# How Old is the Universe? (All Calculations so far)

A Bible-based assumption of a few thousand years old was made by Isaac Newton. Einstein believed in a World that was steady-state, ageless. Since then, somewhere in the centre, data obtained from the Universe places the probable response.

## First Ingredient: Quantum mechanics

In the early 20th century, it was realized that the stability of atomic matter could not be explained using the Maxwell equations of classical electrodynamics. This triumph belonged to quantum mechanics. The hydrogen atom was stable because the possible energy states of the electron in the atom are quantized by the rule

where n is an integer, and m is (approximately) the electron mass.

So when the electron changes energy for some reason, say by absorbing or emitting electromagnetic radiation, it can only absorb or emit light of a wavelength corresponding to the difference in the electron’s quantized energy states. The hydrogen emission spectrum is considered the set of light wavelengths produced by hydrogen gas, and there is a corresponding absorption spectrum. The measurement of the wavelengths in the observed hydrogen spectrum was one of quantum mechanics’ remarkable achievements.

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## Second ingredient: Relativity

Spacetime revolution was the other great revolution that started the 20th century of special and general relativity. In special relativity, as the wavelength light source λem travels away from the observer at some velocity v, the observer sees the light at some other lobs of the wavelength, defined by the theory that the speed of light for all observers is the same. The fractional difference between λem and λobs is called the redshift, denoted by the letter z, and is computed from the relative velocity v between the source and observer by

where c is the speed of light. If the source and observer are moving towards one another, the redshift becomes a blue shift and is given by taking v -> -v in above.

## Age of universe from Hubble constant

Stars are made mostly out of hydrogen and helium, and the emission spectrum of the hydrogen atoms in a star in a far away galaxy ought to be the same as that of hydrogen atoms in a tube of gas in a laboratory on Earth. But that’s not what Edwin Hubble found when he compared the emission spectra of different stars and galaxies. Hubble found that the hydrogen gas’s emission wavelengths were redshifted by an amount proportional to their distance from our solar system. Hubble’s Law relates the redshift z to the distance D through

where the empirical constant H0 is called Hubble’s constant.

Hubble’s observation indicated that with a velocity that increases with distance, the stars and galaxies in the Universe hurtle away from each other, as if the whole Universe was expanding, as in a major explosion. When physicists extrapolated that moves backward in time, it suggested that the Universe started very hot and dense and somehow exploded into the vast cold place that we see today. Hubble’s Law was an empirical observation that demanded, and received, very intense attention from modern theoretical physics after it was first proposed in 1924.

The equation of motion

When physicists want to study a given system, they turn to the motion equations for that system. According to the theory of general relativity, the correct equation of motion for describing a Universe is the Einstein equation

relating the curvature of the spacetime in a given Universe to the distribution of energy and momentum in that Universe. The energy-momentum tensor includes all of the energy from all non-gravitational sources such as matter, electromagnetism, or even quantum vacuum energy, as we shall see later.    The standard cosmological solution to the Einstein equation is written in the form of the Friedmann-Robertson-Walker metric

The function a(t) is called the scale factor because it tells us the size of the Universe. The a(t) scale factor and the k constant are both determined by the Universe’s specific type of matter and/or radiation.

For any value of a(t) or k, the gravitational redshift of light z, due to the changing size of the Universe satisfies

where tobs is the time in the Universe that the light is being observed and tem is the time when the light was first emitted.The Hubble parameter H(t) gives the relative rate of change in the scale factor a(t) by

The observed Hubble constant is just the current value of the dynamically evolving Hubble parameter. The uncertainties of the currently observed value of the Hubble constant have been lumped into the parameter h0.

## How old?

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The inverse of the Hubble constant will approximate a simple calculation of the age of the Universe. The age of the universe turns out to be

Current best estimates of h0 are

the Universe is most likely somewhere between 12 and 16 billion years old, at least according to this calculation process.

But note that time is relative, according to relativity. When the time was a quantity that could be calculated, we can guess the amount of time likely to have passed. But about any processes that might have occurred before the notion of time made sense, we can’t say anything. Quantum gravity could be an everlasting stage of the Universe in some way, and the Big Bang could be considered to be the end of eternity and the beginning of time itself.

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How Old Is the Universe?  