Chapter 1

Principles of cosmology

In the following chapter I provide a brief introduction to some of the main tenets of modern cosmology, building from a homogeneous and isotropic model to the formation of structure which is observed with modern surveys. The former sections are discussed in more detail in Peacock (2000). I discuss cosmology as a modern science, with a brief overview of some of the most promising techniques of measurement pursued in the field. I argue that whilst we are at the advent of ‘precision cosmology’, persistent systematics and an incomplete theory necessitate further study.

1.1 The standard model of cosmology

All cosmological observations to date corroborate Einstein’s Cosmological Prin- ciple, which describes a universe that is statistically homogeneous (the same at all locations) and statistically isotropic (independent of direction) on large scales. Figure 1.1 shows a snapshot of a simulation of the large-scale structure, depicting this principle.

According to the current cosmological model, the Universe underwent a period of rapid expansion 13.8 Gyr ago (Planck Collaboration et al., 2016a). This early period of inflation caused the rapid exponential growth of small causally connected areas, which simultaneously caused the growth of quantum fluctuations in the matter density field which seed early structure formation. The post- inflation, early, hot Universe contains both ionised baryonic matter, dark matter

1 and the dominating radiation, relics of big bang nucleosynthesis during the inflationary period, which are tightly coupled resulting in an opaque Universe. Structure formation on scales smaller than the particle horizon was suppressed until matter-radiation equality, after which the matter content of the Universe dominates. Upon recombination, when ionised nucleii combined with free electrons, the Universe became transparent, allowing the propagation of radiation in all directions and forming the last scattering surface seen today as the cosmic microwave background (CMB). This is a relic blackbody spectrum from the radiation dominated era. During matter domination, the fluctuations, now frozen as classical fluctuations, provide the initial perturbations for gravitational collapse and galaxies and structure form. All the while, the Universe continues to expand such that any two suciently close galaxies will perceive the other to move away from them according to Hubble’s law. First evidence from this came from Slipher (1922), who observed the spectral lines of spiral galaxies to be redshifted with respect to their rest frame wavelengths observed in a laboratory. Interpreted as a Doppler shift, this required a recession velocity proportional to distance (Hubble, 1929). Only relatively recently did the vacuum energy density dominate the total energy density of the Universe and begin to drive the late-time, cosmic acceleration that is observed.

Einstein’s theory of General Relativity is widely accepted as a fundamental theory to describe the geometric properties of spacetime and how they are determined by matter and energy. In a homogeneous and isotropic framework, Einstein’s field equations give rise to the Friedmann equation which e↵ectively describes the evolution of the Universe through radiation and matter dominated eras. This theory holds a ‘cosmological constant’, denoted as ⇤, which can be associated with the vacuum energy and called ‘dark energy’, accountable for the repulsive- gravity e↵ect on cosmic scales that we observe today. Thus the standard model of cosmology, is known as the ⇤CDM model, owing to the inclusion of a cosmological constant, ⇤, that dominates the total energy density today and cold dark matter (CDM) as the main contributor to the matter density.

1.2 Cosmological spacetime

The physics of the expanding Universe is governed by General Relativity (GR). In a gravitational field and in the absence of any other forces, any mass obeys an identical trajectory between two points. GR describes these trajectories through

2 1 Figure 1.1 A 64h Mpc- thick slice through the Horizon simulation showing the matter density field in the past light cone as a function of redshift and look-back time, all the way to the horizon. The Earth is at the vertex and the Big Bang at a look-back time of 13.6 billion years forms the upper boundary. The large-scale structure reveals a pattern that is homogeneous and isotropic. Image drawn from Kim et al. (2009).

3 the geometry of spacetime itself, rather than as a gravitational force experienced by a given particle. An invarient interval of proper time, d⌧ between two distant events in a homogeneous and isotropic Universe can be described by a space- time metric, as the square of the physical distance between two infinitesimally separated points, mapped by coordinates xµ, to give

ds2 = c2d⌧ 2 g dxµdx⌫ . (1.1) ⌘ µ⌫

The value of ds2 should be independent of the coordinate choice and agreed upon by any observer. Here, gµ⌫ is the metric tensor which is determined by solving Einstein’s gravitational field equations and subscripts run over one temporal and three spatial coordinates. This metric tensor describes the geometry of the Universe and is specifically determined by a given matter distribution, according to Einstein’s field equation:

8⇡G Gµ⌫ +⇤gµ⌫ = T µ⌫ , (1.2) c4 where Gµ⌫ is the stress-energy tensor, which determines gravity for all matter and energy, and ⇤is the cosmological constant, discussed in Section 3.5.5. The distortions from Euclidian geometry are sourced by the matter distribution via the energy-momentum tensor, Tµ⌫, which is diagonal for the Universe and composed of energy density and pressure components. The distinction has been made between the left-hand side of this equation, which describes the geometrical properties of space-time and the right-hand side, which describes the properties of matter. As such, this equation can be summarised by “spacetime tells matter how to move; matter tells spacetime how to curve” Wheeler & Ford (1998). On the left hand side, the Gµ⌫ term, geometric information enters through contractions over the Ricci Tensor, Rµ⌫, which describes the curvature of space-time using the metric. 1 Gµ⌫ Rµ⌫ Rgµ⌫ , (1.3) ⌘ 2 where R is the curvature scalar, which is obtained by the remaining contraction of the Ricci tensor. The Newtonian limit to the field equation (v c) is ⌧ 4⇡G 2= (⇢c2 +3p) , (1.4) r c2 where = c2g /2, 2 = is the Laplacian in space, p is the pressure, ⇢c2 00 r i j is the rest frame energy density and ⇢, the mass density. This shows that relativistic fluids have an enhanced gravitational influence in GR due to the

4 pressure contribution to the e↵ective density. For a non-relativistic case, we arrive at Poisson’s equation, defined as

2=4⇡G⇢. (1.5) r

Einstein’s field equation ( 1.3) shows geodesics to be determined by ⇢c2. The vacuum contribution to the energy density must then have an associated energy- µ⌫ momentum tensor, Tv . Assuming this representation is invariant under Lorentz transforms in locally inertial frames.

T µ⌫ = ⇢ c2gµ⌫, (1.6) vac v in an arbitrary frame. Therefore the vacuum contribution enters the field equation in an identical manner to the cosmological constant, possessing a w = 1 2 equation-of-state and constant energy density, ⇢vc . Substitution of the vacuum energy-momentum tensor into Einstein’s field equation,

1 8⇡G Rµ⌫ Rgµ⌫ +⇤gµ⌫ = (T µ⌫ + T µ⌫ ) , (1.7) 2 c4 matter vac

Hence, the vacuum energy density is associated with the cosmological constant,

1 8⇡G Rµ⌫ Rg +⇤gµ⌫ = T µ⌫, (1.8) 2 µ⌫ c4 as long as it is identified with,

⇤ ⇢ = . (1.9) v 8⇡G

The geometry of an expanding universe satisfying the Cosmological Principle, that the Universe is homogeneous and isotropic, is described by a Friedmann- Robertson-Walker (FRW) metric. Under these assumptions, equation 1.33 can be simplified in terms of spherical polar coordinates , ✓ and ,to,

ds2 = c2dt2 + a(t)2[d2 + S2()2(d✓2 +sin2(✓)d2)] , (1.10) k where is the comoving radial distance, a(t) is the dimensionless scale factor 2 and Sk() is the angular diameter distance, which depends on the geometry or global curvature, k. Three cases of a spatially flat, Euclidian universe, an open spherical and a closed, saddle-like universe are specified by a curvature constant

5 of k =0, 1, 1, respectively, defining the angular diameter distance as,

sinh(r)(k = 1) Sk(r)=8r (k =0) (1.11) > > > The radial distance to a galaxy at a: coordinate is s(t)=a(t) and di↵erentiating with respect to time,s ˙ =˙a, which recovers Hubble’s law, (s/s ˙ )=(˙a/a)=H(t). For 1, the metric describes the Hubble expansion. The isotropic velocity ⌧ field, as observed by galaxies at a distance r and with a velocity, v, is observed to obey Hubble’s Law,

v = H0r , (1.12) where H0 is the Hubble parameter today, which can be recast into a dimensionless 1 1 form, h = H/100 km s Mpc .

1.2.1 Expansion dynamics

The Friedmann equation, which governs the evolution of the scale factor with the matter content of the Universe, in its first form is given by

a˙ 2 8⇡G kc2 = H2(a)= ⇢(a) , (1.13) a 3 a2 ⇣ ⌘ Di↵erentiating with respect to time and appealing to an adiabatic expansion of the Universe dE + PdV = 0, we obtain an equation of motion for the scale factor, a¨ 4⇡G 3p = ⇢(a)+ , (1.14) a 3 c2 3⇣p ⌘ where the active mass density, ⇢(a)+ c2 , appears.

