Physical Cosmology Lecture Notes (PHAS0037) January 2024 PDF

Physical Cosmology – PHAS0037 Lecture notes B. Joachimi & A. Saintonge adapted and revised by O. Lahav & J. Sanders January 17, 2024  i The evolution of the wo...

Physical Cosmology – PHAS0037 Lecture notes B. Joachimi & A. Saintonge adapted and revised by O. Lahav & J. Sanders January 17, 2024  i The evolution of the world can be compared to a display of fireworks that has just ended: some few red wisps, ashes and smoke. Standing on a well-chilled cinder, we see the slow fading of the suns, and we try to recall the vanished brilliance of the origin of the worlds. G. Lemaı̂tre, 1931. The effort to understand the universe is one of the very few things that lifts human life a little above the level of farce, and gives it some of the grace of tragedy. S. Weinberg, The First Three Minutes, 1977. ii CONTENTS Contents Preface iii 1 The smooth Universe 1 1.1 Fundamental cosmological observations............................ 1 1.1.1 Olbers’ Paradox..................................... 1 1.1.2 The expansion of the Universe and Hubble’s Law................... 2 1.1.3 Isotropy and homogeneity................................ 4 1.2 General Relativity in a nutshell................................. 5 1.2.1 The equivalence principle and its consequences................... 5 1.2.2 A primer on metrics................................... 7 1.2.3 The FLRW metric.................................... 10 1.3 Dynamics in Newtonian Cosmology.............................. 11 1.3.1 The Friedmann equation................................. 11 1.3.2 Matter components in the Universe........................... 12 1.3.3 The acceleration equation................................ 14 1.3.4 Expansion histories................................... 15 1.4 Redshift.............................................. 16 1.5 Distances, age, and horizons.................................. 19 1.5.1 The comoving distance................................. 19 1.5.2 The angular diameter and luminosity distances.................... 20 1.5.3 The age of the Universe................................. 21 CONTENTS i 1.5.4 The Hubble radius.................................... 21 1.5.5 The particle horizon................................... 22 1.5.6 More on horizons..................................... 23 1.6 How it all began......................................... 24 1.6.1 The need for a Big Bang................................. 24 1.6.2 Problems with the standard Big Bang model...................... 27 1.6.3 Inflation.......................................... 29 1.6.4 Slow-roll models of inflation............................... 31 1.7 Summary and further reading.................................. 34 2 Thermodynamics of the Universe 36 2.1 A primer on thermodynamics.................................. 36 2.1.1 Particles in equilibrium.................................. 36 2.1.2 The blackbody spectrum................................. 41 2.1.3 Decoupling........................................ 43 2.2 Baryogenesis........................................... 45 2.3 The role of neutrinos....................................... 47 2.4 Nucleosynthesis......................................... 50 2.5 Recombination.......................................... 56 2.6 Summary and further reading.................................. 59 3 Structure formation 61 3.1 The non-relativistic wave equation in an expanding Universe................ 61 Bibliography 68 Index 69 ii CONTENTS Contents CONTENTS iii Preface These lecture notes are based on previous generations of the PHAS3137 notes written over the years. We also take inspiration from a number of cosmology and galaxy formation textbooks, with the relevant references given as additional reading at the end of each chapter of these notes. While the notes should be self-contained and provide you with all the information required for this course, we still recommend to make use of the references given, as well as other textbooks you may find useful. This will help you to acquire more background, and to be exposed to the key information expressed in different words, which will facilitate your understanding.  1 Chapter 1 The smooth Universe To take a first stab at a quantitative description of the Universe, we shall assume that the spatial distri- butions of all matter components is smooth, and we will ignore any deviations due to structures such as galaxies, stars, and cosmologists. This will provide us with a smooth background model which forms the basis for studies of physical processes and structure formation on that background. This is a good ap- proximation because any feedback from these ‘foreground’ effects on the background, e.g. the so-called backreaction of massive structures on the evolution of the smooth background, is small. We will begin with some fundamental observations that will help inform a model of the smooth Uni- verse. The physics of this background are governed by gravity, and so should be formulated using General Relativity (GR). However, we will only provide a brief, qualitative overview of GR and use New- tonian physics to motivate equations, supplementing them with GR results without giving proofs. 1.1 Fundamental cosmological observations 1.1.1 Olbers’ Paradox The most basic cosmological observation is that the night sky is dark. While that seems obvious, it is surprisingly difficult to explain if one assumes that we live in a static (in the sense of not evolving with time) and infinite universe. In such a universe one would sooner or later encounter a star along every possible line of sight. The brightness of the star decreases as 1/d 2 with the distance d from Earth (that is the inverse square law), but at the same time the number of stars in a given area of sky increases as d 2 because the volume grows as d 2 for a constant solid angle and constant line-of-sight extent. Assuming for a second that all stars have the same brightness, one would therefore expect to receive the same amount of flux from every direction. If we instead assume that the Sun has close to average brightness for the stellar population in the Universe (recall there are rare, much more luminous 2 1. The smooth Universe giant stars as well as numerous, faint dwarfs), the whole sky would be as bright as the Sun, i.e. have a temperature of about 6000 K. Olbers’ discussion of this puzzle in the 1820’s was neither the first nor particularly ground-breaking, but for some reason his name got attached to it. One could argue that interstellar dust absorbs the light from more distant objects, but as this ab- sorbing material is irradiated from all directions as well, it would soon heat up to the same temperature and also start emitting. The only convincing solution is to postulate a finite or an evolving universe. Convincing evidence for the latter was found in the early twentieth century, as we shall see next. 1.1.2 The expansion of the Universe and Hubble’s Law Hubble discovered in 1929 that galaxies tend to move away from us, independent of direction and the faster the more distant they are; see Figure ??. He calculated the distances to the galaxies using a concatenation of various distance measurements known as the cosmic distance ladder. While the absolute values of the distances were very inaccurate, the relative distances were fortunately more robust, so that the effect could be seen. To obtain the velocities, Hubble measured the spectra of galaxies and identified characteristic absorption and emission features (which could e.g. be from the Lyman and Balmer series of hydrogen) that allowed him to determine a shift in wavelength, ∆λ, of the spectra with respect to laboratory spectra. Interpreting this as a Doppler shift, one can derive the relative velocity between galaxy and observer, ∆v , ∆v ∆λ = , (1.1) c λ0 where λ0 is the laboratory wavelength of the feature and c is the speed of light. Hubble’s law can then be formulated as ∆v = H0 D , (1.2) where D is the distance to the galaxy, and H0 is the Hubble constant, the slope in the proportionality relation between velocity and distance. Modern versions of the cosmic distance ladder and other cos- mological probes agree roughly that the Hubble constant has a value of H0 ≈ 70 km/s/Mpc although there remains some controversy about the exact value. It was recently decided by the International As- tronomical Union to call this very important relation Hubble-Lemaı̂tre Law to acknowledge the important contributions by the latter in predicting it (we will encounter him again in due course). Does the recession of galaxies all around us mean that we live in a special place? To answer this, it is useful to consider the ‘Ant World’, a toy two-dimensional version of the three-dimensional expanding (or contracting) Universe (with some obvious and some more subtle limitations). The Ant World is the surface of a spherical balloon that can be in- or deflated. The ants only know their two-dimensional balloon surface and are unaware that this is embedded in a three-dimensional space, just like we fail to imagine that our space(time) is an at least four-dimensional manifold which may or may not be embedded in an even higher-dimensional space. 