Probing the Expansion History of the Universe using Supernovae and Baryon Acoustic Oscillations

125 Pages·2016·11.59 MB·English

University of The Western Cape

Doctoral Thesis

Probing the Expansion History of
the Universe using Supernovae
and Baryon Acoustic Oscillations
Supervisor:
Prof. R. Maartens
Author:
Co-Supervisors:
Sahba Yahya Hamid Ali
Prof. C. Clarkson
& Prof. M.G. Santos
A thesis submitted in fulfilment of the requirements
for the degree of Doctor of Philosophy
in the
Department of Physics
March 2016
Abstract

The standard model of cosmology (the ΛCDM model) has been very successful

and is compatible with all observational data up to now. However, it remains
an important task to develop and apply null tests of this model. These tests are
based on observables that probe cosmic distances and cosmic evolution history.
Supernovae observations use the so-called ‘standard candle’ property of SNIa to
probe cosmic distances D(z). The evolution of the expansion rate H(z) is probed
by the baryon acoustic oscillation (BAO) feature in the galaxy distribution, which
serves as an effective ‘standard ruler’. The observables D(z) and H(z) are used
in various consistency tests of ΛCDM that have been developed. We review the
consistency tests, also looking for possible new tests.
Then the tests are applied, first using existing data, and then using mock data
from future planned experiments. In particular we use data from the recently
commissioned Dark Energy Survey (DES) for SNIa. Gaussian Processes, and
possibly other non-parametric methods, used to reconstruct the derivatives of
D(z) andH(z) thatare neededto applythe nulltests ofthe standardcosmological
model. This allow us to estimate the current and future power of observations to
probe the ΛCDM model, which is the foundation of modern cosmology.
In addition, we present an improved model of the HI galaxy number counts and
bias from semi-analytic simulations, and we use it to calculate the expected yield
of HI galaxies from surveys with a variety of phase 1 and 2 SKA configurations.
We illustrate the relative performance of the different surveys by forecasting errors
on the radial and transverse scales of the BAO feature. We use the Fisher matrix
method to estimate the error bars on the cosmological parameters from future
SKA HI galaxy surveys. We find that the SKA phase 1 galaxy surveys will not
contend with surveys such as the Baryon Oscillation Spectroscopic Survey (BOSS)
whereas the full ”billion galaxy survey” with SKA phase 2 will deliver the largest
dark energy Figure of Merit of any current or future large-scale structure survey.
Declaration of Authorship

I, Sahba Yahya Hamid Ali, I declare that ”Probing the Expansion History of the
Universe using Supernovae and Baryon Acoustic Oscillations” is my own work,
that it has not been submitted for any degree or examination in any other uni-
versity, and that all the sources I have used or quoted have been indicated and
acknowledged by complete references.The following papers have been published,
based on the work presented in this thesis:
• S. Yahya, P. Bull, M. G. Santos, M. Silva, R. Maartens, P. Okouma and
B. Bassett, ” Cosmological performance of SKA HI galaxy surveys,” Mon.
Not. Roy. Astron. Soc., 450, 3, 2251, March 2015.
• M. G. Santos, D. Alonso, P. Bull, M. Silva and S. Yahya, ”HI galaxy sim-
ulations for the SKA: number counts and bias,” arXiv:1501.03990, January
2015.
• S. Yahya, M. Seikel, C. Clarkson, R. Maartens and M. Smith, ” Null tests
of the cosmological constant using supernovae,” Phys. Rev. D, 89:023503,
January 2014.
• M. Seikel, S. Yahya, R. Maartens and C. Clarkson, “Using H(z) data as a
probe of the concordance model,” Phys. Rev. D, 86:083001, October 2012.
Full name...........S..a..h..b..a. .Y..a..h..y.a....H..a..m....id.. .A...l.i.... Date.........9.. .M..a..r.c..h. .2..0..1.6.. ......
Signed.........................................
ii
iii
Acknowledgements

I would like to thank my family and f riends for their understanding and support.
Without them, I would not have been able to complete my PhD.

