The angular homogeneity scale of the Universe

Transcription

1 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

2 Introduction: a homogeneous or fractal universe? Usual assumption: the Universe is homogeneous and isotropic on large scales

3 Introduction: a homogeneous or fractal universe? Usual assumption: the Universe is homogeneous and isotropic on large scales 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.

4 Introduction: a homogeneous or fractal universe? Usual assumption: the Universe is homogeneous and isotropic on large scales 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

5 Introduction: a homogeneous or fractal universe? Usual assumption: the Universe is homogeneous and isotropic on large scales 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

6 Introduction: a homogeneous or fractal universe? Usual assumption: the Universe is homogeneous and isotropic on large scales 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 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

7 Introduction: a homogeneous or fractal universe? To measure this transition observationally, large survey volume is necessary

8 Introduction: a homogeneous or fractal universe? 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: , 2005

9 Introduction: a homogeneous or fractal universe? To measure this transition observationally, large survey volume is necessary Ideal for photometric galaxy redshift surveys such as DES Due to photometric uncertainty Radial information is lost The Dark Energy Survey Collaboration, arxiv: , 2005

10 Introduction: a homogeneous or fractal universe? To measure this transition observationally, large survey volume is necessary Ideal for photometric galaxy redshift surveys such as DES Due to photometric uncertainty Radial information is lost Must use estimator with only angular info Advantage: Angles are model-independent! The Dark Energy Survey Collaboration, arxiv: , 2005

11 The fractal dimension To study the transition to homogeneity fractality of the galaxy distribution

12 The fractal dimension To study the transition to homogeneity fractality of the galaxy distribution Fractal dimension is also useful to quantify clustering

13 The fractal dimension To study the transition to homogeneity fractality of the galaxy distribution Fractal dimension is also useful to quantify clustering 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

14 The fractal dimension To study the transition to homogeneity fractality of the galaxy distribution Fractal dimension is also useful to quantify clustering 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 As homogeneity is approached J. S. Bagla, J. Yadav and T. R. Seshadri, 2007, MNRAS, 390:829

15 The fractal dimension Angular homogeneity index: Spheres Spherical caps of radius θ

16 The fractal dimension Angular homogeneity index: Spheres Spherical caps of radius θ G 2 ( ) / V ( ) =2 (1 cos ) Non-trivial dependence on θ

17 The fractal dimension Angular homogeneity index: Spheres Spherical caps of radius θ H 2 ( ) d log G 2( ) d log V ( )! 1 G 2 ( ) / V ( ) =2 (1 cos ) As homogeneity is approached Non-trivial dependence on θ

18 The fractal dimension Angular homogeneity index: Spheres Spherical caps of radius θ H 2 ( ) d log G 2( ) d log V ( )! 1 Modelling H 2 (θ): G 2 ( ) / V ( ) =2 (1 cos ) As homogeneity is approached H 2 ( ) = 1+w( ) 1+ w( ) Non-trivial dependence on θ 1 N( ) Fiducial cosmology: ( m,, b,h, 8,n s )=(0.3, 0.7, 0.049, 0.67, 0.8, 0.96)

19 Results I: the theoretical model Projection effects (redshift bin size):

20 Bias: Results I: the theoretical model

21 Cosmological parameters: Results I: the theoretical model H 2 (θ) for 0.5 < z < 0.6 for different values of w H 2 (θ) w = w = w = w = θ (deg)

22 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 )

23 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 (<r) f r n r i (<r)

24 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 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 (<r) f r n r i (<r)

25 Results II: Mock catalogues 100 lognormal mock catalogues with fiducial cosmology, z =0.03 and b= H 2 (θ) for 0.5 < z < 0.6 comparing theory and 100 mock catalogues H 2 (θ) Theory prediction Mock catalogues, estimator 2 Mock catalogues, esitmator θ (deg)

26 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 H 2 (θ) Theory prediction Mock catalogues, estimator 2 Mock catalogues, esitmator 3 2D random walk with α= 0.5 2D random walk with α= D random walk with α= θ (deg)

27 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

28 Thank you!

29 Projection effects revisited: Results II: Statistical uncertainties

30 Results II: Statistical uncertainties Full covariance matrix for both estimators Estimator 2 Estimator 3

31 Photometric redshift uncertainty: Results I: the theoretical model

32 Non-linearities: Results I: the theoretical model H(θ) for 0.5 < z < 0.6 for different treatment of non-linearities H 2 (θ) HALOFIT RPT θ (deg)

33 Cosmological parameters: Results I: the theoretical model H 2 (θ) for 0.5 < z < 0.6 for different values of Ω m H 2 (θ) for 0.5 < z < 0.6 for different values of w H 2 (θ) H 2 (θ) Ω m = 0.3 Ω m = 0.5 Ω m = 0.7 Ω m = w = w = w = w = θ (deg) θ (deg)

34 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 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 (<r) f r n r i (<r) Defining the homogeneity scale: - Arbitrary fraction of homogeneous value (1 or 3), usually 1% - Uncertainties within homogeneity

35 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