1.1 Fundamental cosmological observations 3 Consider a number of ants distributed on the surface of an inflating balloon, none of which move with respect to the surface. Because of the expansion, each ant observes that every other ant around it recedes, with increasing velocity for increasingly distant ants (with distance measured along great circles on the surface). Just like no ant is in any special location on the surface of the balloon, we are not in a special place if we observe the Hubble recession of galaxies from Earth. Note that the recession was observed despite the ants sitting still. Hubble’s Law is not caused by the movement of galaxies but by the increase of space between them. This should not be misunderstood as a stretching of space in the sense that this is an active process (like someone blowing into the Ant World balloon). The space between galaxies increases simply because it did so in the past. Obviously something must have started this process at some point, but the physical mechanism is as yet unknown. This also means that the galaxies themselves are not stretched during expansion (and ants don’t grow as the balloon is inflated). Galaxies and other gravitationally bound systems are decoupled from the expansion of the smooth background. For a quantitative treatment of the kinematics in an expanding universe we introduce comoving co- ordinates. Put the centre of the coordinate system into the centre of the balloon sphere. Every point on the balloon surface can be described as a function of time t by a three-dimensional vector r (t). The position vector of an ant sitting still does not change direction, and its amplitude is only a function of time. Hence we can write r (t) ≡ a(t) x , (1.3) where a(t ) is the scale factor , and x is the comoving coordinate. We have the freedom to choose the normalisation of a and set it to unity today, i.e. a(t0 ) = 1. The position of ants that do not move is uniquely defined by a value of x. They are therefore called comoving observers. Changes of position due to the expansion of the Universe are factored out into the scale factor, as opposed to the intrinsic movement of objects (ants walking about the balloon surface), which is referred to as a peculiar velocity. The comoving coordinate system is very widely used in cosmology. Note however that it is not an inertial frame, which e.g. means that energy is not conserved in this reference frame. To look at Hubble’s Law, we require the object velocities, defined as the time derivative of r (t), ȧ v (r , t) = ṙ = ȧ x = r (t) ≡ H(a) r (t). (1.4) a Here and elsewhere we will use the notation dy /dt ≡ ẏ. In the last step we defined the Hubble parameter as the relative rate of expansion (or contraction if the rate was negative). Since observers/ants can only observe relative velocities, we are interested in ∆v (r , t) = v (r + ∆r , t) − v (r , t) = H(a) ∆r (t) , (1.5) for the separation vector r (t). Equation (1.5) is the generalised form of Hubble’s Law, valid for all times t. It reduces to Equation (1.2) if we specify it to today, t = t0 , and identify H (a = 1) = H0 , i.e. the Hubble constant corresponds to the current expansion rate. 4 1. The smooth Universe Figure 1.1: Left: Example of configurations which are homogeneous (the brick wall) and isotropic (the spokes pattern) but not vice-versa. From http://astro.ucla.edu/∼wright/cosmoall.htm; accessed Nov 2015. Right: a slice of our Universe as seen by the 2dF survey. From http://www.roe.ac.uk/∼jap/2df; accessed Nov 2015. The relative velocity in Equation (1.5) is apparently not bounded by the speed of light and will grow indefinitely for increasing separation. This is fine since relativity only requires that objects do not move faster than c. As we saw before, galaxies do not have to move at all to display the observed recession. However, once ∆v approaches c , these galaxies cannot exchange information with us any more. We will return to this point when we talk about horizons in Section 1.5. 1.1.3 Isotropy and homogeneity The distribution of extragalactic objects around us is obviously not perfectly isotropic, i.e. of the same form in every direction that one looks. Galaxies have a tendency to cluster, and these clusters in turn coalesce into even larger structures known as superclusters. Recent large surveys of galaxies, e.g. with the Sloan Digital Sky Survey, have revealed the galaxy distribution on scales of tens or hundreds of megaparsecs; see Figure 1.1. The line-of-sight distances in this plot were determined with a form of Equation (1.5), assuming a fixed value for the Hubble parameter and its dependence on scale fac- tor, and measuring ∆v via the shift of wavelength in galaxy spectra. One can clearly see the cosmic web, the filamentary structures that mark the large-scale distribution of galaxies. Scales of more than 100 Mpc exceed the largest filaments, so on these scales and beyond it is fair to assume that the galaxy distribution, and consequently the matter distribution, in the Universe appears to be isotropic around us. We have no way of testing if the Universe is also isotropic around other places in the Universe. But one could argue that Earth is not a special place in the Universe, and thus the same result should be obtained by any observer. This is referred to as the Copernican Principle, because we are following Copernicus in displacing Earth from special locations in the cosmos (in his case the centre). Assuming the Copernican Principle, the Universe should be isotropic around every point in the Universe, which implies that it must be homogeneous. Homogeneity means that one can move around the Universe and take snapshots of the matter distribution, which would be indistinguishable, at least in a statistical sense. 1.2 General Relativity in a nutshell 5 Figure 1.2: Illustrations of the equivalence principle. Left: the observed motion of the mass will be identical for a constantly accelerating reference frame and a frame in a homogeneous gravitational field. Right: in both reference frames one observes a deflection of the light ray. The concepts of homogeneity and isotropy are illustrated in Figure 1.1. These constraints imply that the scale factor introduced above can indeed only be a function of time, and not |x |. They will also help us to identify simple solutions for the world models that describe the dynamics of the expanding Universe in the framework of General Relativity. 1.2 General Relativity in a nutshell 1.2.1 The equivalence principle and its consequences In the Newtonian theory of gravity it is an unexplained coincidence that the gravitational mass (the one entering Newton’s equation of gravity) and the inertial mass (the one entering Newton’s second law) are identical. This fact has been verified to high accuracy with precision experiments. Einstein elevated this coincidence to a principle. He set out to generalise his theory of special relativity, which is based on the equivalence of inertial frames, i.e. reference frames at rest or in constant motion with respect to each other, to reference frames that undergo locally constant acceleration. He posited the equivalence principle, stating that there is no way to distinguish a system with constant acceleration from one that is at rest in a constant gravitational field. This implies that inertial and gravitational mass of a test particle are also identical. This is illustrated by one of Einstein’s famous thought experiments; see Fig. 1.2. Put a rocket into space far away from any massive body and accelerate it. An experimenter in the rocket has no means of telling whether she is subject to constant acceleration or stationary in, say, Earth’s gravitational field (which is approximately constant at large distances from the planet’s centre). For instance, in one case a free-falling test mass remains at rest while the rocket floor accelerates towards the mass. In 6 1. The smooth Universe the gravitational field the mass is accelerated towards the centre of the Earth because of gravitational attraction. In both cases the experimenter would make the same observation: the test mass drops to the floor. Such thought experiments can be taken further in order to demonstrate that gravitational acceleration of a test particle is more accurately described by the free fall in a curved spacetime than motion caused by gravitational forces. To see the analogy between the properties of gravity and motion in a curved spacetime (also known as a non-Euclidian spacetime), was Einstein’s key brilliant insight. Let’s now consider photons and the right-hand side of Fig. 1.2. The experimenter in the rocket sends out photons perpendicular to the direction of motion or gravitational acceleration. In the case of the accelerated rocket in empty space an observer at rest would simply see a straight light ray whereas in the reference frame of the rocket, the light ray will appear bent away from the direction of motion. Applying the equivalence principle, this should also occur to the light ray when the rocket is at