# Modern cosmology

Post-publication activity

Curator: Jean-Philippe Uzan

Physical cosmology is the study of the properties and evolution of the large scale structure of the universe. It firstly determines, in the light of astronomical observations, what is there when we average out the matter distribution to the largest scales, secondly what its history has been in the light of our understanding of physical forces, and thirdly what its future is likely to be. Thus it has descriptive and dynamic aspects, supported by theory and observation.

Modern cosmology is a mathematical physics subject, so we present the mathematics used in these studies. However we try to clearly link this in to the underlying conceptual principles on the one hand, and the observational tests of the resulting models on the other.

There is a huge literature on cosmology, and there is no purpose in simply duplicating it here (see e.g. Dodelson, 2003, Durrer, 2008, Mukhanov, 2005, Ellis, Maartens and MacCallum, 2012, Peter and Uzan J-P, 2013 for detailed texts). Our aim is to get the main ideas across to the interested reader in a way that emphasizes key issues, with references to further literature where this can be followed up. To simplify this, we will extensively refer to a recent detailed review by one of us (Uzan J-P, 2016), which is available on line, and where details and further references are given. The big picture is that remarkably simple models, largely based in known physics, succeed in giving an accurate explanation of a vast array of detailed astronomical observations; and they do so in terms of just a few free parameters that are fixed observationally. However major unkowns remain, in particular the nature of dark matter and dark energy, and the supposed inflaton field in the very early universe. In each of these cases we have good observational evidence for our models but no solid link to tested phyiscs that explains them.

## Basic assumptions

As with all sciences, cosmology has at its base certain philosophical methods and assumptions that order its nature. Broadly, these are the assumptions and processes that generically underlie the scientific method. In the particular case of cosmology, a mathematical model of the cosmos is proposed, based in our current understanding of the relevant physics and the data so far. This is then tested in detail against observations, and is adopted only if it explains those observations satisfactorily, with suitable values of its free parameters. However because of the particular nature of its object of study - there is only one Universe accessible to observation - the specific form of the assumptions made is particularly vital in the case of cosmology. This uniqueness means we are unable to carry out observations or experiments that will test important aspects of our proposed models. A key issue then - as in all science, but particularly significant here - is what is the domain of validity of our models, and what kinds of conclusions can we legitimately deduce from them.

### Foundations

The core principles underlying standard cosmology are,

1. Physics is the same everywhere in the universe. This is not obvious, because cosmology covers such vast time and distance scales, going right back to the start of the universe and to immensely larger distances than the Solar System, but without this assumption of universality we could not undertake a scientific study of the universe in the large. This assumption is tested by the success and coherence of the resulting models, for example the precise black body spectrum of the Cosmic Blackbody Radiation confirms that statistical physics and quantum physics were the same 13 billion years ago as they are today.
2. The local physics that applies everywhere determines the dynamics of the universe in the large. That is, there is not a cosmological physics' that applies only on large scales; rather large scale dynamics emerges in a bottom-up way from the combined effects of local dynamics on matter everywhere on small scales.
3. Gravity is the dominant force in the Universe on Solar System and larger scales. This generalizes to galactic and cosmological scales the finding that gravity dominates the Solar System. This would not be true if astronomical objects were electrically charged, for example if electron and proton charges were not of precisely the same magnitude. In that case electromagnetism (the other long range force) would be the dominant force on large scales, because it is a much stronger force than gravity. [1]
4. Gravity in the classical regime is well described by General Relativity Theory (Hawking and Ellis, 1973), so this is the theory that best describes the geometry and dynamics of the observable universe. One represents its evolution by using the Einstein Field Equations to determine the effects of gravity, together with models of the local behaviour of matter which are the source terms for the gravitational field. This assumption, because of the equivalence principle, encodes 1 and 2,

One consequently has to consider a General Relativistic model with geometry and matter descriptions suitable for a cosmological context, and the Einstein Field Equations (see Eq.(7) below) determining the dynamics of the model. As to the nature of the universe,

• The universe is of vast size and age, with the Solar System (and even our galaxy) a tiny speck in a mostly empty spacetime.
• Matter is hierarchically arranged, with planets circulating stars imbedded in star clusters that form galaxies which occur in clusters that again occur in large scale structures with a web-like nature.
• However on the largest scales the universe appears to be spatially homogeneous: as far as we can see, it has no preferred places or centre.
• It is dynamic: it had a hot early stage, structures formed out of an initially very smooth state in a way that was modulated by dark matter of an unknown nature, and at present matter is expanding at a rate characterised by the Hubble constant.
• It may or may not have had a start, and while it is likely that it will continue expanding forever, there are other possibilities because we do not know the nature of the dark energy presently causing its expansion to accelerate.
• Our observational access to the universe is strictly limited by visual horizons and the opaque nature of the very early universe. Our ability to test the relevant physics at very early times is limited by practicalities in terms of what particle accelerators we can construct.

Almost all of this was unexpected. It was assumed by the best physicists in the world who initiated relativistic cosmology in 1917 that the universe was static. Its dynamic nature was generally realised only in 1931 after a famous meeting of the Royal Astronomical Society, Friedmann discovered the expanding $k=+1$ solutions (1922) and the $k = -1$ solutions (1924) of the Einstein equations for a spatially homogeneous and isotropic spacetime. Lemaître independently discovered the $k=+1$ expanding solutions (1927), and related them to the expansion of the Universe via the redshift-distance observations of Slipher, hence offering the first connection to observations. The physical processes of the Hot Big Bang Era and associated existence of Cosmic Blackbody Background Radiation were not generally realised until 1965, even though they had been predicted in 1948 by Gamow, Alpher and Herman. They were not realised by the best minds in the field at the time, even though they are very obvious in hindsight. The existence of the blackbody cosmic background radiation should have been predicted, in terms of physics known at the time, in 1934, when Tolman wrote his book (Tolman, 1934), which includes the radiation dominated version of the Friedmann equation.

### Nature of Spacetime Geometry

The space-time geometry is represented on some suitable averaging scale, which is usually implicit rather than explicit. According to General Relativity (Hawking and Ellis, 1973), spacetime is a 4-dimensional manifold with a Riemannian geometry determined by a symmetric metric tensor $$g_{ab}(x^i) = g_{(ab)}(x^i) (a,b = 0,1,2,3)$$ [2] with normal form $g_{ab} = diag(-1,+1,+1,+1)$ [3] Thus timelike world lines $x^a(s)$ with tangent vector $u^a(s)=dx^a/ds$ have a magnitude $u^a g_{ab} u^b < 0$, null world lines $x^a(v)$ with tangent vector $k^a(s)=dx^a/dv$ have a magnitude $k^a g_{ab} k^b = 0$. Proper time $\tau$ along a timelike world line is determined by the relation $$\tag{1} \tau = \int \sqrt{\left|g_{ab}\frac{dx^a}{ds} \frac{dx^b}{ds}\right|}\,ds$$ and spatial distances along a spacelike world line by the analogous relation.

A connection $\Gamma$ determines the covariant derivative $\nabla$ such that $\nabla_{[ab]}f=0$ for all functions $f$, $\nabla_ag_{bc}=0$, these two relations together determining the connection in terms of the derivatives of the metric. A geodesic curve $x^a(v)$ with tangent vector $K^a=\frac{dx^a}{dv}$ obeys the relation $$\tag{2} K^a \nabla_a K^b = 0,$$ and represents freely falling matter if $K^a$ is timelike ($K^aK_a <0$), or the path of light rays if $K^a$ is null ($K^aK_a =0$).

The curvature tensor $R_{abcd}$ is given by the Ricci identities that hold for an arbitrary vector field $u^{a}$: $$\tag{3} \nabla_{[dc]} u_b =u^{a}R_{abcd}.$$ It satisfies the symmetries $$\tag{4} R_{abcd}= R_{[ab][cd]} = R_{cdab}, \,\, R_{a[bcd]}=0$$ and the integrability conditions $$\tag{5} \nabla_{[a}R_{bc]de} = 0$$ (the Bianchi identities). The curvature tensor has two important contractions, the Ricci tensor $R_{ab}:=R_{\;acb}^{c}$ and Ricci scalar $R:=R_{\;\;a}^{a}.$ Because of (5), the Einstein tensor $G_{ab} := R_{ab} - 1/2 R g_{ab}$ satisfies the identity $$\tag{6} \nabla_{b}G^{ab} =0.$$

### Dynamics: The Einstein Field Equations

The matter present determines the geometry, through the Einstein field equations given by $$G_{ab} ={8\pi G } T_{ab}-\Lambda \,g_{ab}\ , \tag{7}$$ where $G$ is the gravitational constant, $\Lambda$ the cosmological constant (possibly zero), and $T_{ab}$ the matter stress-energy tensor. In essence, this shows how matter (on the right hand side) curves spacetime (on the left), where $\Lambda$ can be regarded as representing a Lorentz invariant matter field. Geometry in turn determines the motion of the matter because the identities (6) together with the equations (7) guarantee the conservation of total energy-momentum: $$\nabla_{b}G^{ab} =0 \Rightarrow \nabla_{b}T^{ab}=0\ , \tag{8}$$ provided the cosmological constant $\Lambda$ satisfies the relation $\nabla _{a}\Lambda =0$, i.e., it is constant in time and space. In essence, this shows how spacetime determines how matter moves because, given suitable equations of state, matter dynamics is controlled by the energy momentum conservation equations (8).

In conjunction with suitable equations of state for the matter, relating the components of the stress-energy tensor $T_{ab},$ the Einstein Field Equations (7) determine the combined dynamical evolution of the model and the matter in it, with (8) acting as integrability conditions. It is crucial that in appropriate circumstances, (7) have the Newtonian gravitational equations as a limit, and so are in accordance with all the well-established results of Newtonian gravitational theory in the contexts where that is an accurate theory of gravitation.

### Matter sources

Clearly the solutions to the Einstein Field Equations (7) depend on the nature of the matter sources present. The stress tensor on the right hand side of these equations should include all matter and radiation fields present. In the cosmological context, there are three significant such sources:

• Matter, usually represented as a perfect fluid with energy density $\rho$ and pressure $p$:

$$\tag{9} T_{ab} = (\rho +p) u_au_b + p g_{ab}$$ where $u^a$ ($u^a u_a = -1$) is the 4-velocity, so $T_{ab}u^au^b=\rho$, $T^a_{\,\,\,\,a} = 3p$. In particular, baryonic matter and Cold Dark Matter (CDM) are represented in this way, with equation of state $p=0$. The equations of motion follow from the conservation equations (8); matter moves on timelike geodesics if $p=0$. Occasionally at times where irreversible processes are taking place, one may include viscosity and heat flux terms.

