<p>The <a href="http://en.wikipedia.org/wiki/Critical_density_%28cosmology%29#Density_parameter">critical density of the universe</a> is 
$$\rho_c=\frac{3H_0^2}{8\pi G},$$
with $H_0=67.8 \frac{\mbox{km}/\mbox{s}}{\mbox{Mpc}}$ the <a href="http://en.wikipedia.org/wiki/Hubble%27s_law">Hubble constant</a> and $G=6.673848\cdot 10^{-11}\frac{\mbox{m}^3}{\mbox{kg }\mbox{s}^2}$ Newton's <a href="http://en.wikipedia.org/wiki/Gravitational_constant">gravitational constant</a>.
Hence with one <a href="http://en.wikipedia.org/wiki/Parsec">parsec</a> $1\mbox{pc}=3.0857×10^{16} \mbox{m}$,
$$\rho_c=\frac{3\cdot (67.8\cdot 10^3\cdot \frac{\mbox{m}/\mbox{s}}{3.0857×10^{22} \mbox{m}})^2}{8\pi\cdot 6.673848\cdot 10^{-11}\frac{\mbox{m}^3}{\mbox{kg}\mbox{s}^2}}=8.635 \cdot 10^{-27}\frac{\mbox{kg}}{\mbox{m}^3}.$$</p>

<p>Ordinary matter content is $4.9\%$, the universe has total density close to critical density, hence ordinary matter density is
$$0.049\cdot 8.635 \cdot 10^{-27}\frac{\mbox{kg}}{\mbox{m}^3}=4.23 \cdot 10^{-28}\frac{\mbox{kg}}{\mbox{m}^3}.$$
(The Hubble constant is not precisely known, so the calculated density depends a bit on the precise value.)</p>

<p><strong>Density of water:</strong>
The density of water is about $10^3 \mbox{kg}/\mbox{m}^3$.
That's a factor $4.23 \cdot 10^{31}$ denser than the average universe.
The space needs to have been reduced by a factor $\sqrt[3]{4.23 \cdot 10^{31}}=3.48\cdot 10^{10}$ in each if the three spatial dimensions. That's the inverse of the <a href="http://en.wikipedia.org/wiki/Scale_factor_%28cosmology%29">scale factor</a>, hence corresponds to a redshift of $z=3.48\cdot 10^{10}$.</p>

<p>Using the <a href="http://adsabs.harvard.edu/abs/2006PASP..118.1711W">Cosmology Calculator</a> on <a href="http://www.astro.ucla.edu/~wright/CosmoCalc.html">this website</a>, the cosmological parameters $H_0 = 67.11$ km/s/Mpc (slightly different to the above value),
$\Omega_{\Lambda} = 0.6825$ provided by the <a href="http://en.wikipedia.org/wiki/Planck_%28spacecraft%29">Planck</a> project, and setting $\Omega_M = 1- \Omega_{\Lambda} = 0.3175$ the age of the universe was <strong>0.022 seconds</strong> at the redshift $z=3.48\cdot 10^{10}$, hence when it was of the average density of water.</p>

<p><strong>Density of air at sea level:</strong>
The <a href="http://en.wikipedia.org/wiki/Density_of_air">density of air</a> at sea level is $1.225 \frac{\mbox{kg}}{\mbox{m}^3}$. The corresponding redshift is about $z=1.423\cdot 10^9$ yielding an age of the universe of <strong>12.68 seconds</strong>.</p>

CC BY-SA 3.0
2014-04-19T01:25:40.517Z
I'm guessing it was still very hot at 12 seconds?
Florin Andrei
CC BY-SA 3.0
2014-04-20T21:50:48.843Z
About 100 gigakelvin, or $10^{11}$  K (http://en.wikipedia.org/wiki/Big_Bang_nucleosynthesis)
Gerald
CC BY-SA 3.0
2014-04-22T06:51:27.007Z
Absolutely amazing! I'm surprised that there's such a difference between the density of air and water.
CJ Dennis
CC BY-SA 3.0
2014-04-18T21:13:49.010Z
2014-04-18T21:13:49.010Z
Gerald
<p>How long did it take after the Big Bang for the average density of the universe to reach one Earth atmosphere at sea level? How about the density of water? Is there a chart of densities that relate to everyday objects (so easier to understand)?</p>

<cosmological-inflation>
What was the age of the universe when the average density was one atmosphere?
CJ Dennis
# What was the age of the universe when the average density was one atmosphere?