rest in Earth’s gravitational field - the ray will bend towards the centre of gravity. This directly leads to the effect known as gravitational lensing. Since photons are massless, this effect cannot be explained properly in Newtonian theory. The deflection of star light by the Sun observed during a total eclipse in 1919 was the first observational proof for General Relativity, which correctly predicted the amplitude of the deflection. To work with General Relativity, we require tools of differential geometry to describe the curved manifolds of spacetime. Moreover, the resulting equations are non-linear, which complicates matters further. To cut a very long story short, here is the centrepiece of the theory, the Einstein field equation 8πG Gµν = Tµν , (1.6) c4 where G is Newton’s gravitational constant. This equation holds for all components µ, ν = 0,..., 3 of the two symmetric tensors G and T. Since there is a correspondence between two-dimensional tensors and matrices, picture them as 4 × 4 symmetric matrices. G contains information on the geometry and is a complicated function of the spacetime metric gµν (see the following section), another symmetric two-dimensional tensor. T is the energy-momentum tensor which contains the sources of gravity. While in Newtonian gravity only mass is a source of gravity, T can additionally contain pressure, momentum, and shear stress (analogous to the shear forces in mechanics). And then there is Einstein’s famous cosmological constant, which has seen more than one dismissal and comeback over the past century. Einstein originally introduced it on the left-hand side of Eq. (1.6) as a second free parameter of his theory (beside Newton’s constant) in order to fabricate solutions that correspond to a static universe. After Hubble’s proof of the expansion of the Universe came Einstein’s legendary statement that this had been the greatest blunder of his life. Nowadays the cosmological constant, usually written as Λ, is back and forms a central element of the standard model of cosmology, this time to explain the accelerated expansion of the late-time Universe. It is quite popular to move the term containing Λ to the right-hand side of Eq. (1.6) and re-interpret its contribution as a new, rather exotic matter component, termed dark energy. We will mostly adopt this latter interpretation as dark energy models allow for flexibility beyond the model that corresponds to the cosmological constant. Keep in mind though that the cosmological constant and the dark energy model with the canonical 1.2 General Relativity in a nutshell 7 equation of state (see below for details) are mathematically equivalent. To describe the dynamics of a test particle in Newtonian gravity, one first has to link the sources of gravity (distribution of mass) to a gravitational force, via Newton’s equation of gravity. Newton’s second law then provides the link from force to particle motion. The corresponding steps in GR look as follows: the Einstein field equations link the sources of gravity to the spacetime metric. The equation of motion is now that of a free-falling particle in a curved spacetime – this is the geodesic equation:1 X µ ẍ µ = − Γαβ ẋ α ẋ β. (1.7) α,β Here, x µ are the elements of a spacetime coordinate vector, and Γ is the Christoffel symbol, which is a function of the metric and its first derivatives. The geodesic equation provides the acceleration of a test particle as a function of the metric and the velocity and position of the particle. A geodesic can be thought of as the generalisation of a straight line in flat, Euclidean space. On the surface of a sphere geodesics correspond to sections of great circles (think of intercontinental aeroplane routes). The world line, i.e. the path through spacetime, of a free-falling particle is another example of a geodesic, hence the name of the equation above. Newtonian gravity has to be the weak-field limit of General Relativity because, when the spacetime curvature is small, as for instance for planetary motion in the Solar system, Newton’s theory yields valid results. We will take advantage of this in the following by using Newtonian arguments to derive the dynamics of world models, which will work on scales much smaller than the curvature scale of the Universe, so that we are in the weak-field limit. 1.2.2 A primer on metrics Special Relativity correctly describes physical processes in reference frames which are in constant mo- tion with respect to each other. Like GR, it introduces the concept of a four-dimensional spacetime, but only in Euclidean space, i.e. special relativity can be thought of as a another limit of GR, when the curvature of spacetime goes to zero. The metric of such a flat spacetime is described by the Minkowski metric,   c2 0 0 0    0 −1 0 0  η=  , (1.8)  0  0 −1 0   0 0 0 −1 where the first element contains time (multiplied by c to get consistent units) and the three remaining ones spatial coordinates. Note that some definitions reverse the signs of the diagonal elements. The 1 Note that the placement of indices as subscripts or superscripts has significance. It identifies the transformation properties of tensors and vectors, associated with the terms covariant and contravariant. We will not work with these concepts in this course. 8 1. The smooth Universe metric tensor takes two spacetime vectors as input and produces a scalar as output. It is the general- isation of the scalar product in Euclidean space which is used to determine the length of vectors and the angle between two vectors. The metric takes on the same job and therefore describes the local properties of space(time), including the curvature (if there is curvature, tangent vectors are plugged into the metric). It is popular in cosmology to not write down the metric in matrix format but instead specify it via the square of the line element, ds, which is independent of the choice of coordinates. In Minkowski space it is given by ds2 = c 2 dt 2 − dx12 + dx22 + dx32 ,   (1.9) where t denotes time and where x1 , x2 , x3 are three-dimensional spatial coordinates. The line element gives the arclength between two points in space(time) which have infinitesimal separation, in this case by dt and dx , and characterises the local geometry like the metric tensor. Note that we actually mean ds2 ≡ (ds)2 and similar in all equations specifying line elements, but this rather sloppy notation is used ubiquitously in cosmology, and we will make no exception. In general, the line element and the metric tensor given in coordinates x1 , x2 ,..., xn in an n-dimensional space are related via n X ds2 = gµν dx µ dx ν. (1.10) µ,ν=1 With this definition one can easily show the equivalence of Eqs. (1.8) and (1.9). Since we have already established that massive objects curve spacetime, we need to understand better how curved metrics are described. For illustration we will now consider the metric on the surface of a three-dimensional sphere, which has constant positive curvature. This curvature can be quantified in terms of the radius of the sphere, R , and all points on the surface obey the relation x12 + x22 + x32 = R 2. (1.11) The differential version of this equation reads 2x1 dx1 + 2x2 dx2 + 2x3 dx3 = 0 (1.12) because R is constant. Solving for dx3 and substituting Eq. (1.11) for x3 , one arrives at x1 dx1 + x2 dx2 dx3 = − q. (1.13) R 2 − (x12 + x22 ) Infinitesimal shifts in the x3 direction are fully specified by the other two coordinates. This is not a surprising result because the surface of a three-dimensional sphere is a two-dimensional space; hence two coordinates should suffice to specify a point in that space. The result becomes a bit more intuitive if we introduce polar coordinates in the (x1 , x2 ) plane for every given x3 : x1 = ρ cos ϕ ; (1.14) x2 = ρ sin ϕ , 1.2 General Relativity in a nutshell 9 with differentials dx1 = dρ cos ϕ − ρ sin ϕ dϕ ; (1.15) dx2 = dρ sin ϕ + ρ cos ϕ dϕ. Eq. (1.13) then simplifies to ρ dρ dx3 = − p , (1.16) R 2 − ρ2 which allows us to write the line element for this metric as ds2 = dx12 + dx22 + dx32 (1.17) 2 2 ρ dρ = (dρ cosϕ − ρ sinϕ dϕ)2 + (dρ sinϕ + ρ cosϕ dϕ)2 + R 2 − ρ2 ρ2 dρ2 = dρ2 + ρ2 dϕ2 + R 2 − ρ2 dρ2 = + ρ2 dϕ2. 1 − (ρ/R )2 The result looks like two-dimensional polar coordinates with an azimuthal coordinate ϕ and a radial coordinate ρ, except that the latter is rescaled according to the curvature of the space, R. One can repeat this calculation for the surface of a four-dimensional sphere, resulting in dρ2   ds2 = 2 + ρ2 dθ2 + sin2 θ dϕ2 , (1.18) 1 − (ρ/R ) for coordinates (ρ, θ, ϕ). Again, we notice that the result looks like spherical coordinates with a rescaled radial component. We will associate this three-dimensional curved metric with the spatial part of a curved spacetime. The curvature of this curved spacetime is defined as K ≡ 1/R 2. In the calculations above we only considered the case of positive curvature, i.e. K > 0, which is also called a closed geometry. For K = 0, Eq. (1.18) reduces to standard spherical coordinates in Euclidean, i.e. flat, space. The geometry of spacetime could also be open, i.e. have negative curvature K < 0. A manifold with constant negative curvature is difficult to picture, but the two-dimensional surface of a saddle near the saddle point has K < 0; see Fig. 1.3. A good test for the local curvature of space is the sum of internal angles of a triangle: in Euclidean space this sum is 180◦ , whereas in closed (open) geometries this sum is larger (smaller). It is customary to introduce a different radial coordinate, χ, which is the same for different types of curvature. The relation between ρ and χ is given by  √ √  1/ K sin( K χ) if K > 0 ;   ρ = fK (χ) ≡ χ if K = 0 ; (1.19)  1/√−K sinh(√−K χ) if K < 0.   