These equations both contain contributions from either the matter density or pressure, highlighting that the expansion history of the Universe is dependent on the dominant contribution to the energy density at a given time. The energy density contains contributions from three di↵erent types of matter: radiation, a matter component which contains dark matter and ordinary baryonic matter, and a vacuum energy contribution

⇢(a)=⇢r(a)+⇢m(a)+⇢v(a) , (1.15)

6 which evolve with the scale factor uniquely. We can define a time-varying equation of state parameter as p p w = = , (1.16) u ⇢c2 where u = ⇢c2 is the energy density and p is the pressure is the component. Radiation, a relativistic gas with an energy density equal to u =3p therefore has w =1/3, pressure-less matter has w = 0, and the cosmological constant description of the vaccuum energy has w = 1. The matter contribution scales with the change in volume as the matter becomes diluted due to the expansion of the Universe. On the other hand, the radiation gains an extra contribution from the scale-factor as the energy density of relativistic particles are redshifted. The relation, for the vacuum energy component assumes the cosmological constant explanation, which is described in Section 3.5.5. Assuming an exactly flat universe, k = 0, with the Friedmann equation (1.13) defines a critical density,

⇢crit,as 3H2 ⇢ = . (1.17) crit 8⇡G The density is more often expressed as a density parameter, ⌦, that incorporates

⇢crit, to give, ⇢ 8⇡G⇢ ⌦ = 2 . (1.18) ⌘ ⇢crit 3H For any massive particles subject to the expansion initiated at an early time, the 2 3 proper rest-mass energy density will dilute with scale factor as ⇢ (a)c a . m / This hold for photons, with the photon wavelengths also stretching with the expansion and the vacuum energy density modelled as a cosmological constant,

⇢v = ⇢⇤, as a perfect fluid, remains constant. The time evolution for each contribution to the total energy budget of the Universe follows, with

2 3 ⇢ (a) c a (1.19) m / 2 4 ⇢ (a) c a (1.20) r / 2 ⇢⇤(a) c =constant. (1.21)

In terms of the dimensionless density parameter, the Friedmann equation can be recast as

kc2 H2(a)=H2 ⌦ (a)+⌦(a)+⌦ (a) (1.22) 0 m r ⇤ a2 2 3 4 2 = H ⇥⌦ a +⌦ a +⌦ +(1⇤ ⌦ )a , (1.23) 0 m,0 r,0 ⇤,0 0 h i

7 where the curvature term has been recast in terms of the total mass density, ⌦, and the time dependence of each density parameter has been detailed explicitly, with the vacuum energy cast as a cosmological constant. For a Universe with a density equal to critical, they are described as spatially flat, with a spatial curvature of k = 0. With this, the Friedmann equation can be rewritten in terms of the density parameter, split into it’s constituents of matter, radiation, a vacuum energy and curvature as,

⌦m(t)+⌦r(t)+⌦⇤(t)+⌦k(t)=1. (1.24)

1.2.2 Redshift and distance in an expanding universe

Light propagates along null geodesics, such that the comoving radial distance of light propagation between two measurements of cosmic time t0 and t1 is given as,

t1 cdt = 0 . (1.25) a(t ) Zt0 0 Consider the emission of two photons in succession at times t0 and t0 + t0, and their subsequent observation at times t1 and t1 + t1, in a homogeneous and isotropic expanding Universe, for a source and observer at positions that are at fixed comoving coordinates. Assuming that the scale factor does not significantly vary between the two photons, the comoving distance of light propagation for the two photons is unchanged and independent of time, such that,

t t 0 = 1 . (1.26) a(t0) a(t1)

If the time coordinate is selected as t 1/⌫, where ⌫ is a measured frequency / shift in spectral lines, then an induced frequency ratio implies a Hubble expansion, observed by the Doppler e↵ect ( 1+ v ), due to a velocity, v, ⇡ c ⌫ a(t ) 1 1 = 0 (1 + z) , (1.27) ⌫0 a(t1) ⌘ ⌘ a where z is the measured redshift. Two suciently close comoving observers see the other to obey Hubble’s law, which can be thought of as the accumulation of repeated Doppler shifts from a chain of fundamental observers between them. As such, measurement of the redshift in observed spectral lines of distant galaxies is determined by the ratio of the scale factor at emission and observation and

8 provides insight into expansion history of the Universe.

As cdt = cdR/(RH)= Rdz/(R H), if we observe a source at redshift, z, the 0 age of the Universe can be deduced as

1 dz t (z)= 0 . (1.28) age (1 + z )H(z ) Zz 0 0 By dating the oldest stellar systems, and within the standard framework, the current best constraint on the minimum age of the Universe (Planck Collaboration et al., 2016a) is t (0) = 13.799 0.021Gyr . (1.29) age ±

Given equation 1.10 and following that Rdr = cdt = cdR/R˙ = cdr/(RH), the comoving distance to a galaxy observed at redshift, z, is therefore

z cdz (z)= 0 . (1.30) H(z ) Z0 0

The angular diameter distance, DA, is defined as the ratio of an object’s physical transverse size to its angular size (in radians). With an observed redshift, one may translate between the comoving distance to sources, , and DA, for a given density composition and for a flat universe by

D (z)= . (1.31) A 1+z

It is also useful to consider the relation between the intrinsic luminosity of a source at emission and the flux density observed today, defined as the luminosity distance

DL(z)=(1+z). (1.32)

1.3 Structure formation

The perturbed FRW spacetime metric may be completely defined in terms of the Bardeen potentials, namely, the Newtonian potential, , which along with density perturbations drives the structure formation of the Universe, and the curvature

9 potential, , along with an expansion scale factor for the Universe, a(t)as,

2 2 2 2 (x,t) 2 2 2(x,t) ds = c dt 1+ 2 + a(t) dx 1 2 . (1.33) " c # " c #

This is a generalised form of the perturbed metric which applies to any metric theory of gravity with a scalar perturbation. Thus, a metric theory of gravity relates the two potentials to the perturbed energy-momentum tensor. Matter density fluctuations or perturbations in an otherwise homogeneous and isotropic and Universe give rise to overdensities and underdensities, galaxy clusters and voids respectively. For an expanding universe in the linear regime it is assumed that the density perturbations, m, in the otherwise homogeneous density field, are small and are characterised as

⇢m(x,t) ⇢¯m(t)) m(x,t)= 1. (1.34) ⇢¯m(t) 

While non-relativistic particles feel only the e↵ects of the derivatives of the Newtonian potential, 2, relativistic particles are equally sensitive to both r potentials such that their equations of motion are characterised by 2( +) r (Jain & Zhang, 2008). In standard GR, the Einstein equation predicts that the two Bardeen potentials are equal for fluids without anisotropic stress (see, for example Weinberg et al., 2013). That is, (x,t)=(x,t) and are both linked to the matter overdensity, m via the Poisson equation,

4⇡G 2 = 2= a2⇢¯ , (1.35) r r c2 m m where ⇢m is the average matter density at the time when the scale factor is a and this links to equation 1.5 as ⇢ =¯⇢mm for the matter component. The Newtonian potential controls the dynamics of non-relativistic particles while the curvature potential deflects light, this equality in GR e↵ectuates an important test of gravity: the dynamical mass equals the lensing mass.

In Fourier space, the Newtonian potential can be written as,

2 2 k (k, a)=4⇡Ga ⇢¯mm(k, a); (1.36) and combined with the curvaturee potential toe give a parameterised Einstein equation, 2 2 k [(k, a)+ (k, a)] = 8⇡Ga ⇢¯mm(k, a), (1.37)

e e e 10 where and are the Fourier transforms of and , respectively.

e e 1.3.1 Correlation function and Power Spectra

For a quantity, x(✓ ), measured on the sky, its angular correlation functions are given by ⇠ ('):= x(✓ )x(✓ + ' ) , (1.38) x h i where the angular brackets combine an average over all positions ✓ with an average over all orientations of the separation vector ' on the sky. Due to the statistical isotropy of the cosmic large-scale structure, it is assumed that the angular correlation function depends only on the absolute value of ' but not on its orientation.

We obtain the angular power spectrum by taking the Fourier-transform of the correlation function ⇠('),

2 i` ' C(`)= d '⇠(')e · . (1.39) Z Here, ` is the two-dimensional wave vector conjugate to the angular separation '. Via Limber’s approximation (detailed in Section 2.3.1), this can be converted into a power spectrum.

1.3.2 The matter power spectrum

In the standard model, the seeds for structure formation are provided by random quantum fluctuations, which expand to cosmological scales outwith the horizon scale, during a period of Inflation. The fractional perturbation to the density field can be described as, ⇢ (1 + ) . (1.40) ⌘ ⇢ ✓ 0 ◆ As a result, matter overdensities form a Gaussian random field (non-Gaussianity is small, Planck Collaboration et al., 2016c), so that structure in the early Universe can be fully described by the matter power spectrum, P, which can be recast 2 3 2 into a dimensionless form as (k)=k P(k)/2⇡ . In the CDM model, the dark matter particle is non-relativistic when it enters the horizon, thereby enhancing structure formation. In this case, structure formation has a ‘bottom-up’ model.

11 In this case, smallest-scale structure forms earliest and as the horizon grows, these may merge to form larger structures, which become causally connected.

The equations of state and interactions of the components of the universe can alter the predictions of perturbation theory. Although the presence of cold dark matter does play a leading role in determining the matter power spectrum, the inclusion of baryons and neutrinos can lead to significant alterations. The transfer function, T (k) is used to describe how the shape of the initial power-spectrum,

k(z), in the dark matter is modified by di↵erent physical processes that occur, to give the power spectrum today, k(z =0),

k(z =0)=T (k) f(z)k(z) , (1.41) where f(z) is the growth rate of structure. Within linear theory dynamics each Fourier mode, ˜(k), evolves independently. Here the logarithmic growth rate is denoted by d ln D f = + , (1.42) d ln a where D is the growing mode. In the matter dominated era, with ⌦(a) + ' 2/3 1 ⌦ 1, a t and H =(2/3)t , the pressureless CDM component evolves as m ' / (t)=D(t)(t0) with the growing and decaying modes given by

2/3 1 D+ =(t/t0) ,D=(t/t0) , (1.43) respectively.