• Radiation, represented either as a perfect fluid (9) with equation of state $p = \rho/3$, or through a kinetic theory description with distribution function

$f(x^i,p^j)$ and stress tensor $T^{ab}(x^i) = \int f(x,p) p^ap^b\pi$, which will almost always not give the perfect fluid form (9). The dynamical equation for the radiation then is the Boltzmann equation ${ \cal L}f = {\cal C}(f)$ representing its interaction with matter (Durrer, 2008). This reduces to the Liouville equation when no interactions take place: ${\cal C}(f)=0 \Rightarrow {\cal L}f=0$.

• A scalar field $\phi$ with=0 potential $V(\phi)$ may be present. This will

have a perfect fluid stress tensor (9) with 4-velocity $u^a = \nabla^a \phi/(\left|\nabla^b \phi \nabla_b \phi\right|)^{1/2}$, and $$\tag{10} \rho =\frac{1}{2}(d\phi /dt)^{2}+V(\phi), \,\, p=\frac{1}{2}(d\phi /dt)^{2}-V(\phi).$$ The equation of motion is the Klein-Gordon Equation $$\tag{11} \nabla^a \nabla_a \phi +dV/d\phi =0.$$ Different kinds of matter dominate at different eras in the history of the universe.

## Geometry of the Observable Universe

Realistic cosmological models are perturbed versions of a background Friedmann-Lemaître -Robertson-Walker model.

### Background models

Robertson-Walker geometries are spatially homogeneous and isotropic universes, representing the geometry of the universe on the largest observable scales (Robertson, 1933). They are everywhere spatially homogeneous and isotropic, and so have no distinguishing feature anywhere. They are characterised by a scale factor $a(t)$ that shows how relative spatial distances between fundamental world lines change as time progresses. In comoving coordinates, the 4-velocity $u^{a}=dx^{a}/dt$ of preferred fundamental observers is $$\tag{12} u^{a}=\delta _{0}^{a},\,\, u^{a}u_{a}=-1$$ and the metric tensor is $$\tag{13} ds^{2}=-dt^{2}+a^{2}(t)d\sigma ^{2},$$ where the 3-space metric $d\sigma ^{2}$ represents a 3-space of constant curvature $k.$ It can be represented by $$\tag{14} d\sigma ^{2}=dr^{2}+f^{2}(r)d\Omega^{2},\,\, d\Omega^{2}:=d\theta ^{2}+\sin ^{2}\theta d\phi^{2},$$ where $f(r)=$ $\left\{ \sin r,r,\sinh r\right\}$ if $k=\left\{ +1,0,-1\right\}$. Spatial homogeneity together with isotropy implies that a multiply transitive group of isometries $G_{6}$ acts on the surfaces $\left\{ t=constant\right\},$ [4] and consequently all physical and geometric quantities are isotropic and depend only on the coordinate time $t.$

These spacetimes are conformally flat, as the Weyl conformal curvature tensor vanishes: $C_{abcd}=0$. Thus there is no free gravitational field, so no tidal forces or gravitational waves occur in these background models. However for realistic matter the Ricci tensor is non-zero and is not proportional to the metric tensor, so the matter world lines and 4-velocity (12) are uniquely determined by the stress tensor $T_{ab}$, and hence, via the Field Equations (7), also by the geometry: $u^a$ is the unique timelike eigenvector of $R_{ab}$. Thus the surface $\{t = const\}$ are geometrically and physically preferred surfaces.[5]

The time parameter $t$ represents proper time (1) measured along the fundamental world lines $x^{a}(t)$ with $u^{a}$ as tangent vector; the distances between these world lines scales with $a(t)$, so volumes scale as $a^{3}(t)$. The relative expansion rate of matter is represented by the Hubble parameter $$\tag{15} H(t)=\frac{\dot{a}}{a},$$ which is positive if the universe is expanding and negative if it is contracting. The rate of slowing down of the expansion is given by the dimensionless decceleration parameter $$\tag{16} q(t):=-\frac{\ddot{a}}{a H^2},$$ positive if the cosmic expansion is slowing down and negative if it is speeding up.

### Topology

It is important to note that the spatial topology of the universe is not determined by the Einstein field equations.

If $k = +1$, the spatial sections of the universe necessarily are spatially closed (this follows from the geodesic deviation equation): they are finite and contain a finite amount of matter. If simply connected they will be spheres $S^3$. However they need not be simply connected, and there are a variety of other possible topologies, obtained by identifying points in the sphere $S^3$ that are moved into each other by a discrete group of isometries.

If $k=0$, then the universe may have simply connected space sections with topology $R^3$. Then there is an infinite amount of matter in the universe. However again they may not have that simply connected topology, and there a variety of other possibilities, for example a torus topology, resulting in the amount of matter in the universe being finite. The same is true for the case $k=-1$, except then there are an infinite number of possible spatial topologies. Classifying them is a major task.

No known principle determines what the spatial topology is. Wheeler and Einstein have argued strongly that the universe should be spatially finite, which means either $k=+1$, or one of the compact topologies in the cases $k=0$ or $k=-1$. However one key point should be noted: the topology cannot change with time. All the spatial surfaces ${\cal S}(t)$ will necessarily have the same topology, because the fundamental world lines form a continuous map between them.

### Perturbed models

To describe structure formation in the universe, one needs to perturb the exactly homogeneous Robertson-Walker geometries. One can perturb cosmological models as in any other case: given a background metric $\bar{g}_{ab}$, define a new metric $$\tag{17} g_{ab} = \bar{g}_{ab} + \epsilon h_{ab}, \,\,|\epsilon| \ll 1$$ and then expand the field equations and conservation equations in powers of $\epsilon$, to obtain the background equations, linear equations, and higher order equations. Thus in order to describe the deviations from homogeneity, one starts by considering the most general form of a perturbed Robertson-Walker metric, \begin{eqnarray}\tag{18} ds^2&=&a^2(\eta)\left[-(1+2A)d\eta^2 + 2B_\nu d x^\nu d\eta +(\gamma_{\mu\nu} + h_{\mu\nu}) d x^\mu d x^\nu\right], \end{eqnarray} where the small quantities $A$, $B_\nu$ and $h_{\mu\nu}$ are unknown functions of space and time to be determined from the Einstein equations (Greek indices, representing spatial dimensions, run from 1 to 3).

### The gauge issue

The problem is however not that simple, because in General Relativity Theory, gravitational dynamics is carried by the dynamic spacetime geometry. Hence there is no given fixed background spacetime $\bar{g}_{ab}$ which we can perturb to find the more realistic lumpy' model $g_{ij}$. Rather, we are faced with modelling a particular almost-Robertson Walker spacetime $g_{ab}$ which can be represented in the form (17) in many different ways, with different background spacetimes $\bar{g}_{ab}$. The problem is related to the fact that we can choose coordinates in the perturbed model in many different ways. Thus for example the perturbed energy density at time $t$ is defined to be $$\tag{19} \delta\rho(x^\nu,t) = \rho(x^\nu,t) - \bar{\rho}(x^\nu,t)$$ where in the first term on the right the coordinates are chosen in the perturbed model, and in the second term they are chosen in the background model. Now the surfaces of constant time in the perturbed model can be chosen in many different ways. For example, we can define them to be surfaces of constant density, labelled by the background density; then by definition, for any given $t$, $\rho(x^\nu,t) = \bar{\rho}(x^\nu,t)$ for all $x^\nu$, which implies $\delta\rho(x^\nu,t)=0$. This gauge choice has made the density perturbations disappear! There are essentially three ways to handle this.

• Careful gauge fixing One can very carefully fix the gauge, i.e. specify fully how the coordinates are chosen in the perturbed model, at each stage of this coordinate fixing, calculating what effect the remaining gauge freedom has on the physical variables by using Lie Derivatives related to this coordinate freedom. This is acceptable if done very carefully. However it has in the past not always been done with due diligence, and many errors have occurred in the literature as a consequence.
• Gauge invariant variables One can combine physical and geometrical variables to give combinations that are invariant when one makes a gauge change (Uzan J-P, 2016,64-67). This is a highly influential and much used method that has been developed in depth, including covering kinetic theory (Durrer, 2008), but the variables so determined do not have a clear geometrical meaning. They have to be interpreted differently in different coordinate systems.
• 1+3 gauge invariant and covariant method In essence, it considers (19) to be defined by a map from the background spacetime into the perturbed spacetime. The gauge freedom is the freedom in that map.

One uses covariantly defined variables with a clear geometric meaning that are invariant under that map. From the vector form of (19), if a variable vanishes in the background it is automatically gauge invariant: $$\tag{20} \{\bar{X}^i(x^\nu,t) = 0\} \Rightarrow \delta X^i(x^\nu,t) = X^i(x^\nu,t)$$ for all choices of the surfaces of constant time in the perturbed model. Therefore a covariant and gauge invariant measure of spatial inhomogenity is the comoving fractional spatial density gradient ${\cal D}_a:= h_a^b \rho_{,b}/\rho$ for an observer with 4-velocity $u^a$ ($u_au^a = -1$), where $h_{ab}:=g_{ab}+u_au_b$ projects orthogonal to $u^a$. The spatial vector ${\cal D}_a$ vanishes in a Robertson-Walker spacetime, and so is gauge invariant by (20). Its dynamic equations follow from the 1+3 covariant equations for generic fluid flows. One can develop this method in detail (Ellis, Maartens and MacCallum, 2012), covering also kinetic theory.

### Scalar, vector and tensor perturbations.

Because of linearisations, one can split the perturbations up into scalar, vector, and tensor parts that (at linear order) evolve independently and so do not interfere with each other. Scalars can generate vector and tensor perturbations, and vector perturbations can generate tensor perturbations. Scalar perturbations represent density and pressure inhomogeneities, vector perturbations represent rotational effects, and tensor perturbations represent gravitational waves.

Scalar perturbations are the most important in cosmology. For structure formation studies, we can use pure scalar perturbations with a metric form $$\tag{21} ds^{2}=a^{2}(t)[-(1+2\Psi )d\tau ^{2}+(1-2\Phi )\gamma_{\mu\nu}dx^{\mu}dx^{\nu}]$$ with conformal time $\tau$ related to proper time $t$ by the transformation $dt=a(\tau)d\tau$. The spatial coordinates are not comoving, in general; that is, the matter moves relative to these coordinates.

