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 – matter domination, current and dark energy domination – can be 54% for its first derivative. This means that the approximations directly transcribed from their physical time equivalents. We note cannot be used for any experiments requiring greater precisions. Cosmology attempts to answer questions about the origin, evolution that each of these approximations can be obtained by entering and future direction of the Universe. The standard model, which appropriate boundary conditions to the following equation: 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 The scale factor is thus a function of the elliptic cosine with 1/4 1/6 1/3 Bang, it describes a period of inflation. Thereafter, the Universe was argument y = 3 ΩΛ Ωm H0η and constant modulus k [3]. understood to expand under the decelerating influence of gravity. The expansion of the Universe is described by the value of the scale Generalizing State Parameter of Dark Energy 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 above calculations have all been made under the assumption the scale factor are described by Friedmann equations, and the first that dark energy is a cosmological constant. However, it could well Fig 3: Evolution of the scale factor. The exact value of the scale of these upon integration yields a(t) itself. be a quintessence, that is, it may have a time-dependent parameter of factor is represented by the solid line, the extrapolation of the However, for problems such as structure formation, cosmologists state. We managed to express conformal time as a function of the approximation for dark energy domination by the dotted line, and prefer to deal with conformal time as opposed to physical time. This scale factor and parameter of state w: that of the approximation for matter domination by the dashed line. factors out the Hubble expansion itself. We define an infinitesimal unit of conformal time as A single equation for all values of time, physical or conformal, where F is the Gauss hypergeometric function. Inverting this greatly aids analysis. We can use the first formula and it's derivatives which in the standard model, with ΩΛ = 0, still yields a simple power function would allow for a direct equation for the scale factor. We to find conformal time values corresponding to moments in law to describe the scale factor in terms of η. used Taylor expansions of F in terms of the gamma function to cosmological history, such as the beginning of matter domination, obtain approximated solutions for lower and upper limits of the switch from decelerated to accelerated expansion, the present Introducing Dark Energy conformal time, but a generalized formula is still in the works [3]. 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 An exact equation for the scale factor will prove useful in the constant to have a non-zero contribution to the density of the analysis of results from upcoming dark energy investigations. Universe. The acceleration is attributed to the action of dark energy. 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 References 15% and blows up at dark-energy domination. [1] Riess, A.G. et al. Observational Evidence from Supernovae for Fig 1: Scale factor against the age of the Universe [2] an Accelerating Universe and a Cosmological Constant. The The cosmological constant is the time-independent, and thus Importance and Application Astronomical Journal (1998) Volume 116, Issue 3, pp. 1009-1038 simplest, way to incorporate dark energy into the Friedmann [2] Wilkinson Microwave Anisotropy Probe. Available from: equations. However, to the best of our knowledge, a single equation The first step in assessing the importance of an exact equation would http://map.gsfc.nasa.gov/media/990350/990350b.jpg for the scale factor as a function of conformal time has yet to be be calculating the errors associated with the approximations. We [3] Sazhin, M. V., Sazhina, O. S., Chadayammuri, U. Scale Factor in derived. Approximated solutions for various epochs of the Universe calculate this to be about 25% for the scale factor itself and about a Universe with Dark Energy. arxiv.org/abs/1109.2258 (2011)