I found some papers discussing ISW as a proof of dark energy, but what is its physical meaning? What is your opinion? And how to calculate it?
http://preposterousuniverse.com/teaching/371/papers06/sherman-371.pdf
The integrated sachs wolf effect (ISW) is the cumulative response of the microwave background to the existence of gravitational perturbations before recombination. It is the source of the observe anisotropy in the MWB and is well understood. In principle, it has no relation to dark energy. Remember, before recombination the amplitude of the dark energy is very small (i.e. the Universe is not accelerating at this time) and so its a very minor correction to the gravitational well effect on the photons.
The Rees-Sciama effect is the full nonlinear effect due to growing density perturbations AT the time of the surface of last scattering. Here dark energy may play a role as accelerated expansion amplifies the time it takes for a photon to cross a density perturbation.
Dear Mohamed, thank you, i just replied your message, please kindly take a look. Thanks
Rees-Sciama also includes effects of nonlinear structures growing all the way through the non-linear phase of growth, which includes the effects of structures at late times, not just at the time of the surface of last scattering. It results from changes in the potential depth of structures growing in mass, all the way up to very late times in the universe. The ISW is also a perturbation on the CMB from the time after the surface of last scattering, i.e., between that surface and us, but is usually used to describe the effect on the CMB from the growth of structure in the linear phase. there are variations in description, ISW, late-time ISW, Rees-Sciama. As I understand it, the typical delineation between ISW and RS is linear growth, vs. non-linear growth of structure being responsible for the CMB perturbation.
Dear Prof. Mohammed El Naschie:
Thank you for your message. Yes i agree with you that mathematics is very important, and there is no advancement in physics without mathematics innovation. But we should also keep it consistent with physical observables.
As you are also an engineer, at least in the past, allow me to offer you this joke:
For engineers, their equations are approximation to reality.
For physicists, reality is approximation to their equations.
For mathematicians, they do not care.
Btw, i am in the middle of preparing a dissertation in cosmology, the idea is to propose a new approach which i call : fractal vibrating string. Perhaps do you know sone friends who perhaps would like to support this idea? For example, i generalized nabla operator to become fractal nablr operator, and i also propose to generalize laplace beltrami operator to become fractal laplace beltrami operator. If you are not interested, just ignore this message, but if you want to read my draft, here it is: https://www.academia.edu/11416689/In_Search_of_Cosmology_based_on_Interpretation_of_Cosmic_Christology_of_the_Johannine_Prologue_Dissertation_Draft_ver._1.1_
I think I didn't answer your question regarding the "physical meaning" of these effects. Physically, they are a result of the frequency shifting of CMB photons due to the change in size of the gravitational potential they are traveling through during that travel time. Meaning, the photon enters the potential, and is blueshifted traveling down into the potential well. The large size of the cosmological potential wells means that as the photon travels, and is still under the influence of the object's gravity, it can grow in mass, deepening the potential. Therefore, when the photon climbs out of the potential, being redshifted by that travel, the corresponding redshift from leaving the deeper potential is larger than the size of the blueshift it encountered when entering the potential. Therefore, on scales relevant to the size of the potentials that are important (for ISW it's large angular scales, for RS it's smaller scales), there is a distortion in the CMB that does not correspond to effects that occur at recombination.
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