The void theory

On tumblr, one of my reader, frankietwohats, ask me the following question:

Can I ask you a random question? I can't remember the name of a theory that argued that the universe wasn't expanding, but instead was stretching. Do you happen to know of it/it's name? It came up in conversation today (well, universe expansion did) and I want to look into it more.

I don't know if this is the theory that you intend, but after a briefly research on Google, i find th Void theory. About it, Esther Inglis-Arkell writes on io9:

There was a time that the earth was considered the center of the universe. Then it got knocked out of the way by the sun, and ever since then the astronomer's mantra was, "We are nothing special." The part of the universe the earth resides in can't be any different than any other part. It's not unique, or remarkable, or even out of the ordinary. Void Theory contradicts all that. Instead of sitting in a typical part of the universe, the earth sits in an unusually empty part; a void. The universe isn't expanding due to some mysterious force. It's just that when light comes from a denser part of the universe and trips across the void, it is altered to make it look like the universe is expanding. Since this exansion is the same when observed from any part of the earth, the earth has to be roughly at the center of this void. Suddenly, the observable universe is geocentric again.

But... what is the void theory?

This diagram reveals changes in the rate of expansion since the universe's birth 15 billion years ago. The more shalow the curve, the faster the rate of expansion. The curve changes noticeably about 7.5 billion years ago, when objects in the universe began flying apart as a faster rate. Astronomers theorize that the faster expansion rate is due to a mysterious, dark force that is pulling galaxies apart.

First of all, following Clifton, Ferreira and Land⁽¹⁾, we must remember that our picture of the universe is based on the following two principles: the spacetime is dynamical, obeying to the Einstein's equations; the Universe is homogeneous and isotropic on large scales, that is a generalisation of the Copernican Principle that the Earth is not in a central, specially favored position.
Now, the exact solution of Einstein's equations was provided by Lemaitre-Tolman-Bondi spacetime \[\text{d} s^2 = -\text{d} t^2 + \frac{Y'^2}{1-K} \text{d} r^2 + Y^2 \text{d} \Omega\] In this model there are four free parameters: the density at the origin, the density and radius at the midpoint, and the radius at which we match to Einstein-de Sitter spacetime⁽²⁾. Instead the void model:

is completely specfied by the radial profile, the Hubble rate at the void centre today, $H_0$, the radiation density today, which is fixed by the CMB mean temperature, $T_0 = 2.725 K$, and the baryon fraction $f_b = \frac{\rho_b}{\rho_m}$. Outside the void we asymptote to EdS.

The curvature for three different types of voids