This is called the Conformal Newtonian Gauge because it uses conformal time $\tau$, and the scalar $\Phi$ corresponds to the gravitational potential of Newtonian gravitational theory. For a pressureless fluid, considering sub-Hubble modes much smaller than the curvature scale, the scalar equations reduce to the comoving Poisson equation $$\tag{22} \Delta\Phi = 4\pi G\rho a^2\delta$$ as in Newtonian cosmology (see Uzan J-P, 2016:(77))with the definition $\delta \equiv \delta\rho/\rho$. In a universe with isotropic stress in this frame, the Einstein equations show that $\Phi=\Psi$.

### Fourier analysis

The perturbations are Fourier analysed in terms of wavelength, represented by a wavenumber $k$. This enables spatial power spectra to be defined, and spatial derivative operators are replaced by multiplication by $k$. The perturbation equations become equations relating the spatial harmonics of physical and geometric variables. Thus one ends up with harmonic components of scalar, vector, and tensor perturbations. Their dynamics is then examined via the linearised Einstein Field Equations, which reduce to a series of coupled ordinary differential equations for the Fourier components of the perturbation variables.

### Justification

Why it is reasonable to use such a perturbed Robertson-Walker model for the observed universe? The key point concerning the observational context is that we can only view the universe from one space-time event, and cannot easily distinguish spatial variation from time variation in this context. We have to test the supposition of Robertson-Walker geometry in that context.

The central point of the argument is that observations of galaxies and radio sources (number counts), as well as of background radiation, show that when averaged on a large enough scale, the universe is isotropic about us. This shows that either (a) the universe is spherically symmetric but inhomogeneous, and we are near the centre; or (b) the universe is spherically symmetric about every point. However a theorem by Walker shows that isotropy everywhere around a preferred family of observers implies the geometry is that of a Robertson-Walker spacetime. Hence unless we are special, being at a very special place in the universe so that we are the only fundamental observers who see an isotropic universe, its geometry must be Robertson-Walker. If we have good enough standard candles at large distances, we can directly test spatial homogeneity from astronomical observations; and we are close to being able to carry out such tests by using supernovae as standard candles.

This argument is greatly strengthened by a theorem of Ehlers, Geren and Sachs, which states that if an expanding family of geodesic observers in a local domain $\cal U$ see isotropic freely moving radiation, the universe is Robertson-Walker in that domain. It can be shown that this result is stable: if the radiation is almost isotropic, the universe is almost Robertson-Walker (Ellis, Maartens and MacCallum, 2012). Now we cannot directly determine whether the background radiation observed from the vantage point of a remote observer, but we can do so indirectly via the kinetic Sunyaev-Zeldovich effect, whereby hot gases in distant objects scatter radiation and so affect the Cosmic Blackbody Radiation spectrum if its temperature is anisotropic at that distant point, also causing polarisation. This currently gives the best limits on large scale inhomogeneity of the universe.

## Dynamics

There are some generic relations one can find for cosmological models, without assuming any symmetries. These reduce to simple forms in the case of Friedmann-Lemaître models with Robertson-Walker geometries.

### Generic

The real universe is nowhere empty, in particular because of the Cosmic Blackbody Background radiation that permeates all of spacetime. Hence a realistic cosmological model always has a geometrically and physically preferred fundamental velocity field $u^a=dx^a/d\tau$ ($u^a u_a = -1$) at each point, (cf. eqn.(12)); specifically, the timelike eigenvector of the Ricci tensor. Using a 1+3 covariant approach, there are a number of exact relations that are true for any cosmological model, whether inhomogeneous, anisotropic, or not (Ellis, Maartens and MacCallum, 2012). The key ones are energy-momentum conservation equations and the Ehlers-Raychaudhuri equation of gravitational attraction. Cosmic time derivatives are denoted by $\dot{f} \equiv f_{,a} u^a$.

### Conservation equations

These are the timelike and spacelike components of (8). For a perfect fluid (9) the energy conservation equation takes the form $$\tag{23} \dot{\rho}+3\frac{\dot{a}}{a} (\rho +p)=0$$ where the fluid expansion $\theta:=\nabla^a u_a= 3\dot{a}/{a}$ defines the variation of the average length scale $a$ along the fundamental world lines. The momentum conservation equation is $$\tag{24} (\rho+p)a_b +{}^{(3)}\nabla_b p=0,$$ where ${}^{(3)}\nabla_b:=(g^{ab}+u^au^b)\nabla_b$ is the spatial derivative orthogonal to $u^a$, and $a^a:=u^b\nabla_bu^a$ is the relativistic fluid acceleration. This shows that $\rho_{\rm inert}:=\rho+p$ can be interpreted, in Newtonian terms, as the inertial mass density.

These equations become determinate when an equation of state $p=p(\rho ,\phi )$ specifies $p$ in terms of $\rho$ and possibly some internal variable $\phi$ such as the temperature $T$ or entropy $S$ (which if present, will have to have its own dynamical equations). The outcome of these equations depends on the equation of state; for example in the case of pressure free matter, $$\tag{25} \{p=0\} \Rightarrow \{\rho = M/{a}^3,\,\, a^a=0\},$$ where $\dot{M}:=u^a\nabla_a M =0$. This shows that pressure-free matter moves on geodesics. In the case of radiation, $$\tag{26} \{p=\rho/3\} \Rightarrow \{\rho = M/{a}^4,\,\, a^a=- {}^{(3)}\nabla_b U\},$$ where the acceleration potential $U$ is defined by $\,\,U:= \frac{1}{4} \log \rho$.

### Field equations

On using the field equations (7), the trace of the geodesic deviation equation for the timelike vector field $u^a$ is the Ehlers-Raychaudhuri equation $$3\frac{\ddot{a}}{a}=-2(\omega ^{2}-\sigma ^{2})-4 \pi G (\rho +3p)+\Lambda -a_{\,\,\,;b}^{b}, \tag{27}$$ where $\omega$ is the vorticity scalar and $\sigma$ the shear scalar. This is the non-linear exact fundamental equation of gravitational attraction for a fluid flow, showing that vorticity tends to resist collapse whereas shear hastens it, $$\tag{28} \rho_{\rm grav}:=(\rho +3p)$$ is the active gravitational mass density that tends to cause matter to collapse (it is positive for all ordinary matter), and a positive cosmological constant $\Lambda$ tends to keep matter apart. The acceleration term can have either sign.

### Background model

The background Friedmann-Lemaître models are given by assuming the Einstein Field Equations (7) govern the dynamics of a spacetime with a Robertson-Walker geometry. The behaviour of matter and expansion in theses universes is governed by three related equations. First, the energy conservation equation (23) relates the rate of change of the energy density $\rho (t)$ to the pressure $p(t).$ In the case of pressure-free matter, (25) holds; in the case of radiation, (26). The Raychaudhuri equation (27) becomes

$$\frac{3}{a}\frac{d^{2}a}{dt^{2}}=-4\pi G(\rho +3p)+\Lambda \tag{29}$$ which directly gives the deceleration due to matter. Its first integral is the Friedmann equation $$3H^{2}=8\pi G \rho +\Lambda -{\frac{3k}{a^{2}}}, \tag{30}$$ where ${{\frac{3k}{a^{2}}}}$ is the curvature of the 3-spaces $\{t=const\}$; this is just the Gauss-Codazzi equation relating the curvature of imbedded 3-spaces to the curvature of the imbedding 4-dimensional spacetime. The evolution of $a(t)$ is determined by any two of (23), (29), and (30), as any two imply the third one. Defining the normalized density parameters$$\tag{31} \Omega_m :=\frac{ 8\pi G \rho }{3H^{2}},\,\, \Omega_\Lambda :=\frac{\Lambda }{3H^{2}},\,\, \Omega_k :=-\frac{k }{H^{2}a^2},$$ the Friedmann equation has the dimensionless form $$\Omega_m +\Omega_\Lambda +\Omega_k = 1 .\tag{32}$$

The behaviour of solutions to this equation for ordinary matter has been known since the 1930s (Robertson, 1933, Tolman, 1934):

• When $\Lambda =0,$ if $k=+1$, then $\Omega >1$ and the universe recollapses;

if $k=-1$, then $\Omega <\ 1$ and it expands forever; and $k=0$ ($\Omega =1)$ is the critical case separating these behaviours, that just succeeds in expanding forever. In the pressure-free case $p=0$, this limiting case is the \textit{Einstein-de Sitter universe} with $$\tag{33} a(t) = a_0 t^{2/3}.$$ There is always a singular start to these universes at time $t_{0}<1/H_{0}$ ago if $\rho +3p>0$, which will be true for ordinary matter plus radiation

• One can in principle get a bounce if both $\Lambda >0$ and $k=+1$.However in practice there is still a singular start to the universe if $\rho +3p>0$ ($\Lambda$ cannot avert this because it is constant, and is small today.)
• When $\Lambda >0$ and $k=+1$, a static solution is possible (the Einstein static universe), and the universe can hover' close to this constant radius before starting to expand (these are the Eddington-Lemaître models).
• The de Sitter universe occurs in three different FLRW forms with different spatial curvatures, which is possible because it has no unique timelike vector field (as it is a spacetime of constant curvature (Hawking and Ellis, 1973): [6] $$\tag{34} k=0, \,\,\,a(t)= a_0 \exp(H_0 t)$$ (a steady-state solution that is geodesically incomplete in the past), $$\tag{35} k=+1,\,\,\,\, a(t) = a_0 \cosh H_0 t$$ (a geodesically complete form with a bounce), and $$\tag{36} k=-1, \,\,\,a(t) = a_0 \sinh H_0 t$$ (a geodesically incomplete singular form).[7]. They are all solutions with $\rho + p = 0$ (perhaps with a cosmological constant $\Lambda>0$), and so are physically unrealistic as exact solutions. However at early or late times, they may be good approximate solutions for a while.
• The Friedmann equation for early times in the Hot Big Bang era of the universe, when radiation dominates the dynamics ($p = \rho/3$) and curvature and the cosmological constant are negligible ($\Omega_k \simeq 0$, $\Omega_\Lambda \simeq0$), gives

$$\tag{37} a(t) = a_0 t^{1/2}$$ (the Tolman solution) and the temperature-time relation $$T=\frac{T_{0}}{t^{1/2}},\,\,\,T_{0}:=0.74\left( \frac{10.75}{g_{\ast }}\right) ^{1/2} \tag{38}$$ with no free parameters. Here $t$ is time in seconds, $T$ is the temperature in MeV, and $g_{\ast }$ is the effective number of relativistic degrees of freedom, which is 10.75 for the standard model of particle physics, where there are contributions of 2 from photons, 7/2 from electron-positron pairs and 7/4 from each neutrino flavor. This relation plays a key role in nucleosynthesis.

