4.4 Redshift Drift: Distinguishing BetweenΛCDM and LTB Models
each∆v, with a standard deviation given by Eq. (4.10).
With these new data points it was performed a GP reconstruction, with the aim of reconstruct the theoretical function of∆v. The best-fit function found if given by
∆v=−0.182z2−0.677z, (4.11) withR2≈0.6685. These results can be found in Figure 4.11.
4.4.2 Lemaître–Tolman–Bondi models
After obtain results for theΛCDM model, we were interested in understandiing if the redshift drift for LTB models varies in a different way when comparing with theΛCDM model. For this reason, in this subsection are presented the results for both CFL and GBH models.
4.4.2.1 Clifton-Ferreira-Land model
Model Parameters
h A w(Mpc)
Fiducial 0.72 0.85 1×103 M1 0.62 0.85 1×103 M2 0.82 0.85 1×103
M3 0.72 0.8 1×103
M4 0.72 0.9 1×103
M5 0.72 0.85 0.5×103 M6 0.72 0.85 1.5×103
Table 4.4: Table with the values for the three initial parameters of the CFL models in study, both the fiducial and the six toy models. The three initial parameters are: the dimensionless Hubble constant,h, the amplitude scale of the curvature,A, and a curvature parameter,w.
The results presented here, as well as the results presented for the shear estimator, have a curvature profile (a) (see Figure 2.1). Taking this into account, we fixed a fiducial model withh=0.72,A=0.85 andw=1×103and created six additional models. Since CFL models have three initial parameters, we created the six models, with parameters up and down, i.e., around, this fiducial model parameters. We present all model parameters in Table 4.4.
In Figure 4.12, it is shown the comparison of the redshift drift for the fiducial and the six additional models considered in here. In each subplot it is represented the variation of only one parameter. We can conclude by looking at that figure that all of the CFL model parameters are sensitive to the redshift drift. Larger differences between models appear at high redshifts, so depending on the quality of future experiments, high redshift observations may be the best strategy to discriminate between models. It is also interesting to see that a bigger change between models happens when the dimensionless Hubble constant varies. To constrain bothAandwit would be necessary very precise data.
As stated before, we are interested in studying the differences betweenΛCDM and LTB models for the redshift drift. Figure 4.13 shows the redshift drift for both types of model, where we only considered the fiducial CFL model. We see an obvious difference between the two models. This is particularly striking at high redshifts (z>1) and that is why our proposed list of redshifts for targeted observations should be able to distinguish between these two classes of models.
4.4 Redshift Drift: Distinguishing BetweenΛCDM and LTB Models
(a) (b)
(c)
Figure 4.12: The redshift drift per year for the fiducial CFL and the eight toy models. The fiducial CFL model has the following parameters: h=0.72,A=0.85 andw=1×103. Each subplot represent only a change in a variable (a) the dimensionless Hubble constant,h, (b) the amplitude scale of the curvature,A, and (c) a curvature parameter,w. The black line represents the fiducial CFL model. The other colors, blue and orange, represent the six toy models as it is shown in the legend of the figure.
Figure 4.14 shows the velocity shift after a period of 10 years for theΛCDM model (black dashed line) and fiducial CFL model (green solid line). We can also see the FLRW theoretical (orange points) and mock (blue points) data. The black solid line represents the GP reconstruction taking into account the FLRW mock data.
Looking at Figure 4.14 it becomes clear that if the Universe follows the FLRW mock data generated here, it would be very easy to discard CFL models in our proposed range of redshift, since, typically, CFL models are severalσaway from the best GP function reconstruction obtained from the mock data.
4.4 Redshift Drift: Distinguishing BetweenΛCDM and LTB Models
Figure 4.13: The redshift drift per year for the fiducial CFL model and theΛCDM model. The black line represents the fiducial CFL model. The blue line represents theΛCDM model.