Inserting this relation into Eq. (1.18), the metric takes on the form   ds2 = dχ2 + fK2 (χ) dθ2 + sin2 θ dϕ2. (1.20) 10 1. The smooth Universe Figure 1.3: Classifications of the curvature of spacetime. In a closed (open) universe, curvature is positive (negative), the total energy density, here denoted by Ω0 , is larger (smaller) than unity, and the sum of internal angles of a triangle is larger (smaller) than 180◦. When the total energy density equals the critical density (Ω0 = 1), curvature is zero and space is Euclidean. Credit: NASA/ WMAP Science Team. 1.2.3 The FLRW metric The possible forms of the spacetime metric of the Universe were first investigated in the 1920’s indepen- dently by Friedmann and Lemaı̂tre2. Their work was quite visionary as they assumed a homogeneous and isotropic spacetime and found solutions that allowed an expanding universe before these facts were observationally established. A decade later Robertson and Walker conducted a more rigorous inves- tigation, establishing that the metric found was the only one possible in a homogeneous and isotropic spacetime. It is also one of the simplest non-trivial solutions of the Einstein field equations. However, it is important to keep in mind that the form of this metric is determined by fundamental geometric as- sumptions, not by the equations of GR – consequently, the metric is even applicable in generalisations of GR. We have now the tools in hand to motivate the form of this Friedmann-Lemaı̂tre-Robertson-Walker metric, in short FLRW metric. Assuming Euclidean space for a moment, we know that the metric should look similar to the Minkowski metric, except that we have to allow for expansion (or contraction) of the spatial part, i.e. ds2 = c 2 dt 2 − a2 (t) dx 2 , (1.21) where a is again the scale factor. Homogeneity demands that a is only a function of time; isotropy 2 Both Friedmann and Lemaı̂tre have interesting biographies that are worth a closer look. 1.3 Dynamics in Newtonian Cosmology 11 requires that to each spatial direction the same time-dependent function is added. Homogeneity also implies that any curvature of the spacetime must be spatially constant. In the previous section we learnt how to express a spatial metric with constant curvature. Putting it all together, we arrive at ds2 = c 2 dt 2 − a2 (t ) dχ2 + fK2 (χ) dθ2 + sin2 θ dϕ2 
. (1.22) This is the FLRW metric with coordinates (t , χ, θ, ϕ), where t is the cosmic time and the remaining three are a choice of comoving coordinates. The only free parameters in this metric are the curvature K and the scale factor a, which is actually a function of time. By inserting the FLRW metric into the equations of GR, one can derive a(t ) and hence the expansion history of the Universe, which we will do next. 1.3 Dynamics in Newtonian Cosmology 1.3.1 The Friedmann equation We will now derive the equations of motion for the scale factor. The result will provide us with the change of a with time, ȧ, as a function of curvature and the densities of the relevant sources of gravity (e.g. mass density). The correct procedure is to plug the FLRW metric (1.22) into the Einstein equations (1.6) and solve for a, given a certain content of the energy-momentum tensor. Instead, we will motivate the key equations using non-relativistic Newtonian dynamics, which will almost provide the correct result. For a more rigorous approach, see e.g. Dodelson (2004). We consider an expanding or contracting sphere of radius r , with the origin of the coordinate system in its centre. Similar to the arguments in Section 1.1.2, this does not violate the Copernican Principle because absolute coordinates will not appear in the final equations. Moreover, since we assume homo- geneity, the centre of the sphere could be positioned anywhere in the Universe without loss of generality. The sphere has mass M which will act as the source of gravity acting on a test particle on the surface of the sphere. According to Newton’s law of gravity, the test particle feels the gravitational acceleration GM r̈ = − , (1.23) r2 where we introduced the mass density of the sphere, ρ(t ). Multiplying by ṙ , this equation can be explicitly integrated, 1 2 GM 4 ṙ = + U = π Gρ(t ) r (t )2 + U , (1.24) 2 r (t ) 3 where U is an integration constant. This is simply a statement of energy conservation – the sum of the kinetic energy of the test particle, per unit test particle mass (left-hand side), and the gravitational po- tential energy per test particle mass (the first term on the right-hand side) is a constant (U ). Introducing comoving coordinates, r (t ) = a(t ) x , and doing a minor amount of algebra, one obtains  2 ȧ 8πG 2U = ρ(t ) + a(t )−2 2. (1.25) a 3 x 12 1. The smooth Universe This is how far we can get in the Newtonian framework. We have missed two important points at this stage. Firstly, special relativity tells us that mass and energy are equivalent, so that it should be the energy density that generates gravity rather than just the matter density (the latter is appropriate for non-relativistic matter). This is fixed by making the replacement ρ(t ) → ϵ(t )/c 2 , where ϵ is the energy density. Secondly, the result above was derived by implicitly assuming Euclidean space, i.e. we have neglected the effects of curvature. The GR calculation tells us that this is fixed by replacing the constant 2U /x 2 → −K c 2 , i.e. the total energy per unit mass of the system, U , is linked to the curvature of the spacetime, K. This results in the Friedmann equation,  2 ȧ 8πG = ϵ(t ) − K c 2 a(t )−2. (1.26) a 3c 2 This equation is central to determining the expansion history. However, it is a single equation linking two unknown functions, a(t ) and ϵ(t ), so we require at least one more equation to solve the system. 1.3.2 Matter components in the Universe In a homogeneous universe the energy-momentum tensor must be diagonal, and isotropy demands that the spatial components are identical (i.e. no direction is preferred or special). This reduces the number of equations that can be extracted from plugging the FLRW metric into the Einstein equations to two. The 00 component leads to the Friedmann equation while the trace of Tµν (or another combination involving the spatial diagonals of Tµν ) yields a second independent equation – the fluid equation which describes the time evolution of the matter/energy density. ‘Fluid’ because the matter components are treated as perfect fluids, which are defined by diagonal energy-momentum tensors that are solely determined by density and isotropic pressure. We will again use Newtonian arguments applied to an expanding or contracting sphere, and once again the derivation is based on the conservation of energy. This time our starting point is the First Law of Thermodynamics, dQ = dE + p dV , (1.27) where dQ is the heat flow into and out of the system, dE is the change in internal energy, p is the pressure, and dV the change in volume. If the Universe is homogeneous, there cannot be any net heat flows from one region of the Universe to another, so dQ = 0. In other words, the expansion of a homogeneous spacetime is an adiabatic process. If we consider changes of these thermodynamic quantities with respect to cosmic time, we can write the previous equation as   ȧ Ė = −p V̇ = −3 p V , (1.28) a where we have used that V = 4/3 πx 3 a(t )3. The internal energy can also be linked to the energy density, ϵ = E /V ,   ȧ Ė = ϵ̇ V + ϵ V̇ = V ϵ̇ + 3 ϵ. (1.29) a 1.3 Dynamics in Newtonian Cosmology 13 Combining Eqs. (1.28) and (1.29), one arrives at the fluid equation, ȧ ϵ̇ = −3 [ϵ(t ) + p(t )] , (1.30) a (t ) or, if expressed in terms of the matter density,   ȧ p(t ) ρ̇ = −3 ρ(t ) + 2. (1.31) a(t ) c In both equations we have made the time dependencies explicit again. Note that in the following we will mostly work with ρ(t ) rather than ϵ(t ), where the matter density should be understood as energy density divided by c 2 , which allows us e.g. to define the density of radiation, ρr , despite photons and highly relativistic particles having zero or negligible rest mass. Now we have got a second equation that contains both ρ(t ) (or ϵ(t )) and a(t ). However, we have also introduced another function, the pressure p(t ), so the system of equations is still under-determined. This can be fixed by additionally providing a relation between matter density and pressure, which is characteristic for every type of matter component and given by the equation of state, p(t ) = w (t ) ρ(t ) c 2 , (1.32) with the equation of state