All these behaviours can be illuminatingly represented by appropriate dynamical systems phase planes (Wainwright, 1997, Uzan J-P, 2016). A particularly useful version by Ehlers and Rindler (Ehlers and Rindler, 1989) gives the 3-dimensional phase plane for models with non-interacting pressure free matter and radiation, as well as a cosmological constant. The special solutions mentioned above (the Einstein static universe, Einstein de Sitter universe, Tolman universe, and de Sitter universe) are all fixed points in these phase planes.

### The Main Epochs

The real universe is filled with a complex mix of matter and radiation, leading to changing behaviour at different times because the nature of the matter present (the right hand side of the Einstein Field Equations) drives the dynamics. The main epochs, represented schematically in Figure 1, are as indicated below:

Table 1: The main epochs in the history of the Universe
Epoch Matter source Events
Start Unknown Creation or bounce 1
Inflation Scalar field dominated Seed perturbations 2
Hot Big Bang Radiation dominated Nucleosynthesis 3
Visible universe Dark Matter dominated Structure formation 4
Accelerating Universe Dark energy dominated Acceleration 5
Future Dark energy or curvature Unknown 6
Figure 1: Graphic of the history of the universe (credit: NASA). The start of the universe is on the left, time increases towards the right, and the vertical scale indicates the size of the universe at each time.

The start of the inflationary era is difficult to determine and very model-dependent. One can estimate it to take place around $10^{-35}$~s. The hot big-bang phase started before the temperature of the radiation bath drops below 100~MeV, i.e. typically earlier than $10^{-3}$~s after the big-bang. The universe becomes transparent after the decoupling between photons and matter. It happens at a redshift of order $1100$, corresponding roughly to $380,000$~years. Note that this is after radiation-matter equality, which occurs at a redshift $1+z=\Omega_{m}/\Omega_{r}\sim 3,600$. The onset of the late time acceleration phase is defined by $\ddot a=0$ which, for a $\Lambda$CDM model, occurs at $1+z=(2\Omega_\Lambda/\Omega_m)^{1/3}\sim 1.7$.

Each epoch has different physical processes happening and consequently different cosmological dynamics. They are discussed in detail in Uzan J-P, 2016, so we will just very briefly discuss them here. The big picture is that the whole history of our visible universe consists of an episode of decelerated expansion, during which complex structures can form, sandwiched in between two periods of accelerated expansion, which do not allow such structures to form.

• Epoch 1: The start of the universe. There may have been a singular beginning to the universe (as in the Tolman case), or a bounce from a previous era (as in a de Sitter universe in the $k=+1$ frame), or an emergence from a static state (as in the Eddington-Lemaître universe). We don't know the physics applicable in the quantum gravity era that would have presumably occurred when the density was higher than the Planck density, so we don't know which of them occurred. Indeed we can only speculate as to what kind of mechanism can create a universe from nothing', for any such proposed mechanism is surely not testable.
• Epoch 2: Inflation is a supposed extremely rapid but short lived accelerating expansion through many efoldings before the hot big bang era. By ((29)) this requires some kind of field $\phi$ such that $\rho_\phi + 3p_\phi < 0$. This is possible for a scalar field $\phi (t)$ with potential $V(\phi )$ obeying the Klein Gordon Equation (11), which in a FLRW universe has the form

$$\ddot{\phi}+3H\dot{\phi} +dV/d\phi =0. \tag{39}$$ This field will have an energy density $\rho_\phi$ and pressure $p_\phi$ given by (10). so the inertial and gravitational energy densities are $$\tag{40} \rho_\phi +p_\phi=(\dot{\phi})^{2}>0,\;\rho_\phi +3p_\phi=(\dot{\phi})^{2}-V(\phi ).$$ Hence the active gravitational mass $\rho_\phi + 3p_\phi$ can be negative in the slow roll' case, that is, when $(\dot{\phi})^{2}<V(\phi).$ Scalar fields can thus cause an exponential expansion $a(t) = \exp H_0 t$ when $\phi$ stays approximately at a constant value $\phi _{0}$ because the friction term $3H\dot{\phi}$ in (39) dominates the $dV/d\phi$ term, resulting in $\rho_\phi +3p_\phi \approx -V(\phi _{0})=const$ and an approximately de Sitter spacetime. This is the physical basis of the inflationary universe idea of an extremely rapid exponential expansion at very early times that smooths out and flattens the universe, also causing any matter or radiation content to die away towards zero. The inflaton field itself dies away at the end of inflation when slow rolling comes to an end and the inflaton gets converted to matter and radiation by a process of reheating.

This epoch has the important feature of providing the basis for a causal explanation of the origin of seeds for structure formation in the early universe. The nature of the inflaton field is not known (it is not tied into any tested particle physics particles or fields).

• Epoch 3: A Hot Big Bang radiation dominated early phase of the

universe, with ionised matter and radiation in equilibrium with each other because of tight coupling between electrons and radiation, and a temperature-time relation given by (38) in this Tolman phase ($a(t) = a_0 t^{1/2}$). Baryosynthesis, neutrino decoupling, and nucleosynthesis took place in this era, with neutron capture processes at a temperature of about $T=10^{8}$K leading to formation of Deuterium, Helium, and a trace of Lithium from protons and neutrons. Predictions of light element abundances resulting from these nuclear interactions agree well with primordial element abundances determined by astronomical observations, up to some unresolved worries about Lithium. It was this theory of nucleosynthesis that established cosmology as a solid branch of physics (Lahav and Liddle, 2015).

The Hot Big Bang phase ended when the temperature dropped below the ionisation temperature of the matter, resulting in matter-radiation decoupling and cosmic blackbody radiation being emitted at the Last Scattering Surface at about $T_{emit}=4000K$. This radiation is observed today as cosmic microwave blackbody radiation (CMB) with a temperature of $2.73K$. A major transition is the baryon-radiation density equality which separates the universe into two eras: a matter dominated later era during which structure can grow, and a radiation dominated earlier era during which the radiation pressure prevents this for many wavelengths, rather giving rise to acoustic waves (baryon acoustic oscillations'). Key aspects are the evolution of the speed of sound with time, and diffusion effects leading to damping of fine-scale structure.

• Epoch 4: The Matter dominated phase of the visible universe. After decoupling of matter and radiation, the radiation density drops faster than the matter density (compare (25) with (26)) and the solution becomes approximately Einstein-de Sitter ($a(t) = a_0 t^{2/3}$). Small density perturbations on the Last Scattering Surface grow by gravitational instability to form structures in the universe in a bottom-up way (small structures form first and then coagulate to form larger structures), with stars forming and nuclear reactions starting in their interiors, leading to stellar nucleosynthesis of elements up to iron and subsequent supernovae explosions whereby these get spread through space to allow formation of second generation stars that can be surrounded by rocky planets.

The gravitational dynamics of these processes is dominated by pressure-free dark matter, which has a density much larger than that of the visible baryons. Its nature is unknown.

• Epoch 5: The accelerating late phase of the visible universe:

In recent times the expansion of the universe has been speeding up rather than slowing down. From eqn.((29)), this means that either the present expansion is dominated by a cosmological constant $\Lambda > 0$, that is $\Omega_\Lambda> \{\Omega_m, \Omega_k\}$, or some dynamic field $\phi$ ("Dark energy") is present such that $\rho_\phi +3p_\phi < 0$ at recent times (similar to what happens in inflation). The data is compatible with a cosmological constant; if there is instead a dynamical field $\phi$, there is no way to test its nature or dynamics in the solar system. [8]

No large scale structure can form in this era, although gravitational collapse (for example to form black holes) can continue in local structures that are already gravitationally bound at the start of this era.

• Epoch 6: The Future. If dark energy is a cosmological constant, which is compatible with the data, then the universe will expand forever, getting ever cooler and more diffuse, with all stars dying out, matter decaying, and a final state of everlasting darkness and cold (this is what used to be called a "Heat Death" in the 1930s to 1970s).

If the dark energy is rather a dynamical field that can decay away, then the future is uncertain. The universe might still expand forever, which will be the case if $k=0$ or $k=-1$, or it might recollapse, which is possible if $k=+1$. [9] In that case there will probably be a future inhomogeneous state with isolated singularities, but re-expansion in some places is a possibility through re-initiation of the mechanism that led to inflation in the previous epoch, leading to a chaotic cyclic model (if the universe were exactly homogeneous, it could bounce and re-expand everywhere, but that is unrealistic). Various other forms of cyclic universe have been proposed (pre-Big Bang models, ekpyrotic models, conformal cyclic cosmology, and so on), but none are based in well-established physics.

The physics is least well understood at very early times (inflation and before), because it cannot be tested in a laboratory or particle accelerators, and so is speculative. It is best understood during the Hot Big Bang era, governed by particle physics, nuclear physics, and statistical physics, up to the time of decouplong, It is less well known at present times and in the future, because we do not know the nature of either dark matter or dark energy.

### Inflation

It is now broadly agreed that there was indeed a period of inflation in the very early universe but the details are not clear:\ there are over 100 different variants, including single-field inflation, multiple-field inflation, and models where matter is not described by a scalar field as, for example vector inflation. As the potential is not tied in to any specific physical field, one can run the field equations backwards to determine the effective inflaton potential from the desired dynamic behaviour. This does not obviously tie the supposed dynamics into any known physical field, unless the inflaton is a non-miminally coupled Higgs field (as was often supposed when the theory was initially introduced). If that were the case, there would be a marvelous link between particle physics and cosmology, whereby collider physics would imply real constraints on cosmological processes in the very early universe.