Many thanks to the Astrophysics group at the University of the Western Cape,

my supervisor Roy Maartens and co-supervisors, Chris Clarkson and Mario G.
Santos, for their provided support.
I would like to thank the South African Square Kilometre Array Project (SKA)
and the German Academic Exchange Service (DAAD) for their financial support
during my PhD.
IamthankfultotheNationalAstrophysicsandSpaceScienceprogramme(NASSP)
for their financial support throughout my Honor and MSc degree, without it I
would not have gotten to this point.
Finally I would like to thank the researchers that I consistently learned from, Phil
Bull, Mathew Smith, Patrice Okouma, Marina Seikel and all the people I had a
productive or inspiring conversation during these years...
Contents

Abstract i
Declaration of Authorship ii
Acknowledgements ii
Contents iv
List of Figures vi
List of Tables ix
1 Introduction 1
2 The Expansion History of the Universe 4
2.1 The Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.1 Luminosity distance and standard candles . . . . . . . . . . 11
2.2.2 Angular diameter distance and standard rulers . . . . . . . 12
3 Large Scale Structure 14
3.1 The growth of structure . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 The power spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3 The correlation function . . . . . . . . . . . . . . . . . . . . . . . . 18
3.4 Large scale structure surveys . . . . . . . . . . . . . . . . . . . . . . 19
4 Null tests of the cosmological constant using supernovae 24
4.1 Null tests of ΛCDM – theory . . . . . . . . . . . . . . . . . . . . . . 24
4.2 Null tests using SNIa data . . . . . . . . . . . . . . . . . . . . . . . 28
4.2.1 Gaussian Processes . . . . . . . . . . . . . . . . . . . . . . . 28
4.2.2 Application to real data . . . . . . . . . . . . . . . . . . . . 30
4.2.3 Mock data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
iv
Contents v
5 Using H(z) data as a probe of the concordance model 40
5.1 Testing ΛCDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.2 Reconstruction and consistency tests from H(z) data . . . . . . . . 52
5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6 Future SKA HI galaxy survey s 62
6.1 Telescope and Survey specifications . . . . . . . . . . . . . . . . . . 63

6.1.1 Flux sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.1.2 Survey specifications . . . . . . . . . . . . . . . . . . . . . . 67
6.2 HI galaxy simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.2.1 HI galaxy number densities . . . . . . . . . . . . . . . . . . 71
6.2.2 HI galaxy bias . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.3 Cosmological Performance . . . . . . . . . . . . . . . . . . . . . . . 77
6.3.1 SKA assumed sensitivities . . . . . . . . . . . . . . . . . . . 77
6.3.2 Fisher forecasts . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.3.3 Comparison with previous results and future experiments . . 83
6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7 Conclusion 90
A Gaussian Processes 92
B Derivation of the consistency tests of w(z) 95
C Fisher matrix and propagation of errors 97
C.1 Wiggles only power spectrum . . . . . . . . . . . . . . . . . . . . . 97
C.2 Errors on D and H . . . . . . . . . . . . . . . . . . . . . . . . . . 101
A
C.3 Error propagation to w . . . . . . . . . . . . . . . . . . . . . . . . . 101
Bibliography 105
List of Figures