parameter w , which can potentially also evolve with time. The parameter w is not a new free function in our system of equations, but given in advance by specifying the properties of matter in the Universe. The different types of matter are categorised as follows. For w = 0 one has pressure-less matter, sometimes referred to as cosmic dust (we will avoid this confusing term which has nothing to do with actual dust in galaxies). Non-relativistic dark matter falls into this category because its velocity is much smaller than the speed of light so that the kinetic energy of dark matter particles is negligible compared to their rest energy. Then there is radiation, which also comprises highly relativistic matter such as neutrinos in the early Universe. This component has radiation pressure, p = 1/3 ρc 2 , so w = 1/3. Finally, if we interpret the cosmological constant as dark energy, i.e. a type of matter, we can include it in the formalism by choosing a negative pressure with w = −1. Dynamical dark energy models also allow for a deviation of the equation of state from −1, and even an evolution of the term. A common parametrisation is w (a) = w0 + (1 − a) wa , with w0 and wa as free parameters. With the equation of state parameter, the fluid equation can be re-written as ȧ ρ̇ = −3 ρ(t ) [1 + w (t )]. (1.33) a(t ) This differential equation can be solved analytically, most easily by recasting the time dependence of the quantities in the equation into a dependence on the scale factor. You can verify yourselves that the solution reads ! 1 da′  Z ′  ρ(a) = ρ0 exp 3 1 + w (a ) , (1.34) a a′ 14 1. The smooth Universe where ρ0 is the matter density today, i.e. at a = 1. For the three types of matter introduced above one then obtains the following density dependencies, ρm (a) = ρm,0 a−3 ; (1.35) ρr (a) = ρr,0 a−4 ; ρΛ (a) = ρΛ,0. For w = −1 the dark energy density is independent of time/ scale factor, as expected for an analogue of the cosmological constant. In the case of dark matter, since no mass is created or destroyed in any given volume, the density should scale inversely with the volume, hence the 1/a3 dependence. This argument also applies to radiation, but there is an additional power of a because the energy of photons and relativistic particles scales as 1/a (bear in mind that actually ρr = ϵr /c 2 ). This decrease is caused by the increase in wavelength due to redshift; see Section 1.4 for details. 1.3.3 The acceleration equation Together, the Friedmann equation and the fluid equation determine the evolution of the scale factor and of the density of the different matter components as a function of time. The two equations can be combined to yield a further relation describing the acceleration of the expansion. This new equation will be useful, but note that it is not independent of the previous two key equations. The Friedmann equation (1.26), now in terms of ρ(t ) and multiplied by a2 , is given by 8πG ȧ2 = ρ(t ) a(t )2 − K c 2 , (1.36) 3 and, by differentiating this equation with respect to time, we obtain the following result 8πG ρ̇ a2 + 2 ρ a ȧ.
2 ȧ ä = (1.37) 3 Now we can substitute ρ̇ from Eq. (1.33) to obtain the acceleration equation ä 4πG  p 4πG =− ρ+3 2 =− ρ (1 + 3 w ). (1.38) a 3 c 3 We see that, for dark matter and radiation with positive equation of state parameter, ä is always negative, i.e. these matter components slow the expansion of the Universe down. An accelerated expansion can only be explained if the dominant matter contribution at that time has w < −1/3, as is the case for the cosmological constant/ dark energy. Historically, the deceleration parameter ä ä a q≡− =− 2 , (1.39) a H (a)2 ȧ was a popular dimensionless parametrisation for changes in the expansion rate. The minus sign rep- resents the built-in expectation that the Universe should slow down in its expansion, which would have yielded positive deceleration q. As the observations of cosmic standard candles proved at the turn of the millennium, q < 0 to everyone’s surprise. 1.3 Dynamics in Newtonian Cosmology 15 1.3.4 Expansion histories To arrive at a comprehensive equation of the expansion history, we have to account for all main types of matter, i.e. dark matter, radiation, and dark energy, in the Friedmann equation. We will also switch to parameterisations that are closer to observables and ubiquitously used in cosmological analyses. This includes a change from ȧ to the Hubble parameter H (a). Moreover, we introduce density parameters, ρi,0 Ωi,0 ≡ , (1.40) ρcrit where ρcrit is the critical density of the Universe. This is the total density of all matter that the Universe needs to have today to be Euclidean, i.e. have K = 0. Writing Eq. (1.26) in terms of ρ and specialising it to a = 1 and K = 0, we find 3H02 kg ρcrit = ≈ 1.88 × 10−26 h2 3. (1.41) 8π G m Here we have encountered for the first time another popular notational quirk in cosmology – the ‘little h’, also sometimes called Hubble parameter (but do not confuse it with H (a)), defined as km H0 = 100 h. (1.42) s Mpc This notation was introduced because the Hubble constant was (and still is) one of the least well deter- mined cosmological parameters, and with this addition results could be easily rescaled as necessary. The density parameters as defined in Eq. (1.40) will generally depend on time or scale factor through ρ. When they appear in the literature, Ωi usually refers to the density parameter of component i today, i.e. at a = 1. We will add a subscript 0 to the symbol if it is necessary to distinguish it from the general a- or t -dependent density parameter; otherwise the parameter will refer to today’s value. Beware that this time-dependent density parameter is normalised by the critical density at the time considered, i.e. 8πG Ωi (a) = ρi (a). (1.43) 3H (a)2 Inserting these definitions and providing the explicit scale factor dependencies of the matter densities derived in the previous section, the Friedmann equation for all matter components reads K c2 H (a)2 = H02 Ωm,0 a−3 + Ωr,0 a−4 + ΩΛ − 2.  (1.44) a Introducing a curvature parameter , ΩK = −K c 2 /H02 , one obtains the following, frequently used form of the Friedmann equation,   Ωm,0 Ωr,0 ΩK H (a)2 = H02 + + ΩΛ +. (1.45) a3 a4 a2 By evaluating this equation at a = 1, we find the constraint ΩK = 1 − Ωm,0 − Ωr,0 − ΩΛ. In other words, in a flat universe the densities of the different matter components today have to add up to the critical density. 16 1. The smooth Universe From the different dependencies on a of the terms in Eq. (1.45) one can deduce that the contributing matter components will have dominated the cosmic expansion history at different times. At some point in the early Universe radiation must have been the prevalent matter component (because of the steepest a scaling), then matter, then curvature provided it is non-zero, and eventually dark energy. The transition times between these regimes are determined by the ratios of the density parameters. We will take a look at their values and the consequences for the history of the Universe in Chapter 2. For now we will take a brief look at the expansion rates that result if one of the contributing terms to the Friedmann equation is dominant. First, we consider the case of strong curvature, where the matter component terms are negligibly small, i.e. this Universe is more or less empty. Then the Friedmann √ equation reduces to H (a)2 = H02 ΩK /a2 , or ȧ = H0 ΩK = const, with ΩK ∼ 1. In this so-called Milne √ model the scale factor therefore grows linearly with time, a = H0 ΩK t and H (t ) = 1/t. An Einstein-de Sitter model is Euclidean and contains only dark matter at the critical density, i.e. Ωm,0 = 1. Before the discovery of the accelerated expansion of the Universe, this model was quite popular, not least because it allows for closed-form analytic solutions of a number of classical problems. The Friedmann equation in this model reads H (a)2 = H02 /a3 , or ȧ2 = H02 /a. Solutions to this differential equations are given by H (t ) = 2/(3t ) and a ∝ t 2/3 , i.e. the scale factor grows somewhat slower than in an empty universe. An analogous calculation can be made for a flat, radiation-dominated universe with Ωr,0 = 1 and Ωm,0 = ΩΛ = ΩK = 0. This results in H (t ) = 1/(2t ) and a ∝ t 1/2. Finally, we consider the late-time Universe in which dark energy or the cosmological constant domi- √ nates the expansion. The Friedmann equation in this de Sitter model reduces to H (a) = H0 ΩΛ = const.  √  The solution for this constant expansion rate is a = a0 exp H0 ΩΛ (t − t0 ) , i.e. for ΩΛ ̸= 0 we expect the scale factor to asymptotically grow exponentially with time. 1.4 Redshift Up to now we have characterised epochs in the history of the Universe via the value of the scale factor. We can replace this by a quantity that has the advantage of being directly accessible to observations, the redshift. Redshift is defined as λobs − λem z≡ , (1.46) λem where λobs is the observed wavelength of a photon while λem is the laboratory wavelength, i.e. the wavelength that the photon had at the time it was emitted. Like Hubble, we can identify characteristic features in the spectra of distant objects (usually galaxies) and compare the observed wavelength of the feature with its rest wavelength. To find the relation between z and scale factor, we consider the FLRW metric for photons. Photons move along special paths, called null geodesics, for which ds = 0, i.e. the arclength of world lines in spacetime is always zero. Why photons pick these special geodesics is best illustrated by considering 1.4 Redshift 17 a Minkowski metric. For ds2 = 0 the Minkowski line element reduces to c 2 dt 2 = dx 2 or d|x |/dt = ±c, which just says that light always travels at velocity c. In a homogeneous and isotropic Universe photons will keep to a radial path (there is nothing to divert them), which corresponds to dθ = dϕ = 0. In this case Eq. (1.22) simplifies to c dt = −a dχ. (1.47) When taking the square root, we have chosen the negative branch (hence the minus sign) because we are interested in events in our past. Now we consider two adjacent crests of a lightwave that has been emitted at redshift z and has travelled to an observer today, so tobs = t0. At emission, the time between the passing of the two crests is ∆tem = λem /c , and a corresponding equation holds for ∆tobs today. We can now integrate the simplified FLRW line element for the first wave crest between the time of emission and observation, Z tobs 1 χem Z dt dχ =. (1.48) c 0 tem a(t ) The corresponding expression for the second wave crest reads Z χem Z tobs +∆tobs 1 dt dχ =. (1.49) c 0 tem +∆tem a(t ) Since the comoving distance to the object that emitted the light is the same in both equations, we can equate them to obtain Z tobs +∆tobs Z tem +∆tem dt dt =. (1.50) tobs a(t ) tem a (t ) The scale factor is constant to good approximation in the small time spans between two wave crests, and so Eq. (1.50) simplifies to ∆tem ∆tobs = (1.51) a(tem ) a(tobs ) Invoking the definition of redshift in Eq. (1.46), we obtain the following important relation between redshift and scale factor, λobs ∆tobs a(tobs ) 1 1+z = = = = , (1.52) λem ∆tem a(tem ) a using a(tobs ) = a(t0 ) = 1 since we observe the photon today, and a(tem ) ≡ a since this is the epoch at which the object with redshift z was observed. In addition we have found that λobs = λem /a, which is commonly interpreted as the wavelength of light being stretched proportional to the scale factor as the Universe expands. This result tells us that we should understand cosmological redshift as an alternative to a in quanti- fying the relative size of the Universe. For instance, at z = 1 the Universe had half its present-day size, and so on. As we shall see below, one can also link the scale factor, and hence redshift, to cosmic 18 1. The smooth Universe time, so that redshift is often used to specify events in cosmic history, as in ”X took place at redshift z ”. Analogous to the Hubble Law, redshift can also be used as a surrogate for distance. Locally, i.e. for z ≪ 1, it is fair to equate redshift with a Doppler shift caused by the recession velocity of galaxies, so vrec z≈ ≪1. (1.53) c This interpretation has to break down when z ∼ 1 or vrec ∼ c. However, it is possible to think of cosmological redshift as a series of Doppler shifts of photons passing between closely spaced comoving observers; see Peacock (1999) for details about this argument. In practice galaxies do not qualify as comoving observers, i.e. they have their own peculiar velocities, for example due to mutual gravitational attraction. The line-of-sight component of this physical motion will affect the observed redshift of the galaxies. In the absence of cosmic expansion, this redshift is correctly interpreted as a Doppler shift, λpec − λem vpec = ≡ zpec , (1.54) λem c where λpec is the observed wavelength if there was no cosmic expansion. This equation also holds for vpec /c ≪ 1 only, but can readily be generalised using the (special) relativistic formula for the Doppler effect. Keep in mind this fundamental difference in the interpretation of the redshift due to peculiar motion, zpec , and the cosmological redshift, z. For clarity, we add a subscript ’cos’ to the cosmological redshift for the remainder of this section. When both effects are present, one observes a photon whose wavelength was first modified by the peculiar motion of the emitting galaxy and subsequently redshifted due to expansion, which results in the observed redshift λobs λobs λpec  vpec  1 + zobs = = = (1 + zcos ) 1 + = (1 + zcos )(1 + zpec ). (1.55) λem λpec λem c How can we convince ourselves that cosmological redshift in an expanding universe is really the correct interpretation of the observed shifts in galaxy spectra? One can think of alternative theories. For instance, in the early twentieth century ’tired light’ was popular for a while, based on the idea that photons lose energy as they travel, and thus increase their wavelength due to some unknown weak interaction. The longer they travelled, the more energy the photons would lose, so that this increase in wavelength would also be proportional to a. A direct observation that is inconsistent with the tired light hypothesis and that supports the expanding FLRW picture is the stretching of Supernova Type Ia lightcurves. We know SN Ia as objects with the same maximum luminosity, but they also have largely identical lightcurves. Their brightness increases sharply after the explosion, peaks, and then declines slowly and with a characteristic functional form. In Fig. 1.4 the typical width of these peaked lightcurves is plotted as a function of redshift. The observational results show that the lightcurves are stretched by a factor 1 + z , in excellent agreement with our prediction in Eq. (1.52) that points in the original lightcurve separated by ∆tem should be observed with a larger separation ∆tobs = ∆tem (1 + z ). Finally, a cautionary remark about energy conservation. We have derived in this chapter that the wavelength of a photon stretches by a factor 1 + z as it travels through an expanding spacetime. The 1.5 Distances, age, and horizons 19 Figure 1.4: The relative width of Supernova Type Ia lightcurves as a function of redshift. The red line is expected for no redshift dependence of the width. The blue line corresponds to a scaling with 1 + z while the black line is the best linear fit to the data. The prediction of an expanding FLRW spacetime fits the data very well. Originally from Goldhaber et al. (2001). energy of the photon is given by E = h c /λ, so it clearly loses energy as it is redshifted. Where does this energy go, and does this violate the conservation of energy? The somewhat disturbing answer is yes. As discussed before, the comoving coordinate system is not an inertial frame, and energy conservation does not hold in that case. More generally, formulating energy conservation in non-Euclidean space is a thorny business, and still a controversial topic in GR. 1.5 Distances, age, and horizons 1.5.1 The comoving distance We can integrate the FLRW metric in Eq. (1.22) along the line of sight to obtain the distance to an object at redshift z , or for a scale factor a, in comoving coordinates. Since we follow the path of the light emitted from this object, we can start from Eq. (1.47). This yields the comoving distance, t0 1 1 z dz ′ Z Z Z Z dt da da χ(z ) = c =c =c =c , (1.56) tem a(t ) aem ȧ a aem a 2 H (a ) 0 H (z ′ ) 20 1. The smooth Universe where we have denoted the scale factor corresponding to redshift z with aem to avoid confusion with the integration variable. The comoving distance can be calculated explicitly by inserting the Hubble parameter as calculated from Eq. (1.45). 1.5.2 The angular diameter and luminosity distances While the comoving distance is measured along the line of sight, one can alternatively consider the distance to two objects at the same redshift that are separated by an angle ∆θ on the sky. In this case we can ignore differences in time and line-of-sight separation (dt = dχ = 0), as well as azimuthal dependence (dϕ = 0). Then the line element corresponds to the physical separation of the objects, ∆s = a(t ) fK (χ) ∆θ. (1.57) If we now define the angular diameter distance as the ratio of physical size or separation over angular size or separation, we obtain fK [χ(z )] DA (z ) =. (1.58) 1+z This definition of distance is matched to the use of standard rulers which are features of a known physical extent ∆s that are observed under an angle ∆θ. Examples of standard rulers are the baryon acoustic oscillation peak in the cosmic microwave background (CMB) and the matter power spectrum, or the size of galaxies predicted by the fundamental plane (see Section ??). Note that, in a closed universe, DA is non-monotonic – it decreases with increasing z ! To obtain the comoving version of this distance, one divides out the scale factor, which results in DA,com (z ) = fK [χ(z )]. (1.59) This is called the comoving angular diameter distance, and it corresponds to the expression fK that we have encountered in the spatial part of the FLRW metric. An alternative name that often appears in the literature is transverse comoving distance, to distinguish it from χ, the