The transfer function is defined in terms of the density perturbations for wavenumber, k, and redshift, z, (see equation 1.34) as

(k, z =0) (k, z = ) T (k)= 1 , (1.44) (k, z = ) (k, z =0) 1 such that at large scales, k 0, the function tends to T = 1 and the power ! spectrum (k) 2 is proportional to the square of the transfer function multiplied h| | i by the initial power spectrum, k(z). The model of the transfer function is influenced by the fact that there is a time delay in the growth of the perturbations between the point when they came through the horizon and began to grow again. In the standard cold dark matter picture, in the radiation-domination era, the baryons are tightly coupled with the photons and share in the same pressure-induced oscillations that lead to acoustic peaks in the Cosmic Microwave

12 Background. This leads to intermediate-scale oscillations in the power spectrum, as well as an overall suppression of power on small and intermediate scales (Eisenstein & Hu, 1998). Before recombination, each constituent has a separate transfer function while afterward, the baryons are essentially pressureless and assume the same transfer function as the cold dark matter perturbations. The overall shape of the power spectrum is determined by the radiation era in linear theory, largely by the dark matter ( R2). Following this period, the linear / behaviour is simply for the amplitude to grow according to D t2/3. + / The shape of the matter power spectrum is well-established to first order and its normalisation is quantified by 8 . It is defined as the root mean square matter 1 density variation averaged over 8h Mpc spheres,

2 2 1 3 2 8 = 3 sin(kR)+kR cos(kR) (k)dk , (1.45) 0 k(kR) Z h i 1 where R=8h Mpc.

With a small density contrast, 1, it can be shown to increase with time in ⌧ proportion to a time-dependent linear growth factor D+(a). The power spectrum, which scales as 2, grows like P D2 on suciently large scales (small k) and / + keeps its initial shape. On small scales, this shape is changed by non-linear e↵ects on the evolution of the density contrast. Notwithstanding this non-linear complication, we write the power spectrum P as a slowly varying shape function 2 times an amplitude. Conventionally, this amplitude is called 8 and set at the present epoch. In terms of the power spectrum linearly extrapolated to the present time, 8 is defined by

2 1 k dk 2 = P (k) W 2(k) , (1.46) 8 2⇡2 8 Z0 where W8(k) is a filter function, or kernel, suppressing all modes smaller than 1 8 h Mpc. Setting the linear growth factor D+(a) to unity at present, the density- fluctuation power spectrum can then be written as

2 2 P(k)=8 D+(a) P (k) (1.47)

13 1.4 Components of the cosmological model

Along with baryonic matter (as well as neutrinos), and radiation, the Universe today pro↵ers dark matter as the dominant form of the matter energy density, as well as a ‘dark energy’ component, associated with a vacuum energy and Einstein’s cosmological constant.

1.4.1 Dark Matter

The current standard model of cosmology assumes that a weakly interacting, cold, non-baryonic dark matter particle is responsible for the following observations:

Galaxy rotation curves: The galaxy rotation curves of spiral galaxies, measured using the rotational speed of stars, flatten at large radii (Rubin et al., 1980). As the density of light-emitting matter decreases towards the visible edge of the galaxy, this contradictory measurement implies that the enclosed mass of the galaxy continues to grow, extending to larger radii than the light-emitting structure. The standard model pro↵ers a dark matter halo with a central baryonic component confined to the disk as the ‘missing mass’.

Galaxy clusters and gravitational lensing: Similarly and prior to this, observations of the velocity dispersion of galaxies in the Coma cluster suggested that there was a ‘missing mass’ beyond the luminous component, required to gravitationally bind the galaxies in the cluster (Zwicky, 1933). Zwicky’s original analysis counted only the stellar component of the galaxies, missing the baryonic, hot intracluster medium. However, more recent X-ray and gravitational lensing (described in Section 1.6.5) measurements of galaxy clusters bolster these initial observations, concluding that the visible components of the largest virialised systems account for only 15 percent of the total mass. Perhaps the strongest support yet for the existence of dark matter comes from the Bullet Cluster (Bradaˇcet al., 2006), shown in Figure 1.3, which is the collision of two galaxy clusters. A gravitational lensing analysis allows the ability to locate the total mass of the Bullet Cluster and it reveals that the majority of the mass is located around the galaxies (represented in purple), not in the gas in between (pink), which is identified by X-ray observations. While this rare event provides evidence that dark matter cannot be accounted for by a non-luminous gas, it also shows that dark matter is weakly-interacting as the purple dark matter halos pass through

14 Figure 1.2 Constraints in the ⌦ ⌦ plane proposed by the latest Planck m ⇤ results, shown along with the Planck lensing reconstruction and the BAO. The Universe is tightly constrained to be flat Planck Collaboration et al. (2016a). Image taken from the NASA press release, with observations by Chandra (X-ray), the Hubble Space Telescope and Magellan (optical). one another, unlike the hot gas.

CMB acoustic peaks: The peaks in the CMB power spectrum are sensitive to the dark matter density in the Universe. In the absence of dark matter, the CMB power spectrum would show much stronger oscillatory features, combined with the e↵ect of radiation pressure on the photon-baryon plasma at early times. The suppressed amplitude of oscillatory features suggests that the height and position of the acoustic peaks are the result of a dominant pressure-less component (Bennett et al., 2003). This is because for a flat universe without dark matter, therefore with the total matter density comprised of baryons, the e↵ect of baryonic acoustic oscillations would be more pronounced as the dark matter doesn’t feel the e↵ect pressure and contribute to the e↵ect. That is, gravity from baryons alone could not have modulated the temperature variations much beyond the first peak in the power spectrum. Therefore, an abundance of cold dark matter was needed to keep the gravitational-potential wells suciently deep. By measuring the ratios of the heights of the first three peaks, the baryonic fraction of matter 2 is determined to be ⌦bh =0.02 (Planck Collaboration et al., 2016a).

15 The search for a dark matter candidate within the ⇤CDM model has, thus far, proved unfruitful. Several candidates seem unlikely: compact brown and white dwarf stars, or massive astrophysical compact halo objects (MACHOs), do not satisfy the theory that dark matter must be comprised of non-baryonic matter, ruled out by mi crolensing (Wyrzykowski et al., 2011); neutrino constraint from the existence of small-scale structures render them insuciently massive to satisfy the observational constraints of (⌦ , ⌦ ) (0.3, 0.05). Aside from the matter m b ⇡ density constraint and the requirement to be non-baryonic, the dark matter particle must not couple electromagnetically and must be weakly interacting, satisfying observations of the Bullet cluster. Furthermore, to ensure that the extent of small-scale structure suppression matches observations (that is, not suppressed as indicted by constraints on massive neutrinos from Beutler et al. (2014)), the candidate must be suciently massive that the thermal velocities are e↵ectively zero and it is ‘cold’. A weakly interacting massive particle (WIMP) and an axion remain viable candidates (see Arcadi et al., 2018, for a review).

1.4.2 The Cosmological Constant

The history of the idea of a cosmological constant is a turbulent one. General relativity intuitively predicts that for a Universe comprised of only matter and radiation, gravity hinders its expansion. The constant was first proposed by Einstein in 1917 in order to realise a static Universe against the e↵ects of gravity and then later dismissed to satisfy an expanding universe. Ironically, it was reintroduced to his equations with the recent observations indicating an accelerated expansion.

The ⇤gµ⌫ term, introduced in Section 1.2 in Einstein’s Field equation ( 1.3) is equivalent to the energy-momentum tensor of a perfect fluid. That is, the cosmological constant is characterised by its constant energy density, ⇢c2 and a negative pressure component, p in an equation of state defined as w = 1, satisfying p = ⇢ c2 . (1.48) v v Due to the fact that the energy density of the cosmological constant is unchanging with time, whereas the energy density due to matter dilutes as the Universe expands, it only began to have a significant contribution to the expansion of the universe relatively recently. The negative pressure is responsible for the repulsive gravity e↵ect. Thus, when the cosmological constant dominates the total energy

16 density of the universe, it gives rise to an acceleration.

The cosmological constant is interpreted as the gravitational signature of the quantum zero-point energy of a vacuum, hence it is often analogous with the vacuum energy ⌦v =⌦⇤. Quantum field theory stipulates that a vacuum is filled with virtual particles that can appear into existence and then annihilate almost instantaneously, giving rise to a vacuum energy with a negative pressure (which is classically infeasible). This energy permeates space homogeneously and has no preferred direction and is therefore, Lorentz invariant. Prior to GR, the absolute energy was irrelevant as Newtonian gravity is determined by only changes in the potential energy, 2⇢. Dimensional analysis dictates that the energy density r 18 theoretically lies on the order of the Planck scale, ⇤= MPl ~c/(8⇡G) 10 ⌘ ' GeV, which yields p 18 4 ⇢v = 10 GeV , (1.49) in natural units (Bousso, 2012).

The Friedmann equation (equation 1.13) predicts a deceleration for the radiation dominated epoch (R(t) t1/2) and the matter dominated (R(t) t2/3), while in / / a flat ⇤dominated universe, there is acceleration according to

R exp(Ht) , (1.50) / and 8⇡G⇢ ⇤c2 H = v = . (1.51) r 3 r 3

1.4.3 Dark Energy

The cosmological constant, ⇤, associated with the vacuum energy, is the least exotic model for dark energy and a key component of the Standard Cosmological Model. In the 1990’s, contradictory evidence emerged: the large scale structure of the universe determined a matter density that was too small to give rise to a flat universe, as was required by data from the cosmic microwave background (Efstathiou et al., 1990). Following that, two independent sources observed Type 1a supernovae to be fainter than expected, revealing an accelerating expansion of the Universe (Riess et al., 1998; Perlmutter et al., 1999). The term ‘dark energy’ was coined to account for the repulsive e↵ect that drives the acceleration. While many competing ideas for the nature of dark energy quickly arose, the

17 Figure 1.3 Constraints in the ⌦ ⌦ plane proposed by the latest Planck m ⇤ results, shown along with the Planck lensing reconstruction and the Baryon Acoustic Oscillations. The Universe is tightly constrained to be flat (indicated by the dashed line). Figure reproduced from Planck Collaboration et al. (2016a). cosmological constant is the most successful. However, almost a century later, persisting flaws hinder the credibility of a cosmological constant.

