What are the latest exciting results from WMAP's three-year data release?

[Garth]

Sorry to take introduce a new element into this thread, but the 3-year WMAP results are just so rich.

There are some ~300 point sources in these data, up from ~200 in the year-1 data. How consistent are these (observed) (extragalactic) point sources with the (observed - SDSS/2dF etc) P(k)?

Hi Nereid! Yes, we have been rather hogging the discussion!

I think that the large-l modes are interesting. Whereas the WMAP2 power spectrum indicated the rise to the third peak it did not continue far enough to mark that peak, WMAP3 does continue into
l > 800 yet does not show the peak at all, its error bars are too large and even then do not cross the predicted curve. What is there seems to 'plateau out'. WMAP has a noise problem at the high-l end. (Hinshaw et al. http://lambda.gsfc.nasa.gov/product/map/dr2/pub_papers/threeyear/temperature/wmap_3yr_temp.pdf page 75.)

That third peak, important to determine \Omega_b, has to be determined by other experiments: Acbar, Boomerang, CBI, VSA.

Garth

 

[SpaceTiger]

The Planck mission was almost certainly designed to look for things we expect from the standard model and, since the standard model hasn't been called into question by WMAP, I wouldn't expect a shift in the Planck design. They're primarily planning to look at angular scales of l < 2000, if I remember correctly, and the interesting range will be between 1000 and 2000, where WMAP hasn't covered. It may seem like a small range, but it's about a million modes on the sky, so there's much to be learned.

If the standard model holds, there aren't a lot of new results to be garnered from the primary anisotropies -- smaller error bars and a possible detection of B mode polarization from gravitational waves. Some of the most interesting results ought to come out of the secondary anisotropies, which include the Sunyaev-Zeldovich effect and extragalactic sources.

There are a lot of WMAP results concerning the galaxy and dust, but I haven't had time to review them. Dr. Spergel gave a talk about it this past Wednesday and it seemed that one of the main results was the confirmation of emission from spinning dust grains.

 

[Chronos]

I think the thrid peak in the power spectrum has been nailed down in the WMAP release, and is well explained by the LCDM model. There will be several more papers on this in the year to come . . . IMO.

 

[Garth]

Bernui, Mota, Reboucas, & Tavakol have today updated their paper
Mapping the large-scale anisotropy in the WMAP data to include WMAP 3 data.

They previously had described another method of measuring large-scale anisotrophies:

Introduction

Here we propose a new indicator, based on the angular pair separation histogram (PASH) method [20], as a measure of large-scale anisotropy. An important feature of this indicator is that it can be used to generate a sky map of large-scale anisotropies in a given CMB temperature fluctuations map. This level of directional detail may also provide a possible additional window into their causes.

With the result:

Conclusions

We have proposed a new method of directionally measuring deviations from statistical isotropy in the CMB sky, in order to study the possible presence and nature of large-scale anisotropy in the WMAP data.
The use of our anisotropy indicator has enabled us to construct a map of statistical deviations from isotropy for the CMB data. Using this σ–map we have been able to find evidence for a large-scale anisotropy in the WMAP CMB temperature field. In particular we have found, with high statistical significance (> 95% CL), a small region in the celestial sphere with very high values of σ, which defines a direction very close to the one reported recently [6, 10].

This result persists even after attempts to explain it away as an artefact of the data processing or foreground cleansing procedures:

We have shown that the results reported here are robust, by showing that the σ–map does not significantly change by changing various parameters employed in its calculation. We have also studied the effects of different foreground cleaning algorithms, or absence thereof, by considering in addition to LILC also the TOH and WMAP CO-ADDED maps. We have found again that the corresponding σ–maps remain qualitatively unchanged. In particular the hot spot on the south-eastern corner of the σ–map remains essentially invariant for all the maps considered here. This robustness demonstrates that our indicator is well suited to the study of anisotropies in the CMB data.

