PLIEGO DE PRESCRIPCIONES TÉCNICAS PARTICULARES

As we saw in section 4.2.1, here all the EFT functions are found to be zero. We thus find the the ΛCDM case, which is known to be stable. Therefore, this case does not give us a model to investigate the stability for. For theΛCDM case, we know that the EFT functions expressed in terms of the phenomenological parameterizations, i.e. γ2, γ3andΩ, are 0 for alla, there is therefore no point in plotting them.

Case II

Here we found analytical equations for all the EFT functions. This speeds up the implemen- tation of the functions in Mathematica, as we can ask Mathematica to compute the deriva- tives of the functions. We plot the results of the evaluation of the viability conditions for different parameters in the available parameter space in figure 5.1.

This model is of particular interest as it obeys the gravitational waves constraint, which has been observationally shown to hold at the present time [27]. We would like our models

-2 -1 0 1 2 -1.0 -0.5 0.0 0.5 1.0 E11 Stability indicator

Figure 5.1:Stability ofµ–ηcase II. The blue line corresponds to the kinetic condition from(5.18), the

orange line corresponds to the gradient condition(5.19). We compute the value of equations (5.18, 5.19) and normalize them so only the information of the sign remains. The blue line is then multiplied by 1/2 to differentiate the two lines. Negative values thus indicate that the condition is not satisfied, while positive values indicate they are. The resolution in the parameter space isdE11 = 0.001, the

range ofa-values considered is(0.01, 0.99), withda =0.05. There is no value ofE11where both the

5.2 Model viability 39

to be able to describe the physics at the present moment, thus obeying this constraint. Un- fortunately, figure 5.1 shows that we can not find stable models in the available parameter space in this case. We will plot the EFT functions for 1 stable point for each case, however as we do not find a stable point here, we also do not plot the EFT functions.

Case III

As we saw in section 4.2.1, equation (4.39), we find an integral in the expression of γ2 in this case. This integral can not be solved analytically. Therefore, we use the numerical integration function, ’NIntegrate[]’, of Mathematica to compute the values required forγ2 in the viability conditions. As Mathematica can not differentiate these integral expression, we do the differentiation by hand and implementγ0₂(a)ourselves. Note that the expression forγ2involves an integral over the scale factor from 0 to 1. The numerical integration runs into trouble for values ofaclose to 0. We therefore integrate fromqtot 1, as was described in section 4.2.1 before, and setq=0.01.

The results are shown in figure 5.2. Note that these are preliminary results. Certain point in the parameter space might let the integral forγ2to diverge, which might trick the code in thinking it found a stable point. Therefore, the areas whereγ2 diverges need to be identified.‡

Case IV

Here, the same applies as with case III of theµ–ηparameterization. We have an integral in our γ2 expression that can not be solved analytically. In fact, we found this to be the case for cases III and IV of theµ–Σ parameterization as well. We apply the same procedure of solving the integral numerically in all cases.

We distinguish between two subcases. One corresponding tof1 =0, and the other which corresponds toc1 =1. The viability results for f1 =0 are shown in figure 5.4 and the results forc1 =1 can be found in figure 5.5.

For the 2 dimensional parameter space of figure 5.4, we could span through the param- eter space systematically and find that we do not find any stable models. In case of the 3 dimensional parameter space of figure 5.5, we use the Monte Carlo method to randomly sample the parameter space and computing the corresponding viability values. This meth- ods is used for all plots where the dimension of the parameter space is 3 or 4. Just as we saw for case III, we use integration from q to 1 over the scale factor in ourγ2 expression (4.39), which in fact is the same expression for both cases. The same problem therefore arises for the divergence of the integral, leading to false positives. The results are therefore prelimi- nary until all regions where divergence might occur are identified. However, so far only 1 stable point was found, which we checked for stability by hand and turns out to be a truly stable point.

-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 5.2:Stability of theµ–ηcase III. The resolution in the parameter space isdc2=dE22=0.1, the

range ofa-values considered is(0.01, 0.99), withda=0.1. Here blue indicates that the conditions are met, orange means that the conditions are not satisfied. The area in dark red is excluded from the parameter space. IfE22 =0in(4.37), we findγ3=0, which is not allowed in this case. (4.39)shows

that we have a division by 0 forc2 = 1. Note thatc2 = 1gives anηwhich is independent of scale,

5.2 Model viability 41

Figure 5.3: Evolution of EFT functions with time for µ–ηcase III using parameters c2 = 0.2and

E22 = −0.2. Note that the EFT functions are dimensionless functions. As imposed by the case III,

Ω(a) =0, however we do find a deviation from theΛCDM case for the other 2 EFT functions. The EFT functions go to 0 fora → 0, which is desired as from CMB observations and BBN, we know that the universe is well described by ΛCDM at those times. This was required by the boundary conditions forγ2and we haveΩ(a) =0in this case, howeverγ3(0) =0was not imposed.