The current observational constraints for these energy density parameters include the supernovae measurements over a broad redshift span (Section 1.6.3), measurements from the large-scale structure (Section 1.6.2) and the temperature anisotropies of the CMB (Section 1.6.1). Figure 1.3 shows the most recent and most precise constraints on the matter and dark energy density parameters, ⌦m and ⌦⇤, respectively, from the Planck Collaboration et al. (2016a), combined with the CMB lensing reconstruction and measurements from Baryon Acoustic Oscillations (Anderson et al., 2014), described in Section 1.6.2. They find, according to a combined posterior derived from also including supernovae and local H0 measurements, the value of the dark energy density as, ⇢v = 12 4 (10 GeV) , or equivalently a vacuum density parameter of

⌦ =0.6911 0.006 . (1.52) ⇤ ±

18 Thus, there is a discrepancy between the theoretical prediction and the observed value of the dark energy density of up 120 orders of magnitude. Weinberg ⇠ (1989) named the enormous inconsistency the ‘cosmological constant problem’. 2 The inferred magnitude of ⇢vc is surprising, or rather, that the observation of a magnitude very much less than the QFT prediction was found, should be expected. The e↵ect of the cosmological constant is a strong suppression of the 2 rate of gravitational collapse, therefore, if ⇢vc was of the magnitude predicted by QFT, we would neither be in existence nor able to measure it (Weinberg, 1987). The more serious quandary was that the observed magnitude was just right for ⇤ domination to occur today. If the cosmological constant problem could be solved by fine-tuning, the need remains to explain the small, non-zero, ⌦v observed and why matter-vacuum equality should occur today.

1.5 Alternative models

1.5.1 The Cosmological Not-Constant

In the absence of a compelling theory, deviations of the background expansion from that of w = 1 are sought. Aside from the cosmological constant, dark energy can be introduced into a cosmological model either by adding a new form of matter-energy to the Universe or by modifying the laws of gravity. One can consider a time-dependent, two-parameter linear model, w(a)=w +(1 a)w , 0 a where a is the cosmic scale factor, w0 is the dark energy equation of state at present and w = dw/da, where a = 1 (Chevallier & Polarski, 2001). a In 1988 a new model for dark energy, “quintessence” (Ratra & Peebles, 1988), emerged by the inclusion of a spatially-homogeneous scalar field that can vary with time, of which the Higgs boson is the first detected example (Higgs, 1964). This gives rise to a dynamical dark energy. By incorporating this new degree of freedom denoted , the vacuum energy density can manifest itself as a scalar

field energy density, ⇢(t). For a scalar field defined by the Lagrangian density, 1 µ L = 2 @ @µ V [(t)], the stress-energy tensor takes the form of a perfect fluid, 2 which specifies the density and pressure in terms of a kinetic term, (˙t) /2, and a potential energy of the field, V [(t)]. Thus, the scalar field dark energy model

19 is characterised by a time-dependent equation of state,

p ˙2/2 V () w(z)= = . (1.53) ⇢ ˙2/2+V ()

In the case of a slowly-varying scalar field, the scalar field energy density is dominated by its potential, ⇢ V [(t)] yielding an equation of state that ⇡ resembles that of a cosmological constant,

V () w = 1. (1.54) ⇡ +V ()

1.5.2 Modified Gravity

An alternative to the inclusion of a new matter-energy component for dark energy, either constant or varying, would be to modify General Relativity itself. In terms of perturbation theory, deviations from GR can be characterised as an addition of parameters to the Poisson equation and the separation of the potentials into two independent quantities. Thus, the original parameterised potentials shown in equations 1.36 and 1.37 are modified to two distinct equations in Fourier space (Amendola et al., 2007):

2 2 k (k, a)=4⇡Ga [1 + µ(k, a)]¯⇢m m(k, a); (1.55)

e f 2 2 k [(k, a)+ (k, a)] = 8⇡Ga [1 + ⌃(k, a)]¯⇢m m(k, a). (1.56)

The generalisatione of the twoe introduced functions, µ and ⌃f, account for a range of modified gravity models, namely ‘fifth force’ models such as f(R) models (de Felice & Tsujikawa, 2009), ‘DGP ’gravity (Dvali et al., 2000) and Galileon models (Nicolis et al., 2009).

While modified gravity models exist, General Relativity is still observed to hold true on the scale of the solar system. As such, any modifications to gravity must be ‘shielded’ in these environments in order to be consistent with the previous constraints. Some existing shielding mechanisms’ are the ‘chameleon e↵ect’, which makes the scalar short ranged, as well as the ‘symmetron mechanism’ and ‘Vainshtein screening’ which both make their kinetic term large to suppress coupling to matter (see Clifton et al., 2012, for a review). Consequently, modified gravity theories must be formulated such that on cosmic scales, the Universe

20 accelerates. For large scale structure, gravity is modified by some fifth force and on small scales, GR is recovered. As there is no leading candidate for a modified gravity model from the vast selection, from an observational perspective it is logical to adopt a model-independent phenomenological approach, such as that shown in equation 1.56.

The recent detection of a gravitational-wave (GW) and a coincident electromag- netic signal from the merging of two neutron stars (Abbott et al., 2017) introduces strong challenges to the observational viability of large classes of gravitational theories for the late Universe. This is because any deviation of the speed of GWs from that of light is now tightly bound, which implies that any dynamics of modified gravity that cause a change in the speed of propagation are limited, reducing the viable theory space. For an assessment of the status of the field, see Amendola et al. (2018).

GR modifications a↵ect the way that the expansion history of the Universe is related to the growth of matter clustering. As such, any discrepancies between observational probes of expansion and of the growth of large scale structure would hint at a deviation from GR. Similarly, modified gravity models can alter the relationship between gravitational lensing and matter clustering, which is controlled by the two Bardeen potentials.

While in standard GR, in a FRW Universe and in the absence of anisotropic stress, the potentials are postulated to be equal, in modified gravity theories, such as f(R) gravity theory, there is a systematic discrepancy between the two, known as a ‘gravitational slip’, ⌘, defined as the ratio of the Bardeen potentials (introduced in equation 1.33), as

⌘ = . (1.57)

1.6 Cosmology measurements

Contemporary cosmology remains consistent with a single framework of con- cordance, ⇤CDM, as the underlying model of the Universe, as described in Section ??, dominated by dark matter and dark energy. This framework can be described by six cosmological parameters. Modern cosmological measurements are driven by the quest to pin down these parameters and ultimately, explain the

21 observations of an accelerated expansion of the Universe.

In order to reveal the properties of dark energy, a combination of probes is demanded as di↵erent probes test di↵erent combinations of parameters and contain di↵erent systematics, breaking degeneracies in the parameter space. Furthermore, we need probes of both the cosmic expansion history, such as Baryon Acoustic Oscillations (BAO) and supernovae (SNe), as well as probes of the growth history like redshift space distortions and weak lensing. This is because, with one probe only, it would be impossible to accurately distinguish between di↵erent physical models of dark energy as these could be interpreted as dark energy or modified gravity. Constraining both sides of the equations breaks the degeneracy as the homogeneous expansion history as well as the growth of perturbations within the expansion are investigated. BAOs and SNe have a↵orded measurements of the expansion rate of the Universe up to a precision of one percent. On the other hand, growth measurements are at best, an order of magnitude poorer. Accordingly, it is the opportune time for redshift space distortions and weak lensing, which I will now describe, to be investigated, with the latter only now being exploited.

1.6.1 CMB

In the era of radiation domination, prior to decoupling, baryons and electrons are tightly coupled to photons due to Thomson scattering and the Universe is opaque. The radiation pressure prevents the collapse of the initial fluctuations in density and pressure on sub-horizon scales, which are seeded by random quantum fluctuations, below a limit. As the radiation pressure expels the baryons outwards against the gravitational e↵ect of recollapse, oscillations are set in motion which represent the battle between the e↵ects of the pressure and gravity on the initial adiabatic perturbation. These acoustic oscillations act as the sources for sound waves that propagate in the photon-electron-baryon plasma of the early Universe, with a speed approximately c/p3.

As the Universe cools suciently, electrons and ions recombine to neutral atoms, and decoupling occurs. That is, photons are freed from baryons and sound speed drops dramatically. An excess of matter was left both at the source of the wave and at the surface where these waves terminated. The matter excesses at these locations left their imprint on the large-scale structure of galaxies and hydrogen gas. The comoving size of the sound horizon at decoupling is an important

22 Figure 1.4 On large scales, cosmology assumes that the Universe is isotropic, but on small scales, observations by the Planck satellite reveal temperature anisotropies in the cosmic microwave background of 5 magnitude 10 ,atthedegreesize,withacomovingsizegivenby the sound horizon at decoupling. It is thought that these temperature perturbations reflect tiny density perturbations seeded by small-scale quantum fluctuations that were amplified by an inflationary period. Image credit ESA and the Planck Collaboration.

23 Figure 1.5 The amplitude of the Planck 2015 Cosmic Microwave Background (CMB) temperature power spectrum fluctuations, or temperature TT variance, D` ,asafunctionofscale,` measured to exquisite precision (blue dots). Given a theoretical model assuming the ⇤CDM parameters, a prediction for this function is made (red curve). The lower panel shows the ‘residual’ between the data and the model (defined by the parameters given in Table 1.1), revealing exquisite agreement. Image taken from the Planck Collaboration et al. (2016a).

24 ‘standard ruler’ - an observable of known comoving size that is established from basic physical processes. Post-decoupling, in the matter domination period, baryons fall into dark matter potential wells (and to some extent, vice versa), which have continued to grow.