Slow rolling takes place when $$\tag{41} \dot{\phi}^2 \ll V,\,\, \ddot{\phi}\ll3 H \dot{\phi}$$ Then the equations of motion reduce to $$\tag{42} H^2 \simeq \frac{8\pi G}{3}V, \,\,\dot{H} \simeq - 4 \pi G \dot{\phi}^2, \,\, 3 H\dot{\phi} \simeq - dV/d\phi$$ giving the almost exponential expansion characteristic of inflation. Key features of this phase are,

• Slow roll parameters, such as

$$\tag{43} \epsilon = \frac{m^2_{PL}} {16\pi} \left(\frac{V'}{V}\right),\,\,\,\eta = \frac{m^2_{PL}} {16\pi} \left(\frac{V''}{V}\right)$$ that characterise the dynamics of inflation,

• The number $N$ of inflationary efolds occurring between the start $a_a:=a(t_i)$ and end $a_f:=a(t_f)$of inflation,

$$\tag{44} N := \ln \left(\frac{a_f}{a_i}\right).$$ This must be large enough to reduce the initial curvature to very close to flat at the end of inflation and also allow the entire Last Scattering Surface to be causally connected, demanding $N \gtrsim 70$ (Uzan J-P, 2016).

• The reheating process at the end of inflation, whereby the inflaton field gets converted to radiation.

The simplest example is chaotic inflation with a massive scalar field $\phi$ so the potential is $V(\phi) = \frac{1}{2}m^2\phi^2$ and the Klein Gordon equation reduces to that of a damped Simple Harmonic Oscillator. If $\phi$ is initially large the Friedmann equation implies that $H$ is also large and the friction term dominates the Klein Gordon equation so the field is in the slow rolling regime. When quantum fluctuations are taken into account, their effects on the dynamics leads to a model in which local inflationary domains occur in a fractal way (chaotic inflation'). However the latest Planck data does not support such a model.

Inflation was initially proposed to solve a series of problems perceived with standard cosmology: the flatness problem', the homogeneity problem, and the cosmic texture relics problem. However the first two are philosophic in nature, as they are based in naturalness assumptions that may not be true. Furthermore Penrose has strongly argued that inflation does not solve the homogeneity problem, rather it assumes it has been solved before the analysis starts. That is because almost all inflationary calculations (such as those sketched above) assume a Robertson-Walker geometry at the start. In that case there is no need to solve the homogeneity issue because homogeneity was presumed from the outset. If one allows generic geometries, inflation is extraordinarily unlikely to take place and result in a smooth universe, because the most likely initial state (that with the highest entropy) is one dominated by black holes. The cosmic textures problem does not arise if the relevant fields do not exists.

The real importance of inflation is that it provides a causal model for how seeds for structure formation in the expanding universe came into being.

### The growth of structure

The real universe is only approximately a Robertson-Walker spacetime. Structure formation in an expanding universe can be studied by using linearly perturbed Friedmann-Lemaître models at early times, plus numerical simulations at later times when the inhomogeneities have gone non-linear. The physical interactions are different in each epoch mentioned above, and impact different wavelengths in crucial ways, leading to a predicted perturbation power spectrum $P(k)$ as a function of comoving wavenumber $k$ as discussed in (Dodelson, 2003, Mukhanov, 2005,Durrer, 2008, Peter and Uzan H-P, 2013).

As an example of how this works, the general non-linear dynamic equations for the normalised spatial density gradient ${\cal D}_a$ follow from taking the spatial gradients of the conservation equation (23) and Raychaudhuri equation ((27)), and substituting the result of the second into the first. On linearising, when $w = p/\rho = const$, $\Lambda = 0$, and spatial curvature $\Omega_k\simeq 0$, the linearised growth equation for density perturbation modes of wave number $k$ is (Ellis, Maartens and MacCallum, 2012) $$\tag{45} {\ddot{\cal D}}_a + (\frac{2}{3}-w)(\frac{3\dot{a}}{a}) \,{\dot{\cal D}}_a -\left(\frac{(1-w)(1+3w)}{2} 4\pi G\rho\right) {\cal D}_a - w \frac{k^2}{a^2}{\cal D}_a=0.$$ This directly gives the general relativistic version of the Jean's length: when there is no expansion: $\dot{a}=0$, and pressure gradients can halt collapse when $$\tag{46} \left|\frac{(1-w)(1+3w)}{2w} 4\pi G \rho\right| < \frac{k^2}{a^2}{\cal D}_a\,,$$ otherwise the inhomogeneity increases exponentially. However this changes dramatically when $\dot{a}\neq 0$. When $w=0\Leftrightarrow p=0$, (45) reduces to $$\tag{47} {\ddot{\cal D}}_a + (\frac{2\dot{a}}{a})\,{\dot{\cal D}}_a - 2\pi G\rho {\cal D}_a =0\,,$$ which directly gives the basic results for pressure free matter: with an Einstein de Sitter background (33), $$\tag{48} {\cal D}_a = c_{1}(t-t_1)^{2/3} + c_{2}(t-t_2)^{-1},$$ the first term being a power law growing mode and the second a power law decaying mode. The first term does not grow fast enough to generate the structures we see today from statistical fluctuations. In a radiation dominated phase, $w = 1/3$ and $$\tag{49} {\ddot{\cal D}}_a + (\frac{\dot{a}}{a}) \,{\dot{\cal D}}_a -\frac{2}{3} \left(4 \pi G \rho - \frac{1}{2} \frac{k^2}{a^2}\right){\cal D}_a=0.$$ which gives damped oscillations if $2\kappa\rho \ll \frac{k^2}{a^2}$ and growing and decaying modes similar to (48) if $2\kappa\rho \gg \frac{k^2}{a^2}$.

Generating fluctuations During inflation, the inflaton field $\varphi$ has quantum fluctuations (as does any matter field) and such fluctuation will generate metric perturbations, as they are coupled by the Einstein field equations. This creates an almost scale-free power law spectrum spectrum of scalar and tensor perturbations at the end of the inflationary era (Mukhanov, 2005). Simple inflation models have three independent parameters describing the primordial power spectrum: the amplitude of scalar fluctuations $A_S$, determined by a scalar potential which is unknown, the scalar to tensor ratio $r$, relating gravitational waves produced to the scalar perturbations, and the scalar spectral index $n_S$, defining to what degree these fluctuations are not scale invariant (as they would be if the expansion were exactly exponential) (Lahav and Liddle, 2015). Here the power spectrum $P_S(k)$ is defined as the variance per logarithmic interval: $(\delta \rho/\rho)^2 =\int P_S(k) d \log k$, and the primordial power spectrum can be written in the form $$\tag{50} P_S(k) = A_S \left(\frac{k}{k_*}\right)^{n_s-1}$$ where $n_s$ is the spectral index. Then a scale-invariant spectrum is given by $n_s=1$; usual inflationary models give $n_s\simeq1$, $n_s<1$. Similarly tensor modes are generated and enjoy a power spectrum $$\tag{51} P_T(k) = A_T \left(\frac{k}{k_*}\right)^{n_t}$$ where $n_t$ is the spectral index. The scalar and tensor spectral indices are given in terms of the inflationary slow roll parameters by $$\tag{52} n_s \simeq 1 - 6 \epsilon + 2 \eta, \,\,\,n_t \simeq = 2 \epsilon$$ and a consistency condition must hold: $$\tag{53} r \simeq 16 \epsilon \simeq = 9 n_t .$$

These fluctuations are then processed by pressure gradients during the hot big bang era, generating baryon acoustic oscillations, and giving a spectrum of primordial perturbations on the Last Scattering Surface with acoustic peaks, that then forms the basis for structure formation through gravitational attraction after decoupling. A transfer function $T(k)$ characterizes how the primordial perturbations of comoving number $k$ are modified by these processes.

As stated above, the real importance of inflation is that it gives a mechanism for the generation of an almost scale-free spectrum of primordial perturbations that lead to structure formation in the universe after decoupling. It is the first theory to give such a causal mechanism for the origin of structure. However one should note that the theory does not fix the amplitude $A_S$ of these fluctuations, because we do not have a unique inflaton field identified which has a potential with a physically determined scale. Thus it has to be determined by observations. Furthermore we do not have a good theory as to how the quantum fluctuations generated during the inflationary epoch become classical by the end of inflation (decoherence does not do the job, as some have suggested, becasue while it gets rid of entanglement it does not get rid of superpositions). This is a major lacuna in the theory.

### Dark energy and Dark matter

Two key realisations arising from detailed studies of structure formation, starting with the linearised theory and then joining the outcomes to numerical simulations, were firstly that one needs Cold Dark Matter as well as baryons for this to work in accordance with observations (hot dark matter would wipe out all but the largest initial fluctuations by free streaming). Secondly that introduction of Dark Energy, possibly a cosmological constant $\Lambda$, was needed to allow a cold dark matter scenario to match observations in an almost flat universe.

### Start of the universe

A key issue as regards the dynamics of the universe is whether it had a start. This has different answers if we consider General Relativity with classical matter, General Relativity with quantum fields, and quantum gravity.

Classical General Relativity and Friedmann-Lemaître models Because of the Raychaudhuri equation ((29)), $a(t) \rightarrow 0$ and a singularity occurs at the start of the current expansion phase of the universe in the standard Friedmann Lemaître models of cosmology, provided the energy condition $$\tag{54} \rho_{\rm grav}:=(\rho +3p)>0$$ is satisfied; and this will be true for all ordinary matter. Note that the present cosmological constant will not be able to prevent this, as it is constant, but the matter would have had a much higher energy density in the early universe. As $a(t) \rightarrow 0$, by the energy conservation equation (23), the energy density of matter will diverge provided $\rho + p > 0$, which will be true for all ordinary matter; [10] and then the Ricci scalar will also diverge, so this is a spacetime singularity. It represents an edge to spacetime, where space, time, and matter come into existence (all timelike geodesics are incomplete towards the past). Physics as we know it comes to an end then because spacetime did not exist.

Classical General Relativity, Singularity theorems A key issue is whether this prediction is a result of the high symmetry of these spacetimes, and so they might disappear in more realistic models with inhomogeneity and anisotropy. Many attempts to prove theorems in this regard by direct analysis of the field equations and examination of exact solutions failed. The situation was totally transformed by a highly innovative paper by Roger Penrose in 1965 that used global methods and causal analysis to prove that singularities will occur in gravitational collapse situations where closed trapped surfaces occur, a causality condition is satisfied, and suitable energy conditions are satisfied by the matter and fields present. Stephen Hawking then showed that time-reversed versions of this theorem would be valid in an expanding universe, because time reversed closed trapped surfaces occur in realistic cosmological models; indeed their existence can be shown to be a consequence of the existence of the cosmic microwave blackbody radiation (Hawking and Ellis, 1973). Thus the prediction of a start to the universe is not dependent on the high symmetries of the Robertson-Walker geometries. According to classical general relativity, they can be expected to occur in realistic classical cosmological models.