(Asked by: username_0 on 2014-04-18)

How long did it take after the Big Bang for the average density of the universe to reach one Earth atmosphere at sea level? How about the density of water? Is there a chart of densities that relate to everyday objects (so easier to understand)?

**Tags:** cosmological-inflation

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## 1 answers

### Answer by username_1 on 2014-04-18. Score: 6 | Accepted answer

The [critical density of the universe](http://en.wikipedia.org/wiki/Critical_density_%28cosmology%29#Density_parameter) is
$$\rho\_c=\frac{3H\_0^2}{8\pi G},$$
with $H\_0=67.8 \frac{\mbox{km}/\mbox{s}}{\mbox{Mpc}}$ the [Hubble constant](http://en.wikipedia.org/wiki/Hubble%27s_law) and $G=6.673848\cdot 10^{-11}\frac{\mbox{m}^3}{\mbox{kg }\mbox{s}^2}$ Newton's [gravitational constant](http://en.wikipedia.org/wiki/Gravitational_constant).
Hence with one [parsec](http://en.wikipedia.org/wiki/Parsec) $1\mbox{pc}=3.0857×10^{16} \mbox{m}$,
$$\rho\_c=\frac{3\cdot (67.8\cdot 10^3\cdot \frac{\mbox{m}/\mbox{s}}{3.0857×10^{22} \mbox{m}})^2}{8\pi\cdot 6.673848\cdot 10^{-11}\frac{\mbox{m}^3}{\mbox{kg}\mbox{s}^2}}=8.635 \cdot 10^{-27}\frac{\mbox{kg}}{\mbox{m}^3}.$$

Ordinary matter content is $4.9\%$, the universe has total density close to critical density, hence ordinary matter density is
$$0.049\cdot 8.635 \cdot 10^{-27}\frac{\mbox{kg}}{\mbox{m}^3}=4.23 \cdot 10^{-28}\frac{\mbox{kg}}{\mbox{m}^3}.$$
(The Hubble constant is not precisely known, so the calculated density depends a bit on the precise value.)

**Density of water:**
The density of water is about $10^3 \mbox{kg}/\mbox{m}^3$.
That's a factor $4.23 \cdot 10^{31}$ denser than the average universe.
The space needs to have been reduced by a factor $\sqrt[3]{4.23 \cdot 10^{31}}=3.48\cdot 10^{10}$ in each if the three spatial dimensions. That's the inverse of the [scale factor](http://en.wikipedia.org/wiki/Scale_factor_%28cosmology%29), hence corresponds to a redshift of $z=3.48\cdot 10^{10}$.

Using the [Cosmology Calculator](http://adsabs.harvard.edu/abs/2006PASP..118.1711W) on [this website](http://www.astro.ucla.edu/~wright/CosmoCalc.html), the cosmological parameters $H\_0 = 67.11$ km/s/Mpc (slightly different to the above value),
$\Omega\_{\Lambda} = 0.6825$ provided by the [Planck](http://en.wikipedia.org/wiki/Planck_%28spacecraft%29) project, and setting $\Omega\_M = 1- \Omega\_{\Lambda} = 0.3175$ the age of the universe was **0.022 seconds** at the redshift $z=3.48\cdot 10^{10}$, hence when it was of the average density of water.

**Density of air at sea level:**
The [density of air](http://en.wikipedia.org/wiki/Density_of_air) at sea level is $1.225 \frac{\mbox{kg}}{\mbox{m}^3}$. The corresponding redshift is about $z=1.423\cdot 10^9$ yielding an age of the universe of **12.68 seconds**.

> **Comments:**
> **username_2**: I'm guessing it was still very hot at 12 seconds?  
> **username_1**: About 100 gigakelvin, or $10^{11}$  K (http://en.wikipedia.org/wiki/Big_Bang_nucleosynthesis)  
> **username_0**: Absolutely amazing! I'm surprised that there's such a difference between the density of air and water.  

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