Now that result has been preliminarly updated by the WMAP3 data:

Finally, we add that after our paper was submitted, the three year WMAP CMB data was released [28]. As a preliminary check, we have calculated the σ−map for the new three year WMAP CO-ADDED map, which is depicted in Fig. 6. As can be seen the hot spot found in the first year σ−map in the south eastern corner of the sky, remains qualitatively unchanged with an axis also in agreement with that found for the first year data. In this way our results are also robust with respect for the three year WMAP CMB data. A complete and detailed analysis of the three year WMAP data using our indicator will be presented elsewhere.

There are three questions to ask:
1. "Is the distribution of anisotropies in the WMAP data non-Gaussian?",
2. "Is there an alignment in the non-Gaussianity?" and
3. "Is any such alignment identifable with local geometry, such as motion through the CMB, the galactic plane etc.?"

The interpretation of the statistical significance of the result depends on the question asked.

Garth

 

[aguy2]

The interpretation of the statistical significance of the result depends on the question asked.

Garth

If you would look at the animation of figure 3 at:
http://www.physics.nmt.edu/~dynamo/PJRX/Results.html
and tried to see it as a possible crude representation of the current cosmological situation, while seeing our streaming galactic cluster somewhat left of center, I think you could see something akin to the observed dipole/octipole structure.

Could the all-matter, observed universe be seen as a pulse/jet and not as an isometric/homogeneous expansion?
aguy2

 

[Garth]

If you would look at the animation of figure 3 at:
http://www.physics.nmt.edu/~dynamo/PJRX/Results.html
and tried to see it as a possible crude representation of the current cosmological situation, while seeing our streaming galactic cluster somewhat left of center, I think you could see something akin to the observed dipole/octipole structure.

Could the all-matter, observed universe be seen as a pulse/jet and not as an isometric/homogeneous expansion?
aguy2

The short answer is no! I find it hard to envisage what geometry your suggested setup is meant to have.

The anisotropies we are talking about are at 10^-5, apart from the dipole, caused by our motion wrt the surface of last scattering, the CMB is remarkably isotropic.

That animation had a vague resemblance to the quadrupole/octopole distribution across the sky but that is all.

Garth

 

[Chronos]

I see problems with Bernui et al.

 

[Garth]

I see problems with Bernui et al.

Such as?

Garth

 

[Garth]

A new paper by Copi, Huterer, Schwarz and Starkmanon the subject of low-l mode correlations: The Uncorrelated Universe: Statistical Anisotropy and the Vanishing Angular Correlation Function in WMAP Years 1-3

We have shown that the ILC123 map, a full sky map derived from the first three years of WMAP data like its predecessors the ILC1, TOH1 and LILC1 maps, derived from the first year of WMAP data, shows statistically significant deviations from the expected Gaussian-random, statistically isotropic sky with a generic inflationary spectrum of perturbations. In particular: there is a dramatic lack of angular correlations at angles greater than sixty degrees; the octopole is quite planar with the three octopole planes aligning with the quadrupole plane; these planes are perpendicular to the ecliptic plane (albeit at reduced significance than in the first-year full-sky maps); the ecliptic plane neatly separates two extrema of the combined ℓ = 2 and ℓ = 3 map, with the strongest extrema to the south of the ecliptic and the weaker extrema to the north.

We have discussed before whether these alignments are just a statistical 'fluke' or whether "more compelling evidence" is required before it is acknowledged that all is not well with the standard \Lambda CDM model interpretation of the WMAP data.

The authors go on:

The probability that each of these would happen by chance are 0.03% (quoting the cut-sky ILC123 S1/2 probability), 0.4%, 10%, and < 5%. As they are all independent and all involve primarily the quadrupole and octopole, they represent a ~10⁻⁸ probability chance “fluke” in the two largest scale modes. To quote [7]: We find it hard to believe that these correlations are just statistical fluctuations around standard inflationary cosmology’s prediction of statistically isotropic Gaussian random a_ℓm [with a nearly scale-free primordial spectrum].

What explanations may there be?

What are the consequences and possible explanations of these correlations? There are several options — they are statistical flukes, they are cosmological in origin, they are due to improper subtraction of known foregrounds, they are due to a previously unexpected foreground, or they are due to WMAP systematics.

