Bang singularity

For a flat universe filled with pressureless matter, the previous lecture found $a(t) = b\,t^{2/3}$, with $t$ counted from the instant $a = 0$. The Hubble rate follows directly:

The constant $b$, which carries the arbitrary notch units, drops out, as any physical answer must. So measuring $H$ today tells you the age.

The Big Bang was in trouble from the very start. Hubble's 1929 expansion rate, about seven times too large, gave ages of only one or two billion years — less than the age of the Earth from geology, and less than the ages of stars from the theory of stellar evolution.

With the modern value Guth uses, from the Planck team in 2013, the flat matter-dominated age is:

But the oldest stars seem to be older. Guth illustrates this with an analysis of globular-cluster stars by Lawrence Krauss and Brian Chaboyer whose age estimate, allowing for time to form the first stars, puts the universe at at least 11.2 billion years. Stellar ages of this kind had already clashed with the flat matter-dominated model before 1998, creating a real tension. Theorists who favoured inflation (Guth counts himself among them) expected a flat universe; flat and matter-dominated meant too young.

The resolution came in 1998 with the discovery of dark energy. The universe is very nearly flat, but it is not matter-dominated today. Including dark energy raises the computed age to the present estimate of about 13.8 billion years, comfortably older than the stars.

The matter density grows as $1/a^3$ toward the past, so at $t = 0$, where $a = 0$, the model has infinite density: a singularity. Guth is careful about what this means.

• Adding radiation to the model (later in the course) does not remove the singularity. Dark energy matters for the future, not the distant past.
• Giving up perfect uniformity does not help either: classical general relativity generally leads to singularities. (Guth credits Stephen Hawking; the singularity theorems are usually attributed to Hawking and Roger Penrose together.)
• But classical general relativity cannot be trusted at such densities. It ignores the uncertainty principle of quantum mechanics, and nobody knows how to combine a quantum description of matter with a classical one of gravity. The smart money is on a quantum theory of gravity that changes the story.

So we do not know how the universe began. In most versions of inflation, our Big Bang was probably not the beginning of everything, but the start of our own region, possibly one of many in a multiverse — and inflation does not answer how it all ultimately began. What certainly happened is the hot, dense, expanding phase we call the Big Bang. The age of the universe means the time since the instant at which the extrapolated scale factor would have been zero. Unknown quantum effects could shift that instant by only a tiny fraction of a second, which makes no difference to the number.

• For a flat matter-dominated universe, $a \propto t^{2/3}$ gives $H = 2/(3t)$ and an age $t = \tfrac23 H^{-1}$.
• Hubble's too-large $H$ made the Big Bang look younger than the Earth; Planck's $H$ gives 9.7 billion years, still younger than the oldest stars.
• The universe is nearly flat but not matter-dominated: dark energy resolves the conflict, giving about 13.8 billion years.
• The $t = 0$ singularity is a feature of the classical model, not necessarily of reality; the age is measured from the instant the extrapolated $a$ vanishes.