CONCLUSIONES, RECOMENDACIONES

5.2. THE CHILDREN MODELS

models for which the evolution and constraint equations are consistent.

0 500 1000 1500 2000

0 200 400 600 800 1000 1200 1400 1600

dU dt

U(t) FLRW

5 tetrahedra 16 tetrahedra 600 tetrahedra 60 tetrahedra 192 tetrahedra 7200 tetrahedra

Figure 5.13: The expansion rate of the universe’s volume against the volume itself for all parent models and their children models.

Figure 5.13 and Figure 5.14 compare the graphs of ^dU_dt against U for each of our models, with U given by (5.2.59) for the subdivided Regge models and by (4.1.16) for the parent models. The graphs in Figure 5.14 focus on the 600-tetrahedral parent and its 7200-600-tetrahedral child, with the bottom plot extended so as to reveal the graphs’ endpoints. At low volumes, subdividing the tetrahedra actually made the Regge approximation worse. We can see this more concretely in Table 5.3, where we list the minimum volumes for each model and their fractional difference from the FLRW minimum. While increasing the number of tetrahedra in the parent models brought the minimum volume closer to the FLRW value, increasing the number in the children models actually brought it further away. In fact, the worst parent model was still more accurate than the best child model.

We also see that all models again diverge from the FLRW model as the uni-verse expanded; however increasing the number of tetrahedra reduces the rate of divergence, and in this sense, increasing the number of tetrahedra improves the Regge approximation. We again believe this lower rate of divergence is the result of more tetrahedra providing a higher resolution approximation that can ‘keep up’ longer with the FLRW model. We also note that each model terminates at large volumes whenever the strut becomes null and that this endpoint gets in-creasingly delayed as the number of tetrahedra is increased. Thus these figures

5.2. THE CHILDREN MODELS

0 1000 2000 3000 4000 5000 6000 7000

0 500 1000 1500 2000 2500 3000 3500 4000

dU dt

U(t) FLRW

600 tetrahedra 7200 tetrahedra

0 10000 20000 30000 40000 50000

0 5000 10000 15000 20000 25000 30000 35000

dU dt

U(t) FLRW

600 tetrahedra 7200 tetrahedra

Figure 5.14: The expansion rate of the universe’s volume against the volume itself for the 600-tetrahedral parent and its 7200-tetrahedral child model. The top graph focuses on the region around the origin while the bottom graph shows both Regge graphs in their entirety.

Model Minimum volume Fractional difference from FLRW

FLRW 102.567937639753 0

5-tetrahedral parent 171.741398309775 0.67442 16-tetrahedral parent 135.70186007972 0.32304 600-tetrahedral parent 105.692461545881 0.03046 60-tetrahedral child 172.637934789289 0.68316 192-tetrahedral child 179.191098180344 0.74705 7200-tetrahedral child 1268.30953855058 11.36556

Table 5.3: The minimum volume and the fractional difference from the FLRW mini-mum for each Regge model and for the FLRW model.

reveal that each child model provides an approximation that starts off worse than its parent but is later much better by virtue of its more robust resolution. Indeed, if one were to extrapolate all graphs past their end-points to very large volumes, the 7200-tetrahedral model would ultimately provide the best performance.

As with the parent models, we can define a 3-sphere radius ˆR(t) analogous to (5.1.18) for Cauchy surfaces of the children models such that ˆR(0) = a(0); that is, ˆR(t) and a(t) match at the moment of minimum expansion; thus, we define R(t) to beˆ

R(t) =ˆ a(0)

v_minv(t), (5.2.64)

where v_min is the minimum value of v and is given by (5.2.51) when θ⁽²⁾ = arccos

2α²−1 4α²−1

= arccos ¹₃

. Such a definition is possible because, as discussed at the end of Section 5.2.1, all dynamical length-scales in the system are related to each other by time-independent scalings such that there is really only one independent dynamical length-scale describing the entire model; therefore any dynamical length-scale, when re-scaled by an appropriate constant, will yield the same ˆR(t). Note that whenn= 5, thenl(t) as given by (5.1.33) andv(t) as given by (5.2.56) will be identical apart from an overall constant factor. When this happens, the two corresponding models will have the same ˆR(t). This happens for the 600-tetrahedral parent model and its 7200-tetrahedral child.