Figure 4.14: The velocity shift for theΛCDM model after a period of 10 years. The orange points represent the theoretical data for theΛCDM model with the error bars estimated from Eq. (4.10). The blue points represent the mock data for theΛCDM model with the error bars estimated from Eq. (4.10). The black solid line represents the GP reconstruction for theΛCDM model using mock data. The black dashed line represents the theoretical velocity shift for theΛCDM model. The green line represents the fiducial CFL model. The grey interval correspond to the confidence level of 1σ. For the confidence level of 1σ was used a fit of a polynomial with a degree two.
4.4 Redshift Drift: Distinguishing BetweenΛCDM and LTB Models
4.4.2.2 Garcia-Bellido-Haugbølle model
To see how the redshift drift changes in GBH models with different initial parameters, it was chosen a fiducial GBH model and then, by performing up and down variations for each parameter, there were created eigth additional models. The initial parameters for each of the eight models, as well as for the fiducial GBH model are shown on Table 4.5.
Model Parameters
h Ωin r0(Gpc) ∆r(Gpc)
Fiducial 0.675 0.3 1.5 0.5
M1 0.55 0.3 1.5 0.5
M2 0.8 0.3 1.5 0.5
M3 0.675 0.005 1.5 0.5
M4 0.675 0.595 1.5 0.5
M5 0.675 0.3 0.05 0.5
M6 0.675 0.3 2.95 0.5
M7 0.675 0.3 1.5 0.005
M8 0.675 0.3 1.5 0.995
Table 4.5: Table with the values for the four initial parameters of the GBH models in study, both the fiducial and the eight toy models. The four intial parameters are: the dimensionless Hubble constant,h, the underdensity at the void center,Ωin, the size of the void,r0, and the transition width of the void profile,∆r. Bothr0and∆rare in units of Gpc.
Moreover, Figure 4.15, shows the redshift drift for all the GBH models considered in this analysis.
As can be seen, the redshift drift is sensitive to two parameters, h andΩin, this means, that through observations of the redshift drift, we may constrain the value for these parameters. Concerning M5, M6, M7 and M8, these models are all on top of the fiducial model, which means thatr0and∆rare parameters that are not sensitive to redshift drift observations.
Furthermore, taking only into account the fiducial model andΛCDM model, we were interested to see if this two models can be distinguished from one another using redshift drift observations. Figure 4.16 shows the redshift drift for these two models. It is clear that both models have a different impact on the redshift drift.
Figure 4.17 shows the velocity shift after a period of 10 years for theΛCDM model (black dashed line) and fiducial GBH model (green solid line). We can also see the FLRW theoretical (orange points) and mock (blue points) data. The black solid line represents the GP reconstruction taking into account the FLRW mock data.
Again, if we consider the redshift range of the observations we propose, see Figure 4.17 one can, for sure, distinguish between these two types of models, because our GP reconstruction confidence intervals exclude one of the models (in this case the GBH model) at manyσ.
4.4 Redshift Drift: Distinguishing BetweenΛCDM and LTB Models
(a) (b)
(c) (d)
Figure 4.15: The redshift drift per year for the fiducial GBH and the eight toy models. The fiducial GBH model has the following parameters:h=0.675,Ωin=0.3,r0=1.5 and∆r=0.5. Each subplot represent only a change in a variable (a) the dimensionless Hubble constant,h, (b) the underdensity at the void center,Ωin, (c) the size of the void,r0, and (d) the transition width of the void profile,∆r. The black line represents the fiducial GBH model. The other colors, blue and orange, represent the eight toy models as it is shown in the legend of the figure.
4.4 Redshift Drift: Distinguishing BetweenΛCDM and LTB Models
Figure 4.16: The redshift drift per year for the fiducial GBH model and theΛCDM model. The black line represents the fiducial GBH model. The blue line represents theΛCDM model.