John Wheeler emphasized that existence of spacetime singularities - an edge to spacetime, where not just space, time, and matter cease to exist, but even the laws of physics themselves no longer apply - is a major crisis for physics:

"The existence of spacetime singularities represents an end to the principle of sufficient causation and to so the predictability gained by science. How could physics lead to a violation of itself -- to no physics?"

It is unclear how to deal with this in physics or in philosophical terms.

Quantum Fields As is shown by inflationary models, it is no longer necessarily true that the energy condition (54) holds once quantum fields are taken into account (see (40)). This means that despite some contrary claims, it is after all possible there are singularity-free models when this is taken into account, because such fields are expected to be important in the early universe. The canonical singularity free inflationary model is the de Sitter universe in the $k=+1$ frame (35); it is also possible to have models that are asymptotic to the Einstein static universe in the past (emergent universes').

Quantum gravity This is of course still a prediction of the classical theory of gravitation, even if the matter source is a quantum field. The real issue is whether there will be a singularity at the start of the universe when we take full quantum gravity into account. It is still not known if quantum gravity solves this issue or not, primarily because we do not know what the true nature of quantum gravity is. There are hints from loop quantum cosmology that quantum gravity might remove the initial singularity, but the issue is not yet resolved. Thus in the end the outcome of the classical singularity theorems is a prediction that there most likely was a quantum gravity era in the very early universe, but not a statement as to whether a physical singularity existed or not.

### Physics Horizons

The problem we are running into here is what we will call the physics horizon. For temperatures larger that $10^{11}$~K, the microphysics is less understood and more speculative than during nucleosynthesis. Many phenomena such as baryogenesis and reheating at the end of inflation still need to be understood in better detail, but we can't test the relevant physics in the laboratory or in colliders at present, or perhaps ever. And no matter how we improve colliders, we have no hope of reaching the Planck energy to explore experimentally the nature of quantum gravity. Thus there are energies beyond which we will never be able to test physics for both experimental and economic reasons. The physics the other side of this horizon will always be speculative. We will be able to make well-informed guessses as to what its nature is likely to be, but we will never be able to directly test such hypotheses. We may be able to test some of their implications for cosmology, such as whether they imply an inflationary era or not; but it is extremely unlikely we will ever be able to prove that what we hypothesize about the relevant physics at such very early eras is the only possibility.

This applies of course specifically to any theories we may have about creation of the universe. We can hopefully extend our knowledge of what happens once the universe exists into theories of what might happen before it exists, but even that sentence does not make sense, because none of before' or exist' or happens' are applicable then, particularly because then' does not exist either.

## Observations and testing

The core feature of any scientific enterprise is the way we can experimentally or observationally test our proposed theories. Cosmology now has an abundance of extraordinarily sensitive observations with which to test its models; nevertheless, there are strict limits to what such tests can achieve.

### The basic observational dilemma

Because of the vast scale of the universe, we can essentially only see it from one spacetime position (here and now'), see Figure 2.

Figure 2: Astrophysical data are mostly located on our past lightcone, shown here in confroemal coordinates, and they fade away with distance (i.e. with larger redshift $z$). Hence we only have observational access to a portion of a 3-dimensional null hypersurface. An astronomical object can be observed only when its worldline (dashed lines) intersects our past lightcone, with all such images being projected onto a 2-dimensional space (the sky'), leaving us with the task of determining the distances of visible obects from this restricted data. Some geological type data about the neighborhood of our Solar system is also available in terms of matter and objects we see nearby

Consequently

• Our access to the universe is a 2-dimensional image of a 4-dimensional spacetime.
• Our first key problem is reliably determining distances of objects we see.
• All observations of distant objects are associated with a lookback time associated with the speed of light, which makes it difficult distinguishing time variation from spatial variation, and means we see them when their properties may be very different from those of similar objects today (source evolution').
• Visual horizons occur, limiting the part of the universe available for observational examination by means of any kind of radiation, now and for the foreseeable future.
• However we do have available at our present spacetime position relics of events in the very early universe, such as elements, stars, and galaxies, which enable us to place limits on physical processes near our past world line at very early times. We will call this geological evidence'.

### Observational relations: background model

Observational relations can be worked out for the background FLRW models, based on the fact that (Kristian and Sachs, 1966) photons move on null geodesics $x^{a}(v)$ with tangent vector $$\tag{55} k^{a}(v)\ =dx^{a}/dv:k^{a}k_{a}=0, k_{\;;b}^{a}k^{b}=0.$$ This shows that the radial coordinate value of radial null geodesics through the origin (which are generic null geodesics, because of the spacetime symmetry) is given by $\{ ds^{2}= 0, \,\,\, \;d\theta = \;d\phi =0\}$ which implies \begin{eqnarray}\tag{56} u(t_{0},t_{1}):=\int _{0}^{r_{\rm emit}}dr=\int _{t_{\rm emit}}^{t_{\rm obs}}\frac{dt}{a(t)}=\int \nolimits _{a_{\rm emit}}^{a_{\rm obs}}\frac{1}{H(t)}\frac{da}{a^{2}(t)}. \end{eqnarray} where $t_0$ corresponds to the emission time $t_{\rm emit}$ and $t_1$ to the reception time $t_{\rm obs}$. Substitution from the Friedmann equation (30) shows how the cosmological dynamics affects $u(t_{0},t_{1}).$ This equation also shows that observational relations will be simplified if we use conformal time $\tau = \int dt/a(t)$ instead of $t$, for then $u=\tau$. Key observable variables resulting are

• observed redshifts $z,$ given by

$$\tag{57} 1+z=\frac{\left( k^{a}u_{a}\right) _{emit}}{\left( k^{b}u_{b}\right) _{obs}}= \frac{a(t_{obs})}{a(t_{emit})},$$

• angular size distance $r_{0}$, giving observed angular size $\alpha$ of objects of physical length $L$ by the relation $L = r_{0} \alpha$. Up to a redshift factor this distance is the same as the luminosity distance $D_{L}$:

$$\tag{58} D_{L}=r_{0}(1+z).$$ This is the reciprocity theorem, which is true for any geometry. A major theoretical result, consequent on the reciprocity theorem, is that radiation emitted with blackbody spectrum for a temperature $T_{emit}$ will be observed as blackbody radiation, but with temperature $$\tag{59} T_{obs}=T_{emit}/(1+z).$$ This is again true for any geometry;hence the black body radiation released from the Last Scattering Surface in the early universe at a temperature of about $1200K$ is observed today as Cosmic Microwave Blackbody radiation with a temperature of about $2.73K$.

• Number counts $N(z_1,z_2)$ for sources of any class visible in a given solid angle $d\Omega$ and redshift range $(z_1,z_2)$. These will be determined by the area distance $d_A = 4 \pi r_0^2$ and the relation between the affine parameter distance $v$ down the past light cone and redshift $z$.

One can work out observational relations for galaxy number counts versus magnitude $(n,m)$ and the magnitude--redshift relation $(m,z)$, which determines the present deceleration parameter $q_{0}$ when applied to standard candles.

Major observational programmes have examined these relations and determined $H_{0}$ and $q_{0}.$ These depend critically on finding suitable standard candles. Galaxies, radios sources, and qso's do not do the job because of their intrinsic variability. However Supernovae of type SN1A have been found to be reliable standard candles, because the rate of decay after the Supernova maximum is related to its intrinsic luminosity. These observations showing that, assuming the geometry is indeed that of a Robertson-Walker spacetime, the universe is accelerating at recent times ($q_{0}<0)$. This means some kind of dark energy is present such that at recent times, $\rho +3p<\ 0$ (Dodelson, 2003, Peter and Uzan H-P, 2013, Ellis, Maartens and MacCallum, 2012).

The simplest interpretation is that this is due to a cosmological constant $\Lambda >0$ (equivalent to a perfect fluid with $p = - \rho$) that dominates the recent dynamics of the universe. Observations of gravitational waves from black hole binary merges have the potential to provide high quality standard candles in the future.

### Observational relations: perturbed model

A power series derivation of observational relations in generic cosmological models is given by Kristian and Sachs (Kristian and Sachs, 1966). A series of further phenomena arise in these cases resulting both directly from the model being inhomogeneous, and also from the natures of the structures that arise and reflect the nature of the inhomogeneous cosmological context that lead to their existence.

• Observational anisotropies in redshifts and number counts allow determination of angular power spectra and angular correlation functions for both matter and background radiation.
• The Cosmic Blackbody Radiation anisotropies characterised by the angular power spectrum allow determination of the radiation power spectrum $P(k)$ with Sachs-Wolfe plateau and acoustic peaks. They are determined by equations for the hierarchy of angular multipoles, which can be written in terms of angular spherical harmonics (Dodelson, 2003, Durrer, 2008), or an equivalent covariant formalism (Ellis, Maartens and MacCallum, 2012).
• Cosmic Blackbody Radiation polarisation measurements allow one to determine the primordial perturbation tensor to scalar ratio, allowing an indirect observation of the effects of gravitational waves.
• Deep redshift surveys of matter sources allow identification of the matter power spectrum $P(k)$, whose Fourier transform is the two-point correlation function, and its baryon acoustic oscillations.
• Redshift space distortions result from gravitational attraction caused by inhomogeneities changing the redshift-distance relation along the line of sight, and put constraints on the growth rate of structure.
• Baryon Acoustic Oscillation features can be measured in both the line-of-sight and transverse directions from number count surveys with redshifts.
• Inhomogeneities lead to both strong and weak gravitational lensing, changing the apparent position of sources in the sky. These are calculated by using the geodesic equations (55) in the perturbed metric (18). Lensing can be used to detect dark matter and can act as a magnifier for galaxies at very large distances if a strongly lensing galaxy or cluster intervenes. Lensing also affects the Cosmic Blackbody Radiation angular power spectrum, leading a smoothing of the acoustic peaks and troughs and the conversion of E-mode polarization to B-modes as well as generation of non-Gaussianity.
• The Sunyaev-Zeldovich effect (scattering of the Cosmic Blackbody Radiation by ionised hot gas in clusters of galaxies) after decoupling acts as sensitive test of anisotropies at the time of scattering, and constrains perturbation amplitude measures.

## The concordance model

The standard model of cosmology is based on nullcone data (electromagnetic radiation) of all wavelengths coming to us up the past light cone, with an associated lookback time, together with geological data, deriving from) massive particles that originated near our past light cone a very long time ago.

### The primary data

The primary data comes from many sources, as follows.

