2-point correlation functions for curvature perturbations

[WannabeNewton]

Does anyone have a good reference or references that go into detail on rigorous/formal developments of 2-point correlation functions for curvature perturbations (and related perturbations) in the cosmological context? I'm using the TASI lectures in inflation, Mukhanov, and Dodelson but none of them go into any detail on the formal statistics of "fluctuations", what exactly in cosmological perturbation theory is randomized across a statistical ensemble and why, how the 2-point correlation is formally defined in terms of random variables, what the random variables are in inflationary cosmology, how the 2-point correlation function relates to "fluctuations" (which I take to mean the variance of a random variable), if "fluctuations" refers to fluctuations across different points in space (or equivalently, different energy scales in Fourier space) as opposed to fluctuations in a random variable at a given fixed point in space, and so on.

So does anyone know of any good resources for these questions, ideally in a cosmological context (but if not I'm also happy with one in a general context or perhaps in the context of statistical mechanics since I have Kardar and Kardar's chapter on probability theory is pretty terrible)? Is Weinberg any good for this? Thanks in advance.

 

[Mordred]

I've seen some papers that have 2 to 4 point perturbation correlations, don't know how accurate they are so I'll post some of the ones I'm aware of

http://arxiv.org/pdf/1306.2992.pdf page 7 for 2,3,4 point correlators
http://arxiv.org/pdf/1210.3461.pdf

"Measuring the Two-point Correlation Function"
http://www.astro.rug.nl/~weygaert/tim1publication/lss2007/computerII.pdf

"We compute the three-point cross-correlation function of the primordial curva-
ture perturbation generated during inflation

http://www.mpi-hd.mpg.de/lin/events/group_seminar/inflation/schmidt.pdf page 5

I haven't completely read this thesis but you may find more in here
Primordial Curvature Perturbations in Inflationary Universe
http://www.ta.phys.nagoya-u.ac.jp/c-lab/shu/thesis/D1.pdf

http://relativity.livingreviews.org/open?pubNo=lrr-2004-3&page=articlesu16.html
http://arxiv.org/pdf/1302.3877.pdf

The Physics of Inflation
http://www.icts.res.in/media/uploads/Talk/Document/baumann_icts_dec2011.pdf

you can probably also find examples in
Encyclopaedia Inflationaris

http://arxiv.org/abs/1303.3787

If I understand correctly how the 2 point correlation function is used would depend on which multi-field inflation model your looking at hope those help there is 74+ inflation models

 

[bapowell]

Does anyone have a good reference or references that go into detail on rigorous/formal developments of 2-point correlation functions for curvature perturbations (and related perturbations) in the cosmological context?

You might try Peacock's "Cosmological Physics" (specifically Chapter 16) or Liddle and Lyth's "Cosmological Inflation and Large Scale Structure". The eprint http://arxiv.org/pdf/astro-ph/9303019.pdf by Liddle and Lyth formed the basis for much of their textbook, so I would recommend having a look there first.

how the 2-point correlation function relates to "fluctuations" (which I take to mean the variance of a random variable), if "fluctuations" refers to fluctuations across different points in space (or equivalently, different energy scales in Fourier space) as opposed to fluctuations in a random variable at a given fixed point in space, and so on.

The 2-point autocorrelation function, [itex]\xi(r)[/itex], determines the probability that two objects will be found separated by a distance [itex]r[/itex]. This quantity must be averaged over the random field (all space). It is therefore a measure of the clumpiness of space -- it can be applied to galaxies, density perturbations, temperature fluctuations, etc. The Fourier transform of the spatial 2-point function is the well-known power spectrum of the random field. If the field is Gaussian random, higher-point functions are zero and all the information regarding the distribution of perturbations is found in the power spectrum.

 

[WannabeNewton]

I've seen some papers that have 2 to 4 point perturbation correlations, don't know how accurate they are so I'll post some of the ones I'm aware of

You might try Peacock's "Cosmological Physics" (specifically Chapter 16) or Liddle and Lyth's "Cosmological Inflation and Large Scale Structure". The eprint http://arxiv.org/pdf/astro-ph/9303019.pdf by Liddle and Lyth formed the basis for much of their textbook, so I would recommend having a look there first.

Thank you very much guys!

The Fourier transform of the spatial 2-point function is the well-known power spectrum of the random field. If the field is Gaussian random, higher-point functions are zero and all the information regarding the distribution of perturbations is found in the power spectrum.

