Deceleration parameter

Encyclopedia

The deceleration parameter in cosmology is a dimensionless measure of the cosmic acceleration of the expansion of space in a Friedmann–Lemaître–Robertson–Walker universe. It is defined by:

where is the scale factor of the universe and the dots indicate derivatives by proper time

. The expansion of the universe is said to be "accelerating" if is positive (recent measurements suggest it is), and in this case the deceleration parameter will be negative. The minus sign and name "deceleration parameter" are historical; at the time of definition was thought to be positive, now it is believed to be negative.

The Friedmann acceleration equation

can be written as

where is the energy density of the universe, is its pressure, and is the equation of state

of the universe.

This can be rewritten as

by using the first Friedmann equation, where is the Hubble parameter and or depending on whether the universe is hyperspherical

, flat

or hyperbolic

respectively.

The derivative of the Hubble parameter can be written in terms of the deceleration parameter:

Except in the speculative case of phantom energy

(which violates all the energy conditions), all postulated forms of matter yield a deceleration parameter . Thus, any expanding universe should have a decreasing Hubble parameter and the local expansion of space is always slowing (or, in the case of a cosmological constant, proceeds at a constant rate, as in de Sitter space

).

Observations of the cosmic microwave background demonstrate that the universe is very nearly flat, so:

This implies that the universe is decelerating for any cosmic fluid with equation of state greater than (any fluid satisfying the strong energy condition does so, as does any form of matter present in the Standard Model

, but excluding inflation). However observations of distant type Ia supernova

e indicate that is negative; the expansion of the universe is accelerating. This is an indication that the gravitational attraction of matter, on the cosmological scale, is more than counteracted by negative pressure dark energy

, in the form of either quintessence

or a positive cosmological constant

.

Before the first indications of an accelerating universe, in 1998, it was thought that the universe was dominated by dust with negligible equation of state, . This had suggested that the deceleration parameter was equal to one half; the experimental effort to confirm this prediction led to the discovery of possible acceleration.

where is the scale factor of the universe and the dots indicate derivatives by proper time

Proper time

In relativity, proper time is the elapsed time between two events as measured by a clock that passes through both events. The proper time depends not only on the events but also on the motion of the clock between the events. An accelerated clock will measure a smaller elapsed time between two...

. The expansion of the universe is said to be "accelerating" if is positive (recent measurements suggest it is), and in this case the deceleration parameter will be negative. The minus sign and name "deceleration parameter" are historical; at the time of definition was thought to be positive, now it is believed to be negative.

The Friedmann acceleration equation

Friedmann equations

The Friedmann equations are a set of equations in physical cosmology that govern the expansion of space in homogeneous and isotropic models of the universe within the context of general relativity...

can be written as

where is the energy density of the universe, is its pressure, and is the equation of state

Equation of state (cosmology)

In cosmology, the equation of state of a perfect fluid is characterized by a dimensionless number \! w, equal to the ratio of its pressure \! p to its energy density \! \rho: \! w=p/\rho...

of the universe.

This can be rewritten as

by using the first Friedmann equation, where is the Hubble parameter and or depending on whether the universe is hyperspherical

Shape of the Universe

The shape of the universe is a matter of debate in physical cosmology over the local and global geometry of the universe which considers both curvature and topology, though, strictly speaking, it goes beyond both...

, flat

Shape of the Universe

The shape of the universe is a matter of debate in physical cosmology over the local and global geometry of the universe which considers both curvature and topology, though, strictly speaking, it goes beyond both...

or hyperbolic

Shape of the Universe

The shape of the universe is a matter of debate in physical cosmology over the local and global geometry of the universe which considers both curvature and topology, though, strictly speaking, it goes beyond both...

respectively.

The derivative of the Hubble parameter can be written in terms of the deceleration parameter:

Except in the speculative case of phantom energy

Phantom energy

Phantom energy is a hypothetical form of dark energy with equation of state \! w Phantom energy is a hypothetical form of dark energy with equation of state...

(which violates all the energy conditions), all postulated forms of matter yield a deceleration parameter . Thus, any expanding universe should have a decreasing Hubble parameter and the local expansion of space is always slowing (or, in the case of a cosmological constant, proceeds at a constant rate, as in de Sitter space

De Sitter space

In mathematics and physics, a de Sitter space is the analog in Minkowski space, or spacetime, of a sphere in ordinary, Euclidean space. The n-dimensional de Sitter space , denoted dS_n, is the Lorentzian manifold analog of an n-sphere ; it is maximally symmetric, has constant positive curvature,...

).

Observations of the cosmic microwave background demonstrate that the universe is very nearly flat, so:

This implies that the universe is decelerating for any cosmic fluid with equation of state greater than (any fluid satisfying the strong energy condition does so, as does any form of matter present in the Standard Model

Standard Model

The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...

, but excluding inflation). However observations of distant type Ia supernova

Type Ia supernova

A Type Ia supernova is a sub-category of supernovae, which in turn are a sub-category of cataclysmic variable stars, that results from the violent explosion of a white dwarf star. A white dwarf is the remnant of a star that has completed its normal life cycle and has ceased nuclear fusion...

e indicate that is negative; the expansion of the universe is accelerating. This is an indication that the gravitational attraction of matter, on the cosmological scale, is more than counteracted by negative pressure dark energy

Dark energy

In physical cosmology, astronomy and celestial mechanics, dark energy is a hypothetical form of energy that permeates all of space and tends to accelerate the expansion of the universe. Dark energy is the most accepted theory to explain recent observations that the universe appears to be expanding...

, in the form of either quintessence

Quintessence (physics)

In physics, quintessence is a hypothetical form of dark energy postulated as an explanation of observations of an accelerating universe. It has been proposed by some physicists to be a fifth fundamental force...

or a positive cosmological constant

Cosmological constant

In physical cosmology, the cosmological constant was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe...

.

Before the first indications of an accelerating universe, in 1998, it was thought that the universe was dominated by dust with negligible equation of state, . This had suggested that the deceleration parameter was equal to one half; the experimental effort to confirm this prediction led to the discovery of possible acceleration.