2.1 The sketch shows the BAO scale, where along the line-of-sight we
can measure H(z), and across the line-of-sight we can measure D .
A
Given dz and θ are the redshift interval and the subtended angle,
respectively [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1 The 2dF galaxy redshift survey map revealing the structure of the
universe with respect to redshift [2]. . . . . . . . . . . . . . . . . . . 16
3.2 The predicted baryon wiggles in the galaxy power spectrum (line)
and the measurements from the galaxy surveys (dots) of 2dFGRS
+SDSS (top), SDSS DR5 LRGS sample (middle) and all of the data
combined (bottom) [3]. . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.1 Gaussian processes reconstruction of D, D(cid:48) (top) and D(cid:48)(cid:48), D(cid:48)(cid:48)(cid:48) (bot-
tom) for Union 2.1 data. The red (solid) line is flat ΛCDM with
Ω = 0.27. The blue (dashed) line is the mean of the reconstruc-
m
tion. Shaded areas give 95% (light) and 68% (dark) confidence
limits of the reconstructed function. . . . . . . . . . . . . . . . . . . 31
4.2 Reconstruction of (1) (top), (2) (middle) and (2) (bottom) for
Om Om OK
Union 2.1 data. Lines and shadings are as in Fig. 4.1. . . . . . . . 32
4.3 Reconstruction of (1) (top) and (2) (bottom) for Union 2.1 data.
L L
Lines and shadings are as in Fig. 4.1. . . . . . . . . . . . . . . . . 33
4.4 (1) (top), (2) (middle) and (2) (bottom) reconstructed using sim-
Om Om OK
ulated DES data, and assuming ΛCDM. . . . . . . . . . . . . . . . 35
4.5 (1) (top), (2) (middle) and (2) (bottom) reconstructed using sim-
Om Om OK
ulated DES data, and assuming evolving w in (4.18). . . . . . . . . 36
4.6 Reconstruction of (1) and (2) for simulated DES data, and as-
L L
suming ΛCDM (top) and the evolving w in (4.18) (bottom). Due
to the degeneracy between w and Ω , the reconstruction of (2)
m
L
for the model with evolving dark energy is consistent with ΛCDM.
However, the inferred values of Ω and Ω differ significantly from
m K
the input value as can be seen in Figs. 4.4 and 4.5. . . . . . . . . . 37
vi
List of Figures vii
5.1 h(z) = H(z)/H (top) and h(cid:48)(z) (bottom) reconstructed from BAO
0
data, using Gaussian processes. Shaded areas represent 68% and

95% confidence levels. The dashed (red) curve is flat ΛCDM with
Ω = 0.27; the solid (blue) curve is the GP mean. Note that while
m  
the BAO data appear to give an inconsistent h(cid:48)(z), this is driven by
the two highest redshift poin ts both of which happen to lie below
the flat ΛCDM curve. . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.2 h(z) = H(z)/H (top) and h (cid:48)(z) (bottom) reconstructed from cos-
0
mic chronometer data, using Gaussian processes. Lines and shad-
ings are as in Fig. 5.1. . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.3 h(z) = H(z)/H (top)andh(cid:48)(z)(bottom)reconstructedfromCC+BAO
0
data, using Gaussian processes. Lines and shadings are as in Fig.
5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.4 (1)(z) reconstructed from cosmic chronometers (top), BAO (mid-
m
O
dle) and CC+BAO (bottom). Where the dashed (red) curve is flat
ΛCDM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.5 (2)(z) reconstructed from cosmic chronometers (top), BAO (mid-
m
O
dle) and CC+BAO (left). The dashed (red) curve is a curved
ΛCDM model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.6 (z) reconstructed from cosmic chronometers (top), BAO (mid-
K
O
dle) and CC+BAO (bottom). The dashed (red) curve is a curved
ΛCDM model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.7 (1) = (1)/(1+z)6 reconstructed from cosmic chronometers (top),
m
L L
BAO (middle) and CC+BAO (bottom). The dashed (red) curve is
a ΛCDM model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.8 (2) = (2)/(1+z)6 reconstructed from cosmic chronometers (top),
m
L L
BAO (middle) and CC+BAO (bottom). The dashed (red) curve is
a ΛCDM model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.9 w(z)reconstructedfromcosmicchronometers(top), BAO(middle–
notethedifferentz range)andCC+BAO(bottom)bymarginalizing
over Ω = 0.275 0.016. The dashed (red) curve is a ΛCDM model. 51
m
±
5.10 h(z) (top), h(cid:48)(z) (middle) and h(cid:48)(cid:48)(z) (bottom) reconstructed from
simulated data, assuming a concordance model. . . . . . . . . . . . 54
5.11 h(z) (top), h(cid:48)(z) (middle) and h(cid:48)(cid:48)(z) (bottom) reconstructed from
simulated data, assuming a model (4.18) with slowly evolving w(z). 55
5.12 (1)(z) (top), (2)(z) (middle) and (z) (bottom) reconstructed
m m K
O O O
from simulated data, assuming a concordance model. . . . . . . . . 56
5.13 (1)(z) (top), (2)(z) (middle) and (z) (bottom) reconstructed
m m K
O O O
from simulated data, assuming a model (4.18). . . . . . . . . . . . . 57
5.14 (1) = (1)/(1 + z)6 (top) and (2) = (2)/(1 + z)6 (bottom) re-
m m
L L L L
constructed from simulated data, assuming a concordance model.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.15 (1) = (1)/(1+z)6 (top) and (2) = (2)/(1+z)6 (bottom) recon-
m m
L L L L
structed from simulated data, assuming a model (4.18). . . . . . . 59
List of Figures viii
5.16 w(z) reconstructed from simulated data, assuming a concordance
model(top)andamodelintroducedin(4.18)(bottom),bymarginal-

