Noise Levels of Data

4.3 Noise Levels of Data

Figure 4.5: The luminosity distance,dL, GP reconstruction as a function of redshift,z. The blue solid line represents the best-fit function obtained by the algorithm. The orange crosses represent theBubbledata with no noise.

R² ×σ_dA(z) Reconstructed function

1º 1 0.0 dL= (1+z)²dA

2º 0.9919 0.1 d_L= (1+z)²d_A 3º 0.9720 0.2 dL= (1+z)²dA

4º 0.9267 0.3 dL= (1+z)²dA

5º 0.9052 0.31 dL=5.181dAz 6º 0.8502 0.4 d_L=5.208dAz 7º 0.5295 1.0 dL=4.528dAz

Table 4.2: The coefficient of determination,R², the standard deviations,σdA(z), and the luminosity distance function,dL(z), for each run of the GP algorithm.

data was created in the following way. At each redshift, we construct a normal distribution ford_A(z) with different standard deviations, σd_A(z). Then, by randomly choosing one point of the distribution, we select it to be our new value for the angular diameter distance, at that redshift. Here we assume that redshift are precise, i.e., have negligible errors.

To pick the standard deviation values,σd_A(z), we considered typical Baryon Acoustic Oscillations (BAO) DESI mock data for the angular diameter distance errors from Ref. [15]. After, we made a GP reconstruction for that data, which can be seen in Figure 4.6. This function is given by

σd_A(z) =−0.038z⁶(0.757−z) +20.182, (4.9) withR²=0.9764. We clearly see that the angular diameter distance errors follow a tendency, they tend

4.3 Noise Levels of Data

Figure 4.6: GP reconstruction of the DESI mock data for the angular diameter distance errors. The grey dots represent the DESI mock data. The black solid line represent the GP reconstruction.

Figure 4.7: The luminosity distance,dL, GP reconstruction as a function of redshift,z. The blue solid line represents the best reconstructed function obtained by the algorithm for a noise level of 0.3σ_dA(z). The orange crosses represent theBubbledata with a noise level of 0.3σ_dA(z).

to increase as the redshift increases. Therefore, we used this function to calculate our standard deviations at each redshift. From now on the values of this function will be known as our 1σd_A(z)value.

Then, we performed several GP reconstruction runs with increasing fractions ofσd_A(z), i.e., a fraction

4.3 Noise Levels of Data

number timesσdA(z). The results of all the runs we performed can be seen on Table 4.2. As we can see on this table, we were unable to reconstruct the luminosity distance function for our 1σd_A(z)realization.

Nevertheless, we were able to find the limit of the noise level for this function. As it is evident from Table 4.2, we see that we were able to reconstruct this function until a noise level of 0.3σd_A(z), with a coefficient of determination of approximately 0.9412.

Figure 4.7 shows the reconstruction for the noise level of 0.3σd_A(z). As we can see, it is quite remarkable that GP is still able to find the true function of the luminosity distance with such deviations at large redshifts.

4.3.2 Shear Estimator

In this subsection, we present our results for the shear estimator,Σs. The shear estimator can be used as a null test of the FLRW models [23, 44]. This is because, for FLRW models, the shear estimator is equal to zero. However, to use this quantity as a null test we must use observables more easy to measure.

If we decide to reconstruct this estimator with observables like those in Eq. (2.41) we must be careful because, this shear estimator is only a null test for FLRW models with a flat curvature, therefore, it can only falsify this kind of models.

Using Bubble, we were able to generate data for the redshift,H_∥andH_⊥, both characteristic of LTB models. Then, using the two Hubble parameters we calculated the shear estimator for each redshift using Eq. (2.40). These results, like in the previous subsection, where obtained using a CFL model with the initial parameters set to(h,A,w) = (0.72,0.85,1×10³)and a curvature profile (a) (see Figure 2.1).

We use our GP implementation to find the best function that describe the data produced by Bubble.

The best function that we found for the shear estimator was in fact the theoretical function for this estimator, which is presented in Eq. (2.40). This function can be seen in Figure 4.8. It is visible from the plot that this function as a sharp behaviour close toz≈0.2 and then stays constant and equal to zero.

This means that atz≈0.5 and beyond the Universe starts to behave like a FLRW model, however, before that we can see the presence of a void.

Then, we created mock data for both H_∥ and H_⊥, taking into account different noise levels. To create mock data for each Hubble parameter, we took a similar approach to the method described in the previous subsection. At each redshift we draw the values of the Hubble functions from random Gaussian realizations with means equal to the model predictions and standard deviations based on the present accuracy of the Hubble function determinations of Table 4.1.

One important aspect of this mock data generation was the criteria we used to choose the values for the standard deviations. We wanted to create mock data that would indicate at which point real cosmological data can be used for GP reconstructions. For that reason, we considered errors for bothH_∥ andH_⊥ as close to present data accuracy as possible. To do it, first, we considered the errors on Table 4.1.

As can be seen in Figure 4.9, we made a GP reconstruction for the errors on Table 4.1. Having that result, we realized that the errors do not follow any specific tendency with redshift. Therefore we decided that it was best to consider a constant value for the standard deviation of the mock data at all redshifts.

The value we took was the median of theσH data points in Figure 4.9 that is equal to 13.75 and from now on we will represent it as our 1σH value of reference.

First, we tried to use a standard deviation of 1σH to create the mock data, and, then, reconstruct the shear estimator. As can be seen on the last entry of Table 4.3, we were not able to reconstruct the shear formula for 1σH. However, on the same table, we can see that we found the limit of the noise level for

4.3 Noise Levels of Data

Figure 4.8: The shear estimator,Σs, GP reconstruction as a function of redshift,z. The blue solid line represents the best-fit function obtained by the algorithm. The orange crosses represent data with no noise.

R² ×σH Reconstructed function

1º 1 0.0 Σs= H_⊥−H_∥

2H⊥+H_∥

2º 0.9774 0.1 Σs= H_⊥−H_∥

2H⊥+H_∥ 3º 0.9450 0.15 Σs= H⊥−H_∥

2H⊥+H_∥ 4º 0.9374 0.16 Σs=0.162H_⊥−0.162H_∥

H_⊥²

5º 0.9089 0.2 Σs=0.265H⊥−0.265H_∥ H_⊥

6º −1.6586 1.0 Σs= H_⊥−1.475 25.222H_∥H_⊥+5.019H_∥

Table 4.3: The coefficient of determination,R², the standard deviations,σH, and the shear estimator,Σs, for each run of the GP algorithm.

this quantity. We discovered that we can reconstruct the shear estimator until a noise level of 0.15σH, values larger than that have too much noise for a successful reconstruction.

The reconstruction of the shear taking into account an error of 0.15σH, can be seen in Figure 4.10.

The coefficient of determination for such reconstruction is approximately of 0.9450.

Please note that both results for 0.16σHand 0.2σHare not bad results in the sense that they have high coefficients of determination. However, if we want to be sure that by using real data we can reconstruct the theoretical expression for the cosmological shear, we have to use data with error levels of 0.15σHor

4.3 Noise Levels of Data

Figure 4.9: GP reconstruction of the Hubble parameter errors of Table 4.1. The grey dots represent the Hubble parameter errors of Table 4.1. The black solid line represent the GP reconstruction. The blue solid line represent the median of the data.

Figure 4.10: The shear estimator,Σs, GP reconstruction as a function of redshift,z. The blue solid line represents the best reconstructed function obtained by the algorithm for a noise level of 0.15σH. The orange crosses represent the data with a noise level of 0.15σ_H.

less.