Perhaps you can help me find some insight for the object because the books I'm using aren't providing it (although Weinberg might be the saving grace). For example say we compute ##\langle \zeta_k \zeta_{k'} \rangle##, with ##\zeta_k## being a curvature perturbation mode in the comoving gauge. This would tell us the correlations between curvature perturbations at different energy scales. But how exactly is this useful? Why exactly we want to know about correlations between curvature modes at different scales?

Also, let's say we instead want a 2-point correlation function in space e.g. ##\langle \Theta(x)\Theta(x')\rangle## where ##\theta## is a CMB temperature perturbation. Then this would be a correlation between the temperature perturbation at two different points. Apart from asking why it's useful, could you explain in what sense the 2-point correlation represents "fluctuations"? I usually think of fluctuations as the variance in ##\theta(x)## for fixed x but I've also been told that due to ergodicity, the variance in the temperature perturbations at a fixed point will be equal to the "fluctuations" in the perturbation when comparing different spatial points; in what formal mathematical sense is the latter a fluctuation? Does it just mean 2-point correlation or is there a more intuitive/deeper meaning? Thanks!

 

[Mordred]

In this you would be far better off if I let Bapowell answer. I know I would likely make mistakes as two point correlation functions in regards to inflation I'm not overly familiar with, and Bapowell is by far better in this arena than I am

 

[bapowell]

For example say we compute ##\langle \zeta_k \zeta_{k'} \rangle##, with ##\zeta_k## being a curvature perturbation mode in the comoving gauge. This would tell us the correlations between curvature perturbations at different energy scales. But how exactly is this useful? Why exactly we want to know about correlations between curvature modes at different scales?

You aren't so much interested in that. When you calculate the correlation function from the power spectrum, cross-terms [itex]k \neq k'[/itex] vanish so that:
[tex]\xi(r) \propto \int \langle|\mathcal{R}_k|^2\rangle e^{-i {\bf k}\cdot{\bf r}} d^3k[/tex]
What the power spectrum of curvature perturbations, [itex]\langle|\mathcal{R}_k|^2\rangle[/itex], tells you is the degree to which fluctuations across space are correlated on the scale [itex]k[/itex]. So, think of [itex]k[/itex] not as referring to the energy but the scale of the fluctuation.

Apart from asking why it's useful, could you explain in what sense the 2-point correlation represents "fluctuations"? I usually think of fluctuations as the variance in ##\theta(x)## for fixed x but I've also been told that due to ergodicity, the variance in the temperature perturbations at a fixed point will be equal to the "fluctuations" in the perturbation when comparing different spatial points; in what formal mathematical sense is the latter a fluctuation? Does it just mean 2-point correlation or is there a more intuitive/deeper meaning? Thanks!

Your quantity [itex]\langle \theta (x) \theta (x')\rangle[/itex], where the [itex]\theta(x)[/itex] is the temperature fluctuation in the direction [itex]x[/itex], measures the amount of correlation between temperature fluctuations in different directions, [itex]x,x'[/itex]. So, it's not that your 2-point function "represents" fluctuations as it is measuring the correlations of temperature fluctuations. But, I feel like you know this and that I am missing the gist of your question. Are you distinguishing "perturbations" and "fluctuations"? You shouldn't -- they are the same thing here.

 

[WannabeNewton]

Are you distinguishing "perturbations" and "fluctuations"? You shouldn't -- they are the same thing here.

Thanks! I was distinguishing between them yes, based on what I've read in Dodelson. For example let ##h## denote a tensor perturbation to the background metric (where I have made implicit the polarization of ##h##) and let ##\tilde{h} = \frac{ah}{\sqrt{16\pi G}}##. Then, as Dodelson puts it, after quantization the variance ##\langle \hat{\tilde{h}}^{\dagger}(k,\eta)\hat{\tilde{h}}(k',\eta)\rangle \propto \delta^3 (k -k')## represents the quantum fluctuations in the tensor perturbation (pp. 158-159). This is in Fourier space so the Fourier transform of this would give us the 2-point correlation function for the tensor perturbation evaluated at different points in space. So Dodelson distinguishes between the perturbation ##h## and its quantum fluctuations which, for this quantum field, is the above variance. This is also my understanding of the term "quantum fluctuation" from QM/QFT.

Also I'm afraid I still don't understand the relationship between the power spectrum and the 2-point correlation function. It's not clear to me from the integral you wrote down. For example for the above, the power spectrum is given by ##P_h = 16\pi G \frac{|v(k,\eta)|^2}{a^2}## where ##v, v^*## represent the coefficients of the creation and annihilation operators of ##\tilde{h}##. What then does ##P_h## tell us about the 2-point correlation between the values of the tensor perturbation at different points in space (at the same conformal time)? Presumably it is this spatial correlation we care about in the end yes? Thanks again.

