(that to correct myself, sorry) it is assumed that the observer is not so massive (does not disturb FRW metric);
and the observer’s quadrupole (along the tangent dimensions) depends on observer’s position on the S^3-sphere.

it is assumed that, after recombination and before formation of
galaxies, the pristine baryon matter was uniform and isotropic
– ie with zero “peculiar velocities”; and it seems that after
clustering the centers of mass of large clusters still should be
“anchored” (having only small “peculiar velocities”);

I think that red/blue shifts have invariant (gage invariant)
sense (and perhaps the Hubble law also may have such a sense –
although the observation accuracy for distances is not so good);
so, if an observer is lucky enough to see several “identical”
massive clusters, “anchors” (on the same distance but in different
directions), and if their red-shifts are some different,
the observer may infer that his/her own “peculiar velocity”
is not zero.

And I have in mind a simple 5D picture (instead of inflation:)

expanding S^3 sphere in R^4 space, R^2=x^2+y^2+z^2+w^2, R ~ c t, and
the perturbation of SO_4-symmetry with the term (quadrupole (!), still high
symmetry SO_2 x SO_2 x P_2…; topological charge in AP can have this
high symmetry, but not SO_4)
x^2 + y^2 - z^2 - w^2

in the field I work in point (i) is practically the same as: “We don’t have a good ideas that work, let’s systematically explore some vague ideas that might work”. And that’s best done when working with a team of people…
Of course, for experimentalists or people who need to do large scale computing this is different. I only need paper and pencil and some limited computing power…

The main difficulty I see with the correlated dipole/quadrupole is that in linear perturbation theory the dipole is gauge dependent because you have to specify the 4-velocity of the observer. (The quadrupole is gauge-dependent only at 2nd order.) So you would have to specify in which gauge your relation is supposed to apply. I’m guessing you would use Newtonian gauge since the synchronous gauge dipole has an enormous contribution from high-k perturbations (what a Newtonian person would call peculiar velocities). This seems to be what you are suggesting (although massive clusters don’t stay fixed in the Newtonian frame, since they “fall” into potential wells with the same acceleration as everything else).

In “standard” cosmology, if you keep 2nd order terms there is a gauge-dependent “kinetic quadrupole” in the CMB that is aligned with the dipole (basically it’s the order v^2/c^2 term in the Doppler formula that results from our peculiar velocity). It’s been calculated and is smaller than the observed quadrupole from WMAP.

Something like A n_z + B(\ksi n_z^2 + n_x^2 - n_y^2),
where \ksi from -1 to +1.

I mean that one could assume that “very large masses” (say,
clusters of galaxies) have zero (own real) velocities — with
respect to “real background”. So it is possible, in principle, to find
the velocity of Earth on this “real background” (different from
the CMB background).

If you have a briliant idea then you will, of course, write it up yourself. Most of the time you don’t have brilliant ideas and usually that leads to collaborations…

I am not aware of collaborations formed due to a lack of ideas. Generally collaborations exist because either (i) there is so much work to do to bring an idea to fruition that no one person could handle it; or (ii) the person who came up with the idea does not have the full range of technical skills or familiarity with hardware, data, or computer codes to implement it.

Those who disbelieve e.g. (i) should consider whether one brilliant physicist plus a pool of unskilled laborers could ever have found the top quark.

There exists some fantastic wavelet mathematical tools that can be applied to the situation in your paper to test the CMB data. Wavelet transform algorithms can be extended to more dimensions, providing directionality features that Fourier methods lack. For example, multidimension-dimensional wavelets can be designed in such a way that they are directionally selective. So, in addition to dilation and translation in one-dimension, one can now also rotate the wavelet, to detect an oriented feature in a 2D image. I proposed to NASA to write an IDL wavelet library to perform this and many other wavelet functions, but it wasn’t accepted, unfortunately. I thought I made my case strongly that this would be very useful to astronomers, but that particular NASA program was competitive (only 1 in 6 was funded). Such multidimensional wavelet tools exist in other languages (C, Matlab), however. Keywords (here I list a collection of wavelets for cosmologists): curvelets, needlets, ridgelets, spherical Haar wavelets, spherical Mexican hat wavelets, 2D and 3D CWTs, a trous wavelet transforms, pyramidal wavelet transforms, and Gabor wavelets on the sphere.