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Testing the cosmological model usinggalaxy voids
Luis Alberto Torres Andrade
Presented in fulfillment of the requirements
of the degree of Doctor of Philosophy
2017
Faculty of Science, Engineering and Technology
Swinburne University
i
AbstractA primary goal of cosmological science is to test dark energy physics by extracting in-
formation on the accelerated expansion of the Universe from two fundamental probes:
the rate of cosmic expansion through distance measurements, and gravitational physics
through the analysis of the growth of large-scale structure. Often, observational probes fo-
cus on a characteristic astrophysical object or clustering feature, such as using Supernovae
Type Ia to build a Hubble diagram for expansion rate studies. In this thesis we focus on
galaxy voids, regions of low density in the cosmological inventory, as tracer objects of the
large-scale structure of the Universe to test the cosmological model.
Using the ZOBOV void finder, we generate a void catalogue from the Galaxy and Mass
Assembly Survey (GAMA) spectroscopic sample, a dense and complete set of approxi-
mately 180000 galaxies covering 180 deg2 in 3 equatorial fields. Specifically, we extract
volume-limited subsamples in two redshift intervals 0 < z < 0.25 and 0.25 < z < 0.5. We
find 454 voids spanning a range of sizes from 5 to 70 Mpc/h.
Using the void catalogue, in chapter 2 we perform expansion rate studies by imple-
menting an Alcock-Paczynski (AP) test. The AP test requires objects which are isotropic
along the line-of-sight (redshift extension) in comparison to the direction perpendicular to
the line-of-sight (angular extension). We generate isotropic objects by stacking the voids
using their geometric centers, defining a density profile as a function of the void-centric dis-
tance perpendicular and parallel to the line-of-sight. Peculiar velocities in galaxy samples
is a contaminating effect for the geometric isotropy. We use matched N -body simula-
tions to study the effect of peculiar velocities and quantify the error in our measurements.
By fitting a functional form of the void profile calibrated from the simulations, we re-
port measurements of the AP parameter F = r(z)H(z)/c, in terms of co-moving distance
r(z) and Hubble parameter H(z), for the effective redshifts z∗ = (0.18, 0.38) with values
F/Ffid = (1.12±0.09, 0.77±0.10) respectively, where Ffid is the prediction of the standardmodel. We hence find a mild tension in the higher-redshift bin favouring a low-density
Universe, but we identify potential systematic errors associated with the survey boundary.
In chapter 3 we generate a statistical Fisher matrix forecast for future galaxy survey
implementations of the Alcock-Paczynski test using cosmic voids. We use a suite of N -
body simulations to calibrate the scaling of the AP parameter accuracy as a function of
the number density of the halo tracers and find almost no evolution of the error with
redshift in the range z < 0.7. Future surveys such as DESI and Euclid will possess a
constraining power exceeding current surveys such as BOSS and eBOSS by more than one
order of magnitude. In an analysis free of other priors, we find that galaxy voids are not
ii
as competitive as baryon acoustic oscillations for constraining cosmological parameters.
We emphasize that this result is generated not considering strong priors on the parameter
Ωm, in particular those coming from CMB experiments such as Planck. This shows that
multiple large-scale structure data sets, in the form of baryon acoustic oscillations or
cosmic voids represent alternatives to probe different directions in the space of cosmological
parameters. We also study the effect of uncertainty in the modelling of peculiar velocities,
finding that the marginalization of this effect degrades considerably the constraining power
of future AP implementations. We determine that the effect of peculiar velocities on F
needs to be calibrated with 1% accuracy in order not to degrade dark energy constraints.
In chapter 4 we develop a measurement of the growth of structure based on the
anisotropic clustering analysis of redshift-space distortions in the two-dimensional void-
galaxy cross-correlation function. We build a model in the quasi-linear regime, based
on the convolution with a kernel function modelling the pairwise velocity dispersion of a
Gaussian streaming model, with effective parameters β = f/b, where f is the growth rate
of structure, b is the galaxy bias and σv as the characteristic velocity dispersion. We use
N -body simulations to evaluate the accuracy of our model, delineating the quasi-linear
regime and the scale-dependence of the velocity dispersion. We report the measurement of
the growth rate parameter for the 2 effective redshift bins β(z∗) = (1.05±0.27, 0.86±0.20).Our measurements are somewhat higher than the prediction of the fiducial cosmology, but
we use our data and simulations to map out potential systematic effects in modelling as a
function of void size and velocity dispersion.
In summary, we advance the study of voids as a cosmological probe through the com-
bination of new measurements, tests using simulations and forecasts. Voids provide an
interesting environment for probing gravitational physics with current and future galaxy
surveys, particularly scenarios which predict density-dependent fifth forces.
iii
Acknowledgements
The day I wrote this page I recovered my hopes. What happened one day before was the
main reason to keep going, to keep hope. These initial lines will make sense someday in
the future when I want to use the words: remember the day before?
I want to use this space to thank my supervisor, Chris Blake, for an amazing and
rewarding PhD experience. Since the first moment, asking for an opportunity inside the
academic world of cosmology, to the final stages of the PhD, his attention and feedback
were beyond the limits of a simple supervisor and reflected his enormous quality as a
human being. Always with simple words to help my spirit travel in this complicated road
called PhD. If I am reaching the point of producing these lines is because the help and
comprehension of my supervisor. It is a pleasure to express it one more time: Chris, thank
you very much for your time and help.
To my associate surpervisors, Felipe Marin, who helped me in the first third of my
PhD, showing patience when both of us were in front of a computer and perspective when
both of us were in front of a cup of coffee. Ixandra Achitouv, who helped me in the final
third of the PhD, showing advice and patience in MCMC techniques, fortran and her
expertise in science related with cosmic voids.
Particular thanks to Jun Koda, for his initial tutorial on how to do a cosmological
numerical simulation and all his work inside the Swinburne Cosmology group.
To my wife Alba Ivette, for all those moments in which we were missing home and
family, and we had visits from sadness with tears as companions, thank you so much for
being my balance and for taking this adventure together. I love you so much. Te amo
mucho esposita y vamos a seguir teniendo esperanza juntos.
To my father, Federico Torres, for the most amazing example of constant work I have
ever seen in my life; to my mother, Olaya Andrade, for always let me feel as the most
loved son of the world , to my sister Tere and his beloved son, Josué, for knowing that
we can count with each other despite the distance and to my brother Mickey, for all those
years as children and for being my little brother. Los quiero mucho a todos y gracias por
todo su apoyo a lo largo de estos años.
To the people and beings that I could not see anymore because of the distance and
the circumstances. Jessica, my favourite cousin; Peluche, my best friend from the animal
kingdom and who literally saw all my process to pursue the dream to professional physics,
next to that lonely desk in those cold nights. I will always remember your company.
To my friend Juan Miguel, because during this time we have begun the path of that
iv
beloved concept called family and for being the best friend the life has given me. To Pablo
Villanueva, as the best friend the Faculty of Sciences has given me. Thanks to my friends
José Antonio, Juan Carlos, David, Gerardo y Enrique for always share your friendship
with me. Thanks to my friend Pablo Galaviz for all his support during my transition to
Melbourne and all those friendship moments during these 4 years.
Thanks to my institution, the Centre for Astrophysics and Supercomputing for its
international atmosphere for students and for the support of having a PhD scholarship.
Special thanks to Liz Thackray for her service to the Centre and to my friend George, who
was always willing to answer my questions about how to use computers in a much better
way. Thanks to the CAS student community for making the Australian Astronomical
Community a very nice place to study.
Tomorrow, it will be a better day. Keep hope.
Luis Alberto Torres Andrade
Melbourne, VIC, Australia
December 1, 2016
v
Declaration
The work presented in this thesis has been carried out in the Centre for Astrophysics
& Supercomputing at Swinburne University of Technology (Hawthorn Campus) between
2012 and 2016. This thesis contains no material that has been accepted for the award of
any other degree or diploma. To the best of my knowledge, this thesis contains no material
previously published or written by another author, except where due reference is made in
the text of the thesis.
