I'm studying the age of the universe for a universe dark energy dominated. Using Friedmann equation $$ \left( \frac{\dot{a}}{a} \right)^2 = H_0^2 \left[ \Omega_R \cdot a^{-4} + \Omega_{NR} \cdot a^{-3} + \Omega_k \cdot a^{-2} + \Omega_{\Lambda} \right] $$ with the conditions $\Omega_R=\Omega_{NR}=\Omega_k=0$, $\Omega_\Lambda=1$, I have found the following expression $$ \frac{\dot{a}}{a} = H_0 \quad \to \quad \frac{da}{a} =H_0 \cdot dt $$ If I integrate from the Big-Bang $(t=0, a(0)=0)$ to the present $(t=T, a(T)=1)$ $$ \int_0^1 \frac{da}{a} =H_0 \cdot \int_0^T dt $$ but I have a singularity in the first integral. Does it mean that is impossible to find a universe full made of dark energy from the Big-Bang?
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2 Answers
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Solving the differential equation for the scale factor $a$ in this case we have explicitly $$ \frac{da}{a} =H_0 \cdot dt\qquad \Longrightarrow\qquad a(t) \propto e^{H_0 t},$$ assumming $H_0$ is constant through time. From the expression above you can conclude for this model there is no point in time where the scale factor was exactly zero. You might then speak of infinite negative time or understand this as a model for some fraction of the cosmological history away from $a=0$.
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Please note that the model of the universe corresponding to ΩR=ΩNR=Ωk=0, ΩΛ=1 has no matter and no radiation and no curvature. This is an empty Euclidean universe with no expansion. Therefore Ho and a are and both constants. Since da/dt = 0, then Ho = 0. If a was a variable rather than a constant, then the integral of da/a is ln a, which is infinity. For da = 0, this integral is zero. Since H0 has units 1/t, T = can be chosen as any constant with a time unit.
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$\begingroup$ As long as there is a cosmological constant there will be expansion and $a$ is cannot be constant. $\endgroup$– ohneValCommented Jan 7, 2022 at 9:37
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$\begingroup$ What is your source for concluding that there is NOT no expansion because there is nothing in this universe model to expand? $\endgroup$– BuzzCommented Jan 7, 2022 at 15:42
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$\begingroup$ There is, that is the whole point of a cosmological constant, it corresponds to negative pressure. Solve the differential equation as you can see. Alternatively have a look at de Sitter space. $\endgroup$– ohneValCommented Jan 7, 2022 at 15:47
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$\begingroup$ I suggest you analyze the equation in which 0<ΩNR<<1, and ΩΛ=1-ΩNR. Then explore what happens as ΩNR approaches zero. $\endgroup$– BuzzCommented Jan 7, 2022 at 15:52
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$\begingroup$ Here is another way to look at this. da/a = Ho dt has the solution a =C e^(Ho t), where C is the constant of integration.. Therefore, when t=0, a = C. When t=T, a = C e^(Ho T). Since a cannot be zero, the limits on the integral of da/a cannot be 0 to 1. The limits will be C to C e^(Ho T). This means you cannot integrate da/a from zero. Also, there is only space that expands, but nothing in space to move apart. $\endgroup$– BuzzCommented Jan 7, 2022 at 16:23