Could the increasing anisotropy of the universe lead to an additional blueshift?

Question

I'm contemplating the possible sources of a wavelength-shift within our universe:

1. The CMB had a lot more energy when it was produced (around 3000 K). Due to the expansion of the universe, it has been heavily redshifted (now: 2.7 K). That is: The light of the CMB has been produced far away and during its movement towards us, it has been redshifted (because of the expansion of the universe). Light is redshifted while traveling, due to expansion of the universe.

2. Distant objects of the universe are redshifted due to their movement away from us due to expansion of the universe. When we plot the redshift over the distance, we see that the redshift doesn't increase linearly with distance. Expansion of the universe must have been slower in former times which leads to dark energy. Light is redshifted when the source, while sending, moves away from us.

Now, I'm wondering whether the increase of anisotropy within the universe has any effect on the wavelength of the photons which travel through it?

During the production of the CMB, the universe is supposed to have been uniform (no galaxies, stars... only plasma soup). Today, the universe is more anisotropic: matter accumulates, builds galaxies and stars and black holes. When light goes into dense regions, it's blueshifted. When it goes into less dense regions, it's redshifted. Let's assume a linear increase of anisotropy. Photons start (source) and end (earth in solar system) in a dense region. Therefore, it's always: Redshift, followed by a Blueshift. Due to the increase of anisotropy, the blueshift is always a bit larger than the redshift. Could it be that this leads to a net blueshift of the wavelength of the photons? A net blueshift that increases with distance? And that has to be added to those points 1. and 2. from above?

Answer 1

It seems to me that what you are pointing out is related to what is known as the (integrated) Sachs-Wolfe effect, which has been considered in the missions that have measured the CMB anisotropies (WMAP/Planck). The temperature fluctuations at the direction $\hat{n}$ contain mainly three contributions (taken from Baumann's book of Cosmology):

$$\frac{\delta T}{T} (\hat{n}) = - (\hat{n}\cdot \vec{v})_\ast + \Big(\frac{1}{4}\delta_\gamma + \Psi \Big)_\ast + \int_{\eta_\ast}^{\eta_0} d \eta \, (\Psi' + \Phi') $$

The first term is the Doppler term (related to what you mentioned in your point 2), and the last two terms are Sachs-Wolfe and Integrated Sachs-Wolfe (ISW). In particular, the ISW encodes the contribution to the anisotropies via gravitational redshift along the evolution of inhomogeneities that form the large scale structure. The contribution is generally small, except at early times (due to radiation) or late times (due to cosmological constant). Since the clustering of matter is nonlinear at the small scales, a net blueshift effect might not be an accurate statement, but the intuition of an effect due to the evolution of the inhomogeneities seems reasonable.