Figure 4.17: The velocity shift for theΛCDM model after a period of 10 years. The orange points represent the theoretical data for theΛCDM model with the error bars estimated from Eq. (4.10). The blue points represent the mock data for theΛCDM model with the error bars estimated from Eq. (4.10). The black solid line represents the GP reconstruction for theΛCDM model using mock data. The black dashed line represents the theoretical velocity shift for theΛCDM model. The green line represents the fiducial GBH model. The grey interval correspond to the confidence level of 1σ. For the confidence level of 1σ was used a fit of a polynomial with a degree two.
Chapter 5
Conclusion
The CP is a major assumption in cosmology. Although statistical isotropy is supported be observa-tions on large scales, the CP homogeneity hypothesis is a fundamental and weakly tested assumption that is at the heart of the most popular models of modern cosmology. Therefore, it is crucial to find ways to test this assumption. In order to do that, one should consider inhomogeneous models. In particular, in this thesis, to test the CP we choose to use the LTB models. Such models are isotropic, spherically symmetric and inhomogeneous models.
However, when studying cosmological models, one must take into consideration the method used to analyse the cosmological data. Usually, in cosmology, data analysis relies on the traditional methods, known as parametric methods. However, these methods are model dependent, therefore, they fit the parameters of known cosmological functions derived from a given model. For that reason and knowing that different classes of models have different functions and different parameters, it is easy to conclude that parameter constraints become model dependent quantities and, therefore, it may be very hard to understand if real progress is being made at understanding the underlying physical process behind the quantities being measured. Thus, how can we know which is the real model that best describes the Universe?
These and other questions took us to study alternative methods to analyse cosmological data. As the computational power increases, so do non-parametric methods. These alternative methods have the advantage of being model independent, as they rely only on data. This means that we do not need to define a modela priori, instead the data will tell us which is the model behind such results.
In this thesis, in order to implement non-parametric methods, we used an AI algorithm known as GP.
AI has been a powerful tool when it comes to analyse cosmological data. This makes the results more reliable since they are not biased by a model. This is essential, especially because present data reveals several tensions that may be a consequence of imposing pre-defined functions, instead of letting the data to provide us with its preferred functions.
The first thing we were concerned with was if non-parametric methods can be as precise as parametric methods. To study such topic we used the SCP Union2.1 SN Ia compilation data of the distance modulus.
Using the parametric approach we were able to find results for theΛCDM parameters with a coefficient of determination ofR2≈0.9930. On the other hand, by using the non-parametric approach we found a function for the distance modulus with a coefficient of determination of R2≈0.9879. Both methods offered high coefficients of determination showing for this exercise that non-parametric methods are comparable with parametric methods and can be an alternative to the traditional methods.
So if we want to use non-parametric methods as an alternative to analyse data, we first have to understand them. Hence, we tried to replicate the results obtained by Arjona and Nesseris in Ref. [41].
Even using the same data, we were not able to exactly replicate their results. However, this is not a surprise given the stochastic nature of GP and since both of us used different GP implementations we naturally obtained different formulas, but they are very close and within the function margins of each other. Note that this does not happen only if one uses different GP implementations, this can also happen when one performs different runs using the same implementation.
Next, we were faced with another important questions. Although non-parametric methods are able to replicate results with the same precision of parametric methods, what would be the limit noise level of data that would guarantee a good reconstruction? Are we already in the presence of such precise data? To answer these questions we reconstruct two cosmological functions: the luminosity distance and the shear estimator. For both cosmological functions we conclude that the quality of the GP reconstruction depends strongly on the noise level of the data. Only low fractions of present noise levels allow the reconstruction of the true theoretical function for the used quantities. More specifically, for the luminosity distance we were able to obtain the theoretical function until we reach noise level of 0.3σdA(z). Regarding the shear estimator, we were able to obtain its function until a noise level of 0.15σH. Thus, we must wait for a more precise generation of cosmological data to use GP as a powerful reconstruction method for these functions.
However, one should note that although the paper of Arjona and Nesseris [41] is from 2020, the data they use is much older, therefore, until this day, the precision of data has increased a lot. For this reason, one future step of this thesis, would be to take into account more precise data that already exists for this analysis. In addition, we would like to extend this analysis to other cosmological functions and for the uncoming galaxy surveys, such as the Euclid survey.
