<p>I can't see the contradiction. One version says average over the regions, the other says integrate over one realisation (containing many regions). I suppose you could argue that one is performing a summation,whilst the other is an integration, but if the regions are small compared with the size of your observable universe then they amount to the same thing.
<span class="math-container">$$\frac{1}{\sum_i V_i} \sum_i \lambda_i  V_i \simeq \frac{1}{V}\int \lambda(r)\ dV$$</span></p>

CC BY-SA 4.0
2019-02-04T23:16:46.167Z
If you have a moment I added some details. Could you check if they are correct please?
MattG88
CC BY-SA 4.0
2019-02-02T08:01:51.360Z
2019-02-02T08:01:51.360Z
ProfRob
<p>I'm studying statistical properties of a cosmological density field and I have some doubts; in the following quote and pictures you can see what my problem is:</p>

<blockquote>
  <p>A fundamental limitation arises in cosmology – because there is only one universe to observe, i.e. there is only one realization of the stochastic process that generates the fluctuations whose consequences we observe. Therefore we cannot measure ensemble averages
  or expectation values, as we would in a repeatable laboratory experiment. What we can do when observing a fluctuation on a given scale <span class="math-container">$\lambda$</span> is to average over many distinct regions of size <span class="math-container">$\sim\lambda$</span>. An ergodic-type hypothesis allows us to replace the ensemble average by a spatial average over these regions. (see pag 307 from <em>Relativistic cosmology</em>-George F.R. Ellis, R. Maartens, M.A.H MacCallum; Cambridge University Press (2012))</p>
</blockquote>

<p>So here we talk about an average over many distinct regions of the Universe (someone call them "fair sample" of the Universe if I understood correctly) and an ensemble average is replaced by a spatial average over these region<strong>S</strong>. Thus I should average the field over region 1, region 2 and so on and I would get many mean values of the field.</p>

<p>But in this presentation things seem different:
<a href="https://i.sstatic.net/6eaOv.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/6eaOv.jpg" alt="enter image description here"></a>
<a href="https://i.sstatic.net/hpyiz.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/hpyiz.jpg" alt="enter image description here"></a></p>

<p>Here instead we talk about an average over an ensemble replaced by a spatial average over <strong>one</strong> realization of the random field (one of the possible  universes I imagine). In this case I have only one realization: our Universe, so I should average my field spatially over the entire Universe. And what about the regions mentioned above?</p>

<p>I'm confused and I would appreciate an explanation.</p>

<p>Thanks.</p>

<p><strong>Addendum</strong></p>

<p>So theoretically if i wanted to get the average of a stochastic field or of a function, we say <span class="math-container">$\delta(\vec x)$</span> or <span class="math-container">$f[\delta(\vec x)]$</span>, in a point <span class="math-container">$\vec x_1$</span> I should do the average over the ensemble of all the universes using (see slide): <span class="math-container">$\int{\delta}\cal P(\delta) d\delta$</span> (it is obviously impossible because we have only one Universe). I have imagined this thing thinking about a stack of planes representing the different universes of the ensemble and drawing the point <span class="math-container">$\vec x_1$</span> in each plane. Now i can take the value of <span class="math-container">$\delta(\vec x_1)$</span> in the first universe, in the second, and so on and finally make the average of these values; if I repeat this procedure for all the points I'll get the average of <span class="math-container">$\delta (\vec x)$</span>, i.e. <span class="math-container">$\langle\delta\rangle$</span> over the ensemble of the universes. Ok it's fine...but it is not possible because I have only one Universe, so I have to appeal to Ergodic Hypothesis (really a <em>spatial</em> version of the Ergodic Hypothesis).</p>

<blockquote>
  <p>An ergodic-type hypothesis allows us to replace the ensemble average by a spatial average over these regions</p>
</blockquote>

<p>It seems that I have to divide the Universe in regions (we can imagine squares for simplicity in two dimensions, I thought the Universe like a chessboard) and I can replace the ensemble averege with a spatial average over any single region. Any square of the chessboard is a fair sample of the Universe, so I have many squares that can "reproduce" the Universe (I divided my sky in many regions and I can consider these squares like the realisations of the "Universe experiment", the outcomes of the experiment). Thus I can do:</p>

<p><span class="math-container">$$\langle\delta\rangle=\frac{1}{\sum_i V_i}\sum_i \langle\delta\rangle_{V_i}$$</span></p>

<p>where <span class="math-container">$\langle\delta\rangle_{V_i}=\int_{V_i}\delta(\vec x) d\vec x^3$</span>.</p>

CC BY-SA 4.0
2019-02-01T11:52:13.003Z
Hi Matt. Roy Maartens is a member of the ICG in Portsmouth. That's headed up by David Wands, who has written papers like [Inhomogeneous and interacting vacuum energy](https://arxiv.org/abs/1209.0563). Personally I think the gist of that is correct, and that it's reasonable to talk of variations in cosmological density. However as far as I know quantum fluctuations and inflation are much more speculative, as is the "ergodic hypothesis" as applied to the  early universe. So I don't know if there's anything factual I can offer. Can you link to the presentation? Maybe somebody else can help.
John Duffield
CC BY-SA 4.0
2019-02-01T12:13:05.297Z
Please don't insert textual information as images!
peterh
CC BY-SA 4.0
2019-02-01T12:24:09.093Z
@JohnDuffield Hi John. http://www.astro.yale.edu/vdbosch/astro610_lecture12.pdf This the link to the presentation that I found on the web to try to understand how to average a cosmological density field or function of the field
MattG88
<cosmology>
Ensemble average VS spatial average for a cosmological density field
MattG88
# Ensemble average VS spatial average for a cosmological density field