How do the authors assess these explanations?

As remarked above it is difficult for us to accept the occurrence of a 10⁻⁸ unlikely event as a scientific explanation.

1.

A cosmological mechanism could possibly explain the weakness of large angle correlations, and the alignment of the quadrupole and octopole to each other. A cosmological explanation must ignore the observed correlations to the solar system, as there is no chance that the universe knows about the orientation of the solar system nor vice-versa. These latter correlations are unlikely at the level of less than 1 in 200 (plus an additional independent ≈ 1/10 unlikely correlation with the dipole which we have ignored). This possibility seems to us contrived and suggests to us that explanations which do not account for the connection to solar system geometry should be viewed with considerable skepticism.
In [16], we showed that the known Galactic foregrounds possesses a multipole vector structure wholly dissimilar to those of the observed quadrupole and octopole. This argues strongly against any explanation of the observed quadrupole and octopole in terms of these known Galactic foregrounds.

2.

A number of authors have attempted to explain the observed quadrupole-octopole correlations in terms of a new foreground [51–56]. (Some of these also attempted to explain the absence of large angle correlations, for which there are also other proffered explanations [57–61].) Only one of the proposals ([53]) can possibly explain the ecliptic correlations, as all the others are extragalactic. Some do claim to explain the less-significant dipole correlations. Difficulties with individual mechanisms have been discussed by several authors [56, 62–65] (sometimes before the corresponding proposal). Unfortunately, in each and every case, among possible other deficiencies, the pattern of fluctuations proposed is inconsistent with the one observed on the sky. As remarked above, the quadrupole of the sky is nearly pure Y₂₂ in the frame where the z-axis is parallel to ˆ w^(2,1,2) (or any nearly equivalent direction), while the octopole is dominantly Y₃₃ in the same frame. Mechanisms which produce an alteration of the microwave signal from a relatively small patch of sky—and all of the above proposals fall into this class — are most likely to produce aligned Y₂₀ and Y₃₀. (This is because if there is only one preferred direction, then the multipole vectors of the affected multipoles will all be parallel to each other, leading to a Y_ℓ0.) The authors of [55] manage to ameliorate the situation slightly by constructing a distorted patch, leading to an underpowered Y33, but still a pure Y20. The second shortcoming of all explanations where contaminating effect is effectively added on top of intrinsic CMB temperature is that chance cancellation is typically required to produce the low power at large scales, or else the intrinsic CMB happens to have even less power than what we observe. Likelihood therefore disfavors all additive explanations [56] (unless the explanation helps significantly with some aspect of structure seen at higher ℓ).

So:

Explaining the observed correlations in terms of foregrounds is difficult. The combined quadrupole and octopole map suggests a foreground source which form a plane perpendicular to the ecliptic. It is clear neither how to form such a plane, nor how it could have escaped detection by other means. This planar configuration means that single anomalous hot or cold spots do not provide an adequate explanation for the observed effects.

3.

The final possibility is that systematic effects remain in the analysis of the WMAP data.

The consequences if indeed these correlations are real?

If indeed the observed ℓ = 2 and 3 CMB fluctuations are not cosmological, there are important consequences. Certainly, one must reconsider [7] all CMB results that rely on low ℓ_s, including the optical depth, τ, to the last scattering surface; the inferred redshift of reionization; the normalization, A, of the primordial fluctuations; σ₈, the rms mass fluctuation amplitude in spheres of size 8h⁻¹Mpc; and possibly the running dn_s/d logk of the spectral index of scalar perturbations (which, as noted in [68], depended in WMAP1 on the absence of low-ℓ TT power).
Of even more fundamental long-term importance to cosmology, a non-cosmological origin for the currently observed low-ℓ microwave background fluctuations is likely to imply further-reduced correlation at large angles in the true CMB. As shown in Section 3, angular correlations are already suppressed compared to \LambdaCDM at scales greater than 60 degrees at between 99.85% and 99.97% C.L. (with the latter value being the one appropriate to the cut sky ILC123). This result is more significantin the year 123 data than in the year 1 data. The less correlation there is at large angles, the poorer the agreement of the observations with the predictions of generic inflation. This implies, with increasing confidence, that either we must adopt an even more contrived model of inflation, or seek other explanations for at least some of our cosmological conundrums. Moreover, any analysis of the likelihood of the observed “low-ℓ anomaly” that relies only on the (low) value of C₂ (especially the MLEinferred) should be questioned. According to inflation C₂, C₃ and C₄ should be independent variables, but the vanishing of C(θ) at large angles suggests that the different low-ℓ Cℓ are not independent.