Luis Alberto Torres Andrade
Melbourne, Victoria, Australia
2016
Contents
Abstract i
Acknowledgements iii
Declaration v
List of Figures viii
List of Tables xix
1 Introduction 1
1.1 The Cosmological Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Existence of Dark Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Large Scale Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Galaxy Voids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.5 Scope of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2 GAMA Survey: Alcock-Paczynski effect using cosmic voids 23
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 The process of finding galaxy voids . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 The ZOBOV void finder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3.1 Density estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.2 Zoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3.3 From Zones to Voids . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3.4 Systematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4 The GAMA Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5 GAMA data sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.6 Alcock-Paczynski measurement . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.7 GAMA AP results and systematics tests . . . . . . . . . . . . . . . . . . . . 44
2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3 Alcock-Paczynski forecasts using galaxy voids in future redshift surveys 59
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2 Fisher matrix method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.3 Fisher method applied to the AP observable from galaxy voids . . . . . . . 64
3.4 Forecasts for present and future surveys . . . . . . . . . . . . . . . . . . . . 69
vii
viii Contents
3.4.1 Summary of surveys . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.4.2 Calibration of the measurement error for each survey . . . . . . . . . 73
3.5 Comparison with Baryon Acoustic Oscillation forecasts . . . . . . . . . . . . 80
3.6 Marginalization of Redshift-Space Distortions . . . . . . . . . . . . . . . . . 87
3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4 GAMA Survey: Void-galaxy cross-correlations 95
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.2 Data Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.3 The void-galaxy cross-correlation . . . . . . . . . . . . . . . . . . . . . . . . 104
4.3.1 Void-Galaxy correlation function modelling . . . . . . . . . . . . . . 108
4.4 Error calibration and velocity profiles around voids . . . . . . . . . . . . . . 111
4.5 Parameter fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.6 Parameter fits varying model assumptions . . . . . . . . . . . . . . . . . . . 121
4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5 Summary and future work 129
5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Bibliography 143
List of Figures
1.1 Illustration of the effect of dark energy on the age of the Universe t0 and
its deceleration rate as a function of redshift q(z). Left panel: age of the
universe in units of H−10 as a function of the Ωm parameter. We show the
flat ΛCDM prediction (solid line) and the open model with ΩΛ = 0 (dashed
line). We display the threshold limit from the age of the oldest stellar
objects. Right panel: deceleration parameter as a function of redshift for
different values of the parameters (Ωm,ΩΛ). The presence of dark energy
introduces a transition from deceleration to acceleration, at a redshift of
about z ' 0.6 for a (0.3, 0.7) cosmological model. . . . . . . . . . . . . . . . 11
1.2 Matter power spectrum as a function of the wavenumber k in units of the
scale of matter-radiation equality keq, obtained by the code CAMB for a
fiducial set of ΛCDM cosmological parameters. We can see the charac-
teristic turnover at the scale keq, which modifies the power-law primordial
spectrum established after the inflationary phase. . . . . . . . . . . . . . . . 14
1.3 Left Panel: Initial measurements of the matter and cosmological constant
densities Ωm,ΩΛ from SNIa (Riess et al., 1998; Perlmutter et al., 1999)
revealing a significant preference for the non-zero ΩΛ an hence an acceler-
ated expansion. Credit Figure: Wang (2010). Right Panel: Complemen-
tary constraints in the same parameter space from different probes such as
SNIa (Blue, displaying the increase in constraining power in comparison to
the initial study), CMB (Yellow), BAO (Green). The combined constraint
is displayed in dark grey, showing a clear preference for the concordance
ΛCDM model. Credit Figure: Kowalski (2008). . . . . . . . . . . . . . . . . 16
ix
x List of Figures
2.1 Different phases of the ZOBOV void finder. Top left: two-dimensional slice
of a numerical simulation of galaxies with dimensions 40×40×5 (Mpc/h)3.Top right: Associated two-dimensional Voronoi tesselation for galaxy po-
sitions, color-coded with the local density estimation, the darker the less
dense. Bottom left: transition from Voronoi cells to zones, identifying the
cores of each zone with the crosses, points from which the growing of the
voids begin. Different colors imply different detected zones. Bottom right:
merging of zones into a void by the watershed argument, with color-code
based on the set of zones that merge into the parent zone, with core parti-
cle labelled with the number 1 on the upper left-hand zone. Credit figure:
Neyrinck (2008) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2 Comparison of different galaxy redshift survey specifications, using the num-
ber density of redshifts per square degree and the area in square degrees.
The order of magnitude of redshifts obtained by different redshift surveys is
also shown. The GAMA survey is focussed toward high density of sources on
the sky. Credit for figure: Ivan Baldry, http://www.astro.ljmu.ac.uk/
∼ikb/research/galaxy-redshift-surveys.html . . . . . . . . . . . . . . 34
2.3 Redshift distribution of the GAMA equatorial fields before the volume-
limited sample is constructed. The histogram is normalized to have unit
area under the curve. The sample variance in the fields can clearly be seen. 37
2.4 Extraction of a volume-limited subsample for the G09 region, for redshift
bins 0 < z < 0.25 and 0.25 < z < 0.5, with cuts in the redshift-luminosity
plane. The dashed line represents the apparent magnitude limit of the
survey 19.8. The darker points are selected for the volume-limited subsample. 38
2.5 Distribution of void sizes in the GAMA data and mock catalogues. The plot
on the left shows the distribution of void radius Rv for the first redshift bin
0 < z < 0.25, with the regions G09, G12 and G15 in red, green and blue
respectively. The grey histrograms correspond to the distribution of 112
mocks. The plot on the right, is the same exposition but for the second
redshift bin 0.25 < z < 0.5 and considering 24 mock catalogues. In both
cases the mock halo catalogues generate the same distribution of sizes Rv
as the data, with void sizes in the interval (5, 40) Mpc/h for the first bin
and (10, 70) Mpc/h for the second bin. . . . . . . . . . . . . . . . . . . . . . 46
List of Figures xi
2.6 Stacked 2D void density profiles for the GAMA regions for the redshift bin
0 < z < 0.25, using stacking method 1. By fitting the functional form of
equation 2.17 to this data, we can measure the stretch factor of each set of
stacked voids. Measurements are shown for the 3 regions and the average. . 47
2.7 Demonstration of the isotropy of voids in the GAMA mocks for the first
redshift bin in real space. In stacking method 1, the density profiles of each
void are simply averaged, and in stacking method 2, the galaxy numbers
and volume elements are combined in order to up-weight the contribution
of voids containing more galaxies. We show the results of the stretch fit for
each mock separately, and also the combined histogram together with the
mean µ and standard deviation σ. We observe that using stacking method
2 reduces the scatter by almost a factor of 2, without introducing significant
bias. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.8 Comparison of the Poisson error computed from the mocks and the error
determined from the standard deviation across the mocks, for the 1D void
profile as a function of r/Rv, for the first redshift bin. The grey lines
represent each individual mock. The dark line represents the mean value of
the grey lines. We find that the Poisson error estimate is about a factor of
2 smaller than the standard deviation from the mocks in the fitting range
r/Rv < 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.9 Comparison of the Poisson error computed from the mocks and the Poisson
error computed from the GAMA data, for the 3 survey regions. The top
and bottom row shows results from the mocks in real-space and redshift-
space, respectively. The grey lines represent each individual mock and the
dark line shows the mock mean. The mocks are a good representation of
the data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.10 Reduced covariance matrix of the one-dimensional void density radial pro-
file, with Cij = Cov(n/nb(ri/Rv), n/nb(rj/Rv)) and the bins extend in the
range 0 < r/Rv < 1.5. We find that off-diagonal contributions are generally
small for r/Rv < 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.11 The standard deviation of the number density across 112 survey conic mocks
for the first redshift bin used in our analysis. The results for the second
redshift bin are similar. The error decreases with increasing r⊥/Rv due to
the cylindrical geometry of the bin. . . . . . . . . . . . . . . . . . . . . . . . 52
xii List of Figures
2.12 Stretch parameter fits to the GAMA mocks. The 4 columns show results
from different fitting ranges fr such that
√(r⊥Rv
)2+(r‖Rv
)2< fr. We in-
vestigate the effect of the stacking method on the stretch measurement,
comparing method 1 (red histogram) with method 2 (blue histogram). The
rows investigate the influence of the survey boundary, showing results stack-
ing up to the boundaries in real space (top row), in redshift space (bottom
row) and outside the survey boundary (middle row). By comparing the top
and bottom rows we see that the introduction of peculiar velocities imprints
an anisotropy in the stretch parameter of approximately 5-10%. . . . . . . . 55
2.13 Representation of our measurements of the stretch parameter from stacked
voids for the 3 equatorial regions of the GAMA survey in 2 redshift bins.
We also show the combined-region results, including the correction for pe-
culiar velocities. We overplot the predictions of 3 flat ΛCDM cosmological
models (Ωm,ΩΛ). We show results for different fitting ranges defined by the
maximum value of r/Rv. Our results favour Universes with a low matter
density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.14 Fits for the matter density Ωm in a flat ΛCDM Universe from the Alcock-
Paczynski measurements. We propagate the measurement of d = F/Ffid in
our two redshift bins into a probability distribution for the Ωm parameter.