(Asked by: username_0 on 2019-02-01)

I'm studying statistical properties of a cosmological density field and I have some doubts; in the following quote and pictures you can see what my problem is:

> A fundamental limitation arises in cosmology – because there is only one universe to observe, i.e. there is only one realization of the stochastic process that generates the fluctuations whose consequences we observe. Therefore we cannot measure ensemble averages
> or expectation values, as we would in a repeatable laboratory experiment. What we can do when observing a fluctuation on a given scale $\lambda$ is to average over many distinct regions of size $\sim\lambda$. An ergodic-type hypothesis allows us to replace the ensemble average by a spatial average over these regions. (see pag 307 from *Relativistic cosmology*-George F.R. Ellis, R. Maartens, M.A.H MacCallum; Cambridge University Press (2012))

So here we talk about an average over many distinct regions of the Universe (someone call them "fair sample" of the Universe if I understood correctly) and an ensemble average is replaced by a spatial average over these region**S**. Thus I should average the field over region 1, region 2 and so on and I would get many mean values of the field.

But in this presentation things seem different:
[![enter image description here](https://i.sstatic.net/6eaOv.jpg)](https://i.sstatic.net/6eaOv.jpg)
[![enter image description here](https://i.sstatic.net/hpyiz.jpg)](https://i.sstatic.net/hpyiz.jpg)

Here instead we talk about an average over an ensemble replaced by a spatial average over **one** realization of the random field (one of the possible universes I imagine). In this case I have only one realization: our Universe, so I should average my field spatially over the entire Universe. And what about the regions mentioned above?

I'm confused and I would appreciate an explanation.

Thanks.

**Addendum**

So theoretically if i wanted to get the average of a stochastic field or of a function, we say $\delta(\vec x)$ or $f[\delta(\vec x)]$, in a point $\vec x\_1$ I should do the average over the ensemble of all the universes using (see slide): $\int{\delta}\cal P(\delta) d\delta$ (it is obviously impossible because we have only one Universe). I have imagined this thing thinking about a stack of planes representing the different universes of the ensemble and drawing the point $\vec x\_1$ in each plane. Now i can take the value of $\delta(\vec x\_1)$ in the first universe, in the second, and so on and finally make the average of these values; if I repeat this procedure for all the points I'll get the average of $\delta (\vec x)$, i.e. $\langle\delta\rangle$ over the ensemble of the universes. Ok it's fine...but it is not possible because I have only one Universe, so I have to appeal to Ergodic Hypothesis (really a *spatial* version of the Ergodic Hypothesis).

> An ergodic-type hypothesis allows us to replace the ensemble average by a spatial average over these regions

It seems that I have to divide the Universe in regions (we can imagine squares for simplicity in two dimensions, I thought the Universe like a chessboard) and I can replace the ensemble averege with a spatial average over any single region. Any square of the chessboard is a fair sample of the Universe, so I have many squares that can "reproduce" the Universe (I divided my sky in many regions and I can consider these squares like the realisations of the "Universe experiment", the outcomes of the experiment). Thus I can do:

$$\langle\delta\rangle=\frac{1}{\sum\_i V\_i}\sum\_i \langle\delta\rangle\_{V\_i}$$

where $\langle\delta\rangle\_{V\_i}=\int\_{V\_i}\delta(\vec x) d\vec x^3$.

**Tags:** cosmology
> **Comments:**
> **username_1**: Hi Matt. Roy Maartens is a member of the ICG in Portsmouth. That's headed up by David Wands, who has written papers like [Inhomogeneous and interacting vacuum energy](https://arxiv.org/abs/1209.0563). Personally I think the gist of that is correct, and that it's reasonable to talk of variations in cosmological density. However as far as I know quantum fluctuations and inflation are much more speculative, as is the "ergodic hypothesis" as applied to the  early universe. So I don't know if there's anything factual I can offer. Can you link to the presentation? Maybe somebody else can help.  
> **username_2**: Please don't insert textual information as images!  
> **username_0**: @JohnDuffield Hi John. http://www.astro.yale.edu/vdbosch/astro610_lecture12.pdf This the link to the presentation that I found on the web to try to understand how to average a cosmological density field or function of the field  

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## 1 answers

### Answer by username_3 on 2019-02-02. Score: 1

I can't see the contradiction. One version says average over the regions, the other says integrate over one realisation (containing many regions). I suppose you could argue that one is performing a summation,whilst the other is an integration, but if the regions are small compared with the size of your observable universe then they amount to the same thing.
$$\frac{1}{\sum\_i V\_i} \sum\_i \lambda\_i V\_i \simeq \frac{1}{V}\int \lambda(r)\ dV$$

> **Comments:**
> **username_0**: If you have a moment I added some details. Could you check if they are correct please?  

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