izing over Ω = 0.275 0.016. . . . . . . . . . . . . . . . . . . . . . 60
m
±

6.1 Dark energy figure of merit versus survey area for SKA1, with dif-
ferent frequency ranges. A Planck CMB prior has been included

in the FOM calculation, but the optimal survey areas are the same
even if the prior matrix is no t included. . . . . . . . . . . . . . . . . 68
6.2 HI galaxy redshift distribution, dN/dz, calculated from simulations
(solid circles) and the corresponding fitting function, (6.5). From
top to bottom, the curves shown correspond to flux sensitivities
S = (0,1,3,5,10,23,100,200) µJy (colour-coded according to
rms
the panel on the right). . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.3 HI galaxy bias for different S . Note that bias values for high flux
rms
rms are uncertain. This has little impact, however, as shot noise
will dominate at these sensitivities. . . . . . . . . . . . . . . . . . . 72
6.4 Number densities for the optimistic, reference, and pessimistic cases
of SKA1 and 2, compared with Euclid. Dashed-dotted lines show
thenumberdensityatwhichn(z)b2(z)P(z,k ) = 1forthevarious
max
surveys, with k 0.2h/Mpc. When dN/dz is above this line,
max
≈
sample variance dominates the shot noise for all k < k ; the point
max
at which it dips below the line is effectively the maximum redshift
of the survey. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.5 Forecast fractional errors on the expansion rate, H(z), and angu-
lar diameter distance, D (z), from BAO measurements with the
A
various surveys. The redshift binning is fixed at ∆z = 0.1 for all
experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.6 Forecast constraints (1 and 2σ contours) on the dark energy equa-
tion of state parameters, w and w , for the reference cases of both
0 a
SKA1 (purple, largest), BOSS (grey, second largest) and SKA2
(green, smallest), compared to a Euclid-like Hα galaxy survey (yel-
low, intermediate). A Planck CMB prior has been included for all
three experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.7 Forecast constraints (1 and 2σ) on w and Ω for the references
0 K
cases of SKA1 (purple, largest), BOSS (grey, second largest) and
SKA2(green, smallest), comparedwithaEuclid-likeHαgalaxysur-
vey (yellow, intermediate). A Planck CMB prior has been included
for all experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
C.1 The power spectrum P(k), the smoothed power spectrum and the
wiggles only function f (k) on the top panel. On the x-axis k in
BAO
Mpc−1h, and on the y-axis the units are Mpc3h−3 . . . . . . . . . . 99
List of Tables

6.1 Telescope specifications . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.2 Survey specifications . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.3 Bias values calculated in each redshift bin of the simulation, as a
function of flux rms. . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.4 Best-fit parameters for the number density and bias fitting func-
tions, (6.5) and (6.7), for different flux limits. S is measured in
rms
µJy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.5 Fitted parameters for the galaxy number density and bias, for the
frequency-corrected S of the various experiments. The flux rms
rms
at the reference frequency, Sref , is in µJy, while the fitted coeffi-
rms
cients are dimensionless. . . . . . . . . . . . . . . . . . . . . . . . . 82
6.6 Forecast 1σ marginal errors and dark energy FOM for the various
SKA1 and 2 reference experiments. A Euclid-like Hα survey, BOSS
forecasts, and a noise-free SKA2 configuration, are shown for com-
parison. (Planck CMB priors have been included in all cases.) . . . 86
C.1 Planck prior matrix for the cosmological parameters of interest. . . 104
ix