Our results (green line) favour a low value of the matter density. As a
comparison, we also include a test case (red line) in which we fit to a
fake dataset generated from the fiducial cosmology with error of magnitude
σd = 0.005. We also display the constraint on Ωm obtained by fitting the
measurements of the AP parameter in 4 redshift bins from the WiggleZ
Dark Energy Survey reported by Blake et al. (2011). . . . . . . . . . . . . . 57
3.1 Mean stretch factor in units of the fiducial value d = F/Ffid, and standard
deviation across 100 mocks, for different values of the number density of
the halo catalog, and for different redshift snapshots (from bottom to top)
in real-space (left column) and redshift-space (right column). In real-space
we see consistency with the value of stretch 1, corresponding to an isotropic
object, while in redshift-space this isotropy is broken due to peculiar veloc-
ities. The error increases at smaller densities due to the lower number of
voids in the stack in comparison to the high-density samples. . . . . . . . . 67
List of Figures xiii
3.2 Mean number of voids detected in the COLA halo catalogs as a function
of the halo number density at redshift 0 (blue points). We see that for the
ZOBOV void finder, for the value of number density n̄ = 10−4(h/Mpc)3,
the number of voids detected is close to 200. When we increase the number
density by 2 orders of magnitude, the number of voids detected grows up to
104. This highlights the advantage of a high-density sample of tracers. As a
comparison, we include the same measurement for the GiggleZ halo catalog
used in Chapter 2 (red points), at the 3 different halo number density values
n̄ = (0.25, 2.0, 5.0) × 10−3(h/Mpc)3. The offset between the 2 lines is thevolume ratio of the N-body boxes V (GiggleZ)/V (COLA) = (1000/600)3. . 68
3.3 Error in the stretch parameter F/Ffid as a function of the number density
of the halo catalogs for different redshift snapshots. The error is smaller at
higher densities, due to the higher number of voids to stack. We can see a
similar error scaling in real-space (left) and redshift-space (right) and the
dependence on redshift is small. We determined an empirical scaling for the
error in the stretch parameter as a function of the number density, shown
as the dashed black line and given by σ(d) = 0.0016n̄−1/2. . . . . . . . . . . 69
3.4 Error in the distortion parameter F/Ffid for the 4 redshift snapshots of
the COLA halo catalogues, for different values of the number density. The
error shows a behaviour independent of redshift, both in real and redshift
space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.5 Number density, in units of (h/Mpc)3, as a function of redshift for different
galaxy surveys. This plot was produced by taking the signal-to-noise factor
nPg in redshift bins for different galaxy surveys and dividing that by the
matter power spectrum at the linear scale k = 0.2h/Mpc, scaling by a
bias factor b(z) characteristic of each survey. The signal-to-noise factor was
taken from Font-Ribera et al. (2014). . . . . . . . . . . . . . . . . . . . . . . 76
3.6 Error in the AP distortion parameter d = F/Ffid as a function of redshift for
different galaxy surveys. This calibration includes the scaling by the volume
of each redshift bin and the number density from the N -body simulations.
We see that the surveys covering the biggest cosmological volume such as
DESI, Euclid and WFIRST produce the tighest constraints, with better
than 2% accuracy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
xiv List of Figures
3.7 Parameter forecasts for current and future surveys for the model Ωm − w.We show the 68% (dotted) and 95% (solid) probability contours in the
Ωm − w plane. We see that current surveys such as BOSS and eBOSScan estimate the cosmological parameters, but future surveys such as DESI
and Euclid will increase the accuracy by almost a factor of 10. The Figure
of Merit (FoM) is calculated as the square root of the determinant of the
Fisher matrix for the parameters (Ωm, w). . . . . . . . . . . . . . . . . . . 78
3.8 Parameter forecasts for current and future surveys for the model w0 − wa.We show the 68% (dotted) and 95% (solid) probability contours in the
w0 − wa plane, marginalising over the parameter Ωm. We see that surveyssuch as BOSS and eBOSS are not powerful enough to constrain the CPL
parameters, while a survey such as DESI has a superior constraining power
defined by the FoM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.9 The forecast accuracy in distance dA(z) and expansion rate H(z) measure-
ments using the BAO method for different surveys. We show the fractional
accuracy achievable for these cosmological observables in the redshift in-
terval listed in table 3.1. While BOSS produces 2-4% accuracy in distance
and expansion rate measurements in ∆z = 0.1 redshift bins, this constraint
will improve to sub-percent accuracy with future surveys such as DESI and
Euclid. Data taken from Font-Ribera et al. (2014). . . . . . . . . . . . . . . 82
3.10 Forecast 68% confidence error ellipses for cosmological parameters using
the BAO method for different redshift surveys. On the left we plot the
(Ωm, w) model and on the right we show the (w0, wa) model. These plots
were produced using the data from Font-Ribera et al. (2014) for the forecast
errors in dA(z) and H(z) for different survey specifications, and evaluating
the Fisher matrix to calculate the covariance ellipses. . . . . . . . . . . . . 83
3.11 Parameter constraints for the (Ωm, w) model from multiple redshift surveys
for voids AP (red), BAO(green) and combined(blue) results. For clarity
we plot only the ellipses enclosing 68% of probability in the 2D parameter
space. We see that with no other prior included, the BAO method has
a better performance to constrain cosmological parameters in comparison
to the void method for all surveys considered in the analysis. When both
probes are combined we achieve an improvement in accuracy for all the
surveys, with DESI and Euclid the best-performing surveys. . . . . . . . . . 84
List of Figures xv
3.12 Parameter constraints for the (w0, wa) model from multiple redshift surveys
for voids AP (red), BAO(green) and combined(blue). For clarity we plot
only ellipses enclosing 68% of probability in the 2D parameter space. Similar
to the (Ωm, w) model, without including priors, the BAO method has more
constraining power than the void method for all the surveys analyzed in this
work. When we combine both probes, DESI and Euclid are the surveys with
the biggest gain in constraining power for this parameterization of dark energy. 85
3.13 Redshift-space correction factor as a function of the number density. We
show the mean value of the quantity dr/dz for 100 COLA mocks, including
the error in the mean. We see that peculiar velocities are responsible for
introducing an anisotropy of about 5-10%. . . . . . . . . . . . . . . . . . . . 89
3.14 Redshift-space correction factor as a function of redshift for different values
of the number density. We show mean values for the quantity dr/dz includ-
ing the error in the mean. The correction is at the level of 5-10% and shows
a slow evolution with redshift. . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.15 Fisher matrix forecasts for the parameter space (w0, wa) for different galaxy
redshift surveys. We show 68% probability contours for the pure AP geo-
metric constraint (red line), the marginalization of a single-parameter RSD
correction (green line) and the multi-parameter case (blue line) all assuming
σ(Pv) = 0.05. We see that surveys like DESI and Euclid can take advantage
of the pure geometric AP method, but the incorporation of the nuisance
RSD parameters generates a significant loss in constraining power. . . . . . 91
3.16 The dependence of the Figure of Merit on the prior in the nuisance pa-
rameter σ(Pv) from redshift-space distortions for the non-evolving (upper
panel) and evolving (bottom panel) case with redshift, for the forecasts in
the parameter space (w0, wa). We have normalized the output relative to
the DESI constraint. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
xvi List of Figures
4.1 Comparison of the halo and void statistics in the data and the mock halo
catalogue with the same survey geometry and selection function. Top left:
we show the histogram of the number of halos in each survey mock real-
ization (gray) and include the number of galaxies for the 3 regions of the
GAMA data for the redshift bin 0 < z < 0.25, showing consistency. Top
right: histogram of the number of voids in each survey mock realization
(gray) and including the number of voids present in the 3 regions of the
GAMA data for the bin 0 < z < 0.25. Bottom row: same as the upper row
but for the second redshift bin 0.25 < z < 0.5. . . . . . . . . . . . . . . . . . 101
4.2 Comparison of the size distribution of voids detected in the GAMA survey
data and in the halo mock realizations. The left-hand plot is for the first
redshift bin 0 < z < 0.25, and the right-hand plot is for the second redshift
bin 0.25 < z < 0.50. We see that after applying the void-finding process
to each mock halo catalogue, the size distribution is similar to that found
in the data, identifying voids with sizes in the interval 5 − 40 Mpc/h forthe first redshift bin and 10 − 70 Mpc/h for the second redshift bin. Themean particle separation is n̄−1/3 ' 5.0 Mpc/h for the first redshift bin andn̄−1/3 ' 13.0 Mpc/h for the second. We see that the void-finder is able toresolve voids with sizes bigger than this scale, as expected. . . . . . . . . . . 102
4.3 In the left-hand plot, we show the halo bias parameter using the method
described in the text as a function of the number density of the halos, at
redshift z = 0. The error in the bias parameter is σb = 0.01 and the error
bars are scaled by a factor of two for visibility. In the right-hand plot, we
show the value of β = f/b for the same halo populations, using the fiducial
growth rate of the simulation at z = 0. . . . . . . . . . . . . . . . . . . . . . 104
4.4 Comparison of different void-galaxy cross-correlation function estimators in
bins of size 3 Mpc/h. The points have an artificial offset of 1 Mpc/h for
clarity. The plot on the left shows that the 3 estimators produce consistent
results, hence contain no relative bias. On the right, we can see that in the
inner parts of the voids, the Landy-Szalay estimator has a bigger variance
that the Peebles-Hauser and Davis-Peebles estimators, while on large scales
the situation is the opposite. Given the small-scale difference, we choose
the Davis-Peebles estimator for our analysis. . . . . . . . . . . . . . . . . . . 105
List of Figures xvii
4.5 Generation of the galaxy and void random catalogue. We plot the number
density as a function of redshift for the GAMA data before extracting the
volume-limited sample in each of the 3 regions, showing the initial high
density of the galaxy sample (red solid line). To alleviate the gradient in
number density we extracted volume-limited subsamples for the 2 redshift
bins of the GAMA data for each of the regions (blue solid lines), produc-
ing a constant selection function. We generated our random catalogues
by modelling this constant selection function with the same geometry as
the GAMA regions (dotted black line). Although we do not require a ran-
dom catalog for voids due to our choice of estimator, we have included the
number density distribution of voids in the 2 redshift bins for completeness. 106
4.6 Void-Galaxy cross-correlation function measurements in 2 dimensions from
the GAMA survey data and mock catalogs. Top row: measurement from
the first (left) and second (right) redshift bin of the GAMA data. Bottom
row: mean value from the mocks with the same number density and survey
geometry as the first (left) and second (right) redshift bin. . . . . . . . . . . 110
4.7 The Void-Galaxy cross-correlation function from the GAMA data (3 re-
gions) for the void sample from the first redshift bin (upper panel) and
second redshift bin (lower panel), in comparison to the mean and standard
deviation of the mocks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.8 The 1D void-galaxy cross-correlation function for different void sizes. Us-
ing a sample of voids from the halo catalog with number density n̄ =
0.01 (h/Mpc)3 we show that small voids have a compensation zone while
large voids do not present a compensation zone. The internal histogram is
a reference to show how many voids are included in each size bin. . . . . . . 114
xviii List of Figures
4.9 Velocity profile around a sample of voids detected in 100 halo catalogs from
the COLA simulation with the number density 5× 10−4(h/Mpc)3. To pro-duce this plot we have measured the velocity and velocity dispersion profile
in spherical shells for each void and stacked them to extract a mean value.
