Now that we have a 'design' and an algorithm for the cosmic balloon, it is easy to apply it some cosmological problems. The first application is about the discussion on the 'cosmic teardrop' or 'cosmic heart'^{[1]} between GK and Physicist during the design phase of the cosmic balloon. Here are some numerically correct illustrations of the scenarios discussed, taken out of a spreadsheet.
The surface of the expanding balloon represents the entire cosmos at any particular time. Figure 1 (right) represents just a slice here, with the circumference of the circle representing 3D space. The interior and exterior of the balloon is a hyperdimension into which the balloon expands or deflates.
When we look back in time (to distant sources), we essentially look 'inside' the balloon  not really inside it, because the balloon was just smaller at earlier times and we are still looking along the surface of the smaller and smaller balloon. The red and blue curves are photons coming from (say) our east and west sides, with the colors just to distinguish between them when they overlap in some scenarios.
This is easily simulated on the cosmic balloon. We just trace the paths of photons backwards along the surface of the shrinking balloon until we find the place where they originated (or until we find the BB!). For every time step, photons move a distance cΔt towards us along the surface of the balloon. Whether they approach or recede from us depends on the expansion rate and their distance from us.
For our present cosmos, even taking its minimum size, the 'heart' formed by the rays is not very remarkable  it is stretched very thin. The photons originated at the time of transparency of the cosmos (the CMB epoch), around 400,000 years after the BB, which is 'at the origin', for all practical purposes.
The balloon radius is 100 Gly, (shown on a linear scale) and the time of photon travel is 13.7 Gy. The 'twogigayearrings' show that the expansion was originally very fast, then slowed down somewhat and is lately picking up speed again. This is understood in terms of the effect of matter domination at earlier epochs and vacuum energy domination at later epochs. The two dotted radial lines represent where the regions of the photon emissions are today  a proper distance of 46 Gly from us. When those photons were emitted (or rather were released to move freely), the source regions were only 42 million light years way from us, just over 1000 times closer than today.
The only way to get a fuller 'cosmic heart' (as discussed in the previous thread) is to slow the expansion down to a crawl. This will happen with a flat Einsteinde Sitter universe, but it takes an extremely long time. Interestingly, there is a matter+vacuum solution to the Friedman equations that does exactly that within a 'reasonable' time (Figure 2, left). With matter energy making up 50% of the critical density and vacuum energy 200% of critical density (case 0.5,0,2.0), the expansion essentially 'stops' at 50 Gly radius  it actually approaches the 50 Gly radius asymptotically. For a period roughly the age of our present universe, the expansion would continue, but then the rate drops off towards zero. This would have made the CMB photons to travel almost the radius of the total cosmos, arriving after some 150 Gy. The halfcircumference of the circle is ~165 Gly.
If we let the time continue for long enough (and if nothing disturbs the nearequilibrium, almost static situation), the 'heart' spreads around the balloon and eventually the origins overlap as shown in Figure 3 (right). For this scenario, it happens after roughly 200 Gy. This scenario means that we could have observed the same region by looking into opposite directions in the sky.
Apart from the first (present day) scenario, these may be hypothetical cases, but it illustrates the power of the cosmic balloon model rather well. It is quite simple to model and results like these flows very easily from it.
As an aside, the requirement for this situation is not so far from the observed case. There is no theoretical reason why the cosmological constant could not have been some three times higher and the matter density around double what we deduce today. However, there are observations that seem to rule out such a set of parameters.
Jorrie
[1] The 'cosmic teardrop' of observable space appears when the expanding universe is presented on a flat spacetime diagram, as shown in Figure 13.2 from Relativity4Engineers (shown in Figure 4, right). The parabola is equivalent to the surface of the hypersphere above.
The main reason for not seeing the 'comic heart' in publications is the fact that one is bound to the hypersphere and cannot show time properly on it. The 'teardrop' view does not suffer from that drawback. However, there are also certain drawbacks in the flat spacetime representation, like how a closed universe would be pictured.
My 'twogigayearring' trick is a way around the time problem. The 'cosmic heart' originally earned its name from taking a flat LCDM cosmos, where the scale of R_{0} is arbitrary and setting R_{0} to a convenient value like the Hubble radius (13.7 Gly). It gives this pretty heart shape trace for the CMB photons (Figure 5, right). Much nicer than the 'streamlined bomb' shape when the proper (minimum) R_{0 }is used, as in the main figure above.
The 'teardrop' is essentially just this 'heart', with the circular coordinates flattened out.
J

Re: Cosmic Balloon Application: the Cosmic Heart