Hubble's Constant And Its Variation With Time

This article on Hubble's Constant is a guest article by Ariana Vlad, senior at the International Computers High School of Bucharest, Romania, where she focuses on studying Physics and Mathematics.

A Great Discovery

One of the cornerstones discoveries in modern Cosmology was the Universal Expansion, which dared the scientists to find a new dynamic model for the Universe. A comprehensive statement about this phenomenon takes the form of Hubble-Lemaître law. It states that for objects situated at distances of tens and hundreds of mega-parsecs from Earth a redshift is observed, resulting from a receding velocity approximately proportional to the distances from the objects to us. In a nutshell, the farther the celestial object, the faster it’s moving away from us.

This proportionality constant is named Hubble's constant, and, when having particular applications in cosmology, its reciprocal is used as Hubble time. 

Hubble's Constant
Edwin Hubble

First Steps

Edwin Hubble had been working on Cepheids and distant nebulae data in the 1920s, a time when it was widely believed that our home galaxy, the Milky Way, constituted the entire Universe. His data on the Andromeda galaxy -correctly- placed the celestial body well outside the Milky Way, while measurements from various distant galaxies became proof of his own law. With one data collecting and analysis experiment, Hubble created the base for the modern Cosmology, as we know it today.

Great physicists of the 20th century worked on this topic. Alexander Friedmann was the first one to derive a set of equations (widely known as the eponym equations) which not only showed that the Universe might be expanding but also presented the explicit formula of the receding speed. Five years later, Lemaître discovered the proportionality between the velocity of a certain celestial object and its distance to Earth. He estimated a potential value for this constant, which was later corrected by Hubble. 

Hubble's Constant Variation
Hubble's constant as derived over time. The first values were calculated using Hubble’s data on distant galaxies. The first and second points are from papers of Georges Lemaître and Howard Robertson, while the fourth one is from Hubble. As the plot suggests, the known value today is around 70 (km/s)/Mpc. 

A Misleading Name

The mathematical formula for Hubble's constant is the time derivative of the scale factor divided by itself. Because of the expansion of the Universe, celestial objects get farther away like two points on an inflating balloon. The scale factor is a time-dependent parameter that relates the distance between the two objects. Because of the scale factor’s expression, the Hubble's constant is also time-dependent. The change of its value over time can’t be directly linked through proportionality with the rate of acceleration of the Universe’s expansion. Mathematically, the relation is more complex. The original name is, therefore, misleading, and this physical quantity is now more appropriately referred to as the Hubble parameter.

Hubble's Constant
Accelerating expansion of the Universe doesn’t imply the Hubble's “constant” is changing toward a bigger or smaller value. Even though scientists suspect the Universe was decelerating its expansion 5 billion years ago, the plot of this parameter is strictly decreasing over 9 billion years, meaning the relationship between the physical parameter and the phenomenon is more complex than one would first believe.

The data collected did not show any variations in the known value because the rate of change is very slow. It would take 200 million years to change its current value by just one unit, which makes the “constant” approach feasible in some physical models.

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