We measure this for each mock catalog (100 of them showed in grey lines).
We show the mean value of the velocity and velocity dispersion profiles with
scatter from the mock catalogs. The black solid line represents the model for
the velocity profile given the measurement of the void-halo correlation func-
tion (eq. 4.11) and the input value β = f/b given the fiducial cosmology,
showing agreement in the quasi-linear and linear regime. For complete-
ness we also show the empirical functional forms for the velocity dispersion
models listed in eq. 4.12 with constant amplitude σv = 200km/sec: error
function model (dashed line) and Lorentzian model (dotted line). . . . . . 117
4.10 Parameter constraints from the GAMA data and the mock mean. We show
the 68%, 95% and 99% confidence regions for the MCMC for the first (left)
and second (right) redshift bin. The measurement from the GAMA data is
in red and the fit from the mean value of the mocks is displayed in blue.
There exists some degeneracy between the growth rate parameter β and
the amplitude of the stochastic velocity σv. The vertical lines represent the
value of β from the mocks calculated using the fiducial cosmology. The
best-fitting χ2 values are shown: they are low for the case of the mock
mean, because we use the error in the data for the purposes of this test. . . 120
4.11 Parameter fits to the GAMA data (black points) and mock mean (red
points) as a function of the minimum fitted scale Rc. In the left column we
show results from the first redshift bin and in the right column we show the
second bin. The error bars in the parameter measurements correspond to
the standard deviation over the mocks. . . . . . . . . . . . . . . . . . . . . . 122
4.12 Parameter constraints from the GAMA void-galaxy cross-correlation func-
tion, showing the effect of the assumed velocity dispersion model. We show
the 68%, 95% and 99% confidence regions for the MCMC for the first (left)
and second (right) redshift bin. The measurement assuming a constant ve-
locity dispersion is in green, the error function model in red and the Hamaus
et al. parameterization in blue. The vertical lines represent the value of β
from the mocks calculated using the fiducial cosmology. . . . . . . . . . . . 123
List of Figures xix
4.13 Parameter constraints for different ranges of void size, where we show 68%
and 95% confidence regions for the MCMC. The left column represents the
first redshift bin, with the data in the upper row and the mock in the lower
row. The right column shows results for the second redshift bin, with the
data at the top and the mock at the bottom. . . . . . . . . . . . . . . . . . 124
4.14 Measurements of β from the GAMA void sample. Fitting results are shown
for the different cases considered in this section, varying velocity dispersion
model, in comparison to those obtained by other surveys. . . . . . . . . . . 125
4.15 Measurements of the growth rate f from the GAMA void sample. We
converted the β measurement into a growth rate measurement using f = bβ,
using the bias values calibrated from the simulations. Results are shown
for the different cases considered in this section, varying velocity dispersion
model, in comparison to those obtained by other surveys. The solid line
represents the ΛCDM growth rate parameterization Ωm(z)γ with γ = 0.55. 126
List of Tables
1.1 Solutions to the evolution of density and scale factor for different single-
component models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1 Summary of the 3 GAMA fields used in this thesis. . . . . . . . . . . . . . . 35
2.2 Volume-limited subsamples extracted from the GAMA equatorial regions.
This table lists the absolute magnitude threshold, redshift range, number
of galaxies Ngal and the resulting number density n̄. . . . . . . . . . . . . . 36
2.3 Compilation of the parameter fits for the 2 redshift bins of the GAMA
data for the 4 fitting range ranges shown in Figure 2.11. Measurements are
shown for the parameters defining the void profile in Equation 2.17. We
also show the best-fitting χ2 value and number of degrees of freedom for
each case (where the latter varies with the number of bins included by the
fitting range). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.1 Survey specifications used in this work. Data was extracted from Font-
Ribera et al. (2014). LRG: Luminous Red Galaxy, ELG: Emission Line
Galaxy. The product of the bias factor b(z) and the growth factor D(z) is
characteristic of the galaxy tracer used in each survey. . . . . . . . . . . . 71
3.2 Compilation of errors in the cosmological parameters forecast for each sur-
vey from the Fisher matrix analysis, including the FoM for the two dark
energy models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.3 Compilation of results for the Fisher matrix analysis of the model (Ωm, w).
For the redshift surveys considered in this work, we show the error in the pa-
rameters of the model and the Figure of Merit for the void method (columns
2-4), BAO method (columns 5-7) and the combination (columns 8-10). . . 83
3.4 Compilation of results for the Fisher matrix analysis for the model (w0, wa).
For each redshift survey considered in this work, we present the error in
the parameters of the model and the Figure of Merit for the void method
(columns 2-5), BAO method (columns 6-9) and the combination (columns
10-13). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
xxi
xxii List of Tables
4.1 Summary of the halo mock samples used in this chapter for analysis. We
quote the number density n̄ of the four samples considered, the mean halo
separation given by n̄−1/3, the halo mass thresholds used to establish the
catalogue, and bias parameter determined by the power spectrum fits de-
scribed in the text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.2 Summary of the galaxy sample. We show the number of galaxies Ng in each
region and redshift bin before and after the extraction of the volume-limited
subsample, for which we quote the corresponding number density n̄. We
also show the mean separation n̄−1/3 between galaxies. . . . . . . . . . . . . 100
4.3 Summary of the void sample. We show the number of voids Nv in each
region and redshift bin with the corresponding number density n̄v of the
void sample and mean separation n̄−1/3v between voids. . . . . . . . . . . . . 100
4.4 Estimators for the auto-correlation function from counts of data D and
randoms R. For clarity, we absorb the normalization factors into the pair
counts. We tested 3 of these estimators for the void-galaxy cross-correlation.107
4.5 Compilation of the parameter fits for the 2 redshift bins of the GAMA data
and the mock test. Fitting scales represent the interval for the quantity
(σ2i + π2j )
1/2. We compare the error in the parameter β calculated with the
MCMC procedure, with the scatter of the measurement of this parameter
over the mocks. For the velocity dispersion amplitude, we quote the best
fit value from the MCMC with the error derived from the scatter over the
mocks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
1Introduction
1.1 The Cosmological Model
In any physical science, the observations from experiments and theory advance together
to produce understanding. The establishment of the cosmological model is an example of
this process, where sometimes progress was led by theory, and sometimes by observation
(see Longair (2013) for a historical perspective of modern cosmology).
The first modern cosmological models, in the absence of observational data, were based
on invoking mathematical symmetries to describe the universe as a whole. These sym-
metries establish the cosmological principle: when averaged on sufficiently large scales,
beyond individual galaxies or clusters of galaxies, the universe is homogeneous (its prop-
erties are invariant under translations in space) and isotropic (invariant under rotations).