The Cosmic Microwave Background (CMB) is the relic blackbody spectrum from the radiation dominated era, created from the surface of last scattering at z ⇡ 1100. It has redshifted with the expansion

T (z)=T0(1 + z) (1.58) and the observed temperature today is found to be T =2.718 0.021K 0 ± (Planck Collaboration et al., 2016a). At the moment when matter domination had just begun, the recombination of ionised particles created during Big Bang nucleosynthesis first caused the Universe to become neutral and transparent to radiation because light was no longer being scattered o↵free elections. The Planck satellite observed this radiation field and determined that it can be described by a thermal Planck spectrum across the sky, after the subtraction of Galactic foregrounds (Planck Collaboration et al., 2016a). They have shown to extremely high precision that it is isotropic on large scales, but with temperature 5 variations of the order T 10 on small scales, as shown in Figure 1.4. With ⇠ the onset of the formation of structure, matter-radiation coupling dampened temperature fluctuations, causing temperature shifts in the radiation on small scales and radiation climbing out of gravitational potential wells creating large scale temperature shifts. The measurement of the temperature power spectrum of the CMB is shown in Figure 1.5 as the temperature variance, D`, which can be described by d (T/T)2 C D = h i = `(` +1) ` . (1.59) ` d ln ` 2⇡ 1 as a function of angular scale, ✓ ` , where C is the angular power spectrum, ⇡ ` the corresponding dimensionless spatial statistic, found by taking the Fourier transform of the CMB power spectrum. The positions and amplitudes of the peaks seen contain some of the most precise measurements of cosmological information. The position of the main peak, ` 220, corresponds to the angular ⇡ size of the sound horizon at last scattering, which is of a known comoving size. When the angular scale is combined with matter and baryon densities, the sound horizon and physical scale of the acoustic peaks can be inferred. This gives the distance to last scattering, which depends on curvature and dark energy. For a

25 Parameter Value Error H 67.8 0.9 0 ± ⌦⇤ 0.69 0.01 ⌦ 0.31 ± 0.01 m ± 8 0.815 0.009 n 0.968 ±0.006 s ±

Table 1.1 Cosmological parameter constraints from the CMB analysis by Planck Collaboration et al. (2016a). Results are best fit values taken from Planck TT+lowP+lensing alone, assuming a flat ⇤CDM model. constant dark energy w = 1 case, flatness can be accurately measured. This probe, with unprecedented precision from the Planck experiment, provides the benchmark measurements of many cosmological parameters in the flat ⇤CDM framework, which are listed in Table 1.1. While the primary fluctuations of the CMB power spectrum are only weakly sensitive to dark energy, secondary e↵ects like the Integrated-Sachs-Wolfe e↵ect and the lensing of the CMB by large-scale structure provide more constraining power on that parameter.

1.6.2 Large-scale structure

Complementary to the CMB, low-redshift probes from spectroscopic data provide a wealth of information, such as BAO, galaxy clustering and redshift-space distortions (Blake & Glazebrook, 2003; Eisenstein et al., 1999).

BAOs are the imprint of oscillations in the photon-radiation field, or the pressure- gravity battles, that occurred up to the point of recombination, at which point they froze due to decoupling. These battles are scarred as oscillations on the CMB matter power spectrum and furthermore, their pattern is imprinted in the three-dimensional distribution of large-scale structure. While the first BAO peak measured in the CMB matter power spectrum corresponds to the angular diameter distance of the sound horizon at matter-radiation equality, it shows up as excess correlations in the matter density at the characteristic distance of the sound horizon at decoupling, at the redshift of the matter. By comparing the two, it is possible to infer cosmology through angular diameter distances.

More specifically, before a sound wave was frozen at recombination, it traveled a comoving distance s 150Mpc (computed from CMB-given cosmological ⇡

26 parameters). Viewed transversely, the 150Mpc ruler subtends an angle, ✓, related to the angular diameter distance, DA(z), to an object at redshift, z,by

z cdz s =(1+z)D (z)✓ = ✓ 0 (1.60) A H(z ) Z0 0 given a flat Universe. As a result, enhanced correlations of galaxies, as tracers of the matter distribution, as measured by spectroscopic surveys are observed for galaxy pairs separated by z, such that s cz/H(z). This provides a ⇡ means of measuring both the angular diameter distance, DA(z), and therefore the integrated expansion history, as well as the Hubble expansion rate, H(z). Baryon Oscillation Spectroscopic Survey (BOSS) most recently measure the Hubble parameter today from BAO in conjunction with supernovae data to be 1 H =67.3 1.0kms Mpc 1 (Alam et al., 2017b). The reduced availability of 0 ± galaxy spectroscopic redshifts reduced beyond z 2 limits high-redshift BAO ⇠ measurements with galaxies as the tracer. Interestingly, the Ly-↵ forest provides the possibility to resolve this and probe the expansion history at higher redshifts (Croft et al., 2016).

Mapping the large-scale structure (LSS) of the Universe is possible by considering the clustering of galaxies, measuring either the angular projected 2-dimensional distributions or the 3-dimensional mapping in redshift-space (Peacock et al., 2001). In a perfect FRW universe, the observed spectroscopic redshift would provide an unbiased estimate of the radial comoving position. However, measurements in redshift space are a↵ected by redshift-space distortions (RSD), as illustrated in Figure 1.6. That is, the observed redshift of a galaxy, zobs, has contributions from the Hubble flow, which gives a cosmological redshift, zcos,as well as peculiar velocities, which arise from the gravitational collapse of large-scale structure, zpec (Kaiser, 1987),

(1 + z ) (1 + z )(1 + z ) . (1.61) obs ⇡ cos pec

Peculiar velocities are simply deviations in the motion of galaxies from the Hubble flow due to the gravitational attraction or repulsion to local over- or under-densities. For non-relativistic peculiar motion, czpec = vpec. For density perturbations given by equation 1.34, the peculiar velocity, v(r) is directly proportional to the gravitational acceleration.

As the two contributions to the observed redshift cannot be distinguished, galaxies are said to be ‘redshift-space distorted’. On large scales, the correlated inflow of

27 Figure 1.6 Redshift space distortions in a mock galaxy catalogue with ⇤CDM cosmology. The left-hand side shows the distribution of galaxies, in real-space, for a patch of sky out to 800Mpc. On the right-hand side is the mirror image of the same galaxies, but in redshift-space. Image taken from the Heymans (2017)

28 matter to overdense areas compress objects along the line-of-sight. Conversely, coherent outflow from underdense areas stretch the objects in the line of sight direction (Peacock et al., 2001). On smaller scales, velocity dispersions in collapsed objects stretch structures along the line of sight. As this interferes only with the redshift and not the galaxy’s position on the sky, the object is only stretched radially. This is exemplified by the ‘Fingers-of-God’ e↵ect, which describes the line-of-sight stretching of galaxy over-densities due to their virialised motion (Jackson, 1972). Therefore, the distribution of galaxies in redshift-space is distorted in overdense regions in a way that the density field along the line-of- sight is compressed, which amplifies the large-scale clustering in an anisotropic matter. While the e↵ect of RSD hinders the determination of a true distance, it provides a means of measuring the growth of structure over time.

The anisotropic e↵ect of the redshift space distortion is to introduce a directional dependence and remove the isotropy of the power spectrum from Pgg(k)to

Pgg(k, µ). Therefore, the redshift-space power spectrum under the assumption of linear theory is 2 2 2 Pgg(k, µ)=b (1 + µ ) Pmm(k) , (1.62) where Pmm is the real-space matter power spectrum and µ is the cosine of the angle of the Fourier mode to the line-of-sight (Kaiser, 1984). It is assumed that measured galaxies trace the total underlying matter through linear biasing on large scales with a proportionality constant of b. That is, the galaxy overdensity

g, varies linearly with the underlying matter density m,

f .v = f = = . (1.63) r m b g g

The factor is introduced as a redshift-space distortion parameter which governs the clustering amplitude on the angle to the line-of-sight. This factor is defined as, f/b, (1.64) ⌘ where the growth rate of structure, f(⌦m), can be expressed in terms of the growth factor D(a) at a particular cosmic scale factor, a and the growth factor is defined in terms of the amplitude of a single perturbation as (a)=D(a)(1) in equation 1.42.

Measurements of this quantity have revealed cosmological parameters that are consistent with constraints from other galaxy surveys and the cosmic microwave

29 Figure 1.7 The (⌦m, ⌦v) posterior from the JLA compilation of 740 SN1a extending to z =1.Theplottedcontoursshowthe68% and 95% confidence contours assuming a the ⇤CDM model and non- zero spatial curvature. The blue contours represent the JLA SN1a compilation, green shows the combination of Planck temperature and WMAP polarisation measurements of the CMB fluctuation, red is the combination of the measurements with BAO analyses. The black dashed line corresponds to a flat universe. Image taken from Betoule et al. (2014). background data that point to the benchmark cosmological constant dark energy model in which the expansion rate accelerates at lower redshifts (Beutler et al., 2011; Blake et al., 2012; Betoule et al., 2014).

However, a shortcoming of this technique stems from the fact that galaxy surveys trace the light from baryonic matter, measuring the galaxy power spectrum and we do not fully understand how this translates to the matter power spectrum, despite advancements in our understanding from N-body and hydro-dynamical simulations. As such, these analyses are limited to large-scales by the galaxy bias parameter, b.

