The angular homogeneity scale of the Universe - Centre for ...

The angular homogeneity 

scale of the Universe 

Alicia Bueno Belloso 

ITP, Heidelberg 

Work in collaboration with: 

David Alonso, Juan García-Bellido, Eusebio Sánchez and Javier Sánchez 

in preparation 

2 nd of September 2013 

DAMTP, Cambridge

Introduction: a homogeneous 

or fractal universe 

• Usual assumption: the Universe is homogeneous and isotropic on large scales 

Alicia Bueno Belloso The angular homogeneity scale of the Universe 2 nd September 2013




• Homogeneous regime is not realised on small scales, due to the form of the 

spectrum of matter perturbations and to their evolution via gravitational collapse. 







• The primordial spectrum of metric perturbations is predicted to be almost 

scale-invariant by inflation, still expect some level of inhomogeneities 









• Different groups have argued that the Universe might not reach homogeneity 

on large scales, and that it behaves like a fractal… 

Pietronero. et al., 1997, Critical Dialogues in Cosmology, 24 

F. Sylos Labini, et al., Europhys. Letters 2009 









• Different groups have argued that the Universe might not reach homogeneity 

on large scales, and that it behaves like a fractal… 

• … while other groups claim the opposite result!! 

Pietronero. et al., 1997, Critical Dialogues in Cosmology, 24 

F. Sylos Labini, et al., Europhys. Letters 2009 

Scrimgeour, M., et al., 2012, MNRAS, 425, 116 

Nadathur, S. 2013, MNRAS, 434, 398 




• To measure this transition observationally, large survey volume is necessary 





• Ideal for photometric galaxy redshift surveys such as DES 

The Dark Energy Survey Collaboration, arXiv:0510346, 2005 






Due to photometric uncertainty 

Radial information is lost 







Due to photometric uncertainty 

Radial information is lost 

Must use estimator with only angular info 

Advantage: Angles are model-independent! 



The fractal dimension 

• To study the transition to homogeneity 

fractality of the galaxy 

distribution 





distribution 

• Fractal dimension is also useful to quantify clustering 





distribution 


• How do we specify this 

Correlation integral 

C 2 (r) = 1 N 

NX 

nP (n; r, N) / r 3 

n=0 

D 2 (r) ⌘ d log C 2(r) 

d log r 

! 3 

Volume 

As homogeneity 

is approached 





distribution 


• How do we specify this 

C 2 (r) = 1 N 

NX 

nP (n; r, N) / r 3 

Correlation integral 

• Departures from D 2 = 3 due to: 

- Clustering 

- Shot noise 

n=0 

D 2 (r) ⌘ d log C 2(r) 

d log r 

! 3 

Volume 


is approached 

J. S. Bagla, J. Yadav and T. R. Seshadri, 2007, MNRAS, 390:829 



• Angular homogeneity index: 

Spheres 

Spherical caps of radius θ 




Spheres 


G 2 (✓) / V (✓) =2⇡(1 cos ✓) 

Non-trivial dependence on θ 




Spheres 


H 2 (✓) ⌘ d log G 2(✓) 

d log V (✓) ! 1 

G 2 (✓) / V (✓) =2⇡(1 cos ✓) 


is approached 





Spheres 


H 2 (✓) ⌘ d log G 2(✓) 

d log V (✓) ! 1 

• Modelling H 2 (θ): 

G 2 (✓) / V (✓) =2⇡(1 cos ✓) 


is approached 

H 2 (✓) = 1+w(✓) 

1+ ¯w(✓) 


1 

¯N(✓) 

Fiducial cosmology: (⌦ m , ⌦ ⇤ , ⌦ b ,h, 8,n s )=(0.3, 0.7, 0.049, 0.67, 0.8, 0.96) 


Results I: 

the theoretical model 

• Projection effects (redshift bin size): 


• Bias: 

Results I: 



• Cosmological parameters: 

1.005 

Results I: 


H 2 (θ) for 0.5 < z < 0.6 for different values of w 

1.000 

0.995 

0.990 

H 2 (θ) 

0.985 

0.980 

0.975 

0.970 

0.965 

w = - 0.4 

w = - 0.8 

w = - 1.0 

w = - 1.2 

0.960 

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 

θ (deg) 


Measuring H 2 (θ) 

• Complications when measuring D 2 (θ) or H 2 (θ): 

- Non-homogeneous radial selection function 

- Imperfections (fiber collisions, star contamination, CCD saturation…) 