And what of the standard model's interpretation of the data?

This does not seem reasonable to us — that one starts with data that has very low correlations at large angles, synthesizes that data, corrects for systematics and foregrounds and then concludes that the underlying cosmological data is much more correlated than the observations — in other words that there is a conspiracy of systematics and foreground to cancel the true cosmological correlations.
This strongly suggests to us that there remain serious issues relating to the failure of statistical isotropy that are permeating the map making, as well as the extraction of low-ℓ Cℓ.
At the moment it is difficult to construct a single coherent narrative of the low ℓ microwave background observations. What is clear is that, despite the work that remains to be done understanding the origin of the observed statistically anisotropic microwave fluctuations, there are problems looming at large angles for standard inflationary cosmology.

So it is the standard model that is a conspiracy theory!

As I have said several times, we must not forget the interpretation of the precise WMAP data is model dependent and that model is looking more problematic as time goes on...

Garth

 

[Garth]

A paper today Anomalies in the low CMB multipoles and extended foregrounds that explains the low-l mode anomalies by an extended forground centred on the Local Supercluster (LSC).

We discuss how an extended foreground of the cosmic microwave background (CMB) can account for the anomalies in the low multipoles of the CMB anisotropies. The distortion needed to account for the anomalies is consistent with a cold spot with the spatial geometry of the Local Supercluster (LSC) and a temperature quadrupole of order DeltaT_2^2 ~ 50 microK^2. If this hypothetic foreground is subtracted from the CMB data, the amplitude of the quadrupole (l=2) is substantially increased, and the statistically improbable alignment of the quadrupole with the octopole (l=3) is substantially weakened, increasing dramatically the likelihood of the "cleaned" maps

A solution has been found?

CONCLUSIONS
We have presented circumstantial evidence that an extended foreground near the dipole axis could be distorting the CMB. The subtraction of such a foreground increases the quadrupole, removes the (anomalous) quadrupole-octopole alignment, and dramatically increases the overall likelihood of the CMB maps. Possible physical mechanisms that could account for this foreground are the Sunyaev-Zeldovich effect [25] and the Rees-Sciama effect [27], although it should be noted that both options only work in extreme situations that are probably unrealistic. Another possibility is that a combination of effects is responsible for the foreground. However, if the Sunyaev-Zeldovich effect due to the LSC’s gas is indeed responsible for the foreground, it could be directly observed by the Planck satellite [53] within the next few years.

(emphasis mine)
So we should know in a few more years...

Or is this an example of the Copi, Huterer, Schwarz and Starkmanon suggestion (my post # 59) that:

there is a conspiracy of systematics and foreground to cancel the true cosmological correlations.

Garth

 

[oldman]

What is it about the WMAP results that tells us the universe is flat?

I've been browsing back in this thread and the "What is CMB" thread to answer this question, and gather (mainly from Space Tiger and Garth) that a flat geometry for the universe is deduced from a best fit of theory (treating many factors?) to the high frequency peaks (of an assumed power-law spectrum of evolved primordial density fluctuations?).

Is this anywhere near correct?

I'd like to understand just what is it about the flat geometry that produces the good fit to the data. Perhaps the sensitivity of the Sachs-Wolfe effect to geometry, or something more subtle or quite different?

I realize from the threads that analysing the spectrum is a highly technical matter. But I'd love a simple explanation.