line-of-sight comoving distance. They are identical in a flat universe, where fK (χ) = χ, but spatial curvature leads to differences between them. Another important physical distance is the luminosity distance which is derived from standard can- dles, i.e. objects with a known luminosity L for which a flux S is observed. Invoking the inverse square law, we can relate those two quantities. Imagine a sphere whose surface touches the observer position around an isotropically radiating sphere. The flux through that sphere, i.e. at the position of the observer, is given by L S= , (1.60) 4πDL2 where DL denotes the luminosity distance. Naively, one would expect this definition of distance to be equivalent to the angular diameter distance. However, in a non-Euclidean, expanding spacetime the 1.5 Distances, age, and horizons 21 inverse square law does not hold. Therefore, if we still insist on using Eq. (1.60) as a definition of distance, we will generally get DL ̸= DA. There are two reasons for this: firstly, the light emitted from the source is redshifted, so that the energy arriving at the observer is decreased by a factor of a. Secondly, expansion causes time dilation, which implies that the arrival rate of photons is also decreased by a factor of a. Taken together, this leads to the following equation, DL (z ) = (1 + z )2 DA (z ). (1.61) This is called the Etherington relation, which actually holds in much more general circumstances than the ones we have discussed here. It tells us that locally the two distances agree, but that at higher redshift the luminosity distance is systematically larger than DA , i.e. sources are systematically fainter than expected in Euclidean space because of redshift and time dilation. 1.5.3 The age of the Universe We can use the definition of the Hubble parameter to obtain an expression of cosmic time that has passed since the Big Bang, i.e. the age of the Universe. Writing ȧ 1 da H (a) = = , (1.62) a dt a we can integrate over cosmic time until today, i.e. Z t0 Z 1 Z 1 da 1 da t0 = dt = = p , (1.63) 0 0 H (a) a H0 0 Ωm,0 a−1 + Ωr,0 a−2 + ΩΛ a2 + ΩK where in the last step we have inserted the Friedmann equation. We have already looked at a few special cases with analytic solutions to this equation in Section 1.3.4. To recap the Einstein-de Sitter case, we found H (t ) = 2/(3t ), which can be transformed into t0 = 2/(3H0 ). Generally, 1/H0 provides a typical timescale for the age of the Universe and is thus called the Hubble time (note the units of H0 ). For h = 0.7 the Hubble time is 13.5 × 109 years, which is quite close to the value one obtains for today’s best-fit cosmological model. This provides a direct test of our cosmological model. If we find one object that is clearly older than the predicted age of the Universe, the model has to be wrong. Among the oldest objects we know are globular clusters in the outskirts of the Milky Way. The oldest stars in them are calculated to have lifetimes of about 13 billion years to the present day, which is compatible with our model, under the assumption that these stars formed not long after the Big Bang. 1.5.4 The Hubble radius The Hubble constant not only provides us with a typical timescale, it can also be employed to construct a typical length scale, called the Hubble radius and given by c /H0 ∼ 4Gpc today. It is a rough estimate of the size of the visible Universe. More precisely, the sphere defined by the Hubble radius corresponds 22 1. The smooth Universe to the region in which the recession velocities of objects are smaller than the speed of light. To see this, plug the Hubble radius into Eq. (1.5) for a = 1. The Hubble radius can be defined at every epoch using the Hubble parameter. The comoving version is given by c χH (a) = , (1.64) a H (a ) where, as usual, we divided by a to go from the physical to the comoving quantity. Objects outside the Hubble radius recede faster than light, so even if that object sends a light signal straight at us, we will not receive it as the photons are actually receding from us (with c − vrec ). One would naively expect that the Hubble radius therefore constitutes a horizon for us. By horizon, cosmologists refer to a dividing surface between regions that are in causal contact (currently, in the past, or in the future), and those that know nothing about each other. Surprisingly, it turns out that the Hubble radius is not a horizon – we can receive information from objects that currently recede faster than c. We will come back to that point below. One would expect that χH (a) grows with time, and indeed it generally does, but not always. The comoving Hubble radius shrinks if the denominator in Eq. (1.64) increases with time. This implies d [a H (a)] = ä > 0 , (1.65) dt i.e. the comoving Hubble radius decreases when the Universe undergoes accelerated expansion, as is currently the case. This will have important consequences for another proposed phase of accelerated expansion in the very early Universe; see Section 1.6. 1.5.5 The particle horizon If the Hubble radius is not the horizon that delineates regions that have been in causal contact with us, what is? We want to define the comoving radius of a sphere that encompasses all regions that could have sent a signal to the observer within the lifetime of the Universe. This is not limited to cases of signals that can reach us right now, so that the actual horizon has to be larger than the Hubble radius at any given time. Consider the comoving distance in Eq. (1.56) again. It has got a maximum when tem → 0, i.e. the furthest object we can ever hope to see is an object that emitted light at the beginning of the Universe. The comoving distance to such an object is defined as the (comoving) particle horizon, t dt ′ Z χhor (t ) = c. (1.66) 0 a (t ′ ) Using the Einstein-de Sitter case as an example again, one finds χhor (t0 ) = 2c /H0 with a(t ) = (t /t0 )2/3. So, in this case the particle horizon today is just twice the Hubble radius, and in other models there are similar prefactors of order unity (but always larger than 1). Transforming the integral above into one over the scale factor and inserting Eq. (1.64), one obtains 1.5 Distances, age, and horizons 23 Figure 1.5: Minkowski diagram of our Universe. We are located along the vertical worldline of zero comoving distance, at a = 1. Shown are our past lightcone, the Hubble sphere defined by the Hubble radius, and the event horizon. The particle horizon at a given scale factor is given by the intersection of the dashed curve with the line a = const. We will never be able to receive signals from objects in the grey region, while we may do so for objects in the hatched region although these objects currently recede faster than light. From Davis & Lineweaver (2004). the following relation between particle horizon and Hubble radius, Z a ′ da χhor (a) = χH (a ′ ). (1.67) 0 a′ So, χhor is the integral over all values χH had up to that epoch, where early contributions are weighted more strongly due to the 1/a factor. For instance, if the Hubble radius grows strongly, now or at some point in the past, so does (or did) the particle horizon, while the horizon still grows in phases when the Hubble radius shrinks, albeit more slowly. 1.5.6 More on horizons Consider the Minkowski diagram in Fig. 1.5. Time, or equivalently the scale factor, is plotted on the ordinate axis, while a comoving spatial coordinate is shown on the abscissa. Note that we should actually have three spatial dimensions, but since we assume isotropy, all separations in this plot can be understood as radii of spheres. Our worldline is a vertical line at comoving distance 0, and currently we reside at a = 1. The plot shows (one half of) our past lightcone, the region that encompasses all objects that have been in causal contact with us. As one would expect this region grows the further one goes back in time. This definition sounds very much like the one of the particle horizon, and indeed they are closely linked. The dashed line in the figure provides us with χhor at each epoch, and it corresponds to the size of the past lightcone of that epoch at t = 0. In other words, follow the vertical line from the intersection of the dashed line and the a = 1 line down to t = 0 and you will find the size of the past lightcone. A further vertical line illustrates the maximum width of the lightcone at a = 1/2. 