µ – Σ parameterization

In section 3.4.1 we found that the QSA imposed a condition on theµ–ηparameterizations, removing the scale dependence of µ. The µ–Σparameterization does not suffer from this problem; we therefore expect the number of free parameters forµ–Σto be larger than for µ–η. In section 4.2 we identified the free parameters for both parameterizations, where we found the number of free parameters for µ–Σ to be larger indeed. Due to the larger parameter space, we expect the viability conditions to be satisfied more easily for theµ–Σ parametrization when comparing the two parameterizations case wise.

Case I

This case is analogous to case I for µ–η. The EFT functions are zero, leading to a stable

ΛCDM model.

Case II

This case is of particular interest to us as γ3 = 0, which makes it satisfy the gravitational wave constraint supported by recent observations, while still allowing for stable models. In the analogous case of theµ–ηparameterization such stable models were not found, as was shown in figure 5.1. Figure 5.8 show stable models and a well defined allowed parameter space.

-1. _-0.8 _-0.6 _-0.4 _-0.2 0. 0.2 0.4 0.6 0.8 1. -1. -0.8 -0.6 -0.4 -0.2 0. 0.2 0.4 0.6 0.8 1. c₂ E22

Figure 5.4: Viability result of µ–η case IV with f1(a) = 0 for the evaluation of the 2 dimensional

parameter space ofE22 andc2. As in 5.2, blue indicates that the conditions are met, orange means

that the conditions are not satisfied. As in figure 5.2, the area in dark red is excluded from the parameter space. IfE22=0, we findγ3=0via(4.42), which is not allowed in this case.(4.43)shows

that we have a division by 0 forc2 = 1just as we saw forµ–ηcase III. Note thatc2 =1gives anη

which is independent of scale, which is undesired. We have a resolution in the parameter space of dE22 =dc2=0.05. The range of values forawas(0.01−0.99)with∆a=0.05. As there are no blue

5.2 Model viability 43

Figure 5.5: Viability result ofµ–ηcase IV withc1=1. We find 1 stable points, indicated by the blue

dot, out of35 000randomly sampled parameter points. Using(4.39)and (4.46), we again find that E22 = 0and c2 = 1need to be excluded from the computations. We consider values of awithin (0.1, 1)withda=0.1.

.

Figure 5.6: Evolution of EFT functions with time for µ–η case IV using parametersc1 = 1, c2 =

−0.82797,E11 =−0.969489andE22 =−0.58106. As the behavior of all 3 EFT functions was difficult

to capture in 1 plot, we consider the full evolution ofγ2(a)with time in figure 5.7. We again find that

the EFT functions go to 0 fora→0, which was imposed by the boundary conditions forγ2, though

Figure 5.7: Evolution ofγ2(a)with time forµ–ηcase IV using parametersc1 = 1,c2 = −0.82797,

E11=−0.969489andE22 =−0.58106.

Figure 5.8: Stability of theµ–Σcase II. We have a 3 dimensional parameter space with parameter

E11,E22andc1, indicated on the axes of the figure. The range ofavalues considered is(0.01, 1)with

5.2 Model viability 45

Figure 5.9:Evolution of EFT functions with time forµ–Σcase II using parametersc1= −0.8,E11 = 0.7andE22=−0.8.γ3(a)is fixed to 0 for case II, the other EFT functions are free to vary. Despite the

freedom to vary, we find that the functions go to 0 fora→0.

Case III

Just as we saw for µ–η case III, this case gives an integral for γ2 which can not be solved analytically. Therefore, we use the numerical integration function, ’NIntegrate[]’, of Math- ematica to compute the values required for γ2 in the viability conditions. For this choice of EFT functions, we do not find any stable models after 2400 random samples using the Monte Carlo method, where we consider values of awithin(0.1, 1)withda =0.1. This can be explained by the fact that we require the models to have some very specific physics, i.e. a gravitational waves speed not equal to the speed of light while havingΩ = 0. It is not unthinkable that such a constraint does not yield any stable models in this parameterization.

Case IV

This is the most general case in our work in terms of the number of free parameters we find. We have 4 free parameters, giving us a 4 dimensional parameter space to sample. This is hard to plot, which is why we show the c2 parameter using the color bar. The results are shown in figure 5.10. We find stable models, but a clear pattern in the parameter space is not easily identified in contrast to figure 5.8. We can see however that negative values of c1are favored and negative values ofc2seem to correlate with positive values ofE22, while positive values ofc2are correlated with negative values ofE22. Unfortunately, the numerical computations in this case suffer from the same false positive problem as we saw inµ–ηcase III and IV. Therefore, we can not be entirely sure that the points we found are true stable models until the areas whereγ2diverges are identified.

Figure 5.10: Stability of theµ–Σcase IV. We have a 4 dimensional parameter space. We plotted 3

parameters in the dimensional axes, i.e. E11,E22andc1, and plotted the values of the fourth param-

eter,c2, as the color of the dots. We consider values ofawithin(0.1, 1)withda= 0.1. We find 171

stable models out of 6900 random parameter samples. Note that negative values ofc1 are favored

and negative values ofc2seem to correlate with positive values ofE22, while positive values ofc2are

correlated with negative values ofE22.

Figure 5.11:Evolution of EFT functions with time forµ–Σcase IV using parametersc1=−0.328844,

c2 = −0.0240837,E11 = −0.215057andE22 = 0.950477. All EFT functions are free to vary in this

case, however we find thatΩstays close to itsΛCDM form for the given parameter combinations. We again see that the function go to 0 fora → 0, which was only imposed forγ2 here though its

Chapter

6