 

[hellfire]

As fas as I know, the main data to infer about geometry is the angular size of the first peak. The first peak is a primary anisotropy, the greatest of acoustic nature. Its physical size is determined by the size of the particle horizon at decoupling. The relation between the observed angular size of the first peak and its physical size depends on the geometry of space and on the distance to the last scattering surface. Thus, to infer about geometry from the observed angular scale of the firs peak it is needed an assumption about the size of the particle horizon at decoupling as well as about the distance to the last scattering surface. For example, a positive curvature implies that the physical scale of the first peak should be smaller than in a flat space.

 

[marcus]

hellfire, oldman,
maybe we could do a calculator experiment about this!

we know, because of the temperature, that the CMB is at z = 1100.

we could convert that to distance using two assumptions, with Ned Wright's calculator: we could assume flat and we could assume some nonflat case.

then we would get two different figures for the area of the lastscatter sphere---two different ideas of the actual physical size of the universe at the time of decoupling.

this seems doable (using Wright's calculator) with some simple arithmetic

from the temperature we could deduce the speed of sound in that medium, and we would have two separate cases of what the size of the medium is---maybe we could get some intuition about what hellfire says about the size of the first acoustic peak.

then, we would expect that the angular size we calculate in the FLAT case would match the observed CMB power spectrum---i.e. fit the mottled way it looks. and the angular size we calculate from the NONFLAT case would NOT match the observed CMB picture. so we would be doing a crude imitation of the professional CMB analyst routine.

hands dirty CMB interpretation you can do in your own kitchen. I would like to see it, if anyone's game.

 

[hellfire]

The impact of the curvature in the relation between actual size and measured angular size of the first peak is given through the definition of the angular diameter distance d_A (for which you have started a thread recently).

In general, one has that the actual size l relates to the angular size \theta:

l = d_A \theta

In the cosmological calculator you get, for example for z = 1100:

d^{SCDM}_A = 24.08 Mly
d^{OCDM}_A = 74.26 Mly

For:
- The flat model SCDM (standard cold dark matter model): \Omega = 1, with \Omega_m = 1
- The open model OCDM (open cold dark matter model): \Omega = 0.3, with \Omega_m = 0.3

If you assume a measured angular size of about 1° for both models, you see immediately that the actual size of the first peak would be smaller in the flat model than in the open model.

This may help to illustrate the angular diameter distance in different models.

But the problem is that both models produce a different angular size for the first peak (it cannot be assumed that both are 1° today). To calculate the actual size of the first peak one should proceed as you propose, i.e. the size of the sound horizon should be calculated. How to do this I don't know.

Afterwards, aplying then the formula \theta = l / d_A, you could calculate the expected angular size in sky of the first peak for both models.

 

[oldman]

The impact of the curvature in the relation between actual size and measured angular size of the first peak ...

Thanks, Marcus and Hellfire, for your answers and calcs. It seems that finding out from the WMAP results that the universe's geometry is flat is much easier than I thought. Viva Euclid!

 

[hellfire]

I have read that it is usually assumed that the speed of sound during recombination is equal to c_s = c/ \sqrt{3} (would be nice if someone could check this). The sound horizon is:

s = \int_0^{t_{rec}} dt \frac{c_s}{a}

this would mean that it is 1 / \sqrt{3} times the size of the particle horizon at recombination. In my calculator I have an output of the particle horizon for a specific redshift, but I see now that I have made a silly mistake there and that this output field is incorrect. I will see if I can correct this today. Then we would have the values s^{OCDM}, s^{SCDM}.

Next step would be to know how to go from the value of s to the size of the first peak...?

 

[hellfire]

For the \LambdaCDM model we might try to do the calculation backwards. First, the angular diameter distance for z = 1100 is:

d^{\Lambda CDM}_A = 41.34 Mly

Now, if I assume that the size of the first peak is equal to the sound horizon size l = s (?), then, applying l = d_A \theta, with \theta = 1°, I would get:

s^{\Lambda CDM} = 0.7 Mly

This value seams meaningful to me. But the point is this value for s should arise from the horizon formula I put above. Then, inserting in l = d_A \theta it should lead to the angular size of the first peak. I have been trying to modify my calculator to give such an output but I did not succeed; the value for s I am getting with the calculator is about 833 Mly.

