It has been observed that the universe is expanding, for half of its history at a decelerating rate (after a very early period of inflation), but for the past six billion years or so, at an accelerating rate, see for example,  and  . The determination that expansion is occurring is a direct result of solving Einstein’s equations  . However, the solution to these equations shows that expansion decelerates through the full history of the universe. Thus, without some modification, there is no way to fit the observations from supernova data that demand a new view: a universe expanding at an increasing rate during the past several billion years  .
The current approach to resolving the lack of congruence between general relativity and observation has been to add a cosmological constant to the equations, creating a repulsive force. The resulting concordance ΛCDM (Lambda Cold Dark Matter) model  posits an unknown form of energy with negative pressure and an energy density 123 orders of magnitude different from theoretical expectations. The ΛCDM paradigm reproduces most observations, although so far no plausible candidate for dark energy has emerged and some issues remain  . In particular, recent local measurements of the Hubble constant  are up to 3.4σ higher than the value derived from Planck’s observations  of the cosmic microwave background.
In addition to positing dark energy, some have tried to account for the differences by modifying gravitational theory (for a review, see  ). However, Einstein’s theoretical predictions continue to be proven right.
Another approach has been based on the fact that local inhomogeneities inﬂuence the overall expansion rate and can produce acceleration (for a review, see  and  ). Some have argued that these effects are inconsequential (see, for example,    ), while some have argued the effect is real  . Buchert et al.  have disputed the general applicability of the Green and Wald  proof.
Yet homogeneity in the universe is still not well understood. For example,  have shown homogeneity in our universe over vastly larger volumes and scales than prior studies. Solutions to the general relativity equations in inhomogeneous scenarios are also still not well understood. For example, Giblin, Mertens and Starkman  have used numerical solution approaches to show a stronger impact from inhomogeneities, beyond what is expected from simpler mathematical solutions. Finally, Rácz et al.  , have shown multiscale statistical solutions for a universe with many void regions that show accelerated expansion.
The debate continues. What remains clear is that inhomogeneities produce acceleration, and yet we know neither the magnitude of the inhomogeneities in the universe nor how to solve Einstein’s equations of general relativity accurately in this complex “Swiss-cheese” universe.
Is there a way to maintain our concept of the homogeneous isotropic universe at very large scale which has served us so well and still generate an accelerated expansion effect from inhomogeneity beyond the Hubble sphere or even the observable universe?
This paper proposes an approach that preserves our isotropic homogeneous universe but surrounds it, at scale on the order of the observable universe, by another isotropic homogeneous section―perhaps best described as a thick shell universe versus the traditional, more locally inhomogeneous Swiss-cheese universe discussed above.
This geometry violates the Copernican principle. However, the geometry maybe appropriate because repeated observations of the Cosmic Microwave Background (CMB) show an anisotropy, perhaps centered around the plane of the ecliptic    .
In Section 2, we state the hypothesis. In Section 3, we develop the mathematical model for such a universe and in Section 4, we numerically solve the equations to explore how well the model can account for the observed acceleration. In Section 5, we discuss how such a universe could develop naturally as our assumed ΛCDM model universe if we posit different initial conditions.
The geometric hypothesis used thus far for the universe envisions an isotropic spherical expanding space. According to this hypothesis, the expanding universe is always converting its initial kinetic energy into potential energy. This naturally results in continuous deceleration of the universe throughout its history.
As we have seen above, one can add repulsive energy or modify the equations of gravitational theory to produce acceleration. But can the same result be achieved by altering the geometry and mass distribution at the beginning of the universe? What if our isotropic spherical expanding space universe was surrounded by a shell of different density, and this shell was beyond the observable universe? For a shell with decreasing mass, the modified equation for the expansion of the spherical space inside the shell would show an acceleration term due to the decreasing mass of the shell. Can this account for the observations of an accelerating universe?
3. Theoretical Development
3.1. Standard Universe
For the case of an isotropic spherical expanding universe it has been shown that the key aspects of the solution can be understood with purely Newtonian dynamics, as it generates almost the identical Friedmann equation  . In general relativity, the universe and space expand together; in the Newtonian treatment, we imagine a homogeneous sphere of matter expanding isotropically into existing empty Euclidian space. The sphere has an edge, a center of symmetry and a fixed mass.
Assume a sphere of mass MT and radius rs that expands with time. We consider the effect on a small mass at the very edge of the sphere. The gravitational force on the small mass is caused by the same constant mass MT enclosed at all times. Equating the gravitational force to mass times acceleration and solving for the velocity vs of the small mass, we obtain  :
where G is the gravitational constant and k1 is a constant representing twice the total energy of a unit of mass. The rate of change of the radius rs of the sphere (our universe) is equal to vs.
It is well known that Equation (1) has three solutions, depending on k1  : a closed universe that will expand and then contract for positive value of k1; a zero-curvature universe that will expand into a “whimper” (parabolic expansion) for k1 equal zero; and a more open universe that will also continue to expand (hyperbolic expansion) for a negative value of k1. In all cases, expansion is decelerating right from the beginning as rs increases, driving vs lower and lower.
3.2. Proposed Universe
We now proceed to develop the solution for a different proposed universe. We will continue to use Newtonian mechanics, as we maintain our isotropic spherical geometry and Newtonian mechanics has been shown to lead to the same result as general relativity for a “spherical symmetric dust solution”  .
Our proposed universe now consists of a similar homogeneous isotropic uniform sphere but with a shell of uniform density surrounding it, as depicted in Figure 1. We make no assumptions about the thickness of the shell versus the sphere’s radius and we allow the structure to expand. We imagine a mass existing just outside the shell at rs to determine the behaviour of the outside radius, and we consider the expansion of the inner radius of the shell to see how the radius r of our universe develops with time. In this universe, we allow M1 and M2 to be different but their sum, MT, the total mass, to be constant.