Still regarding the shear estimator, one of our future steps would be to rewrite this null test with observables more easy to measure so we could be able to perform such test.
Moreover, after a deep study of GP, we focused on another important part of this thesis. We were interested in distinguishing between FLRW and LTB models, so, we investigated the potential of using observations of the redshift drift for this purpose. We created data for what would be expected for the ΛCDM model if we could do such observation. Then, we compared with both CFL and GBH models and conclude that the LTB models used would be discarded easily from the FLRW mock data if we perform this type of observations. The redshift drift is a remarkable way to distinguish cosmological models and the power of such observation would be an amazing step to test the CP.
Since our LTB models can be easily discarded if we observe the redshift drift as it is proposed in Chapter 4, one future concern we have is to extend the range of LTB models to understand if all of them behave so distinctively from FLRW models. If so, the study of such models can be abandoned by performing this simple observation.
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Appendices
Appendix A
Hubble Parameter Code
In this Appendix, there is our code for the Hubble Parameter, which was used in Chapter 4 (Sec.
4.2.1).
1 #!/ usr / bin / env p y t h o n 3 2 # * c o d i n g : utf 8 * -3 """
4 C r e a t e d on Fri Nov 26 1 5 : 0 8 : 0 0 2 0 2 1 5
6 @ a u t h o r : M a r i a G o n c a l v e s 7 """
8
9 f r o m g p l e a r n . g e n e t i c i m p o r t S y m b o l i c R e g r e s s o r 10 i m p o r t m a t p l o t l i b . p y p l o t as plt
11 i m p o r t n u m p y as np 12 i m p o r t g r a p h v i z 13 f r o m s y m p y i m p o r t * 14 i m p o r t m at h
15 16 17
18 # R e a l d a t a 19 h _ d a t a = []
20 z _ d a t a =[]
21 s i g m a =[]
22 23
24 f = o p e n ( ’ z - h ( z ) ’ , ’r ’) 25 i =0
26 for l i n e in f :
27 if i <= 35:
28 s = l i n e . s p l i t ()
29 h _ d a t a . a p p e n d ( f l o a t ( s [ 1 ] ) ) 30 z _ d a t a . a p p e n d ( f l o a t ( s [ 0 ] ) ) 31 s i g m a . a p p e n d ( f l o a t ( s [ 2 ] ) )
32 i +=1
33 e l s e:
34 b r e a k
35
36 f . c l o s e () 37
38 39
40 z = np. r e s h a p e ( z_data , (36 ,1)) 41
42 43 44 45
46 # S y m b o l i c R e g r e s s o r C l a s s
47 gp = S y m b o l i c R e g r e s s o r ( p o p u l a t i o n _ s i z e =100 , g e n e r a t i o n s =1000 ,
48 t o u r n a m e n t _ s i z e =10 , s t o p p i n g _ c r i t e r i a = 0 . 0 0 0 1 ,
49 c o n s t _ r a n g e =( -1.0 , 1.0) , i n i t _ d e p t h =(2 , 10) ,
50 i n i t _ m e t h o d = ’ h a l f and half ’ ,
51 f u n c t i o n _ s e t =( ’ add ’ , ’ sub ’ , ’ mul ’ , ’ div ’) ,
52 m e t r i c = ’ rmse ’ , p a r s i m o n y _ c o e f f i c i e n t =0.01 ,