Could anyone find out the value of the particle horizon and sound horizon at z = 1100 for the \LambdaCDM model?

 

[marcus]

hellfire, I want to echo the appreciation expressed by oldman.
Your demonstration of the down-and-dirty-nitty-gritty of the first acoustic
seems to satisfy oldman (at least for now) and although I have not
followed all your steps I feel generally better about it too.
sometime I hope we locate an online tutorial article about this,
but for now we seem to have gotten a better handle on it.

 

[oldman]

Your demonstration ... of the first acoustic
seems to satisfy oldman (at least for now) and ...I feel generally better about it too.

Marcus, in this thread Hellfire and yourself seem to agree that the main flat-geometry-indicator, if I may call it that, is the angular width of the first acoustic peak. Fine -- I think I grasp the explanations you both so kindly gave.

Yet in your most recent post in the thread "WMAP 3 and spatial closure" (#108) you quoted a statement: "...However, altering the geometry of universe mainly affects the positions of the CMB acoustic peaks,..." This seems to imply that it is the position of the peaks rather than a width that is the main flat-geometry-indicator. This confuses me again.


The WMAP results are rich and important to grasp. A list of the main conclusions, each with a statement of which feature/s of the results they are attributed to would be very illuminating for the uninformed. Tabulated along lines like:

Geometry is flat..... deduced from first peak angular width

Baryonic matter is 4%...deduced from fit to peaks l > 150

Dark energy is 75% ... deduced from distance between 3rd and 4th peaks

and so on, perhaps. (The entries above are of course a fiction ... I have no idea of how to draw such a table up).

The tutorial article you mentioned sounds like a good idea!

 

[Garth]

Geometry is flat..... deduced from first peak angular width

This is a statement agreed by all in the community.

My personal beef - this statement more generally should be:

"Geometry is conformally flat...deduced from first peak angular width"
as the WMAP data is angular in nature and conformal transformations are angle preserving.

Garth

 

[hellfire]

Marcus, in this thread Hellfire and yourself seem to agree that the main flat-geometry-indicator, if I may call it that, is the angular width of the first acoustic peak. Fine -- I think I grasp the explanations you both so kindly gave.

Yet in your most recent post in the thread "WMAP 3 and spatial closure" (#108) you quoted a statement: "...However, altering the geometry of universe mainly affects the positions of the CMB acoustic peaks,..." This seems to imply that it is the position of the peaks rather than a width that is the main flat-geometry-indicator. This confuses me again.

Both are basically the same. You can convert from the position of any multipole \ell to its angular scale \theta with:

\ell \sim \frac{180^{\circ}}{\theta}

 

[oldman]

Both are basically the same. You can convert from the position of any multipole \ell to its angular scale \theta with:

\ell \sim \frac{180^{\circ}}{\theta}

Thanks, Hellfire. But how does this change of label along the x-axis change "width" into "position" ? The two are qualitatively different. I'm stupid today.

 

[hellfire]

Take a look to the picture in wikipedia. You can see that the x-axis is the number of the multipole moment \ell. The peak at \ell ~ 200 tells you that the power is strongest at that value. This is the first peak. When we talk about the angular width of the first peak we are not talking about the width of the peak in this picture, but about the conversion of \ell to \theta I gave you before. E.g. a smaller angular scale \theta of the first peak would just mean that it would be located more to the right, at higher values of \ell. This means that curvature shifts the position of the first peak to the left or to the right in this picture.

 

[oldman]

Thanks again. I understand now.

 

[hellfire]

I have a question about that picture in wikipedia, may be someone can answer.

The y-axis corresponds to the power of the multipole. It is written as a function of C_{\ell} or as \mu K^2. I do not see how both magnitudes are equivalent:

The \mu K^2 means that the y-axis gives the deviation from the mean temperature for a given multipole.