The outer radius rs of the sphere is given by Equation (1), as the enclosed mass and spherical geometry are the same as before and the different distribution of mass does not affect the solution. Thus, the outside of this sphere decelerates with time, depending on k1.
To understand the behaviour of the inner radius r, we proceed to calculate the gravitational force from the enclosed sphere of mass M1 on a very thin section of the shell of mass m at radius r and equate it to the rate of change of its momentum. The sphere behaves as if all its mass, M1, is at its center. Thus, the force on the section of the shell of mass m is given by
Figure 1. Proposed universe.
The rate of change of momentum of the thin section is
where v is the rate of change of radius r. We assume that the mass of the thin section is proportional to M2, an assumption that is more accurate if M2 is changing slowly (Alternatively, one can assume that the density of the thin section is proportional to the density of the shell. The author obtained similar results with this assumption). Thus, Equation (3) becomes:
where α is a constant.
Combining Equations (2) and (4) we obtain:
As the sphere expands, it gains mass:
where is the density of the shell. Thus, the change in M1, which is equal to the negative of the change of M2 (as the total mass remains constant), is given by
We can calculate straightforwardly from M2, rs and r as follows:
We are left with the following set of equations to solve together:
•Equation (1)―providing rs as a function of time, with MT and k1 being free variables to fit the observed data;
•Equation (5)―providing r as a function of time, with M1, r and v (r’s rate of change) at some time being free variables to fit the data (in particular, we can obtain the rate of change of r over r at the current time from the Hubble constant); and
•Equation (7)―combined with Equation (8)―providing the increase of M1 and decrease of M2.
Thus, to run the equations we allow MT, k1, rs and M1 to vary for the best fit (to the ΛCDM universe with dark matter). We set the current rate of change in r over r to the current value of the Hubble constant.
It is clear from Equation (5) that for a decreasing M2, there could well be periods where the acceleration term (the positive term on the right-hand side, when M2 is decreasing) exceeds the deceleration term (the negative term on the right-hand side, due to gravity from M1), producing the appearance of an accelerating universe. It is also clear that the value of the acceleration term can be determined independently from the deceleration term by varying the input parameters above.
We solve the set of equations numerically and plot the results in Figure 2. Figure 2 shows time along the x axis, with time zero being today. The y axis shows the scale factor of the universe, scaled to a value of 10 today. Displayed are curves for:
•A standard all-matter universe at critical mass (dash-dot line);
•The ΛCDM universe (solid black line) with a Hubble constant of 67.8 km s?1 Mpc?1 and a matter component ΩM of 0.3089 and ΩΛ of 0.6911  ; and
•The proposed universe r (dotted line) and rs (dashed line).
We vary the parameters to provide the best fit possible of the proposed universe (r) to the ΛCDM curve. We keep the proposed universe “flat” (although technically the acceleration term here has nothing to do with dark energy). The best fit is achieved with a current matter component ΩM of 0.38 (note that ΩM of 0.38 and ΩΛ of 0.62 lie on the edge of the 95% confidence interval for CMB and supernova data  ). This fit best fit is achieved for:
•MT 1.086 times critical mass;
•rs of 1.803 the current scale factor value; and
•k1 of ?9.547 times the value of the mass term in Equation (1) at the current time, i.e., an open universe.
One can see that the fit is very good, and that the main effect is on the age of the universe, although in the proposed universe, the deceleration and acceleration components are a different function of radius than in the ΛCDM universe―in particular, the “cosmological” constant or acceleration parameter is not constant.
Figure 2. Scale factor vs. time for different universe models.
Figure 3. Ratio of acceleration to deceleration components.
A useful graph for seeing the difference in the nature of the acceleration and deceleration components between the proposed model and the ΛCDM model is shown in Figure 3, which displays the ratio of acceleration to deceleration components as a function of time for the proposed universe in Figure 2 and the ΛCDM universe. When the ratio is one, the universes flip from decelerating to accelerating. Clearly the fit in Figure 2 is achieved with different varying terms that integrate to almost the same result for times in the past 10 billion years. The proposed universe is gentler in terms of its switch from deceleration to acceleration.
The results described above have been achieved by fitting the data for the past several billion years and then running the model back in time. What emerges is the following picture:
•Initially, the mass of the universe is concentrated in the shell and the core is almost empty. However, the core quickly gains mass at the expense of the shell, and for the large majority of the universe’s existence, the mass of the shell, M2, slowly decreases, producing the effect required for acceleration. The masses of the two regions of the universe versus time are shown in Figure 4.
•Except at the very beginning of the universe, the density of the core is higher than the density of the shell, and the ratio of core density to shell density increases with time. At the current time, the core is 65 times denser than the shell.
•The ratio of r to rs varies very slowly after the initial period, getting larger, as expected, in the past several billion years, as shown in Figure 5.
The acceleration effect is being produced by a universe like the one we currently imagine but surrounded outside its visible extent by (currently) a very low-density thick shell.
Figure 4. M1 and M2 as a function of time.
Figure 5. Ratio of r to rs as a function of time.
It is demonstrated herein that by altering the geometry of the assumed expanding universe, one can achieve an initially decelerating universe and later in time an accelerating universe without inventing dark energy―simply by retaining visible and dark matter. Furthermore, the universe remains an expanding sphere that obeys Hubble’s law. The process develops naturally from the initial conditions and explains how the universe has been accelerating for the past several billion years or so with no dark energy.
Three key pieces of work remain. The first is to perform a detailed fit to the experimental data, in particular the supernova data, by varying the parameters described herein. The second is to reproduce the theoretical result with the general relativity equations, and the last is to fully explore the initial conditions that make this approach work.
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.
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