Hubbles law

The Hubble–Lemaître Law is a foundational observational principle in physical cosmology which states that galaxies are receding from the Earth at velocities proportional to their distances. This linear relationship between recessional velocity and distance provides one of the strongest lines of evidence for the expansion of the universe and underpins the modern Big Bang model. The law expresses the general behaviour of galaxies participating in the Hubble flow, the large-scale motion of cosmic structures driven not by movement through space but by the expansion of spacetime itself.
The law is often illustrated using the analogy of raisins in a rising loaf of bread, where each raisin moves away from every other as the dough expands. In this model, a raisin twice as far from a fixed point moves away twice as fast, mirroring the linear relationship central to Hubble–Lemaître dynamics.

Historical Background and Early Theoretical Foundations

Prior to Edwin Hubble’s 1929 observations, several scientists had already laid the mathematical groundwork for an expanding universe. In 1922 Alexander Friedmann derived what are now known as the Friedmann equations, demonstrating that the universe could be either expanding or contracting. The equations were solutions to Einstein’s field equations of general relativity under the assumption of a homogeneous and isotropic universe, consistent with the cosmological principle. Friedmann’s solutions included a calculable expansion rate, corresponding to what would later be termed the Hubble constant.
Work by Vesto Slipher in the 1910s provided the first significant observational data. Beginning in 1912, Slipher measured Doppler shifts of spiral nebulae—now known to be galaxies—and found that most exhibited redshifts, indicating motion away from the Earth. Although he did not interpret these measurements in cosmological terms, his data later formed the empirical basis for demonstrating cosmic expansion. By 1917, Slipher had measured dozens of such redshifts.
In the early 1920s Carl Wilhelm Wirtz published analyses showing that fainter, smaller galaxies tended to have larger redshifts, suggesting a link between distance and recessional velocity. These findings anticipated the proportional relationship later formalised by Hubble.
The Belgian physicist and astronomer Georges Lemaître independently derived the concept of universal expansion in his 1927 paper, in which he identified a proportionality between galaxy distance and recessional velocity. He also estimated a numerical value for this proportionality. His work, originally published in French in a low-circulation journal, did not initially receive broad attention.

Development of the Observational Law

The classical formulation of the law was published by Edwin Hubble in 1929. By combining Slipher’s redshift measurements with his own observations of distances to galaxies—established largely through the identification of Cepheid variable stars following the method pioneered by Henrietta Swan Leavitt—Hubble plotted recessional velocity against distance. His data revealed a roughly linear trend: galaxies farther away exhibited greater recession velocities. From this he determined an approximate value for the proportionality constant, initially about 500 km/s per megaparsec. Although this value was later found to be significantly higher than the accepted modern estimate, the linear relationship itself proved robust.
Hubble’s original diagram visualising this relation is now known as the Hubble diagram. Its positive linear slope provides a clear and simple representation of cosmic expansion and remains a key tool in observational cosmology.
In the English translation of Lemaître’s 1927 paper published in 1931, an equation relating to the expansion rate was omitted. Subsequent research revealed that Lemaître himself had removed the constant in the translation, leading to early under-recognition of his contribution to the discovery of universal expansion.

Mathematical Formulation and Definition of the Hubble Constant

The Hubble–Lemaître Law is expressed mathematically as:
v = H₀D
where:

• v is the galaxy’s recessional velocity,
• D is its proper distance from the observer, and
• H₀ is the Hubble constant, the proportionality constant describing the current rate of expansion.

Distance in this context refers to proper distance, which changes over cosmic time because of the dynamical nature of spacetime. This differs from comoving distance, which remains fixed with respect to the cosmic coordinate grid.
Although the Hubble constant is labelled a constant, the Hubble parameter—its general, time-dependent form—varies as the universe evolves. The constant therefore represents only the present-day value of the parameter. This distinction highlights an inherent subtlety: the expansion rate of the universe is not fixed but changes over time.
The units of the Hubble constant are conventionally expressed in km/s per megaparsec (km/s·Mpc⁻¹), signifying the velocity at which an object recedes per megaparsec of distance. Simplifying the units reveals a quantity with the dimensions of a frequency, making its reciprocal the Hubble time, approximately 14.4 billion years, representing the timescale for expansion assuming a constant rate.
The constant may also be interpreted relatively. For instance, the approximate expansion rate means that on cosmic scales an unbound structure expands by around 7% per billion years at the current epoch.

Cepheid Variables and Distance Measurements

A major breakthrough enabling the formulation of the law came from Hubble’s ability to measure distances to spiral nebulae. Using observations from the Mount Wilson Observatory, he identified Cepheid variable stars in these nebulae. Cepheid variables follow a well-defined relation between luminosity and pulsation period, enabling determination of their intrinsic brightness and thus distance.
These measurements demonstrated that many nebulae lay far beyond the Milky Way, resolving the long-standing Great Debate of 1920 between Harlow Shapley and Heber Curtis regarding the scale of the universe. This discovery allowed the nebulae to be correctly reclassified as external galaxies.

Redshift Measurements and the Role of Slipher and Humason

Hubble’s correlation between recessional velocity and distance was possible only through the redshift data collected primarily by Vesto Slipher and later by Milton Humason. Slipher’s spectral data provided the velocity component of the relationship, while Humason contributed new high-precision redshift measurements for fainter and more distant galaxies.
The observed redshift is interpreted as a cosmological redshift, not a classical Doppler shift. It arises from the stretching of light waves as space itself expands. At sufficiently large distances, the cosmological component dominates any peculiar velocities associated with local gravitational interactions.

Abandonment and Revival of the Cosmological Constant

The publication of Hubble’s results had profound consequences for theoretical physics. Einstein, who had initially modified his field equations by adding the cosmological constant to enforce a static universe, abandoned the constant after the discovery of expansion. He later referred to this modification as his “greatest mistake”. With a dynamic universe confirmed, the simple, unaltered Einstein field equations were recognised as describing an expanding cosmos.
However, in the late twentieth century and early twenty-first century, the cosmological constant returned to prominence as a possible explanation for dark energy, the phenomenon driving the accelerating expansion of the universe.

Interpretation and Significance

The Hubble–Lemaître Law provides the first and most direct observational evidence for an expanding universe. The linear relationship between redshift and distance indicates that space is expanding uniformly on the largest scales. This expansion is not due to galaxies physically moving through space at great speeds, but rather to the metric expansion of space itself, carrying galaxies apart.
The law served as a cornerstone for the development of the Big Bang theory, offering insight into the origin and evolution of the cosmos. It continues to underpin modern observational cosmology, especially in precision measurements of the Hubble constant aimed at resolving the current “Hubble tension”—the discrepancy between values derived from early-universe observations and those inferred from the late universe.