Scale Factor in a Universe with Dark Energy
Mikhail V. Sazhin1, Olga S. Sazhina1, Urmila Chadayammuri2
1
Sternberg Astrophysics Institute, Moscow State University, Russia
2
Brown University, Providence, Rhode Island
The Friedman Equations
Cosmology attempts to answer questions about the origin, evolution
and future direction of the Universe. The standard model, which
explains most of the observed properties of the Universe, starts with
a Big Bang – an explosion of an infinitely dense point or singularity
– about 13.7 billion years ago. Within the first 10-32 seconds after the
Bang, it describes a period of inflation. Thereafter, the Universe was
understood to expand under the decelerating influence of gravity.
The expansion of the Universe is described by the value of the scale
factor, a(t), which measures how the distance between two points in
space changes as a function of time. The first two time-derivatives of
the scale factor are described by Friedmann equations, and the first
of these upon integration yields a(t) itself.
However, for problems such as structure formation, cosmologists
prefer to deal with conformal time as opposed to physical time. This
factors out the Hubble expansion itself. We define an infinitesimal
unit of conformal time as
which in the standard model, with ΩΛ = 0, still yields a simple power
law to describe the scale factor in terms of η.
Introducing Dark Energy
– matter domination, current and dark energy domination – can be
directly transcribed from their physical time equivalents. We note
that each of these approximations can be obtained by entering
appropriate boundary conditions to the following equation:
The scale factor is thus a function of the elliptic cosine with
argument y = 31/4ΩΛ1/6Ωm1/3H0η and constant modulus k [3].
Generalizing State Parameter of Dark Energy
The above calculations have all been made under the assumption
that dark energy is a cosmological constant. However, it could well
be a quintessence, that is, it may have a time-dependent parameter of
state. We managed to express conformal time as a function of the
scale factor and parameter of state w:
Fig 3: Evolution of the scale factor. The exact value of the scale
factor is represented by the solid line, the extrapolation of the
approximation for dark energy domination by the dotted line, and
that of the approximation for matter domination by the dashed line.
where F is the Gauss hypergeometric function. Inverting this
function would allow for a direct equation for the scale factor. We
used Taylor expansions of F in terms of the gamma function to
obtain approximated solutions for lower and upper limits of
conformal time, but a generalized formula is still in the works [3].
A single equation for all values of time, physical or conformal,
greatly aids analysis. We can use the first formula and it's derivatives
to find conformal time values corresponding to moments in
cosmological history, such as the beginning of matter domination,
the switch from decelerated to accelerated expansion, the present
and dark energy domination (t → ∞). We thus set the domain of
conformal time for the entire history of the Universe as:
In light of the 1998 discovery of the accelerated expansion of the
Universe [1], however, we can no longer assume the cosmological
constant to have a non-zero contribution to the density of the
Universe. The acceleration is attributed to the action of dark energy.
An exact equation for the scale factor will prove useful in the
analysis of results from upcoming dark energy investigations.
NASA's WFIRST and ESA's Euclid missions, both expected to
launch around 2019, will be mapping the expansion of the Universe
to a greater accuracy and covering a greater time interval than ever
before. WFIRST will be using three separate methods – Type Ia
Supernova, Baryon Acoustic Oscillations and Weak Gravitational
Lensing, while Euclid will be using the latter two. Comparing
observed results to theoretically predicted ones will allow us to
determine the parameter of state of dark energy, thus helping us
better understand its nature.
Fig 2: The above graph plots the percentage deviation of the scale factor for an
arbitrary w from that for a cosmological constant. Thus it is currently about
15% and blows up at dark-energy domination.
Fig 1: Scale factor against the age of the Universe [2]
The cosmological constant is the time-independent, and thus
simplest, way to incorporate dark energy into the Friedmann
equations. However, to the best of our knowledge, a single equation
for the scale factor as a function of conformal time has yet to be
derived. Approximated solutions for various epochs of the Universe
54% for its first derivative. This means that the approximations
cannot be used for any experiments requiring greater precisions.
Importance and Application
The first step in assessing the importance of an exact equation would
be calculating the errors associated with the approximations. We
calculate this to be about 25% for the scale factor itself and about
References
[1] Riess, A.G. et al. Observational Evidence from Supernovae for
an Accelerating Universe and a Cosmological Constant. The
Astronomical Journal (1998) Volume 116, Issue 3, pp. 1009-1038
[2] Wilkinson Microwave Anisotropy Probe. Available from:
http://map.gsfc.nasa.gov/media/990350/990350b.jpg
[3] Sazhin, M. V., Sazhina, O. S., Chadayammuri, U. Scale Factor in
a Universe with Dark Energy. arxiv.org/abs/1109.2258 (2011)