Introduction to Cosmological Constant, Vacuum Energy, and Dark Energy

Over a century ago, a solution to the Einstein field equations was derived, allowing models of the entire universe to be created. These models suggested that the universe should eventually contract due to its own gravity. Einstein, who believed the universe should be static and unchanging, added a constant scalar term to the field equations to create such a static model. This term, known as the cosmological constant (Λ), modified the metric of spacetime (gμν) to account for this.

Exact Formulation of the Cosmological Constant

The Einstein field equations, modified with the cosmological constant, are given by:

R}_{mu u}-12Rg}_{mu u}8πGc4T}_{mu u}

In this equation, the

R}_{mu u}

are the Ricci curvature tensor,

R is the Ricci scalar,

g}_{mu u} is the spacetime metric, and

T}_{mu u} is the energy-stress tensor containing all information on mass and energy. Essentially, the modified spacetime curvature models the gravitational field due to mass and energy, altered by the cosmological constant.

Resurrection of the Cosmological Constant

Shortly after the 1920s, Hubble observed that the universe was expanding, not static, which led to the abandonment of the cosmological constant. Einstein referred to this addition as ‘his greatest blunder’. However, in 1998, the cosmological constant was resurrected when it was observed that the universe was not only expanding but accelerating. This acceleration cannot be modeled without including the cosmological constant (Λ).

The Friedmann-Lemaitre-Robertson-Walker metric from solving the Einstein equations for an isotropic, homogeneous, spherically symmetric universe of a perfect fluid provides the Friedmann equations:

a2˙a28πGρ3 Λ23H2

a2¨a2-4πGρ3-3pΛc23c4

where a is the scale factor, ρ is density, p is pressure, and H is the Hubble parameter. The second equation is the acceleration equation, where density and pressure (which are always positive) reduce acceleration, while Λ increases it.

Physical Meaning of the Cosmological Constant

Physically, it is not known what Λ actually is. It was termed 'dark energy' because it acts somewhat like negative density or pressure, akin to 'anti-gravity' negative energy. However, since it is a scalar constant independent of the energy-stress tensor, the similarity ends here.

Vacuum Energy: A Candidate for Dark Energy

Vacuum energy is the ground state of a quantum field; it need not be zero despite being the ground state. Since this is also directly proportional to volume, it can be considered to have a constant density and is a good candidate for 'dark energy'. However, calculations have shown that vacuum energy is off by a factor of 10120, making it the worst prediction in physics. Despite this, many still believe there is a connection between dark energy and vacuum energy.

In conclusion, both the cosmological constant and vacuum energy are intriguing candidates for dark energy. While the cosmological constant provides a theoretical framework, the vacuum energy offers a quantum mechanical interpretation, both still waiting for a more complete understanding of the universe's dark side.