These assumptions have since been supported by observational tests of large-scale
homogeneity and isotropy. The most powerful evidence is the observed high degree of
isotropy of the Cosmic Microwave Background (CMB) radiation (Planck Collaboration
et al., 2016), the photon thermodynamic remnant of the primordial plasma of the early
universe, once a dipole anisotropy due to our local motion is subtracted.
The cosmological principle was the cornerstone in the development of relativistic cos-
mology, although prior to the discovery of the CMB lacked observational support. Indeed,
as relativistic cosmology was first being developed during the decade 1920-1930, including
the discovery of the expansion of the universe by Hubble, there was a debate about the
existence of extragalactic objects, the famous Shapley-Curtis debate. The other important
assumption in the model is the Copernican Principle: the equivalence of all cosmological
observers.
In the history of the development of the cosmological model, two main relativistic
cosmologies have been proposed. The steady state model postulated a perfect cosmological
1
2 Chapter 1. Introduction
principle (Hoyle & Narlikar, 1964), in which the properties of the cosmos were also invariant
with time. This was succeeded by the Big Bang model, in which the Universe expands
from an initial hot, dense state. Observational evidence for the hot Big Bang included:
• Measurement of a redshift of the spectra of galaxies, implying a universal recessionor cosmic expansion.
• Existence of the Cosmic Microwave Background Radiation and its measurement asa black body with a thermodynamic temperature of 2.7 K, for a review see Peebles
et al. (2009).
• Big Bang Nucleosynthesis: a physical model for the generation of light elements suchas hydrogen, helium and deuterium and their primordial abundances, for a review
see Fields & Sarkar (2006).
This evidence, along with many other observations, has delivered support to the homo-
geneous and isotropic Big Bang model. There have been efforts to develop models of
non-homogeneous cosmologies (see Clarkson & Maartens, 2010 for an example) but their
complexity results in computational difficulties to calculate key quantities related to the
evolution of cosmological perturbations, such as CMB and matter power spectra.
The Big Bang model requires additional ingredients in the early and late universe to
solve two cosmological problems. First, due to the existence of cosmological horizons,
there are regions of the cosmos that were in causal contact in the past but today appear
not to be in causal contact anymore. Considering the size and age of the Universe and
the high degree of isotropy of the CMB, a physical communication between these regions
is required. The proposed solution to this problem is a phase of exponential inflation in
the early universe (Guth, 1981; Liddle & Lyth, 2000), such that at early times regions
were in causal contact, and after the exponential inflation they were separated. The
inflationary phase establishes a model for the initial conditions of the Universe, in which
structure emerges from the growth of fundamental quantum seeds to eventually become,
through gravitational amplification, the galaxies that we observe today. Secondly, as will
be discussed further in Section 1.2, the Universe appears to be entering a new phase of
accelerating expansion due to some form of dark energy.
The gravitational interaction is the strongest force of nature on large scales, implying
that the theory of gravity is the most important component of the cosmological model.
This mathematical description is provided by General Relativity; for a general review of
these details see Carroll (2003), and Hobson et al. (2006). The starting point is the line
1.1. The Cosmological Model 3
element ds2 that measures the distance between any two events in space-time, given by
ds2 = gµνdxµdxν , (1.1)
where gµν represents the metric tensor and xµ are the space-time coordinates. The metric
encodes the geometry of space-time, including curvature in response to the presence of
mass-energy. The equations describing the manner in which the matter reacts to the
curvature of space-time are the Einstein equations (Carroll, 2003), in which we include
the term for the cosmological constant Λ, discussed in more detail below:
Gµν := Rµν −1
2Rgµν = 8πGTµν − Λgµν , (1.2)
where the left-hand side of this equation represents the Einstein tensor Gµν , calculated
purely from the metric, and the first term on the right-hand side contains the energy-
momentum tensor Tµν , the object representing mass-energy in General Relativity. We
work in units where the speed of light c = 1. In the cosmological model we hypothesize
that the matter-energy is a perfect fluid, parametrized by the value of the energy density
ρ and pressure p and given by (Hobson et al., 2006),
Tµν = (ρ+ p)UµUν + pgµν , (1.3)
where Uµ is the 4-velocity of the fluid element.
With this basis, we impose the additional assumptions of homogeneity and isotropy,
creating a co-moving coordinate system expanding with the Universe as a function of
proper time t. It can be shown that the general form of space-time metric that satisfies
the properties of homogeneity and isotropy is the Friedmann-Robertson-Walker (FRW)
metric (Hobson et al., 2006),
ds2 = −c2dt2 + a2(t)[
dr2
1− κr2+ r2dΩ2
], (1.4)
where (r, θ, φ) represent comoving spherical polar coordinates with dΩ2 = dθ2 + sin2 θdφ2
at proper time t. The quantity a(t) is the scale factor, which is determined from the field
equations. The curvature parameter κ is a constant whose sign determines the global
geometry of the Universe, being positive, zero or negative if the geometry of the Universe
is closed, flat or open, respectively.
Applying the Einstein equations to the FRW metric, it can be shown that the dynam-
4 Chapter 1. Introduction
Domination w ρ(a)/ρ0 a(t)
Matter 0 a−3(tt0
)2/3Radiation 1/3 a−4
(tt0
)1/2Curvature -1/3 a−2
(tt0
)Λ -1 constant exp
(√Λ3 t
)
Table 1.1 Solutions to the evolution of density and scale factor for different single-component models.
ical equations for the scale factor are the Friedmann equation for the expansion rate
ȧ2 =8πG
3ρa2 − κ+ Λ
3a2, (1.5)
where dots indicate a derivative with respect to cosmic time t, and the term for the energy
density ρ includes the contribution of all sectors such as matter and radiation, and the
acceleration equation
ä = −4πG3
(ρ+ 3p) a+Λ
3a. (1.6)
From the conservation of mass-energy extracted from the null divergence of the energy-
momentum tensor, it can be shown that we recover the density equation
ρ̇+ 3ȧ
a(ρ+ p) = 0. (1.7)
Out of the 3 dynamical equations, only two are independent. The closure of the system
of variables is achieved by the equation of state, given by the ratio
w =p
ρ. (1.8)
For a single-component model with constant equation of state, in the hypothesis of a flat
Universe for the cases of matter, radiation and cosmological constant, the solutions for the
density as a function of scale factor, and scale factor as a function of time, are displayed in
table 1.1. The Universe is well-described by a multi-component model of these contents.
Tests of the cosmological model depend on the definition of a cosmological distance.
1.1. The Cosmological Model 5
From the FRW metric, we can define a proper distance as an integral of the radial compo-
nent of the metric, taking into account the curvature. Distances are conventionally defined
using comoving coordinates. That is, if x is the comoving vector position of a galaxy, the
physical vector position r is a scaled version in time, by the scale factor a(t) given by
r = a(t)x. (1.9)
The apparent radial velocity of this galaxy with respect to the origin (neglecting peculiar
velocities which change the co-moving coordinate x) is given by
v = ȧ(t)x =ȧ
ar. (1.10)
Equation 1.10, which describes a velocity proportional to distance, is called the Hubble
law and the quantity
H(t) =ȧ
a, (1.11)
is called the Hubble parameter. Its present value H(t0) = H0 is given by (Planck Collab-
oration et al., 2016)
H0 = 67.8± 0.9 km sec−1 Mpc−1. (1.12)
Conventionally, quantities are expressed as a function of unknown H0 using the dimen-
sionless parameter h = H0/(100km/sec/Mpc).
In observational cosmology, the wavelength of light increases with the expansion of the
Universe as described by the scale factor a. We define the redshift z of a source emitting
light, e.g. a galaxy, by the ratio of wavelengths
z =λo − λeλe
, (1.13)
where λo is the observed wavelength and λe is the emitted wavelength in the rest frame
of the source. The light travels on a geodesic trajectory (satisfying ds2 = 0) for the line
element of the metric. Cosmological expansion induces a time dilation factor between the
times of emission te and reception to of the signal, which is inversely proportional to the
scale factor. If we consider that these time intervals are used to measure frequencies and
hence the wavelength of light, the definition of redshift z in terms of the scale factor takes
the form
z =a0a− 1, (1.14)
where a0 is the value of the scale factor at the present epoch, which is conventionally set
6 Chapter 1. Introduction
to unity. The measurement of the density of each component, e.g. matter or radiation,
in units of the critical density ρc = 3H20/8πG needed to produce a precisely flat Universe
with zero curvature, is the density parameter Ω defined as
Ωi =ρiρc. (1.15)
In terms of the density parameters for each component and the redshift, the observational
Friedmann equation, assuming a constant equation of state for dark energy, can be cast
in the form
H(z) = H0
[Ωr(1 + z)
4 + Ωm(1 + z)3 + Ωκ(1 + z)
2 + Ωw(1 + z)3(1+w)
]1/2, (1.16)
where r,m,w stands for radiation, matter and dark energy, respectively. The contribution
of the curvature to the density parameter is given by Ωκ = κ/H20 .