Supernovae Supernovae provide good standard candles, determining the deceleration parameter and hence showing the universe is accelerating at recent time (Figure 3), and so dark energy must be present, in agreement with the conclusion from structure formation studies.

Figure 3: A Hubble diagram obtained by the joint lightcurve analysis of 740 SNeIa from four different samples. The top panel depicts the Hubble diagram itself with the best fit (black line); the bottom panel shows the residuals. 3(b) (right) Constraints on the cosmological parameters obtained from this Hubble diagram (together with other observations: Cosmic Background Radiation (green), and Baryon Acoustic Oscillations added in (red). The dot-dashed contours correspond to the constraints from earlier Supernova data.

Discrete sources and the Lyman-$\alpha$ forest Number counts of very large numbers of sources with associated redshifts determine the matter power spectrum over a wide range of scales. This can be compared with measurements of 2-point angular correlation functions, as well as Lyman-$\alpha$ forest measurements of the intergalactic medium. These both reveal the Baryon Acoustic Oscillation peaks that were imprinted on the Last Scattering Surface (Figure 4). Their angular size on the LSS constrains the background cosmological model. [11]

Figure 4: 4(a) (left) Matter power spectra measured from the luminous red galaxy sample and the main galaxy sample of the Sloan Digital Sky Survey. Red solid lines indicate the predictions of the linear perturbation theory, while red dashed lines include nonlinear corrections. 4(b) (right) Two-point correlation function for objects aligned with the line of sight, measured with quasars ($2.1\leq z\leq3.5$) and the intergalactic medium traced by their Lyman-$\alpha$ forest, as function of comoving distance $r$. The effective redshift is $z=2.34$ here.

Background radiation Background radiation at all wavelengths provides important constraints on cosmological models, however the most important is spectrum and angular power spectrum of the relic Cosmic Blackbody Radiation which at present is microwave radiation with a temperature of $2.725K$. The spectrum of this radiation is a perfect black body spectrum, confirming the origin of this radiation from a plasma with matter-radiation equilibrium in the early universe. The Cosmic Blackbody Radiation angular power spectrum has a series of acoustic peaks corresponding to the matter acoustic peaks, extremely well modeled by structure formation theories based on an initial almost scale-free primordial spectrum (Durrer, 2008, Peter and Uzan H-P, 2013), see Figure 5(a).

Figure 5: 5(a) Angular power spectrum of the Cosmis Microwave Background temperature (left), and 5(b) $E$-mode -polarisation anisotropies of the radiation, as measured by the Planck Satellite (right). From Ref. Ade et al., 2015

This radiation will be polarised because of interactions with matter, and the Cosmic Blackbody Radiation E-mode polarisation spectrum also has acoustic peaks (Figure 5(b)). These spectra, and particularly the positions and relative heights of the peaks, importantly constrains inflationary universe models (Ade et al., 2015). There will also be B-mode polarisation if gravitational waves are present; presence or absence of such modes is a key test of inflationary models. The present best values provide evidence against a quadratic inflaton potential, as is required for the simplest version of chaotic inflation.

Relics The geological' relics from the early universe, giving information on conditions near our world line in the very distant past, [12] are of various types:

• Baryons, giving evidence of an epoch of baryogenesis, whose details are not understood;
• Chiral asymmetry of neutrinos, again for reasons that are not understood;
• Helium and Lithium abundances, well understood in terms of the process of nucleosynthesis in the early universe, but with some open questions about the details of the abundance of Lithium;
• Galaxies and clusters of galaxies, giving evidence of the nature of the galaxy formation process. This is broadly understood in terms of a bottom-up process of spontaneous structure formation facilitated by Cold Dark Matter, but detailed issues remain to be sorted out, for example why the density in the cores of galaxies do not have a cusp-like nature and why there are so few dark matter halos.

### The concordance parameters

The concordance model is characterised by a set of parameters that are determined by taking all these observations into account. "A simple inflationary model with a slightly tilted, purely adiabatic, scalar fluctuation spectrum fits the Planck data and most other precision astrophysical data" (Ade et al., 2015). Not all groups use precisely the same set, but a very useful comprehensive survey is given by the Particle Data Group (Lahav and Liddle, 2015) with measured values updated annually. The main ones given by the Planck group (Ade et al., 2015), using data from all sources, are shown in Tables 2 and 3.

Table 2: Background model parameters.
Background Mode Measured value
Hubble Parameter $H_0$ $H_0 = (67.8\pm 0.9) km/s/Mpc$
Total matter density $\Omega_m$ $\Omega_m h^2 = 0.134. \pm 0.006$
Baryon density $\Omega_b$ $\Omega_b h^2 = 0.0221 \pm 0.00034$
Radiation density $\Omega_r$ $\Omega_r h^2 = 2.47 \times 10^{-5}$
Neutrino density $\Omega_\nu$ not measured
Cosmological constant $\Omega_\Lambda$ $\Omega_\Lambda = 0.707 \pm 0.010$
Spatial curvature $\Omega_k$ $\Omega_k =0.0008 \pm 0.0040$
Ionization optical depth $\tau$ $\tau = 0.066 \pm 0.016$
Equation of state of dark energy $w$ $w = - 1.019 \pm 0.080$

We know the neutrino background must be present, even though we cannot measure it. The reionization optical depth parameter $\tau$ provides an important constraint on models of early galaxy evolution and star formation. It is determined by the EE spectrum in the multipole range $\ell = 2-6$. This is an example of how good limits on background model parameters follow from the observations related to the perturbed model. Other parameters, such as the age of the universe ($t_0 = 13.799 \pm 0.021$ Gyr), follow from those given here.

Table 3: Perturbation parameters.
Perturbations
Density perturbation amplitude $\sigma_8$ $\sigma_8 = 0.8159 \pm 0.0086$
Density perturbation spectral index $n_s$ $n_s = 0.9667\pm 0.0040$
Tensor to scalar ratio $r$ $r \leq 0.113$
Angular size of the sound horizon at recombination $\theta_*$ $\theta_* =(1.04093\pm0.00030)\times 10^{-2}$

Here the perturbation amplitude $A_S$ is represented in terms of a quantity $\sigma_8$, which is the Root Mean Square matter fluctuations today in linear theory on a scale of $8 h^{-1} Mpc$. These values directly place limits on the slow roll parameters (43) via (52), and hence constrain inflationary models (Ade et al., 2015). In particular they give an upper limit on the energy scale of inflation. The tensor-to-scalar ratio $r$ gives no statistically significant evidence for a primordial gravitational wave signal, hence ruling out the quadratic potentials underlying chaotic inflationary models. An important future effort will be getting more precise limits on this ratio by observations of B-mode Cosmic Microwave Backround polarisation.

Neutrino parameters Cosmological models with and without neutrino mass have different primary Cold Dark Matter power spectra, so one obtains limits on the number of neutrinos and on neutrino masses. The Planck team state (Ade et al., 2015), "There is no evidence for additional neutrino-like relativistic particles beyond the three families of neutrinos in the standard model. Using Baryon Acoustic Oscillation and Cosmic Microwave Background data, we find $N_{eff} = 3.30 \pm 0.27$ for the effective number of relativistic degrees of freedom, and an upper limit of $0.23$ eV for the sum of neutrino masses". This is a striking demonstration of how cosmological observations can help determine parameters of the standard model of particle physics.

Density Parameters The density parameters of the concordance model determined by the various observations are shown in Figure 6. The left hand diagram shows how the parameters $\Omega_{\Lambda}$ and $\Omega_m$ are restricted by data, and the right hand diagram shows the resulting proportions of different kinds of matter/energy in the universe.

Figure 6: 6(a) (Left) Constraints on the cosmological density parameters obtained from the Hubble diagram, structure formation studies, and CMB measurements before Planck. Credit: ESO. 6(b) (Right) Fractions of the different energy components after Planck. Credit: https://darkmatterdarkenergy.com/tag/dark-energy/.

The supernova data represent direct measurements of the parameters via the geometry of the background models. The Baryon Acoustic Oscillations and Cosmic Microwave Background data represent measurements of these parameters via the effects of the background model on structure formation, through the coefficients in (45). The data are compatible with each other, but it is the latter that provide the tightest limits. The amount of baryonic matter is determined by nucleosynthesis theory and observations (Uzan J-P, 2016), which also place limits on the number of neutrinos. The key point resulting is that the dominant dark matter is not ordinary baryonic matter: its nature is unknown. Furthermore the nature of dark energy is also unknown. However these two components dominate the dynamics of the universe (Figure 6(b)).

### A data rich subject

The overall conclusion is that due to a great many very advanced large scale observational projects, cosmology is now a very data rich subject with many observations supporting the concordance model (see Figures 3-6). A variety of different kinds of data all agree on the same basic model, which gives it much more credibility than if there were just a few items supporting the overall model. There are unresolved issues about the physics involved, but that is because this is a work in progress.

## Causal and visual horizons

A key finding is the existence in cosmology of limits both to causation, represented by particle horizons, and to observations, represented by visual horizons. Event horizons relate to the ultimate limits of causation in the future universe, that is whether $u(t_{0},t_{1})$, given by (56), converges as $t_{1}\rightarrow \infty$. They will exist in universe models that accelerate forever, but are irrelevant to observational cosmology.

### Particle horizons

The issue is whether $u(t_{0},t_{1})$ given by (56) converges or diverges as $t_{0}\rightarrow 0.$ For ordinary matter ($a(t)$ is given by (33)) and radiation ($a(t)$ is given by (37)), it converges. Then

$$\tag{60} u_{ph}(t_1) := \lim\limits_{t\rightarrow 0}u(t_0,t_1)$$ gives the comoving particle horizon size at time $t_1$, representing a limit to how far causal effects can have propagated since the start of the universe. Matter at a greater comoving distance $r$ cannot have had any causal contact whatever on the particle at the origin, because this is the furthest that any causal effect travelling at the speed of light can have moved since the start of the universe. Thus it is an absolute limit to causal effects. Much confusion about their nature was cleared up by Rindler in a classic paper (Rindler, 1956, with further clarity coming from use of Penrose causal diagrams for these models (Hawking and Ellis, 1973). These show that particle horizons occur if and only if the initial singularity is spacelike. There are many statements in the literature that such horizons represent motion of galaxies away from us at the speed of light, but that is not the case; they occur due to the integrated behaviour of light from the start of the universe to the present day.