53 p _ c r o s s o v e r =0.5 , p _ s u b t r e e _ m u t a t i o n =0.01 ,
54 p _ h o i s t _ m u t a t i o n =0.01 , p _ p o i n t _ m u t a t i o n =0.01 ,
55 p _ p o i n t _ r e p l a c e =0.01 , m a x _ s a m p l e s =1 ,
56 f e a t u r e _ n a m e s = ’ z ’ , w a r m _ s t a r t =False,
57 l o w _ m e m o r y =False, n _ j o b s =1 ,
58 v e r b o s e =1 , r a n d o m _ s t a t e = 2 3 )
59
60 # Fit the G e n e t i c P r o g r a m a c c o r d i n g to z and h _ d a t a 61 gp . fit ( z , h _ d a t a )
62 63 64 65 66 67
68 # o p e r a t i o n s c o n v e r t i o n for p r i n t i n g f i n a l f u n c t i o n 69 c o n v e r t e r ={ ’ add ’: l a m b d a x , y : x + y ,
70 ’ sub ’: l a m b d a x , y : x - y , 71 ’ mul ’: l a m b d a x , y : x * y , 72 ’ div ’: l a m b d a x , y : x / y } 73
74
75 # f i n a l f u n c t i o n s i m p l i f i e d
76 f u n c t i o n = s y m p i f y ( str ( gp . _ p r o g r a m ) , l o c a l s = c o n v e r t e r ) 77 p r i n t( f u n c t i o n )
78 79
80 # h p r e d i c t e d by the a l g o r i t h m t a k i n g i n t o a c c o u n t z 81 h = gp . p r e d i c t ( z )
82
83 # c o e f f i c i e n t of d e t e r m i n a t i o n 84 p r i n t( gp . s c o r e ( z , h _ d a t a )) 85
86 87 88 89
90 # 1 s i g m a i n t e r v a l c a l c u l a t i o n 91 dfp =[]
92 dfn =[]
93 for i in r a n g e ( len ( z )):
94 eps = 0 . 0 0 0 1 95 d e l t a _ f =1
96 c = m a t h . erf ( 1 / ( m a t h . s q r t ( 2 ) ) ) 97
98 f1 = ( 1 / 2 ) * ( ma t h . erf (( d e l t a _ f + h [ i ] - h _ d a t a [ i ])/
99 ( m a t h . s q r t ( 2 ) * s i g m a [ i ]))
100 + m a t h . erf (( delta_f - h [ i ]+ h _ d a t a [ i ])/
101 ( m a t h . s q r t ( 2 ) * s i g m a [ i ]) ) )
102 103
104 w h i l e abs ( f1 - c ) > eps : 105
106
107 if f1 > c :
108 delta_f - = 0 . 0 0 1
109
110 if f1 < c :
111 d e l t a _ f + = 0 . 0 0 1
112
113 f1 = ( 1 / 2 ) * ( m a t h . erf (( d e l t a _ f + h [ i ] - h _ d a t a [ i ])/
114 ( m a t h . s q r t ( 2 ) * s i g m a [ i ]))
115 + m a t h . erf (( delta_f - h [ i ]+ h _ d a t a [ i ])/
116 ( m a t h . s q r t ( 2 ) * s i g m a [ i ]) ) )
117 118 119
120 dfp . a p p e n d ( d e l t a _ f + h [ i ]) 121 dfn . a p p e n d ( h [ i ] - d e l t a _ f ) 122
123 124 125 126
127 ## p o l i n o m i o CI 128
129 zz = z . r a v e l () 130
131 # c o e f f i c i e n t v a l u e s
132 p o l y _ d f p = np. p o l y f i t ( zz , dfp , deg =2) 133 p o l y _ d f n = np. p o l y f i t ( zz , dfn , deg =2) 134
135
136 # p o l y n o m i a l p o i n t s
137 p _ d f p =np. p o l y v a l ( p o l y _ d f p , zz ) 138 p _ d f n =np. p o l y v a l ( p o l y _ d f n , zz ) 139
140 ###
141 142 143 144 145 146 147
148 h _ l c d m =[]
149 for i in r a n g e ( len ( z )):
150
151 h _ l c d m . a p p e n d ( 6 9 . 2 7 * m a t h . s q r t ( 0 . 6 8 4 7 + 0 . 3 1 5 3 * ( 1 + z [ i ] ) * * 3 ) ) 152
153 154
155 h _ p a p e r =[]