30 1.6.3 Type 1a Supernovae

As aforementioned, first indications of the acceleration came from Type 1a Supernova (SN1a) (Riess et al., 1998; Perlmutter et al., 1999). These are extremely luminous ‘standardisable candles’- a population of sources thought to possess a common intrinsic luminosity, provided the timescale over which their flux density decays is used to remove the dependence of the intrinsic luminosity on progenitor mass. This corrects for the stretch of their light curve and colour e↵ects. Measurements of their relative luminosity distances, DL(z) as a function of redshift (equation 1.32) can provide constraints on the matter and vacuum energy density parameters, (⌦m, ⌦v). By precisely mapping the distance-redshift relation up to redshift z 1, SN1 are, at this stage, the most sensitive probe of ⇡ the late-time expansion history of the universe. Observations of DL(z)ofover 700 SN1a from the JLA sample out to a redshift of z = 1 are shown in Figure 1.7, taken from Betoule et al. (2014). This analysis revealed that assuming a flat geometry, the Universe’s expansion is accelerating and there is no evidence of a dynamical dark energy. Combining these results with constraints from the CMB by Planck Collaboration et al. (2016a) and WMAP9 (Hinshaw et al., 2013), as well as BAO measurements at redshifts z =0.106, 0.35 and 0.57 from Beutler et al. (2011), Padmanabhan et al. (2012) and Anderson et al. (2012), respectively, reveal a posterior that is fully consistent with a flat universe and a dark energy equation of state parameter of w = 1.018 0.057. ±

SN1a can also be used to constrain the Hubble parameter, H0. Measurements of their relative luminosity distances, DL(z) at low redshift may be extended to high redshift, forming a ‘distance ladder’. The classical route to an accurate value of H0 currently has two distinct components: firstly the calibration of stellar luminosities and distances to nearby galaxies (traditionally using Cepheid variables, identified optically and anchored to geometric parallax measurements of stars within the Milky Way), extended to high redshift, thereby forming a ‘distance ladder’ by the calibration of SN1a providing further distance measures. Most recently, Riess et al. (2018) measured this parameter to be H =73.52 0 ± 1 1.62kms Mpc 1, using Hubble Space Telescope Milky Way Cepheids, calibrated against Gaia DR2 parallaxes.

31 1.6.4 Cluster abundances

Seeded by the most over-dense dark matter fluctuations in the early universe, clusters have grown in density over time by the evolution of their gravitational potential wells. Less dark matter in the Universe would lead to fewer massive clusters. Similarly, reducing the dark energy fraction in the Universe would result in fewer massive clusters today. A measurement of the cluster abundance in the Universe, or the number of clusters of a certain mass as a function of redshift is sensitive to the growth of structure in the Universe, more specifically, the normalisation, 8 and matter density parameter, ⌦m. This cosmological probe is challenged by the diculty in ensuring that all clusters are detected, as well as the means of measuring cluster masses, observable-to-mass relations. Samples of clusters are assembled based on several observables such as the density of galaxies in optical/IR observations, the projected density map measured by weak lensing, the X-ray emission from cluster hot gas, and the thermal Sunyaev Zel’dovich e↵ect Sunyaev & Zeldovich (1972).

1.6.5 Gravitational lensing

Einstein’s theory of GR predicts that as a light ray passes through a gravitational field, it experiences a curvature of its path, such that ‘mass bends light’. The confirmation of this prediction, discriminating between Newtonian gravity and GR, was facilitated by a solar eclipse in 1919, visible from both Principe, an island in the Gulf of Guinea and Sobral, Brazil. This eclipse took place when the Sun was directly in front of the Hyades star cluster. Arthur Eddington compared the observed stellar positions during totality with reference positions taken without the proximity of the sun and observed that the apparent positions of the stars were altered. The gravitational field of the sun had bent the path of the light rays from the Hyades stars. This result was bolstered by subsequent optical and radio observations to reveal that the observed angular shift in stellar positions matches the predicted alteration due to the mass of the Sun, in the context of GR. From this, ‘gravitational lensing’ was born, named due to the similarity of the e↵ect with that of an optical lens (see Section 2.1.1 for further details).

Gravitational lensing is the distortion of light from distant objects due to the ubiquitous large-scale density fluctuations of the matter distribution of the Universe. These perturbations or inhomogeneities in the large scale structure

32 create tidal gravitational fields that generate di↵erential deflections in light bundles. Photons, as relativistic test particles, are sensitive to both the curvature potential of space, produced by density perturbations, as well as the Newtonian potential, (equation 1.5). Their deflections induce an additional ellipticity on the galaxy image, called a ‘shear’ as well as a magnification of the image. These are statistically measured in the deep surveys by looking at the correlations mapped in the apparent shapes of background galaxies behind foreground structures. As galaxies are primarily comprised of dark matter, in our concordance model, lensing provides a direct probe of dark matter, allowing the invisible to be ‘seen’ and the mapping of the cosmic structure. Lensing is a powerful cosmological probe as it can detect mass independently of its nature and requires no information of how the luminous matter traces the underlying dark matter distribution, the galaxy bias. Furthermore, it allows the inference of the properties of dark energy, which suppresses the degree to which matter clumps due to gravity and the evolution of structure with time. Furthermore, unlike the other probes, it is most sensitive to the non-linear power spectrum.

Gravitational lensing has three applications. In this Chapter, I provide a brief introduction to each of them.

Microlensing

A brown dwarf, or massive dark object, passing in front of a background star, acts as a gravitational lens of the star light, causing a magnification. This transient e↵ect is detected as an amplification or a sharp peak in the intensity of the light from the star. As the alignment between the source, the lens and the observer changes, the apparent brightness of the source boosts and diminishes, creating a characteristic light curve. Microlensing is useful as a robust probe of the structure and mass distribution in the Milky Way (??) and it can be applied to the search for MACHOs and other dark transient objects, as well as to the search for extrasolar planets. A MACHO with a transient in front of a star would produce the characteristic light curve but it itself would not be seen. As such, a successful detection requires ruling out other potential causes of a change in brightness, for example the intrinsic variability of the star. The MACHO (Alcock et al., 2000), EROS and OGLE collaborations working to identify microlensing events in the Milky Way halo, set a limit on the contribution of compact bodies to the Galactic dark matter halo at less than 20 percent, contradicting the hypothesis that our

33 halo consists of MACHOs that are entirely comprised of dark matter dark matter (Hawkins, 2015).

Strong lensing

Strong gravitational lensing refers to the case when a massive object, for example a galaxy cluster, lenses background galaxies such that multiple sheared and magnified arc-like images of them are produced. These e↵ects are apparent for the case of galaxy cluster Abell 1689, shown in Figure 1.8. A modelling of the strong lens system allows for the determination of the surface mass density of the lens, and in the case where a quasar is multiply imaged, the determination of the Hubble constant through time delays. Strong lensing was first observed in strongly distorted galaxies a few decades ago (Walsh et al., 1979; Young et al., 1981; Hewitt et al., 1988; Soucail et al., 1988; Lynds & Petrosian, 1986). This technique has the power to constrain cosmology directly as the e↵ect is dependent on angular diameter distances, which are defined by the geometry of the Universe. By measuring the time delay between arcs originating from the same time-varying source, one can measure the value of the Hubble parameter today, H0 (see Bonvin et al., 2018, for a thorough summary).

Weak lensing

Weak gravitational lensing, was first detected (Tyson et al., 1990; Brainerd et al., 1996), as the small but spatially coherent distortions in the shapes of high-redshift galaxies. The model-independent aspect of lensing, along with the advancement of survey technology, motivates the recent boom of lensing research. There are several reviews on the topic (Mellier, 1999; Bartelmann & Schneider, 2001; Hoekstra et al., 2002; Mellier & van Waerbeke, 2002; Hoekstra & Jain, 2008; Kilbinger, 2015).

The weak lensing signal is of order of a few percent of the intrinsic ellipticity of the galaxies when due to another galaxy or ten percent for galaxy clusters. These measurements of the correlation between the distorted shape or ‘shear’ of background galaxies and the positions of the foreground structures are collectively called ‘galaxy-galaxy lensing’ or shear-position lensing. Through their shear, lensing galaxies imprint a weak, tangential distortion pattern on the images of the lensed galaxies in the vicinity. This technique is profitable as it can be used

34 Figure 1.8 Strong lensing around galaxy cluster Abell1689. Image credit to NASA/ESA/STSci. to understand the masses of the lensing galaxy halos and for mass reconstruction of foreground structures (Brainerd et al., 1996; Seitz & Schneider, 1997; Hoekstra et al., 1998). Figure 1.9 shows a depiction of the tangential distortion of some background galaxies’ light paths due to a foreground structure.

The measurement allows for the constraint of the potential depth and size, or density profile, of the dark-matter halos in which the lensing galaxies reside. From this one can learn about the total mass of the dark matter halo and in turn, derive relations between the halo mass and properties of the observed galaxy, known as the galaxy-halo connection. (see, for a review Wechsler & Tinker, 2018). More recently, galaxy-galaxy lensing is useful in constraining cosmology in conjunction with other probes (Mandelbaum et al., 2005).

35 Figure 1.9 Background shows an illustration of weak gravitational lensing. The light from distant galaxies travels passed massive structures of dark matter (shown here as grey spheres). The dark matter’s gravitational potential curves the local space-time and the path that the light follows, which distorts the images of the background galaxies as we observe them. The amount of distortion depends on the amount and distribution of dark matter along the light path. Galaxies that are near to each other in the sky therefore appear to have correlated alignment. Image credit; APS/Alan Stonebraker. The inset shows the idealised pattern of tangential shear seen in the background galaxies (turquoise) around an isolated, spherically- symmetric overdensity or foreground ‘lens’ (purple).

36 Figure 1.10 AnumericalsimulationfromS.Colomni(IAP)highlightsthe distribution of dark matter in the cosmic web, where the brighter regions have a higher density. The e↵ect of weak lensing is heavily dramatised to show how a propagating light ray is distorted and deflected along is path.

37 When the shear of a galaxy is e↵ectuated by the entire cosmic web, the phenomenon is referred to as ‘cosmic shear’. This is depicted in Figure 1.10, where a light ray’s path is distorted by the large scale structure. In an empty, flat universe, or in the case of complete density homogeneity, the light rays from distant galaxies would be undisturbed and follow a straight path. Otherwise, each ray experiences a small deflection by the gravitational fields of foreground structures, such that nearby galaxies appear coherently distorted according to the distribution of dark matter along their line of sight. For weak lensing, it is assumed that deviations from purely random galaxy orientations arise due to lensing.