• Solve this complications 

N (r) ⌘ 1 XN c 

N c 

use random catalogues 

i=1 

n d i (






N (r) ⌘ 1 XN c 

N c 


• 3 estimators: 

1. Use only spheres within the survey volume 

2. Use only spheres within the survey volume + random catalogues 

with observational effects from data 

3. Consider also spheres outside survey volume corrected with random 

catalogues 

i=1 

n d i (

Results II: 

Mock catalogues 

• 100 lognormal mock catalogues with fiducial cosmology, z =0.03 and b=1 

1.002 

H 2 (θ) for 0.5 < z < 0.6 comparing theory and 100 mock catalogues 

1.000 

0.998 

H 2 (θ) 

0.996 

0.994 

1.0005 

1.0000 

0.9995 

0.992 

0.9990 

0.990 

5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 

Theory prediction 

Mock catalogues, estimator 2 

Mock catalogues, esitmator 3 

0.988 

0 1 2 3 4 5 6 7 8 9 10 

θ (deg) 


Results II: 

Mock catalogues vs fractals 

• 100 fractal realisations using a 2D random walk 

1.00 

P (✓ d < ✓) = 

H 2 (θ) for 0.5 < z < 0.6 comparing mocks and fractal realisations 

( 1 ✓ < ✓0 

⇣ 

1 cos ✓ 

1 cos ✓ 0 

⌘ ↵ 

✓ ✓ 0 

0.98 

0.96 

0.94 

H 2 (θ) 

0.92 

0.90 

0.88 

0.86 

Theory prediction 

Mock catalogues, estimator 2 

Mock catalogues, esitmator 3 

2D random walk with α= 0.5 



0 1 2 3 4 5 6 7 8 9 10 

θ (deg) 


Conclusions 

• Usual assumption of homogeneity on large scales can be tested with large 

volume galaxy surveys 

• Introduced angular homogeneity index H 2 (θ) to use with photo-z surveys 

Advantage: Angular measurements are model-independent!! 

• Modelled H 2 (θ) theoretically and studied dependence on several effects 

• Built several estimators to measure H 2 (θ) and tested them on 100 

mock catalogues 

results fit theoretical model and are compatible 

with homogeneity scale found in the 3D case 

• Found that we can distinguish fractal models from homogeneous universes 


Thank you! 


• Projection effects revisited: 

Results II: 

Statistical uncertainties 


Results II: 

Statistical uncertainties 

• Full covariance matrix for both estimators 

Estimator 2 Estimator 3 


• Photometric redshift uncertainty: 

Results I: 



• Non-linearities: 

Results I: 


H(θ) for 0.5 < z < 0.6 for different treatment of non-linearities 

1.00 

0.99 

0.98 

0.97 

H 2 (θ) 

0.96 

0.95 

0.94 

0.93 

0.92 

0.91 

0.90 

HALOFIT 

RPT 

0.1 1 10 

θ (deg) 


• Cosmological parameters: 

Results I: 


H 2 (θ) 

0.985 

0.980 

H 2 (θ) for 0.5 < z < 0.6 for different values of Ω m 

w = - 0.4 

1.005 

1.005 

H 2 (θ) for 0.5 < z < 0.6 for different values of w 

1.000 

1.000 

0.995 

0.995 

0.990 

0.990 

H 2 (θ) 

0.985 

0.980 

0.975 

0.975 

0.970 

0.970 

0.965 

Ω m = 0.3 

Ω m = 0.5 

Ω m = 0.7 

Ω m = 0.9 

0.965 

w = - 0.8 

w = - 1.0 

w = - 1.2 

0.960 

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 

θ (deg) 

0.960 

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 

θ (deg) 







N (r) ⌘ 1 XN c 

N c 


• 3 estimators: 

1. Use only spheres within the survey volume 

2. Use only spheres within the survey volume + random catalogues 

with observational effects from data 

3. Consider also spheres outside survey volume corrected with random 

catalogues 

i=1 

n d i (

Conclusions 

• Usual assumption of homogeneity on large scales must be corroborated 

observationally 

• Best way to study transition to homogeneity 

large volume galaxy 

surveys 

• Introduced angular homogeneity index to use with photo-z surveys 

Advantage: Angular measurements are model-independent!! 

• Modelled H 2 (θ) and studied dependence on several effects 

• Built several estimators to measure H 2 (θ) and tested them on 100 

mock catalogues 

• Results obtained fit theoretical model and are compatible with the 

homogeneity scale found in 3D of ~100Mpc/h 

• Found that we can distinguish fractal models from homogeneous universes

The angular homogeneity scale of the Universe - Centre for ...

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