On the other hand, the C_{\ell} indicates that the power is calculated making use of the two-point correlation function. To my understanding this would mean that the y-axis gives the deviation from a Poisson distribution of anisotropies for a given multipole.

This understanding seams to be incorrect, because this seams not to be equivalent to a temperature deviation.

 

[SpaceTiger]

The y-axis corresponds to the power of the multipole. It is written as a function of C_{\ell} or as \mu K^2. I do not see how both magnitudes are equivalent:

The former is the quantity being plotted and the latter are its units. In general, the power spectrum is just the Fourier transform (in the spherical expansion) of the two-point correlation function. The latter, I believe, is just:

C(\theta)=&lt;\delta T (\vec{e_1}) \delta T (\vec{e_2})&gt;

where the \delta T are the deviations from the mean temperature at a given point in the sky. This, of course, has units of temperature squared. Since the anisotropies are on micro-Kelvin scales, the units of the angular power spectrum are also given in \mu K^2[/tex]. I think the other scaling factors are chosen to emphasize the acoustic peaks.<br /> <br /> If the anisotropies are Gaussian (that is, described by a Gaussian random field), then the power spectrum is a complete description of them. As best we can measure, the anisotropies are indeed Gaussian, as predicted by inflation. Inflation also predicts small deviations from Gaussianity, but we&#039;re not yet at the level where we can detect that.

 

[hellfire]

Thanks for your answer, but I still don't get it. According to my knowledge the two-point correlation function measures the deviation from an homogeneous distribution of anisotropies. If the distribution of l = 200 or 1° anisotropies is homogeneous through the sky, wouldn't this mean that the correlation should vanish, independently from the fact that these have a higher temperature than the average? What is C(\theta) telling us exactly?

 

[SpaceTiger]

Thanks for your answer, but I still don't get it. According to my knowledge the two-point correlation function measures the deviation from an homogeneous distribution of anisotropies. If the distribution of l = 200 or 1° anisotropies is homogeneous through the sky, wouldn't this mean that the correlation should vanish, independently from the fact that these have a higher temperature than the average?

Suppose I filled the sky with fluctuations (both hot and cold spots) that had typical sizes of order 1°. What would we expect from the correlation function at angular scales of 0.5°? If I look at just one point in, say, a cold spot, then most of the points at distances 0.5° away should also be cold (the size of the fluctuation is larger than the angular scale we're probing). This means that, for this one point, the quantity I quoted above should be positive (the product of two negative temperature fluctuations). In the hot spots, both temperature fluctuations will be above the mean, so the correlation function will again be positive. Thus, averaged over the whole sky, we expect the correlation function at 0.5° to be positive.

This is not the case at much larger angular scales, however. If I look again at a point in a hot spot and compare it to a point 5° away, I will be just as likely to run into a hot spot as a cold spot. Thus, the correlation function at 5° should come out to zero (or nearly zero) when averaged over the whole sky.

What is C(\theta) telling us exactly?

It's telling us about the relative amplitudes of fluctuations of different angular sizes. C(\theta) is somewhat more difficult to interpret than the power spectrum because, as you can imagine, a sky full of 1° fluctuations will produce correlations at all scales less than about a degree. When you combine this with fluctuations at smaller scales, it becomes difficult to distinguish fluctuations of different sizes. The power spectrum, however, tells you directly about the relative contributions of fluctuations at various scales (in this case, expressed in terms of the spherical wavenumber, l). The more power there is at a given l, the larger the amplitude of fluctuations at that scale.

 

[hellfire]

It's clear now, thanks!

 

[Harry Costas]

Hello All

Have a look at
http://astro.uwaterloo.ca/~mjhudson/research/threed/

Loacation of our super clusters of galaxies

 

[marcus]

Hello All

Have a look at
http://astro.uwaterloo.ca/~mjhudson/research/threed/

Loacation of our super clusters of galaxies

these look like nice pictures, I have not watched the animations yet (haven't checked that they are online)

thanks.