Cosmic shear provides a statistical map of the cosmological structure, that is the voids, filaments and halos along the line of sight. Therefore, as it is a direct way to measure the total matter distribution of the Universe, it allows the ability to constrain cosmological parameters, specifically, the matter density parameter

⌦m, and the amplitude of clustering 8, though the two are partially degenerate Kilbinger et al. (2013). As such, a combined parameter is used to quantify lensing results (Jain & Seljak, 1997), defined as,

⌦m S8 = 8 . (1.65) r 0.3 With lensing, one can test the cosmological model through observations of large- scale geometry and the expansion rate of the Universe, as well as the formation of inhomogeneities in the Universe. Weak lensing by modern imaging surveys exploit both of these routes.

Cosmic shear is a measure of the correlation between galaxy shapes and alignments, or shear-shear lensing, as a function of their angular separation on the sky. Tomography, that is, making these measurements at di↵erent redshift slices, enables a further gain of information Hu & Tegmark (1999) and this can be extended to 3-dimensional lensing Heavens (2003); Heavens et al. (2006). An example of a 2D measurement, namely, the two-point shear correlation function, is shown in Figure 1.11, along with predictions for a ⇤CDM model with di↵erent values for ⌦m and 8. At smaller separations, galaxies’ alignments are more correlated as these galaxies light rays have been distorted by the same large scale structures. For a universe with a higher matter density parameter, and similarly a universe with a higher amplitude of clustering, there is more clumpy dark matter along the line of sight to cause a lensing e↵ect and therefore the

38 Figure 1.11 The cosmic shear signal, called a two-point shear correlation function, ⇠+,whichindicateshowcorrelatedgalaxiesalignments are, as a function of angular separation on the sky, ✓.Thedi↵erent lines indicate the predictions for a flat ⇤CDM cosmology with di↵erent values of the matter density parameter, ⌦m and clustering amplitude, 8.Thebluedatapointsrepresentameasurementfrom the Kilo-Degree Survey. Figure reproduced from (Heymans, 2017). signal is predicted to be stronger. The measurement, namely, the two-point shear correlation function, is on the order of 1 percent, smaller than the intrinsic, randomly-oriented galaxy ellipticities or “shape noise” and therefore demands the statistical treatment of millions of images of background galaxies in order to obtain small statistical errors.

Cosmic shear measurements have advanced since the first detection (Van Waerbeke et al., 2000; Bacon et al., 2000; Wittman et al., 2001; Rhodes et al., 2001), to the state-of-the-art imaging surveys today that are at the level of precision to be competitive with other cosmological probes. Along the way, the earliest ‘Stage 1’ surveys measured cosmic shear over tens of square degrees, such as SDSS (Hirata et al., 2004), VIRMOS-Descart (Van Waerbeke et al., 2005), GEMS (Heymans et al., 2005a), CTIO (Jarvis et al., 2006) and COSMOS (Massey et al., 2007; Schrabback et al., 2007). Stage 2 surveys - deeper and now at the order of a hundred square degrees- include the Canada-France Hawaii Telescope Lensing Survey (CFHTLenS; Heymans et al., 2012b), Deep Lens Survey (DLS; Jee et al., 2013), SDSS (Lin et al., 2012; Hu↵& Graves, 2014) and the Red Cluster Squence Lensing Survey (RCSLenS; Hildebrandt et al., 2016b). The on- going Stage 3 generation survey at least 1000 square degrees and constraining cosmological parameters with power competitive with other cosmological data: Kilo-Degree Survey (KiDS; Kuijken et al., 2015), the Dark Energy Survey (DES;

39 DES Collaboration et al., 2017a) and the Hyper Supreme-Cam (HSC; Aihara et al., 2017) survey. Growing observational power has necessitated an evolution of methodology to ensure that both observational and astrophysical systematic errors do not dominate or bias the measurements. These ‘systematics’ can arise from instrumental e↵ects (for an overview, see Mandelbaum, 2015), the measurement of coherent shears from images and the estimation of redshift distributions from the photometric redshifts, as well as from the insucient understanding of the astrophysics of galaxies.

1.7 Concordant cosmology?

It is clear that cosmological parameters measured by Planck Collaboration et al. (2016a) are measured to great precision and find exquisite consistency with the ⇤CDM cosmological model. However, these results di↵er with preliminary ones from SPT (Henning et al., 2018) at a 2 level, as well as those from ⇠ the previous CMB experiment, WMAP9 (Hinshaw et al., 2013), stemming from di↵erences in the ⇤CDM model fits to their angular power spectra. Translation of these di↵erences to the set of 6 cosmological parameters reveal a6-level discrepancy, as reported by Larson et al. (2015). Furthermore, some measurements using other probes appear to be in tension with the cosmological parameters predicted by Planck. Examples include local measurements of the Hubble constant (see Freedman, 2017, and references therein), cluster abundances (Planck Collaboration et al., 2016d), the high redshift BAO scale measurements from the Ly↵ forest **cb do you know this ref**(?) and the amplitude of the mass fluctuations inferred from weak galaxy lensing. It is worth noting that it is quite remarkable that measurements of the very early Universe, since the surface of last scattering of the CMB, even resemble those from the Universe after 13.8 billion years of evolution. If the model that propagates the cosmological parameters forward is correct, we expect predictions from measurements at di↵erent epochs to be the same.

With new methods and technology, the accuracy in measurement of the Hubble constant has vastly improved. However, strong lensing results reveal a value for this parameter that is consistent with supernovae measurements but noticeably higher than that predicted by Planck and other early universe measurements. This 2 3 ‘tension’ in results is coherently summarised in Figure 1.12 and in ⇠ Freedman (2017) for supernovae and early universe techniques.

40 Probe Experiment H0 Source CMB WMAP 70.0 2.2 (Bennett et al., 2013) CMB Planck 66.9±0.6 (Planck Collaboration et al., 2016a) CMB SPTpol 71.3±2.1 (Henning et al., 2018) SN1a SH0ES 73.2±1.7 (Riess et al., 2016) Time-delay H0LiCOW 71.9±2.7 (Bonvin et al., 2017) BAO 67.4±1.3 (Aubourg et al., 2015) lensing+ BBN + BAO DES-Y1+BBN 67.2±1.2 (DES Collaboration et al., 2017b) ±

Table 1.2 Measurements of the Hubble constant.

This recent tension either hints at new physics or the presence of unrecognised uncertainties in the techniques. For supernovae measurements, the calibration of the extragalactic distance scale has recently been bolstered by the European satellite, Gaia, establishing enhanced accuracy by eliminating challenges with the absolute zero-point calibration of the extragalactic distance scale, via geometric parallax measurements of Cepheids in the Milky Way (Riess et al., 2018). However, it is possible that local measurements of the Hubble parameter su↵er from unknown systematic errors. For the distance ladder methods, systematic e↵ects may arise from metallicity variations and photometry biases in high-density regions of the sky. Similarly, it is plausible that with systematics are present in the CMB analysis, currently observed discrepancies between measurements at large spatial scales (low `) and small spatial scales (high `) are not well understood (Addison et al., 2016) (Planck Collaboration et al., 2017).

It is notable that this trend, albeit less significantly, crops up in other cosmology probes: the number of clusters inferred from reconciling the Planck primary CMB parameters with the Planck SZ cluster counts is in 2 disagreement ⇠ with that obtained by weak-lensing calibrations of Planck SZ cluster masses (Planck Collaboration et al., 2016d; Makiya et al., 2018). While this is not a significant di↵erence at present, it will prove interesting as the precision of cluster measurements improves with the current Atacama Cosmology Telescope (ACT; Thornton et al., 2016), the South Pole Telescope (SPT; Carlstrom et al., 2011) experiments.

Bearing some similarity to measurements of the Hubble parameter, some recent tomographic cosmic shear results from the CFHTLenS (Heymans et al., 2012b), KiDS (Hildebrandt et al., 2017) and DES (Troxel et al., 2017) find varying degrees of shifts or ‘discordance’ with their results from those of Planck (Planck

41 Figure 1.12 Recent measurements of the Hubble parameter as a function of the publication date. Measurements shown in blue were determined from the late-time universe, employing a calibration bared on the Cepheid distance scale, whereas measurements in red represent those which involve the CMB. The most recent measurements show atensiongreaterthan3.FiguretakenfromFreedman(2017).

Collaboration et al., 2016a) reporting consistently lower values for the S8 parameter. Almost all lensing analyses, (bar that from Jee et al., 2013), point to a universe that is less matter-dense and less matter-clumpy today, than that predicted by Planck. Figure 1.13 demonstrates this for both 450 square degrees of KiDS data (KiDS-450) and CFHTLenS, with the level of discrepancy reported at 2. ⇠ Furthermore, lensing surveys have evolved from cosmology with cosmic shear to ‘combined-probe’ analyses, incorporating galaxy-galaxy lensing and galaxy clustering or redshift-space distortions measurements (van Uitert et al., 2017; Joudaki et al., 2018; K¨ohlinger et al., 2017; DES Collaboration et al., 2017a; Amon et al., 2017) in order to break degeneracies in cosmological parameter space. Figure 1.14 shows that in most cases, these combinations do not resolve the small discrepancies in the parameters from those predicted by Planck, for

42 Figure 1.13 (a) Constraints on the cosmological parameters ⌦ and m 8 (b) ⌦ S ,obtainedfromcosmicshearcorrelationfunctions m 8 measured using 450 square degrees of data from the Kilo-Degree Survey (blue), over-plotted on the results from the Canada-France- Hawaii Telescope Lensing Survey (grey) and independent CMB constraints from CMB (red) and prior to Planck, taken from Hildebrandt et al. (2017).

