The deceleration parameter ''
'' in cosmology is a
dimensionless
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
measure of the
cosmic acceleration of the
expansion of space
The expansion of the universe is the increase in distance between any two given gravitationally unbound parts of the observable universe with time. It is an intrinsic expansion whereby the scale of space itself changes. The universe does not ex ...
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
In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The proper time interval b ...
. The expansion of the universe is said to be "accelerating" if
(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 expected to be negative, so a minus sign was inserted in the definition to make
positive in that case. Since the evidence for the
accelerating universe
Observations show that the expansion of the universe is accelerating, such that the velocity at which a distant galaxy recedes from the observer is continuously increasing with time. The accelerated expansion of the universe was discovered duri ...
in the 1998–2003 era, it is now believed that
is positive therefore the present-day value
is negative (though
was positive in the past before dark energy became dominant). In general
varies with cosmic time, except in a few special cosmological models; the present-day value is denoted
.
The
Friedmann acceleration equation can be written as
:
where the sum
extends over the different components, matter, radiation and dark energy,
is the equivalent mass density of each component,
is its pressure, and
is the
equation of state
In physics, chemistry, and thermodynamics, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or inter ...
for each component. The value of
is 0 for non-relativistic matter (baryons and dark matter), 1/3 for radiation, and −1 for a
cosmological constant
In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant,
is the constant coefficient of a term that Albert Einstein temporarily added to his field e ...
; for more general
dark energy
In physical cosmology and astronomy, dark energy is an unknown form of energy that affects the universe on the largest scales. The first observational evidence for its existence came from measurements of supernovas, which showed that the univer ...
it may differ from −1, in which case it is denoted
or simply
.
Defining the critical density as
:
and the density parameters
, substituting
in the acceleration equation gives
:
where the density parameters are at the relevant cosmic epoch.
At the present day
is negligible, and if
(cosmological constant) this simplifies to
:
where the density parameters are present-day values; with Ω
Λ + Ω
m ≈ 1, and Ω
Λ = 0.7 and then Ω
m = 0.3, this evaluates to
for the parameters estimated from the
Planck spacecraft
''Planck'' was a space observatory operated by the European Space Agency (ESA) from 2009 to 2013, which mapped the anisotropies of the cosmic microwave background (CMB) at microwave and infrared frequencies, with high sensitivity and small an ...
data.
(Note that the CMB, as a high-redshift measurement, does not directly measure
; but its value can be inferred by fitting cosmological models to the CMB data, then calculating
from the other measured parameters as above).
The time derivative of the
Hubble parameter
Hubble's law, also known as the Hubble–Lemaître law, is the observation in physical cosmology that galaxies are moving away from Earth at speeds proportional to their distance. In other words, the farther they are, the faster they are moving a ...
can be written in terms of the deceleration parameter:
:
Except in the speculative case of
(which violates all the energy conditions), all postulated forms of mass-energy yield a deceleration parameter
Thus, any non-phantom universe should have a decreasing Hubble parameter, except in the case of the distant future of a
Lambda-CDM model
The ΛCDM (Lambda cold dark matter) or Lambda-CDM model is a parameterization of the Big Bang cosmological model in which the universe contains three major components: first, a cosmological constant denoted by Lambda ( Greek Λ) associated wit ...
, where
will tend to −1 from above and the Hubble parameter will asymptote to a constant value of
.
The above results imply that the universe would be 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
The Standard Model of particle physics is the theory describing three of the four known fundamental forces ( electromagnetic, weak and strong interactions - excluding gravity) in the universe and classifying all known elementary particles. It ...
, but excluding inflation). However observations of distant
type Ia supernova
A Type Ia supernova (read: "type one-A") is a type of supernova that occurs in binary systems (two stars orbiting one another) in which one of the stars is a white dwarf. The other star can be anything from a giant star to an even smaller wh ...
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 the negative pressure of
dark energy
In physical cosmology and astronomy, dark energy is an unknown form of energy that affects the universe on the largest scales. The first observational evidence for its existence came from measurements of supernovas, which showed that the univer ...
, in the form of either
quintessence or a positive
cosmological constant
In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant,
is the constant coefficient of a term that Albert Einstein temporarily added to his field e ...
.
Before the first indications of an accelerating universe, in 1998, it was thought that the universe was dominated by matter with negligible pressure,
This implied that the deceleration parameter would be equal to
, e.g.
for a universe with
or
for a low-density zero-Lambda model. The experimental effort to discriminate these cases with supernovae actually revealed negative
, evidence for cosmic acceleration, which has subsequently grown stronger.
References
{{reflist
Physical cosmology