We now examine the behaviours of ^d_dt^Rˆ versus ˆR(t) and of the correspond-ing 3-sphere volumes. Figure 5.15 shows the relationship between ^d_dt^Rˆ and ˆR(t)

5.2. THE CHILDREN MODELS

0.0 0.5 1.0 1.5 2.0 2.5 3.0

1 2 3 4 5 6

d(Radius) dt

Radius(t)

FLRW 5 tetrahedra 16 tetrahedra 600 tetrahedra 60 tetrahedra 192 tetrahedra 7200 tetrahedra

Figure 5.15: A combined graph of the radius’ expansion rate against the radius itself, using ˆR(t) as the Regge models’ radius.

for each of the Regge models and the relationship between ^da_dt and a(t) for the FLRW model. Figure 5.16 shows the relationship between the corresponding 3-sphere volumes and their rates of expansion. We note that in both figures, the graphs for the 600-tetrahedral model and its 7200-tetrahedral child coincide, as expected. The definition of ˆR(t) has clearly removed any variability in the initial performance of the Regge models; we now see that the Regge models with more tetrahedra consistently outperform those with fewer. The rate of divergence from FLRW is again reduced and the graphs’ end-points further delayed as the num-ber of tetrahedra is increased. Thus in these graphs, increasing the numnum-ber of tetrahedra clearly improves the Regge approximation.

As mentioned at the start of Section 5.2.3, we were unable to obtain a consis-tent set of Hamiltonian constraints when we varied the Regge action for the child model locally. We suspect this may be due to an inappropriate specification of strut-lengths, thereby causing the model to be over-constrained. As we remarked at the end of Chapter 4.4.1, we do not in general have complete freedom to spec-ify all struts independently of each other, unlike lapse functions in the ADM formalism; therefore, our choice of having all strut-lengths be equal may not be appropriate. There may instead be some other choice of strut-lengths that would lead to a consistent set of constraint equations. We note though that so long as the strut-lengths remain interdependent, which we expect to be the case, there

0 200 400 600 800 1000 1200

0 200 400 600 800 1000

dU dt

U(t)

FLRW 5 tetrahedra 16 tetrahedra 600 tetrahedra 60 tetrahedra 192 tetrahedra 7200 tetrahedra

0 500 1000 1500 2000 2500 3000 3500 4000 4500

0 500 1000 1500 2000 2500 3000 3500

dU dt

U(t)

FLRW 5 tetrahedra 16 tetrahedra 600 tetrahedra 60 tetrahedra 192 tetrahedra 7200 tetrahedra

Figure 5.16: A combined graph of the 3-sphere volume’s expansion rate against the volume itself, using ˆR(t) as the Regge models’ 3-sphere radius. The top graph focuses on the region around the origin while the bottom graph shows all Regge graphs in their entirety.

5.2. THE CHILDREN MODELS

will also be some constraint between the tetrahedral edge-length ratios α and β similar to (5.2.27); this would follow from a similar reasoning to that which led to (5.2.27). Indeed, one could perhaps view the set of three constraint equations as follows: one equation determines the evolution of the tetrahedral edge-lengths, while the other two determines the evolution of the two non-independent sets of struts, one equation for each set. After determining the lengths of these two sets, one could then deduce the constraint between α and β.

Nevertheless, we may still run into the same problem encountered when lo-cally varying the parent model. There, we discovered that for the evolution equation (5.1.46) to be a first integral of the Hamiltonian constraint (5.1.35), the constraint equation (5.1.45), obtained from varying the diagonals, must also be satisfied. However these latter constraints required the model to behave in an unphysical manner, and thus, the model broke down. We suspected the un-derlying cause to be the breaking of Copernican symmetries from introducing the diagonals. With the children models, we should also expect the diagonals to break Copernican symmetries, possibly rendering for example the mid-point vertices inequivalent to each other. However, even if the local model were not vi-able, it would, through the chain-rule relationship of (5.1.47), still point towards an alternative but also viable global model, and we believe the properties of this new model to be worthy of further investigation. Indeed, it may possess some desirable advantages over our current child model, such as, for instance, having a Hamiltonian constraint that is unconditionally a first integral of the evolution equation. At this point though, we shall leave a more thorough examination into the viability and properties of such models, global and local, to future study.

Regge models of closed lattice universes

With a better understanding of the CW formalism obtained from modelling the Λ-FLRW universe, we shall now apply it to the lattice universe; our focus however will be on modelling closed lattice universes. The CW formalism naturally lends itself to modelling such universes since the Cauchy surfaces of the three parent models themselves form three of the closed Coxeter lattices listed in Table A.1 of Appendix A. Our ultimate interest will be to perturb one of the lattice masses and study the resulting universe’s behaviour, as this would help bring us closer to approximating the actual universe’s matter distribution; but before doing this, it is necessary to first test the formalism on the standard lattice universe itself.

From our investigations in the previous chapter, we can draw on some lessons to guide us in our application of the formalism here. First, we saw that locally varying the skeleton in the previous chapter led to an unviable model, and we believed this arose from a contradiction between assuming that all edge-lengths in the Cauchy surface should remain identical and the actuality that the diag-onals introduced broke symmetries between the edges in the surface. We shall therefore only consider global models in this chapter, though we shall again take the continuum time limit of all models, where δt_i → 0. Secondly, we saw that in the continuum time limit, the Hamiltonian constraints, obtained by varying the Regge action with respect to the struts, were first integrals of the evolution equations, obtained by varying with respect to the tetrahedral edges. As a