Figure 1.14 (a) Constraints on the cosmological parameters ⌦m-8,obtained from combined measurements of cosmic shear (green), galaxy- galaxy lensing and redshift-space distortions, from 450 square degrees of data from the Kilo-Degree Survey and the overlapping BOSS and 2dFLenS spectroscopic surveys (purple), overplotted on the results from independent CMB constraints from Planck, taken from Joudaki et al. (2018). (b) Similarly, from the Dark Energy Survey (DES Collaboration et al., 2017a).

43 Method Experiment S8 Source CMB Planck-TT+LowP 0.852 0.024 Planck Collaboration et al. (2016a) CMB Planck-TT+LowP 0.821 ± 0.028 Spergel et al. (2015) CMB WMAP-9 0792 ± 0.052 Hinshaw et al. (2013) WL KiDS-450 0.745 ± 0.039 Hildebrandt et al. (2017) WL DES 0.782 ± 0.039 Troxel et al. (2018) ± WL KiDS-450 0.782 0.039 K¨ohlinger et al. (2017) WL CFHTLenS 0.737 ± 0.027 Joudaki et al. (2017a) 3 2 KiDS+BOSS+2dFLenS 0.742 ± 0.035 Joudaki et al. (2018) 3⇥2 KiDS+GAMA 0.801 ± 0.027 van Uitert et al. (2017) 3⇥2 DES 0.782 ± 0.027 DES Collaboration et al. (2017a) ⇥ ±

Table 1.3 Measurements of the S8 parameter either KiDS-450 or the first year of DES data, DES-Y1. It is noteworthy that current weak-lensing analyses, as well as supernovae experiments, have attempted to shield themselves against experimenter’s bias that may hold a preference to agree or disagree with Planck through redundancy and ‘blinding’. Both KiDS and DES have taken multiple approaches for the challenging steps of the analysis, in order to check for consistency and have created fake contaminated datasets in order to conduct a ‘blind’ analysis. In Table 1.3, we summarise the measurements on this parameter from several techniques and teams. While these di↵erences are not necessarily significant, the consistency in weak-lensing measurements preferring a lower S8 is interesting and heightens anticipation for the growing lensing datasets. This increased precision necessitates an enhanced understanding of the current challenges to lensing datasets: are we measuring a gravitationally induced alignment of galaxies, or are they intrinsically aligned? Do we understand the e↵ect of baryons on the non-linear scales or the matter power spectrum and could this e↵ect be responsible for any discrepancy in observations? Are we doing our data analysis incorrectly, in our assumption of a Gaussian likelihood (Sellentin et al., 2018)? In addition to theoretical systematics, a shift in cosmological parameters is degenerate with measurement ‘systematics’, such as those that arise in estimating the redshift distribution of the lensed galaxies (for example, Hildebrandt et al., 2017), in understanding the selection biases that arise when eliminating galaxies that are too dense or ‘blended’ for their shapes to be distinguished (Harnois-Deraps et al., 2018), in calibrating the shear of galaxies and in understanding and correcting for the e↵ect of the point-spread function from the atmosphere and telescope. It is possible that the shift away from Planck’s

44 cosmological parameters may be resolved by future lensing experiments due to improvements in methodology that addresses either theoretical or observational systematics.

The apparent discrepancies in measurements of cosmological probes may signal the need for an alternative physics to connect our understanding of the very early universe and late-time universe, or indicate unrecognised systematics in either one or both of the probes, or a statistical fluke or cosmic variance. It is conceivable that the standard model requires new physics that can simultaneously resolve these discordances. Some possibilities include a decaying dark matter, an evolving dark energy, an additional source of dark radiation in the early Universe, modified gravity or deviations from flatness. An additional neutrino or tensor modes other relativistic species, would have the e↵ect of increasing the expansion rate at early times, and could explain the discrepancy. However, the analysis of the Planck Collaboration et al. (2016b) CMB experiment with the addition of external probes does not favour any of these new additions. Reanalyses of CFHTLenS with massive neutrinos or tensor modes also failed at significantly resolving its tension with Planck (MacCrann et al., 2015). On the other hand, Di Valentino et al. (2016) find that a phantom-like dark energy component would resolve tension between the Planck dataset and the supernovae measurement by Riess et al. (2016) in a combined analysis. Similarly, investigations into extended cosmological models that resolve the tensions with Planck and the KiDS cosmic shear analysis (Joudaki et al., 2017b) also found a phantom dark energy seemed plausible. However, for the KiDS combined-probe analysis (Joudaki et al., 2018), a tightening parameter space from the galaxy-galaxy lensing and redshift-space distortions meant that this model was no longer favoured in a model selection sense.

With no one non-⇤CDM theory providing hope of an attempt at reconciliation between the cosmological measurements, datasets are expanding for enhanced precision in order to understand if these tensions persist and gain insight into dark energy, the cornerstone of the ⇤CDM model. The volumes surveyed by imaging experiments will tremendously increase over the next decade with the Large Synoptic Survey Telescope1 (LSST Ivezic et al., 2008), Euclid2 (Laureijs et al., 2011) and the Wide Field Infrared Survey Telescope 3 (WFIRST Spergel et al., 2015). These Stage 4 datasets jump 10-fold from the 1000 square ⇠ 1http://www.lsst.org/ 2http://sci.esa.int/euclid/ 3http://wfirst.gsfc.nasa.gov/

45 degree area to span 10, 000 degrees. LSST’s 6.7m e↵ective area telescope will ⇠ span the entire Southern sky, covering 18, 0000 square degrees, continuously, ⇠ accumulating depth to r 27.5 magnitudes, competitive with HST COSMOS- ⇠ depth in 6 bands. Euclid is set to image 15, 000 square degrees with a 1.2m telescope in one broad optical band with high-resolution space-based observations and accompanying near-infrared data to r 24.5. These surveys are poised to ⇠ measure the dark energy equation of state with 1% precision when combined with data from the CMB. The complementarity of the two datasets renders their synergy extremely powerful (Jain et al., 2015; Rhodes et al., 2017). Later on, WFIRST will also collect weak-lensing data from space, using a 2.4m telescope with a wide-field instrument.

In tandem, new spectroscopic datasets, the Dark Energy Survey Instrument4 (DESI) and 4MOST5 are on the horizon. On the CMB side, the South Pole Telescope6 and Atacama Cosmology Telescope7 are collecting data and anticipating CMB-Stage 4 and Simons Observatory experiments. The Square Kilometre Array (SKA; Brown et al., 2015) will contribute radio observations to the lensing field.

This spells the advent of the era of “precision cosmology” for lensing, large scale structure and CMB measurements. Further observations aid in addressing cosmology’s outstanding concerns, but more importantly motivate a need to ensure accuracy in the measurements by rigorous testing for systematic error and refining of methodologies.

1.8 Thesis outline

We have introduced the successes of contemporary cosmological measurements in comparison to the standard ⇤CDM model of cosmology. However, there exists outstanding problems, in particular, the directly observed but unexplained accelerated expansion of the Universe and the suggestions that cosmological parameters inferred from the early Universe may be in contention with that from the late-time Universe. Further analyses are required to make progress on understanding these problems, in particular through combined-probe analyses

4http://desi.lbl.gov/ 5https://www.4most.eu/cms/ 6https://pole.uchicago.edu/public/publications.html 7https://act.princeton.edu

46 that use weak-lensing, in order to both break degeneracies in parameter space and probe the evolution of structure in the Universe. In tandem with that approach and perhaps more importantly, it is crucial to investigate plausible sources of systematic error in cosmological measurements.

The remainder of this work presents a ***would love to summarise it into one sentence- analysis into the robustness and potential of weak-lensing?*** In the following chapter, I review the gravitational weak lensing theory and its application to galaxy-galaxy lensing and cosmic shear measurements geared towards understanding cosmology. I also outline the main steps for converting the raw pixel images into measurements of shear and eventually, cosmological parameters, in the context of the Kilo-Degree survey, along with the required systematic tests. Chapter 3 exemplifies these methods, presenting the analysis of a new lensing dataset: the secondary KiDS i-band sample. In the theme of redundancy, I used this dataset as a consistency check for the fiducial KIDS data and to test for unaccounted for systematics. In doing so, I lay down the framework for comparing lensing datasets that overlap on the sky. Same-sky datasets provide a unique opportunity for assessing the level of systematics in each dataset by comparison. In Chapter 4, I present the results of a combined-probe analysis that measures the EG statistic, a combination of measurements of weak- lensing, galaxy clustering and redshift-space distortions, thereby producing a new constraint on the matter density parameter. Similar to those described in the previous section, this analysis gives a value for the cosmological parameter that is in tension with Planck. As many weak lensing analyses show some preference for a lower ⌦ cosmology, this discordance becomes more interesting and m 8 may beg the question of the level of systematics associated with weak lensing as a cosmological probe. Ultimately, weak lensing and CMB results would be combined in order to most eciently test dark energy, but this is not robust while constraints are in ‘tension’. Rather than debating whether this is tension is significant, or questioning what level of concordance is required for weak-lensing constraints to be combined with those from Planck, I pioneer the approach of rigorously testing weak lensing analyses through comparison of data and methodology, in order to fuel more confidence in results from this technique. As current weak-lensing surveys begin to overlap on the sky, a unique stringency test of consistency is set up. The comparison of independent datasets at a more fundamental level tgab cosmological parameters is a key step before one considers combining their cosmological constraints for enhanced precision. This becomes increasingly important at the advent of a planned synergy of LSST, Euclid and

47 WFIRST. The main conclusions of this thesis and future prospects of the work are discussed in Chapter 5.

48