You're correct that if some of the [tex]<a_{lm} a_{l'm'}^*>[/tex] can beAndrew Pontzen wrote:No, that's incorrect. The 'designer' theory in 1004.2706 allowsBoud Roukema wrote: It seems to me that 1004.2706 makes the assumption that the [tex]a_{lm}[/tex]'s

are drawn from (statistically) independent Gaussian distributions. Is

this correct?

complete freedom in the covariance matrix, meaning the a_{lm}'s can be

correlated in any way whatsoever. The restrictions are just that the

theory is Gaussian and has zero mean.

non-zero, then the [tex]a_{lm}[/tex]'s are not statistically independent

distributions :). So, let me try again to understand

your work.

Are you assuming that all the statistical information is contained in

the [tex]C_l[/tex]'s together with the covariances [tex]<a_{lm} a_{l'm'}^*>[/tex], under

the assumption that each of these is Gaussian distributed, and that

there are no further correlations e.g. the n-covariances for [tex]n >2[/tex] ?

I do not see where the higher n>3-point spherical harmonic

correlations are included in your work.

Complete freedom in the covariance matrix does not mean that you

consider

*spatially*-Gaussian perturbation theories in FLRW

models. For example, maybe I didn't go into enough detail above when I

wrote "If you're looking at the same point in comoving space when

you're observing widely separated sky positions,...". For example,

for one of the popular candidates for a best fit to the WMAP data,

i.e. the [tex]S^3/I^*[/tex] FLRW model (the Poincare dodecahedral space -

astro-ph/0310253), a single physical spatial point is in some cases

seen 12 times on the sky (considering naive SW only). So even when the

amplitudes of the comoving

*spatial*amplitudes are distributed

according to Gaussian distributions that are statistically independent

from one another, you would need a lot of n>3-covariance matrices to

represent all this information using [tex]a_{lm}[/tex]'s. That doesn't

necessarily make a [tex]C_\theta[/tex] approach into a complete tool. [tex]C_\theta[/tex]

would probably also need [tex]n\gg 1[/tex]-point angular correlations if the

aim were to represent the full information of a spatially-Gaussian

model. [Copi et al do not claim that [tex]C_\theta[/tex] represents all the

statistical information for spatially-Gaussian models; they state the contrary

(IV.C 1st paragraph, last sentence).]

So your statement that the "restrictions are just that the theory is

Gaussian and has zero mean" seems to me to be incomplete and ambiguous. You do seem

to be restricting yourself to simply-connected FLRW models, plus

a few particular multiply-connected FLRW models in which ignoring the

n>2-point [tex]a_{lm}[/tex] cross-correlations is accurate.

Moreover, in II, just below Eq (4), you state that you use a finite

sum with [tex]l\le30[/tex], i.e. in effect you set [tex]a_{lm}=0 \;\; \forall \, l > 30[/tex],

in all numerical calculations, and in IV.B. 2nd paragraph, you assume

an infinite flat model for covariances for [tex]l > 10[/tex].

Hiranya Peiris wrote:Andrew Pontzen wrote: No, that's

incorrect. The 'designer' theory in 1004.2706 allows complete freedom

in the covariance matrix, meaning the a_{lm}'s can be correlated in any

way whatsoever. The restrictions are just that the theory is Gaussian

and has zero mean.

And the obvious requirement that the covariance matrix of the theory

is positive definite.

Could you please give or cite a proof that the covariance matrix

in FLRW models is always positive definite, i.e. without the assumption

of comoving spatial simple-connectedness? Given that the definition of

positive definiteness is for multiplying a matrix by an arbitrary

vector, it's not obvious to me that this is correct. If it is incorrect,

then this is another restriction resulting from assuming simple

connectedness. If it is correct, then it would be useful to have

that proof available.

Hiranya Peiris wrote:I didn't see that it had responded to any of the points made inBoud Roukema wrote:I agree that 1004.5602 is a very clear review of

the issue, and gives clear responses to several recent papers.

1004.2706. We would obviously welcome a discussion of these points

here, if the authors are watching.

Please see Copi et al. 1004.5602's 4th and 5th paragraphs and the

above comments. In addition to the references given by the authors,

you might try a few early review papers (gr-qc/9605010,

astro-ph/9901364) or if you are impatient, a short pedagogical review

(astro-ph/0010185). Copi et al do not want to restrict themselves

necessarily to FLRW models, but FLRW models are an obvious start.

Comoving space must be a 3-manifold to a good approximation.

Please also see IV intro (2 paragraphs), IV.C 1st paragraph, 2nd and 3rd

sentences.

Maybe consider it this way. It is correct that all the information in

a function [tex]f(\theta,\phi)[/tex] on [tex]S^2[/tex] is contained in the infinite set

of [tex]a_{lm}[/tex]'s, given some restriction on the class of possible

functions f, and using theorems about vector spaces regarding

orthonormal basis sets, linear independence, etc. However, this does

not guarantee that the physically valid information in f is

represented in the [tex]a_{lm}[/tex]'s in a simple way. It may

*seem*

that ignoring n>3-point correlations between [tex]a_{lm}[/tex]'s and ignoring

e.g. [tex]l>20[/tex] is simple. However, [tex]S^2[/tex] is not 3D comoving space. What

is simple in comoving space (Gaussian amplitudes of projections of a

function on comoving space [tex]M[/tex] onto a set of orthonormal

eigenfunctions) may be complicated on [tex]S^2[/tex].

Where is simplicity more important: in the physical model or in the observational model?

It would help if you could be clearer about the difference betweenAndrew Pontzen wrote:(Of course, a theory which is non-Gaussian on

large scales but highly Gaussian on small scales might come along and

evade our limits. I'll eat my hat...)

Gaussianity and independence of distributions of the amplitudes of

spatial eigenmodes, versus the same properties of [tex]Y_{lm}[/tex]'s. These

are not equivalent conditions. FLRW models which are spatially

Gaussian at small length scales and can be modelled as spatially

Gaussian at large scales are well-known. If sky map analysis is

done by forcing onto [tex]a_{lm}[/tex]'s, then in some ways these

are non-Gaussian and have [tex](n\ge2)[/tex]-correlations of [tex]a_{lm}[/tex]. Since you appear

to be unware of these, please see Copi et al. 1004.5602's 4th and 5th paragraphs,

papers they cite elsewhere in their text, "early" review

papers (gr-qc/9605010, astro-ph/9901364) or if you are impatient, a

short pedagogical review (astro-ph/0010185). Detailed calculations

require a lot of work. Please see Fig. 4 of astro-ph/0412569 for an

example.

You might also like to see some

*a priori*(i.e. published before [tex]\sim[/tex] 1990) references:

- Friedmann 1924, http://cdsads.u-strasbg.fr/abs/1924ZPhy...21..326F
- Lemaitre 1931, http://cdsads.u-strasbg.fr/abs/1931MNRAS..91..483L especially see the first paragraph
- Robertson 1935, http://cdsads.u-strasbg.fr/abs/1935ApJ....82..284R e.g. page 297, where the default k>0 model is a multiply-connected model, although
Howard Robertson wrote:we are still free to restore this second [simply-connected] possibility - but the decision on observational ground is presumably far beyond our present resources.