### Visual horizons

We cannot see all the way to the particle horizon, because the early universe was opaque. The comoving visual horizon is the most distant matter we can detect by electromagnetic radiation of any kind (Ellis, Maartens and MacCallum, 2012). It is given by the comoving coordinate value $$\tag{61} u_{vh}(t_1) := u(t_{dec},t_1)$$ at time $t_1$, where $t_{dec}$ is the time of decoupling of matter and radiation (corresponding to the Last Scattering Surface). It necessarily lies inside the particle horizon. Its physical size at time $t_1$ is $d_{vh}(t_1) = a(t_1)u_{vh}(t_1)$. The visual horizon size can be 42 billion light years in an Einstein de Sitter model with a Hubble scale of 14 billion years, because of the changing expansion rate of the universe given by (33).

There will be corresponding horizons for neutrino observations, arising from the neutrino decoupling time, and gravitational waves, corresponding to the time of ending of gravitational wave equilibrium with other matter and radiation in the very early universe. However cosmological observations to those distances by directly detecting neutrinos and primordial gravitational waves would appear very unlikely. For practical purposes the observational limit is given by the visual horizon, the comoving matter comprising that horizon being the matter seen by COBE, WMAP, and Planck satellites (Figure 7).

### Primeval particle horizons

The size of the particle horizon at the time of decoupling represents the largest scale at which matter (see in Figure 7) can have been causally connected at that time. It is given by $$\tag{62} u_{pph} := \lim\limits_{t\rightarrow 0}u(t_0,t_{dec}), \,\, d_{pph} = a(t_{dec})u_{pph}.$$ It is much smaller than the Last Scattering Surface as a whole if we have a Tolman model (37) at very early times all the way back to the start of the universe, which is the horizon problem: what can have caused the matter on the Last Scattering Surface to be so uniform, when it consists of a large number of causally unrelated regions? A partial solution is provided by inflationary universe models, when the expansion at these very early times was almost exponential, as in (34).

However one must be cautious here: (62) is valid only in a Robertson-Walker geometry, or approximately in an almost Robertson-Walker geometry, and will be inapplicable if the universe is not spatially homogeneous to begin with. Furthermore, having causal contact is necessary but not sufficient: one also needs a mechanism to create uniformity. So this solution' largely assumes the result it wants. In a genuinely inhomogeneous model, the problem remains open, and inflation does not provide a solution.

### Small universes

As mentioned above, complex topologies are possible for all three spatial curvatures, resulting in altered number counts and Cosmic Microwave Background Radiation anisotropy patterns if the smallest identification scale is less than the size of the visual horizon. Then we live in a "small universe" where we have seen right round the universe since last scattering. This would result in several observational signatures, in particular there would be identical circles of temperature fluctuations in the Cosmic Microwave Background Radiation sky that would depend explicitly in the specific spatial topology.

Figure 7: The image of the Last Scattering Surface obtained by the Planck Satellite, representing fluctuations at the level of one part in $10^{-5}$. The matter seen here comprises our visual horizon (we cannot see any more distant matter). Harmonic analysis of this data leads to the angular power spectra shown in Figure 5. Note that we cannot see this matter as it is today: we see it as it was $13.8$ billion years ago. Credit: Planck.

The simplest such models have been ruled out by the Planck observations, but some complex topologies might still be viable. Checking all such possibilities is a massive observational task. The Scholarpedia article "Cosmic Topology" by J-P Luminet discusses these possibilities. One should note that if the size of the universe is 15% larger than the size of the observable universe, one cannot distinguish observationally a large universe from an infinite universe.

## More general models

The Robertson-Walker models have an exceptionally simple geometry. Other geometries have been explored, and this is an important exercise, for one cannot put limits on anisotropy and inhomogeneity if one does not have anisotropic and inhomogeneous models where one can compute observational relations which one can compare with the data. One also needs to explore the outcomes of alternatives to standard General Relativity Theory to see if they can do away with the need for dark energy or dark matter.

### The Bianchi models and phase planes

Following the work of Gödel, Taub, and Heckmann and Schücking, there is a large literature examining the properties of spatially homogeneous anisotropically expanding models (Ellis, Maartens and MacCallum, 2012), generically invariant under 3-dimensional continuous Lie groups of symmetries. Special cases (Locally Rotationally Symmetric models) allow higher symmetries. This family of models allows a rich variety of non-linear behaviour, including

• highly anisotropic expansion at early and late times, even if the present behaviour is nearly

isotropic;

• different expansion rates at the time of nucleosynthesis than in

Friedmann-Lemaître models, leading to different primordial element abundances;

• complex anisotropy patterns in the Cosmic Microwave Background sky;
• much more complex singularity behaviour than in Friedmann-Lemaître models, including cigar singularities, pancake singularities (where particle horizons may be broken in specific

directions), chaotic (mixmaster') type behaviour characterised by `billiard ball' dynamics, and non-scalar singularities if the models are tilted.

Dynamical systems methods can be used to derive phase planes showing the dynamical behaviour of these solutions and the relations of families of such models to each other (Wainwright, 1997). If a cosmological model is generic, it should include Bianchi anisotropic modes as well as inhomogeneous modes, and may well show mixmaster behaviour. These models are discussed in the Scholarpedia article "Bianchi universes, by Pontzen.

### Lemaître -Tolman-Bondi spherically symmetric models

The growth of inhomogeneities may be studied by using exact spherically symmetric solutions, enabling study of non-linear dynamics. The zero pressure such models are the Lemaître-Tolman-Bondi exact solutions (Krasinski, 1997, Ellis, Maartens and MacCallum, 2012) where the time evolution of each spherically symmetric shell of matter is independently governed by a Friedmann equation. The solutions generically have a matter density that is radially dependent, as well as a spatially varying spatial curvature and bang time. These models can be used to study how a spherical mass with low enough kinetic energy breaks free from the overall cosmic expansion and recollapses to form a local bound system. They can have a complex singularity behaviour, but this is unrealistic as the early universe will not be pressure free. Near the singularity they can however be velocity dominated.

What is important is that these models show that any observed source magnitude ($m,z$) and number count $(N,z)$ relations can be obtained in a suitable Lemaître -Tolman-Bondi model where one runs the Einstein Field Equations backwards to determine the free functions in the metric from observations. This can be done for any value whatever of the cosmological constant $\Lambda$, including zero (Ellis, Maartens and MacCallum, 2012). This opens up the possibility of doing away with the need for dark energy if we live in an inhomogeneous universe model rather than a Friedmann-Lemaître universe, where the data usually taken to indicate a change of expansion rate in time due to dark energy are in fact due to a spatial variation. However although the supernova and number count observations can be explained in this way, detailed observational studies based in the kinematic Sunyaev-Zeldovich effect show this possibility is unlikely.

### Other geometries

Szekeres Models These are non spherically symmetric Lemaître -Tolman-Bondi generalisations that can be used to study more general exact dynamical and observational behaviour than the Lemaître -Tolman-Bondi models (Krasinski, 1997, Ellis, Maartens and MacCallum, 2012).

Wheeler-Lindquist type models These models are quite different than perturbed FLRW models. They are based in the realisation that most of the universe is in fact empty space, so instead of using perturbed FLRW models one attempts to patch empty Schwarzschild solutions together in such a way that the average distance between them changes with time. One can then show that this distance obeys a Friedmann like equation and examine observational properties of such models.

### Other dynamics

Other dynamics than general relativity might be applicable on the cosmological scale. Many such possibilities have been examined, in particular $f(R)$ models, where the Lagrangian is modified by replacing the scalar curvature $R$ by a function $f(R)$, but none are compelling at the present time. This class of models span only a subclass of the more general scalar-tensor theories.

### Multiverses

Some cosmologists propose a multiverse exists: it is suggested the observed expanding universe domain (bounded by the visual horizon) is just one of billions of such domains, in each of which different physics or different cosmological parameters obtain, i.e. the other domains do not just represent more of the same (an extension beyond the horizon of the same physics and geometry). The reasons for proposing this are primarily,

• This is claimed to be a result of known physics, which is alleged to lead to chaotic inflation, and it is then assumed that some mechanism related to string theory ensures that different vacua, and hence different local physics, are realised in these domains;
• As a plausible explanation of anthropic fine tuning of various parameters so as to allow life to come into existence, and in particular as an explanation of the small value of the cosmological constant $\Lambda$.

Because of the existence of visual and physical horizons discussed above, the supposed existence of these other universe domains is not observationally testable in the usual sense. Nevertheless it is claimed by some that the observed small value of the cosmological constant (about 100 orders of magnitude less than the value suggested by quantum field theory) should be taken as adequate proof that they exist. However this is not the majority view of working cosmologists.

Claims of infinities Some multiverse enthusiasts insist that there are not just a finite number of other such domains but an infinite number, each containing an infinite number of galaxies. This is certainly not an observationally testable scientific claim, nor does it inevitably follow from established local physics. It should be regarded as metaphysical speculation rather than established science.

## Cosmological successes, puzzles, and limits

Standard cosmology is a major application of GR showing how matter curves spacetime and spacetime determines the motion of matter and radiation. It is both a major success, showing how the dynamical nature of spacetime underlies the evolution of the universe itself, with this theory tested by a plethora of observations (Ade et al., 2015, Uzan J-P, 2016), and a puzzle, because the nature of three major elements of the standard model is unknown:

• Dark matter,
• Dark energy,
• The Inflaton.

While much of cosmological theory (the epoch since decoupling) follows from Newtonian gravitational theory, this is not true of the dynamics of the early universe, where pressure plays a key role in gravitational attraction: thus for example Newtonian Theory cannot give the correct results for nucleosynthesis. The theory provides a coherent view of structure formation (with a few puzzles, for example the core-cusp problem), and hence of how galaxies come into existence. A worry about differing values for the Hubble constant obtained by different methods (Addison et al. 2017) needs to be resolved. It raises the issue that the universe not only evolves but (at least classically) had a beginning, whose dynamics lies outside the scope of standard physics because it lies outside of space and time.

Limits to physical cosmology The domain of validity of the models has been discussed above. We can only be sure of their validity as geometric and dynamic models as regards the matter within the domain we can see, curtailed by the visual horizon. We cannot be certain of the physics that held sway in the very distant past.

Only physics issues The models discussed here inform but cannot resolve philosophic issues; for example they have nothing directly to say about purpose or meaning. The only reason we mention this is because some scientists do indeed claim they can resolve such issues on the basis of our current understandings of physical cosmology. This is a regrettable attempt to extend the implications of these models way beyond their domain of validity.

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