The acceleration of the expansion of the Universe is described by the deceleration
parameter q given by the dimensionless equation
q(z) = − äa
1
H(z)2. (1.17)
Cosmological distance measurements must be related to observed quantities (for a
review of cosmological distances see Hogg (1999)). Considering the light we receive from
an object, we can define the luminosity distance. If L is the luminosity of a source locatedat a radial comoving distance r, and F is the flux, then the luminosity distance is givenby
dL =(L
4πF
)1/2. (1.18)
The area of a spherical surface enclosing the source is 4πr2 and the light experiences red-
shift by the expansion. This allows us to conclude that F ∝ (a/r)2 and so the luminositydistance is related to the radial comoving distance by dL = (1 + z)r.
We can also define angular-diameter distance. If we take Lp as the proper physical
diameter of a light source subtending an angle θ on the sky, then we define the angular-
diameter distance dA as dAθ = Lp. The 3 distances are related to each other as follows:
dL(z) = (1 + z)r(z) = (1 + z)2dA(z). (1.19)
In the limit of low redshifts, these definitions converge to the same value, cz/H0. The
angular-diameter distance does not diverge at high redshift because of the term (1 + z)−2.
1.2. The Existence of Dark Energy 7
The radial comoving distance is determined by the light path along the null geodesic and
is given by (Wang, 2010)
r(z) =c
H0
1√|Ωk|
Sκ
(√|Ωk|
∫ z0
dz′
E(z′)
), (1.20)
where the function Sκ(x) = {sin(x), x, sinh(x)} if the curvature parameter κ is positive,zero or negative, respectively.
1.2 The Existence of Dark Energy
The observational evidence that the Universe is filled with dark energy is the discovery of
its accelerated expansion. This discovery was confirmed by the High-z Supernova Search
Team (Riess et al., 1998) and the Supernova Cosmology Project (Perlmutter et al., 1999)
by observing type Ia supernovae (SNIa) as standard candles. Riess et al. (1998) used 16
supernovae in the redshift interval 0.16 < z < 0.62 and 34 supernovae at low redshifts to
conclude that the expansion of the universe has entered a current phase of accelerating
expansion, reporting a deceleration parameter q0 < 0 at 2.8σ significance and that the dark
energy density parameter ΩΛ > 0 at the 3σ significance level. Additionally, Perlmutter
et al. (1999) analyzed 42 SNIa in the redshift interval 0.18 < z < 0.83, complemented
by a low-redshift sample (z < 0.1) from the Calán-Tololo Supernova Survey to report a
measurement of the matter density parameter Ωm = 0.28 with a statistical precision of
about 30%. On the assumption that the spatial geometry of the Universe is flat, this
implies that the present-day energy density is dominated by ΩΛ. More recent studies,
based on much larger supernovae samples and a range of other cosmological probes, have
confirmed these results and suggested that dark energy comprises around 70% of the
energy density of the present Universe (Planck Collaboration et al., 2016). Dark energy
with an equation of state w = −1 is also known as a cosmological constant, equivalent tothe final term introduced in equation 1.2.
The accelerating expansion of the Universe currently lacks a physical explanation,
and this is labelled as the dark energy problem. Two possible explanations are that this
phenomenon is generated by the presence of a new component in the matter-energy of the
Universe, which exerts an effective negative pressure, or by a modification to the theory
of gravitational interactions described by Einstein gravity, on cosmological scales. The
solution to this problem represents one of the biggest open issues in research in physics at
the present, with profound implications for fundamental theory.
In the historical context, the cosmological constant was first introduced by Einstein
8 Chapter 1. Introduction
as a mechanism to achieve a static model of the universe, by introducing this parameter
in an ad hoc manner into the field equations. Einstein abandoned this approach after
observations revealed the Universe was expanding, not static. However, the cosmological
constant provides a phenomenological mechanism for producing accelerating expansion
and its inclusion is strongly favoured by modern observations.
We now outline how observations of type Ia supernovae can be used to establish ac-
celerating expansion. The model for progenitors of SNIa reflects two main alternatives: a
single degenerate, with a white dwarf accreting material from the companion star; and a
double degenerate, with a binary white dwarf system generating a merger. (Hillman et al.,
2015). In any scenario the mass of the final system crosses the Chandrasekhar limit, close
to 1.4M�, where no more pressure from the degenerate electron gas is able counter the
gravitational force, generating a collapse with a subsequent increase in core temperature,
igniting a nuclear reaction and the release of energy in the form of a violent explosion.
This leads to a sufficiently reproducible process in which the peak in brightness is
almost constant, and the scatter in this relation may be further reduced through calibration
based on other parameters such as the light-curve stretch. Hence these objects may be
used as standard candles to calibrate the distance-redshift relation, given that the flux
from the object scales with distance according to F ∝ d−2. In terms of the magnitudescale, the distance modulus µ is defined by the equation
µ = m−M = 5 log10[dL(z)
Mpc/h
]− 5 log10 h+ 25, (1.21)
where m represents the apparent magnitude of the object, M the absolute magnitude,
dL(z) is the luminosity-distance in Mpc/h as a function of redshift and h is the Hubble
parameter in units of 100 km/sec/Mpc. The observed SN magnitudes can be linked to the
distances predicted by a theoretical model, and hence constrain cosmological parameters
of the model, such as the deceleration parameter q0. Figure 1.1 displays the behaviour of
the deceleration parameter as a function of redshift for different cosmological models.
Since the initial confirmation of the accelerating expansion, the size of SNIa samples
has increased: Riess et al. (2004) analyzed the Gold data set of 157 SNIa and Kowalski
et al. (2008) studied the Union data set of 307 SNIa. Betoule et al. (2014) presented the
JLA sample, comprising 740 spectroscopic redshifts of SNIa, including a calibration study
of the light curves.
The anisotropies in the temperature field of the CMB can also be used to confirm the
existence of dark energy, via several effects. First, the locations of the acoustic peaks in the
angular power spectrum of temperature fluctuations are changed by the presence of dark
1.2. The Existence of Dark Energy 9
energy, which modifies the distance to the last-scattering surface. Second, evidence is pro-
vided by the Integrated Sachs-Wolfe (ISW) effect, which refers to the correlation between
CMB anisotropies and low-redshift large-scale structure, observed at low multipoles. The
origin of the effect is the time variation of the graviational potentials when calculating
the scalar contribution to the temperature anisotropies. In the matter-dominated era, the
potentials are approximately constant in time; dark energy introduces time variation of
the potentials and a Sachs-Wolfe contribution to the anisotropy. Third, CMB lensing has
recently provided further evidence of dark energy by tracing low-redshift structure and
geometry. For a textbook exposition of CMB anisotropies see Dodelson (2003) and for a
recent review of the main aspects of CMB physics see Wands et al. (2016).
Even stronger evidence of dark energy is obtained by combining CMB and large-scale
structure measurements. As we discussed in the next section, the clustering of galaxies
and growth of structure is consistent with a low matter density, Ωm ' 0.3. However,measurements of the anisotropies of the CMB, in particular the location of the first acoustic
peak, confirm the cosmic geometry corresponding to a flat universe (Hinshaw et al., 2013;
Planck Collaboration et al., 2016), implying that the total density of the universe is close
to the critical value. This requires the existence of dark energy to complement the low
matter density.
A final argument we mention is the comparison of measurements of the Hubble pa-
rameter, which maybe used to estimate the age of the universe, with age measurements
from old stellar objects such as globular clusters (Chaboyer et al., 1996). Without dark
energy, the perverse result is obtained that the Universe is younger than its contents, as
illustrated by Figure 1.1.
The simplest explanation of the presence of dark energy is a cosmological constant.
Physically, this may be associated with a quantum vacuum energy, although the theoretical
prediction of its value is discrepant from observations by 120 orders of magnitude, which is
known as the cosmological constant problem (Weinberg, 1989). Another solution to alter
the matter-energy sector of the Universe is via a scalar field ϕ or quintessence model (Ratra
& Peebles, 1988). Extensions of this approach include K-essence (Armendáriz-Picón et al.,
1999; Chiba et al., 2000), the tachyon field (Bagla et al., 2003), the dilaton field (Lu et al.,
2004), the Chaplyin gas (Kamenshchik et al., 2001) and even considering the dark energy
phenomenology as manifesting from extra dimensions (DGP model, Dvali et al. (2000)).
See Copeland et al. (2006) and references therein for a review of theoretical modelling
of the dark energy problem. A commonly-used phenomenological parametrization of a
time-varying dark energy equation of state was suggested by Chevalier & Polarski and
10 Chapter 1. Introduction
Linder (CPL, Chevallier & Polarski (2001); Linder (2003)):
w(a) = w0 + wa(1− a) (1.22)
where w0, wa are constant parameters, such that measuring wa 6= 0 would constituteevidence for evolving dark energy, and a is the cosmic scale factor.
