I invite all interested readers to help design the 'perfect cosmic balloon', compatible with all of cosmology  if not, then at least with most it! The recent debate on this Blog, pitching the kinematical against the hyperspherical (balloon) cosmic models, made it clear that there are lots of uncertainties about what the balloon model can and cannot represent.^{[1]}
First, let's do the system engineer's thing and write a brief high level specification for the perfect cosmic balloon, trying to keep it "as simple as possible, but not simpler" (A. Einstein).
1 Specification
1.1 General
The twodimensional balloon surface (black) shall represent all of 3Dspace. This means we disregard the third spatial dimension. The extra inside/outside dimension (pink) shall represent hyperspace. It is not accessible to us  just a visualization aid. It simply provides a direction for the surface to expand and curve into.
Formally, the hyperradius (R) may tend to infinity, or even have an imaginary value (iR), but for this exercise we will stick to real, finite values of R. Our observable universe (blue) is a limited circular patch on the surface with a radius R_{O}, determined by how far light could have traveled since the balloon blowup started.
1.2 Specifics
1.2.1 Mass
The surface of the balloon shall be able to hold massive particles (green) in a frictionless manner, i.e., they shall be able to move freely across the surface, but never be able to 'fly' off the surface, even if the radius should suddenly shrink. These particles shall represent the ordinary and dark matter of the cosmos.
1.2.2 Radiation
The surface shall also be able to hold photons (red), always traveling at the speed of light along the local surface and hence have energy that depends on wavelength. These photons shall represent cosmic radiation energy and hence also the cosmic microwave background (CMB).
1.2.3 Vacuum (Change note: reply #243)
Vacuum energy (cosmological constant) shall be included in any inflating/deflating mechanism of the balloon.
1.2.4 Total Energy
It may be assumed that the energy of expansion is included in the algorithm that inflates/deflates the balloon.
1.2.5 Momentum
It may be assumed that the momenta of the skin and all energy on it are included in the algorithm that inflates/deflates the balloon.
2 Design
Assume a balloon material that retains its elasticity over a reasonable range of balloon radii (at least enough range to illustrate the principles). A sensor system measures the balloon radius (R) directly and also determines the rate of change (ΔR/dt). A pump/reservoir/valve system supplies or withdraws gas to/from the balloon at a rate that will keep ΔR/dt = R H, where H is a function of R and the energy density makeup of the cosmos to be simulated. H, the time variable Hubble parameter, is obtained from:
H^{2} = H_{0}^{2} [(1Ω_{0})/a^{2} + Ω_{m}/a^{3} + Ω_{r}/a^{4} + Ω_{Λ}]
where H_{0} is the (present) Hubble constant, Ω_{0} = Ω_{m} + Ω_{r} + Ω_{Λ} is the present total energy density parameter, a=R/R_{0} the expansion factor, Ω_{m} the present matter energy density parameter, Ω_{r} the present radiation energy density parameter and Ω_{Λ} the present vacuum energy density parameter.
The operating equation is then:
(ΔR/dt)^{2} = R^{2}H^{2} = R^{2}H_{0}^{2} [(1Ω_{0})/a^{2} + Ω_{m}/a^{3} + Ω_{r}/a^{4} + Ω_{Λ}]
'Guest' has proposed the following neat little high level algorithm for this system in reply #242:
1. Set gas flow direction valve for inflating the balloon.
2. Continuously measure the radius R of the balloon and calculate (ΔR/dt)^{2} = R^{2}H^{2} = R^{2}Ho^{2} ((1Ω)/a^{2} + Ω_{m}/a^{3} +Ω_{r}/a^{4} + Ω_{Λ}), with all the constants given and a = R/Ro, where Ro is a value that corresponds to the R for which the parameters are given.
3. Calculate switch = (1Ω)/a^{2} + Ω_{m}/a^{3} +Ω_{r}/a^{4} + Ω_{Λ}. If switch goes negative, even temporarily, change the gas flow direction valve for permanently deflating the balloon.
4. Measure ΔR/dt, square the result and compare it with the calculation of (ΔR/dt)^{2}.
5. If squared result is smaller than the calculation, increase gas flow rate.
6. If squared result is larger than the calculation, decrease gas flow rate.
7. Repeat from 2 until exit condition is reached.
3 Tests
First a general discussion is given and then specific simulations and 'tests' (to follow).
3.1 General
Since the balloon is ensured to follow the Friedman equations (at least theoretically), paper 'tests' would not be very meaningful. It is proposed that the 3 cases (de Sitter, Einsteinde Sitter and Lambdacolddarkmatter (LCDM)) be simulated and the results shown here for comparison to other simulations.
3.2 The de Sitter model
The de Sitter expansion curve for a flat (Ω_{m} = Ω_{r} = 0, Ω_{Λ} = 1) universe is obtained by integration of the following simplified form of the above expansion equation:
ΔR/dt = RH_{0}/978
where the factor 1/978 is a conversion of Ho from km/s/Mpc to 1/Gy.
This is also called case (0,0,1), as on tthe figure below. It is clearly an exponential expansion curve. A Hubble constant of 72 km/s/Mpc was used for the curve.
The age of such a universe would have been about 105 Gy, read off where the curve intersects the 100 Gly radius line (which is taken as a=1). De Sitter did not intend this to be a model of the real universe, but as a tool to investigate expansion dynamics.
3.3 The Einsteinde Sitter model
The first workable attempt to model the real universe came when Einstein and de Sitter made the assumption that the universe is flat and for all practical purposes contains only matter, i.e., Ω_{m} = 1, Ω_{r} = Ω_{Λ} = 0. This means that the Friedman equation reduces to: (ΔR/dt)^{2} = (RHo/978)^{2} Ω_{m}/a^{3}, giving the parabolic curve below, again for Ho = 72 km/s/Mpc.
The curve intersects the 100 Gly radius line at around 9 Gy age, making such a universe uncomfortably young! It used to be no problem when Ho was still believed to be around 50 km/s/Mpc, but not any more.
3.4 The Lambdacolddarkmatter (LCDM) model
This is the 'standard' model at the present time, comprising about 26% matter (ordinary plus dark matter), a tiny amount of radiation energy, with the bulk of the energy (74%) made up of vacuum energy (the cosmological constant). The full equation must be used here: (ΔR/dt)^{2} = (RHo/978)^{2} ((1Ω)/a^{2} + Ω_{m}/a^{3} +Ω_{r}/a^{4} + Ω_{Λ})
This caused an expansion curve that started out with a decreasing rate, slowly turning over to an increasing expansion rate at around 7 Gy.
This model universe has a present age of around 13.7 Gy, which is quite comfortable.
4 Conclusion
While it may be impossible to 'design and build' a laboratory sized cosmic balloon that will 'automatically' have the properties of the real cosmos, it is definitely possible to construct one that can follow the Friedman equations, at least for a short period of time. Section 2 (Design) describes such a device and its method of operation.
Such a balloon can serve as a 'crutch' to lean on in discussions of cosmological principles. At least it is a little more 'tangible' than the presumed dark matter and dark energy of the real cosmos. In a next Blog entry I will attempt to use it to show how some cosmological issues can be explained.
Jorrie

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