How to measure the CDM power spectrum?

[MykDee]

Hi all! It's my first post, but I follow this forum since a couple of years. Now I have a big question for you: how to measure the CDM (or $$\Lambda$$CDM) power spectrum (better, the variance $$\propto k^3 P(k)$$) from the observations? What parameters are necessary?

 

[Chalnoth]

Hi all! It's my first post, but I follow this forum since a couple of years. Now I have a big question for you: how to measure the CDM (or $$\Lambda$$CDM) power spectrum (better, the variance $$\propto k^3 P(k)$$) from the observations? What parameters are necessary?

Well, you measure the power spectrum by looking at the CMB. Leaving out the gory details of how we determine what part of our observations are CMB and what parts are stuff between us and the CMB, this basically just consists of computing the spherical harmonic transform of the sky map, and then computing the power spectrum from that:
(omitting the normalization factor which I don't remember offhand)

$$a_{lm} = \int s(\theta, \phi) Y_l^m (\theta, \phi) sin(\theta) d\theta d\phi$$
$$C_l = \sum_m |a_{lm}|^2$$

Once we have this power spectrum, we then need to infer the model parameters that produced it, typically using Bayesian inference. This basically involves computing the power spectrum $$C_l$$ that we would get for a variety of different parameter choices and comparing them to the measured $$C_l$$.

 

[MykDee]

Well, you measure the power spectrum by looking at the CMB. Leaving out the gory details of how we determine what part of our observations are CMB and what parts are stuff between us and the CMB, this basically just consists of computing the spherical harmonic transform of the sky map, and then computing the power spectrum from that:
(omitting the normalization factor which I don't remember offhand)

$$a_{lm} = \int s(\theta, \phi) Y_l^m (\theta, \phi) sin(\theta) d\theta d\phi$$
$$C_l = \sum_m |a_{lm}|^2$$

Once we have this power spectrum, we then need to infer the model parameters that produced it, typically using Bayesian inference. This basically involves computing the power spectrum $$C_l$$ that we would get for a variety of different parameter choices and comparing them to the measured $$C_l$$.

Ok, thank you. Now, suppose that I've measured the $$C_l$$ and then the $$P(k)$$. This power spectrum is valid also for the galaxies? I mean, the P(k) calculated via CMB, does represent both the large and the small structures in the Universe, or for those structure (like galaxies or cluster of galaxies) we can infer the P(k) from other things (e.g. galaxy redshift surveys)?

 

[mikeph]

The present power spectrum is the original power spectrum at decoupling multiplied by the transfer function, which describes how the power in each mode k evolves over the course of time. The original power spectrum is purely representative of density and velocity perturbations just after the big bang. The transfer function T(k) from decoupling to now depends on what has happened to the universe over this time interval, and will involve an integration over two separate domains, the matter-dominated and the radiation dominated, as the perturbations evolve differently in different types of universe. The transfer function is also a function of k, because modes inside and outside the horizon will grow differently.

I don't remember the exact way they grow, though. We did a little maths on it but I could never follow the entire derivation... perturbation theory in general relativity is probably the nastiest set of formulas I've ever seen.

edit-

To try to address your question a little more: The P(k) describes the power contained in different length scales, represented by k, so it contains information about structures over very large and very small scales if you want. But the key is the CMB power spectrum originated long before galaxies or stars formed, a snapshot of the universe as a hot plasma.

 

[Chalnoth]

Ok, thank you. Now, suppose that I've measured the $$C_l$$ and then the $$P(k)$$. This power spectrum is valid also for the galaxies? I mean, the P(k) calculated via CMB, does represent both the large and the small structures in the Universe, or for those structure (like galaxies or cluster of galaxies) we can infer the P(k) from other things (e.g. galaxy redshift surveys)?

It's not quite so simple, as MikeyW alludes to. The basic issue is that you have to take the dynamics of the matter into question. This means that the power spectrum of the CMB is related to the power spectrum of nearby galaxies, but it's not a simple mapping. You also have to take into account how the various forms of matter interact (dark matter and baryonic matter being the most important for this analysis). Generally such analyses are extremely error-prone at small scales where nonlinear effects dominate. But at larger scales the predictions match observation dramatically well.

 

[MykDee]

The present power spectrum is the original power spectrum at decoupling multiplied by the transfer function, which describes how the power in each mode k evolves over the course of time. The original power spectrum is purely representative of density and velocity perturbations just after the big bang. The transfer function T(k) from decoupling to now depends on what has happened to the universe over this time interval, and will involve an integration over two separate domains, the matter-dominated and the radiation dominated, as the perturbations evolve differently in different types of universe. The transfer function is also a function of k, because modes inside and outside the horizon will grow differently.

I don't remember the exact way they grow, though. We did a little maths on it but I could never follow the entire derivation... perturbation theory in general relativity is probably the nastiest set of formulas I've ever seen.

edit-

To try to address your question a little more: The P(k) describes the power contained in different length scales, represented by k, so it contains information about structures over very large and very small scales if you want. But the key is the CMB power spectrum originated long before galaxies or stars formed, a snapshot of the universe as a hot plasma.

I agree with you, and indeed this is what I would do if I'm asked something like "how do you calculate mathematically the power spectrum?". My answer would be: I know the primordial P(k), a power law with spectral index n = 1, and I also know the form of the transfer function (that goes to 0 for small scales (large k)). The final P(k) is the convolution between the initial power spectrum and the transfer function. Okay, nothing difficult here.

But my question was a little bit different: I wanted to know how to calculate the form of the spectrum, on both large and small scales, using only observations. In other words: I look at the CMB, or at some catalogue of galaxy survey: is it possible to calculate the P(k) from this direct observations or not? How?

From the answers of Chalnoth (thank you) I've understood that looking at the spectrum of the CMB makes it possible to calculate the parameters that are required to the CDM model; that model will generate a P(k). Is this right?

 

[Chalnoth]

From the answers of Chalnoth (thank you) I've understood that looking at the spectrum of the CMB makes it possible to calculate the parameters that are required to the CDM model; that model will generate a P(k). Is this right?

Sort of. The P(k) isn't directly observable in the CMB, because we see a projection of P(k) upon the last scattering surface. The mapping from what we see on the last scattering surface and P(k) is generally nontrivial. So instead we work purely in $$C_l$$, whose structure can be directly derived from theory.

For the nearby matter, we actually have full three-dimensional information, and thus can actually compute P(k). This can be done, for instance, by assuming that certain galaxies trace the underlying power spectrum of matter. So just by measuring the power spectrum P(k) of, say, luminous red galaxies we can obtain a measurement of the power spectrum of the matter (and there's quite a lot of work going into understanding just how well various galaxies do trace the underlying matter).