The alternative approach for explaining accelerating expansion is to modify the grav-
itational sector of the theory. One such approach, which changes the Einstein-Hilbert
action of General Relativity through generalizations of the curvature term, is known as
f(R) gravity (Capozziello et al., 2003; Carroll et al., 2004). Extensions to the gravitational
model can be reflected in the appearance of new particles or degrees of freedom, mimicking
the presence of a fifth force. In order to satisfy existing tests of General Relativity, such
theories must include a screening mechanism to ameliorate these effects in the regimes
where General Relativity has been tested. Examples of these phenomena include scalar-
field inspired models such as the Chameleon Field (Khoury & Weltman, 2004) and the
symmetron model. For a review of modified gravity, descriptions of the different screening
mechanisms and the implications for cosmology see Joyce et al. (2016).
The dark energy problem has motivated many previous, current and future observa-
tional projects to constrain its evolution with cosmic epoch, using observables that probe
both cosmic expansion and growth of structure. The Dark Energy Task Force report
(DETF, Albrecht et al. 2006) was a useful milestone in reviewing the constraining power
and complementarity of these methodologies. In the next section we describe in more
detail the main strategies that have been used to study the dark energy problem from the
perspective of large-scale structure, the field of this thesis.
1.3 Large Scale Structure
The development of new astronomical technology, in particular advances in multi-fibre
spectroscopic instrumentation with telescopes, has accelerated the development of galaxy
redshift surveys in the last decade. Initial examples of such surveys were the 2-degree Field
Galaxy Redshift Survey (Colless et al., 2001) at the Australian Astronomical Observatory
and the Sloan Digital Sky Survey (York et al., 2000) at the Apache Point Observatory.
More recently, the WiggleZ Dark Energy Survey (Drinkwater et al., 2010) obtained about
220000 Emission Line Galaxy redshifts over 1000 deg2 in the redshift interval 0 < z < 1,
and the Baryon Oscillation Spectroscopic Survey (Eisenstein et al., 2011) gathered 106
redshifts in the interval 0 < z < 0.7 with a much wider angular coverage of 10000 deg2.
1.3. Large Scale Structure 11
0.0 0.2 0.4 0.6 0.8 1.0
Ωm
0.5
1
1.5
2
H0t 0
ThresholdStellar
ΛCDM
Open
0.0 0.2 0.4 0.6 0.8 1.0
z
2.0
1.5
1.0
0.5
0.0
0.5
1.0
1.5
2.0
q(z)
(0. 3, 0. 7)
(1. 0, 0. 0)
(0. 0, 1. 0)
Figure 1.1 Illustration of the effect of dark energy on the age of the Universe t0 and itsdeceleration rate as a function of redshift q(z). Left panel: age of the universe in unitsof H−10 as a function of the Ωm parameter. We show the flat ΛCDM prediction (solidline) and the open model with ΩΛ = 0 (dashed line). We display the threshold limit fromthe age of the oldest stellar objects. Right panel: deceleration parameter as a functionof redshift for different values of the parameters (Ωm,ΩΛ). The presence of dark energyintroduces a transition from deceleration to acceleration, at a redshift of about z ' 0.6 fora (0.3, 0.7) cosmological model.
12 Chapter 1. Introduction
The goal of all these studies is to map out the large-scale structure of the universe and hence
learn about the physical mechanisms responsible for the clustering pattern of galaxies, the
growth of structure over time and cosmic geometry. This will be extended in the future by
projects such as the Dark Energy Spectroscopic Instrument (Levi et al., 2013), the Euclid
satellite (Laureijs et al., 2011) and the Square Kilometre Array (Maartens et al., 2015).
The mathematical description of the large-scale structure of the Universe is fundamen-
tally based on the overdensity field (or density contrast) δ(~r ), given by
δ(~r ) =ρ(~r )− ρ̄
ρ̄, (1.23)
where ρ(~r ) is the density (of matter or galaxies) as a function of position, and ρ̄ repre-
sents the averaged background density. Using linear perturbation theory in an expanding
Universe, the evolution of the matter density contrast follows the equation (see Huterer
et al. (2015) for a review of the growth of structure)
δ̈ + 2Hδ̇ = 4πGρδ. (1.24)
In the initial conditions created by inflation and in linear theory, the density field is a
random Gaussian field, such that only the first two moments of the density field are
different from zero (Mo et al., 2010) and all the information is captured by 2-point statistics
of the field.
Because the linear theory cosmological information is encoded in the 2-point statistics,
it may be captured by simple clustering estimators. In particular, given the density
contrast of two tracers, e.g. two kinds of galaxy populations, or tracers from numerical
simulations such as dark matter halos, we define the correlation function of the tracers t1
and t2 as
ξt1t2(r) = 〈δt1(x)δt2(x + r)〉, (1.25)
where this definition includes an auto-correlation function when the tracers are the same.
From the hypothesis of homogeneity and isotropy (e.g., neglecting redshift-space effects),
it follows that the correlation function only depends on the magnitude of the radial sep-
aration of the tracers. The correlation function measures the clustering in excess of a
random distribution of points which would possess ξ = 0 (Amendola & Tsujikawa, 2010).
Equivalently, the Fourier transform of the density field may be taken and statistics of
the Fourier amplitudes δ(~k ) analyzed. The Fourier space counterpart of the correlation
1.3. Large Scale Structure 13
function is the power spectrum P (k) (Wiener-Khinchin theorem), given by the equation
〈δ(k)δ(k′)∗〉 = (2π)3δD(k− k′)P (k), (1.26)
where δD is the Dirac delta function. The Fourier amplitudes are hence independently
sampled from the power spectrum. For the analysis of fluctuations smoothed on a given
scale R, we process the power spectrum with a filter function. In particular, σ2R, the
variance of the density field when smoothed on a scale R, is given by
σ2R =1
(2π)3
∫P (k)W (kR)2d3k, (1.27)
where the functional form for the window function W (kR) depends on the smoothing filter
(e.g. Gaussian, top-hat).
The power spectrum is a very important observable of large-scale structure analysis
because it can be predicted using theory of the early Universe. Its form at the present
epoch depends on the physics of the transition from the radiation-dominated phase of
expansion to the matter-dominated phase, considering the interaction among different
particle species such as baryons, neutrinos and dark matter, which is described by a
transfer function T (k), such that
P (k, z) = Pprim(k)T (k)2D(z)2, (1.28)
where Pprim(k) = Akn refers to the primordial power spectrum generated after the exit of
the inflationary phase and parametrized by the Harrison-Zeldovich index n and a constant
amplitude A, T (k) is the transfer function normalized to unity at large scales (k → 0), andD(z) is the growth factor obtained by solving equation 1.24. The transfer function encodes
the scale of matter-radiation equality keq. Analyzing perturbations in the early universe
as a function of scale, the battle between gravity (to collapse) and radiation pressure (to
expand) is mediated at the characteristic Jeans scale LJ , which marks a characteristic
turnover scale in the processed power spectrum after radiation domination. This is shown
in figure 1.2, which displays a matter power spectrum calculated using the CAMB code
(Lewis et al., 2000) which performs a full numerical solution of the early-Universe physics.
The scale of matter-radiation equality (and the shape of the power spectrum) depend
on the combination of parameters Ωmh2, although because of the factor h included in
measurements of distances in Mpc/h, the combination becomes Ωmh, known as the shape
parameter. In detail, the transfer function also depends on the other constituents of the
14 Chapter 1. Introduction
10-2 10-1 100 101 102
k/keq
101
102
103
104
105
P(k
)[M
pc/h]3
ΛCDM
Figure 1.2 Matter power spectrum as a function of the wavenumber k in units of the scaleof matter-radiation equality keq, obtained by the code CAMB for a fiducial set of ΛCDMcosmological parameters. We can see the characteristic turnover at the scale keq, whichmodifies the power-law primordial spectrum established after the inflationary phase.
Universe such as baryons and neutrinos. This implies that the power spectrum is sensitive
to this information, such as particle properties of the relativistic species (Beutler et al.,
2014a), due to the damping of power at small scales caused by massive neutrinos.
Another scale imprinted in the clustering of matter, as well as the turnover at matter-
radiation equality, is the sound horizon at last-scattering (or more precisely, baryon drag).
In the early universe, baryons and photons are tightly coupled in a hot primordial plasma.
