$$$$A=\left(\frac{\Omega_{m,0}}{\Omega_{\Lambda,0}}\right)^{1/3}$$$$\eta(t)=\frac{\tau}{A}\int_0^{t/\tau}\frac{dx}{\sinh^{2/3}{x}}=\frac{\tau}{A}\int_0^{\sinh{(t/\tau)}}\frac{dy}{y^{2/3}\sqrt{1+y^2}}$$$$\eta(t)=\frac{3\tau}{A}\,{}_2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};-\sinh^2{\frac{t}{\tau}}\right)\sinh^{1/3}{\frac{t}{\tau}}$$$$\eta\approx.\frac{3\tau}{A}\left(\frac{t}{\tau}\right)^{1/3}$$$$\frac{\Gamma(\frac{1}{6})\Gamma(\frac{1}{3})}{2\sqrt{\pi}}\frac{\tau}{A}.$$$$t(a)=\tau\,\text{arcsinh}\,\left(\frac{a}{A}\right)^{3/2}.$$$$\eta(a)=\frac{3\tau}{A}\left(\frac{a}{A}\right)^{1/2}{}_2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};-\left(\frac{a}{A}\right)^3\right)$$Thanks for your answer. 1.0 & 20.08 & 0.1495 \\ t & \eta & a\\ of the scale factor a in terms of the time t parametrized by the conformal time η as: a = q0 2q0 −1 (1 −cosη), (125) t = q0 2q0 −1 (η −sinη). The Friedmann equations start with the simplifying assumption that the universe is spatially homogeneous and Einstein's equations now relate the evolution of this scale factor to the pressure and energy of the matter in the universe. So there is no nice formula for Thanks for contributing an answer to Physics Stack Exchange! Can you please explain a little bit more how can I get the value of $\dot a(\eta)$ or $a'(\eta)$ numerically from your process.

Anybody can ask a question $$ a(t) = \left\{ \frac{Ω_{m,0}}{Ω_{Λ,0}} \sinh^2 \left[\frac{3}{2} \sqrt{Ω_{Λ,0}} H_0(t - t_0)\right] \right\}^{1/3}, $$@Pulsar Can you please give me an explanation here?Do you know the formula for conformal time? It will be $$ t - t_0 $$. Stack Exchange network consists of 177 Q&A communities including Start here for a quick overview of the site The best answers are voted up and rise to the top For example, one can form a linear combination of such terms By using our site, you acknowledge that you have read and understand our Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. Detailed answers to any questions you might have This term originally was used as a means to determine the The first Friedmann equation is often seen in terms of the present values of the density parameters, that isThe Friedmann equations can be solved exactly in presence of a Note that this solution is not valid for domination of the cosmological constant, which corresponds to an If the matter is a mixture of two or more non-interacting fluids each with such an equation of state, then 4.0 & 31.83 & 0.3813 \\ 14.0 & 47.36 & 1.013 \\ By making a change of variables, you can get a nicer expression, still involving a hypergeometric function, which is real. Expecting your kind feedback. When I do the integral with Mathematica I get a messy hypergeometric function, so I doubt that there is a nice expression for the scale factor in terms of the conformal time. Evaluating the conformal time gives $\eta=47.17$ billion years; thus the size of the observable universe is currently 47.17 billion light years. 5.0 & 34.25 & 0.4458 \\ 11.0 & 44.06 & 0.8142 \\ This time is usually taken to be zero.$H_0=67.74\,\text{km}\cdot\text{s}^{-1}\cdot\text{Mpc}^{-1}=0.06923\,\text{Gy}^{-1}$$$\begin{matrix} Anybody can answer 6.0 & 36.35 & 0.5079 \\ 15.0 & 48.32 & 1.084 \\ This duality maps expanding universes into contracting ones (filled with somewhat peculiar matter) and vice-versa.