<p>Is it possible for the Hubble parameter of an /expanding/ universe made of radiation, matter and cosmological constant to be increasing with time? I'm trying to figure out if any form of scale factor <span class="math-container">$a(t)$</span> would make <span class="math-container">$\frac{\dot{a}}{a}$</span> increases with time, which would mean the critical density (<span class="math-container">$\frac{3c^2}{8 \pi G}H^2)$</span> will have to be increasing in an expanding universe. Would that be possible at all?</p>

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2022-12-09T12:55:24.053Z
Unless I misunderstand your question, it's not possible. From the Friedman equation, you can express the relation between the Hubble parameter $H$, the density parameters $\Omega_{r,m,k,\Lambda}$, and the scale factor $a$ as
$$
\frac{H^2(a)}{H_0^2} = 
\frac{\Omega_\mathrm{r}}{a^4} +
\frac{\Omega_\mathrm{M}}{a^3} +
\frac{\Omega_k}{a^2} +
      \Omega_\Lambda,
$$
where $H_0$ is the present-day value of $H$.

For an expanding Universe, $a\rightarrow\infty$, so $$
H\rightarrow\sqrt{\Omega_\Lambda},
$$
which is constant for a cosmological *constant*.
pela
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2022-12-10T19:03:05.637Z
Interesting point. Yeah I've convinced myself from a math point of view, any power law $a\sim t^n$ will always have $\dot{a}/a$ decreasing with time, and if it's exponential function (cosmological constant only universe) then it'll be constant, which is exactly your conclusion from a different perspective.
ABC
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2022-12-10T20:07:10.630Z
This actually triggers another question, but it's too long for the comment, so I'll post as a new question.
ABC
<cosmology><expansion><hubble-constant>
Critical density and Hubble parameter
ABC
# Critical density and Hubble parameter

(Asked by: username_0 on 2022-12-08)

Is it possible for the Hubble parameter of an /expanding/ universe made of radiation, matter and cosmological constant to be increasing with time? I'm trying to figure out if any form of scale factor $a(t)$ would make $\frac{\dot{a}}{a}$ increases with time, which would mean the critical density ($\frac{3c^2}{8 \pi G}H^2)$ will have to be increasing in an expanding universe. Would that be possible at all?

**Tags:** cosmology, expansion, hubble-constant
> **Comments:**
> **username_1**: Unless I misunderstand your question, it's not possible. From the Friedman equation, you can express the relation between the Hubble parameter $H$, the density parameters $\Omega_{r,m,k,\Lambda}$, and the scale factor $a$ as $$ \frac{H^2(a)}{H_0^2} =  \frac{\Omega_\mathrm{r}}{a^4} + \frac{\Omega_\mathrm{M}}{a^3} + \frac{\Omega_k}{a^2} +       \Omega_\Lambda, $$ where $H_0$ is the present-day value of $H$.  For an expanding Universe, $a\rightarrow\infty$, so $$ H\rightarrow\sqrt{\Omega_\Lambda}, $$ which is constant for a cosmological *constant*.  
> **username_0**: Interesting point. Yeah I've convinced myself from a math point of view, any power law $a\sim t^n$ will always have $\dot{a}/a$ decreasing with time, and if it's exponential function (cosmological constant only universe) then it'll be constant, which is exactly your conclusion from a different perspective.  
> **username_0**: This actually triggers another question, but it's too long for the comment, so I'll post as a new question.  

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## 0 answers