 

[Mordred]

ah I got what your after, your into his harmonic oscillator section, I found looking at other references helped (PS one trick I learned is sometimes you can copy a latex then post a search on it lol,greatly helps me)

this article should help

Cosmology Part II: The Perturbed Universe
http://icc.ub.edu/~liciaverde/Cosmology2.pdf

the whole paper is good but Dodelsons is covered in section 2.5.8 onward, though you will also need the previous sections

 

[bapowell]

So Dodelson distinguishes between the perturbation ##h## and its quantum fluctuations which, for this quantum field, is the above variance. This is also my understanding of the term "quantum fluctuation" from QM/QFT.

Interesting. I'm familiar with Dodelson's text, but was unaware that he makes this association. I agree that there is some sense in referring to the variance [itex]\langle h^\dagger h\rangle[/itex] as a "quantum fluctuation", but this is not conventional. When the term "fluctuation" comes up in the study of cosmological perturbations, it is generally taken to be synonymous with "perturbation". The variance above is the power spectrum, and you'll frequently hear it referred to as the "power spectrum of fluctuations" or something similar. So yeah, some semantic shenanigans here.

What then does ##P_h## tell us about the 2-point correlation between the values of the tensor perturbation at different points in space (at the same conformal time)? Presumably it is this spatial correlation we care about in the end yes? Thanks again.

Yes, that's right. For example, for the CMB we want the spatial temperature correlation. The physical process is: field fluctuations --> temperature perturbations. We set out to calculate the behavior of the field fluctuations, which naturally takes place in k-space. The power spectrum should be viewed as a fundamental quantity arising from the problem of QFT in the inflating spacetime. Once we have the power spectrum, we use it to compute what's happening in space, like the CMB temperature correlation function.

 

[WannabeNewton]

Thanks to both of you! So would it be fair to say that the power spectrum is a means to an end? I ask because while it's clear what the physical meaning of the spatial 2-point correlation function is, it is not clear to me what the physical meaning is of the power spectrum. For example is there an easy way to interpret the ##P_h## from above or is it to be viewed as purely a mathematical object in the interim before calculating the spatial 2-point correlation? What exactly is ##\langle h^{\dagger}_{k'}h_k \rangle## representing if ##\langle h^{\dagger}_{k'}h_k \rangle \propto \delta^3(k - k')## in Fourier space?

Also sorry for the questions but there's one more thing confusing me that I still can't grasp even after reading Mordred's link above and Linde's review. Say we're looking at ##\langle \Theta(x)\Theta(x')\rangle## where ##\Theta## is a temperature perturbation. In what sense is ##\Theta(x)## a random variable or a random process?

 

[bapowell]

So would it be fair to say that the power spectrum is a means to an end? I ask because while it's clear what the physical meaning of the spatial 2-point correlation function is, it is not clear to me what the physical meaning is of the power spectrum.

That depends on your perspective, I'm sure. As a Fourier transform, the power spectrum is the spectral decomposition of the spatial correlation function. It's just as physical -- just a different way of looking at things. But, yes, for cosmologists using observations to constrain theories, we calculate the power spectrum as a means to arriving at the spatial correlations.

Also sorry for the questions but there's one more thing confusing me that I still can't grasp even after reading Mordred's link above and Linde's review. Say we're looking at ##\langle \Theta(x)\Theta(x')\rangle## where ##\Theta## is a temperature perturbation. In what sense is ##\Theta(x)## a random variable or a random process?

So, I'm guessing [itex]\Theta(x)[/itex] is the temperature fluctuation at x? If so, the short answer is that the temperature fluctuation at a given point is a random variable because the underlying quantum field fluctuations that generated it are random variables. Specifically, the temperature fluctuation in a direction [itex]{\bf n}[/itex] can be decomposed
[tex]\Theta({\bf n}) = \sum_{\ell, m} a_{\ell m} Y_{\ell m}({\bf n})[/tex]
where the [itex]Y_{\ell m}[/itex] are spherical harmonics (this decomposition is motivated by the spherical geometry of the last scattering surface), and the [itex]a_{\ell m}[/itex] are Gaussian random variables that can be connected directly the power spectrum.

 

[WannabeNewton]

Awesome, thanks again! I appreciate the help.

 

[Mordred]

I might be off here but Dodelson has two gravitational wave functions. page 130 what he refers to as the large scale mode and the small scale mode. The equation 5.63, page 130, so the section you on relates the tensor perturbations of each the large scale mode and the small scale mode, and then he will also have 2 scalar modes. Each will have their own thermodynamic influences. At least that's the understanding I got from reading it.

I see you got the answer from Bapowell lol