As the material falls into clumps due to gravity, radiation pressure launches spherical
sound waves which propagate until the coupling is ended at recombination. At the time
of decoupling, a ring of baryons is left surrounding each dark matter overdensity at a
characteristic scale given by the sound horizon at the baryon drag epoch (this is slightly
different to decoupling due to the presence of a pressure term affecting the baryons and
different optical depths for baryons and photons (Wang, 2010)). Since matter accumulates
1.3. Large Scale Structure 15
in the central overdensity and baryon shell, this sound horizon scale creates a standard
ruler in the matter power spectrum consisting of a series of baryon acoustic oscillations
(BAO) (Blake & Glazebrook, 2003; Seo & Eisenstein, 2003). The pattern of oscillations
in Fourier space becomes a single peak in configuration space (correlation function) at the
characteristic scale of the sound horizon, at about 100 Mpc/h, which may be calibrated
by analysis of the CMB anisotropies. The BAO feature, which allows measurements of
cosmological distances in units of this standard ruler, has been utilized in observational
studies by Eisenstein et al. (2005) for the SDSS LRG sample, Percival et al. (2007, 2010),
for SDSS, Beutler et al. (2011) for the 6-degree Field Galaxy Survey, Blake et al. (2011c)
for the WiggleZ Dark Energy Survey and Anderson et al. (2012, 2014) for BOSS. The
BAO signal has been also identified in the Lyα forest from quasar surveys in the redshift
interval 2 < z < 4 (Busca et al., 2013; Slosar et al., 2013).
Standard ruler measurements from BAOs may be improved by the method of density-
field reconstruction (Eisenstein et al., 2007). The motivation for this technique is that at
scales smaller than the BAO scale, the bulk displacements of galaxies from their original
positions in the density field erases the small-scale oscillations in the power spectrum
and hence partially washes out the BAO feature. However, these displacements may be
estimated from the density-field itself, allowing the galaxies to be retracted to their original
positions and the peak to be sharpened. Padmanabhan et al. (2012) reported one of the
first applications of this method to the BAO measurement at z = 0.35 from the SDSS
DR7 LRG sample.
A key issue with interpreting clustering measurements from galaxy redshift surveys
is to relate the properties of the galaxy density field to those of the underlying matter
density field predicted by theory. This connection can be represented by a generic bias
relation of the form
δg = ψ(δm) = bδm (1.29)
where g,m refers to galaxies and matter respectively, and the last equation refers to the
linear bias model, applicable to large scales with constant bias parameter b. In its more
general incarnation in the halo model (Cooray & Sheth, 2002), the most general galaxy
bias relation is considered to be scale-dependent, non-linear, non-local and stochastic, as
opposed to the linear, constant, local and deterministic relation parametrized by b. The
linear bias relation implies an amplitude re-scaling of the power spectrum at large scales
as Pg(k) = b2Pm(k). For an extensive review of galaxy bias see Desjacques et al. (2016).
We now consider two other applications of large-scale structure for testing the cosmo-
logical model. First, we mention the gravitational lensing of light: the correlated distortion
16 Chapter 1. Introduction
Figure 1.3 Left Panel: Initial measurements of the matter and cosmological constant den-sities Ωm,ΩΛ from SNIa (Riess et al., 1998; Perlmutter et al., 1999) revealing a significantpreference for the non-zero ΩΛ an hence an accelerated expansion. Credit Figure: Wang(2010). Right Panel: Complementary constraints in the same parameter space from differ-ent probes such as SNIa (Blue, displaying the increase in constraining power in comparisonto the initial study), CMB (Yellow), BAO (Green). The combined constraint is displayedin dark grey, showing a clear preference for the concordance ΛCDM model. Credit Figure:Kowalski (2008).
1.3. Large Scale Structure 17
in the shape of background galaxies as their light passes through the intervening matter
distribution, which is sometimes called cosmic shear. The distortion depends only on the
overall matter distribution, which represents an advantage of the weak lensing method: to
trace mass independently of galaxy bias.
The mathematical formalism of weak lensing relates the source plane to the image plane
through a Jacobian transformation of the coordinates which depends on the convergence
factor κ and the complex shear (γ1, γ2). The cosmic shear observables are sensitive to the
multipole power spectrum of the convergence field (Weinberg et al., 2013), defined by a
weighted projection of the matter power spectrum along the line of sight
Pκ(`, z) =
∫ z0g(z) · P (`/r(z), z)dz, (1.30)
where in this equation ` represents the multipole, the power spectrum is evaluated at
wavenumber k = `/r(z) where r is the co-moving distance, and the kernel function g(z)
is the combination of the source distribution and weak lensing geometry. Weak lensing
observables hence depend on both the geometry of the Universe and the growth of struc-
ture. This dual sensitivity is one of the powerful advantages of the weak lensing method.
Surveys like the Canada-France-Hawaii Telescope Lensing Survey (CFHTLenS) (Heymans
et al., 2012) have produced cosmological measurements from weak lensing data, examples
include e.g. Simpson et al. (2013) and Heymans et al. (2013). Weak lensing surveys face
challenges to control systematic errors arising from shape measurements, intrinsic align-
ments, the source galaxy redshift distribution, and modelling non-linear matter clustering
(e.g. Hildebrandt et al. (2016)). However, if these challenges can be overcome then weak
lensing promises to deliver some of the most accurate measurements of the cosmological
model in the future.
Second, large-scale structure surveys allow tests of the cosmological model via the num-
ber density distribution of galaxy clusters. The abundance of clusters may be predicted
from theoretical mass function models as an integral of the number density weighted by
the volume element
Nc ∝ Ωobs∫Mf (z)
dV
dzdΩdz, (1.31)
where Ωobs is the solid angle of observation and Mf is the cumulative mass function,
which can be determined for a given cosmological model using relations calibrated by
numerical simulations such as Sheth & Tormen (1999). The sensitivity to the volume
element exemplifies the geometric nature of this observable. A compilation of results from
galaxy clusters related to dark energy are presented in the review by Weinberg et al.
18 Chapter 1. Introduction
(2013). The main challenge for galaxy cluster cosmology is to accurately calibrate the
mass-observable relation and hence establish the mass sampling of the observations.
In the next section we will discuss how regions of low density, or cosmic voids, are
being incorporated into large-scale structure studies to extract geometry and growth of
structure observables, and hence extend the methods available to analyze the dark energy
problem.
1.4 Galaxy Voids
In the characterization of the cosmic web, high-density regions or clusters, filaments, walls
and voids are the elements that constitute the topology of the Universe. Although there
is no clear definition of what precisely constitutes a void, which is a challenge for the field,
voids are associated with regions which are largely empty of galaxies, containing matter
density which is a small fraction of the average background density. In quantitative terms,
the density contrast of a void is expected to be in the interval (−1,−0.5) in the inner partsof the void. The first discovery of a void was reported by Gregory & Thompson (1978),
for a description of historic context see Thompson & Gregory (2011). Early observational
studies revealed the existence of an individual void in the region of Boötes (Kirshner
et al., 1981), and subsequent studies by de Lapparent et al. (1986), using data from the
CfA survey (Geller et al., 1987), identified a catalogue of voids with characteristic sizes of
about 25 - 50 Mpc/h. More recent galaxy redshift surveys have enabled void catalogues
to be dramatically extended e.g. Sutter et al. (2012b); Nadathur & Hotchkiss (2014).
Voids are also readily identified in N -body simulations as regions with almost no pres-
ence of dark matter (Icke, 1984). Sheth & van de Weygaert (2004) proposed treating voids
in a similar manner to collapsed high-density structures, by determining their abundance
in the formalism of excursion set theory.
Peebles (2001) has highlighted the absence of galaxies in voids as a possible prob-
lem for the ΛCDM cosmology. The issue in question is how voids populations fall on the
morphology-density relation, by which early-type galaxies are correlated with high-density
regions and late-type star-forming galaxies prefer low-density environments. In this con-
text, Peebles (2001) described a tension between observations and the predictions of the
cold dark matter structure formation scenario. Voids have also been studied within dark
matter particles, halos and simulated galaxies from semi-analytic models (Benson et al.,
2003).
Void samples are highly dependent on the void-finder used in the analysis. Many
algorithms have been proposed for identifying voids in different tracer populations of the
1.4. Galaxy Voids 19
density field and tested using both galaxy datasets and numerical simulations, e.g. Hoyle
& Vogeley (2004); Ceccarelli et al. (2006); Patiri et al. (2006); Neyrinck (2008); Paz et al.
(2013). Colberg et al. (2008) presented a void-finder comparison using a single simulated
galaxy dataset from semi-analytic models. In section 2.2 we describe the approaches of
different void-finders in more detail.
Voids can be used to probe the cosmological model in a number of ways. Ryden
(1995) proposed using voids as spherical objects for an Alcock-Paczynski test (Alcock &
Paczynski, 1979) to measure the deceleration parameter q0. This idea was extended by
Lavaux & Wandelt (2012) by considering, instead of individual voids as spherical objects,
void stacks to increase the isotropy signal. This work also introduces the measurement
of void radial profiles in one and two dimensions as a working example to implement the
Alcock-Paczynski measurement, evaluating on the way the impact of peculiar velocities.
Other avenues to probe the cosmological model using cosmic voids include Park & Lee
(2007), who introduce ellipticity measureme