24 1. The smooth Universe The figure also shows another horizon, the comoving event horizon. It corresponds to our past lightcone in the limit t → ∞, or t → tmax for cosmological models that recollapse eventually and thus have a maximum time. Objects outside the event horizon (the dark grey area in the figure) will never be able to send a signal to us, so these parts of the Universe will never interact with ours. If the Universe behaved completely different in those regions, we will never know (and thus we also do not care, when trying to understand the physics in our part of the Universe). The comoving radius of the event horizon at a given time t is given by Z tmax dt χevent (t ) = c. (1.68) t a (t ) Note the different integration limits compared to the definition of the particle horizon. Figure 1.5 also displays the comoving Hubble radius (which defines the Hubble sphere, which is the label used in the figure). It initially grows with time but eventually begins to decrease. This occurs when the Universe enters its late-time phase of accelerated expansion due to the increasingly dominant dark energy. The hatched area comprises regions in which objects currently cannot send us signals (because they recede faster than light) but they may have done so, or may do so, at some point in our cosmic history. Part of the hatched area lies in our current past lightcone, which means that for a < 0.4, or z > 1.5, a substantial fraction of the objects that we observe recede superluminally. So, how is this possible? The photons that these objects emitted initially receded from us. But in this phase of cosmic expansion the Hubble radius increased with time, so some of these photons entered the Hubble sphere whence they were able to reach us. Meanwhile the objects that emitted these photons moved to even greater distances and therefore still recede faster than light. 1.6 How it all began Since we observe an expanding universe, spacetime must have been more compact in the past. The Friedmann equation formally even allows a → 0 – the Big Bang. This Big Bang is a key concept in our standard model of cosmology, which however has some major shortcomings that necessitate the introduction of another key ingredient in the earliest stages of the expansion of the Universe, the period of inflation. 1.6.1 The need for a Big Bang Let us consider the possible expansion histories of the Universe in more general terms, ignoring the very early stages of radiation domination, i.e. we will assume Ωr,0 → 0. Using ΩK ≈ 1 − ΩΛ − Ωm,0 , the Friedmann equation in the version of Eq. (1.45) can then be simplified into the following form,       2 2 1 1 1 1 H (a) ≈ H0 Ωm,0 − 2 + ΩΛ 1 − 2 + 2 , (1.69) a3 a a a 1.6 How it all began 25 Figure 1.6: Possible expansion histories for a given Ωm,0 and ΩΛ. The grey region at the bottom contains universes that re-collapse eventually, while the grey region in the top left corner corresponds to bounce models without a Big Bang. The diagonal line represents a spatially flat universe. From Peacock (1999). 26 1. The smooth Universe with two free parameters, Ωm,0 and ΩΛ. We can (numerically) solve this Friedmann equation for every parameter combination and build a map of the characteristic of the resulting expansion histories, as was done in Fig. 1.6. The diagonal line marks flat cosmologies with Ωm,0 + ΩΛ = 1, and our Universe lies on that line at Ωm,0 ≈ 0.3. In the grey region at the bottom there exists a maximum scale factor after which the Hubble parameter will become negative, and the Universe recollapses. This happens for ΩΛ < 0 and for matter-dominated universes with large density. For most solutions of the Friedmann equation, incl. the Universe we live in, the expansion will continue forever. At the top left there is another grey region though, which is of particular interest for us now. To gain some insight, we pick one particular point in that region for which we can obtain solutions analytically. We are interested in the values of a at which the expansion rate becomes zero, usually indicating a turnaround in the dynamics of spacetime (it could also be a saddle point). Assuming Ωm,0 = 0, we find from Eq. (1.69), s 1 alim = 1− , (1.70) ΩΛ which, for ΩΛ > 1 has a real solution with alim < 1, i.e. the turnaround in this model was at some time in the past. This means that the Universe collapsed down to scale factor alim , then ‘bounced’ to enter the current phase of expansion, hence the term bounce models for this class of cosmologies. If a finite matter density is allowed, one can also find solutions with ȧ = 0 that correspond to saddle points, which means that such universes would remain at a nearly constant scale factor for a long period of time. Such scenarios are covered by loitering models. For these models one would either observe a maximum redshift (which corresponds to alim ) or a redshift at which a particularly large number of objects would be found (zL in the figure). For a conservative minimum matter density of Ωm,0 = 0.1, this characteristic redshift would have to lie at z = 2 or lower. Today we observe galaxies at much higher redshifts, with no conspicuous excess number of objects at any redshift, so that bounce and loitering cosmologies can be ruled out with confidence. This is important because these are the only models for which the scale factor does not go to zero at early times. Since these are clearly excluded, the Universe must have formally started in a singularity with zero spatial extent (a = 0) and infinite matter density. As we argued in Section 1.4, the energy of photons scales with 1/a. This argument can be extended to all particle ensembles, so that their energy, or equivalently, their temperature, also has to diverge for a → 0. This initial state is called the Big Bang, after Fred Hoyle who coined the term with an ironic undertone because he did not believe in this scenario. However, this was before another smoking gun for the existence of a hot early state of our Universe was discovered in the form of a relic radiation known as the cosmic microwave background. Actually, we have no way of telling whether this divergent initial state really happened. As the tem- perature increases going back in time, we have to consider increasingly exotic states of matter and interactions between them. The fundamental forces unify at these very high energy scales. At some point we simply run out of physical theories to describe the expected processes. An obvious example is the unification of gravity with the other three fundamental forces, which is still an unsolved problem. At 1.6 How it all began 27 the same time, it is intuitively clear that both gravity and the unified forces governing quantum physics will play important roles at this stage. Hence, beyond some very early, dense, and hot state of the early Universe, it is open to speculation what happened. If the Universe started in a point, at least formally, does that not single out a special place, breaking our assumption of the Copernican principle? No, because the Big Bang took place at every point in space, or no point, depending on view point. To understand this, visit the Ant World once again. Their balloon started in a singularity in the centre of the sphere described by the balloon. Keep in mind though that the ants are oblivious of anything beyond the two-dimensional surface they live on. They are unaware that their Universe is a manifold embedded into three-dimensional space. For a > 0 the point where the Big Bang took place is not part of the space formed by the balloon surface, so the ants will never find this point. On the other hand, the whole balloon was concentrated in the singularity at a = 0, and in this sense all points on the surface correspond to the spatial location of the Big Bang. Beware though that our Universe is not embedded into a five- or higher-dimensional space. The FLRW spacetime is an abstract manifold that successfully evades human imagination. 1.6.2 Problems with the standard Big Bang model While the FLRW-based cosmological model seems to do an excellent job at describing the Universe we live in, there are some major problems which indicate that we are missing at least one key ingredient. In the following we will discuss four of these issues. The first problem originates in the seemingly innocent observation that the CMB has the same temperature no matter in which direction we look. In the 1990s the COBE satellite demonstrated impressively that TCMB,0 = 2.73 K everywhere in the sky. Let us calculate the angle on the sky subtended by the particle horizon, defined as θhor (z ) = χhor (z )/χ(z ) at redshift z , i.e. one divides the comoving size of the particle horizon by the comoving distance to the redshift under consideration (we have used the small angle approximation for the tangent because this angle will prove to be quite small). At zCMB ∼ 1000, we are in the matter-dominated phase of the expansion (see next chapter for details), so we can safely ignore dark energy, radiation, and curvature when evaluating Eqs. (1.56) and (1.66), using H (a) = H0 Ωm,0 a−3/2. This results in p (1 + zCMB )−1/2 θhor (zCMB ) = ≈ 1.7◦. (1.71) 1 − (1 + zCMB )−1/2 This means that with a model based on FLRW cosmology, parts of the CMB separated by more than 1.7◦ have never been in causal contact at the time the CMB is formed because they are outside the particle horizon. So how is it possible that CMB photons from opposite sides of the sky (θ = 180◦ ) have been able to fine-tune their energies to yield exactly the same energy distribution and temperature? This puzzle is called the horizon problem. A second, ostensibly harmless observation is that our Universe is very close to spatially flat, or in other words, the total matter density today is close to the critical density. Remember that the total matter density is composed of dark matter, radiation, and dark energy, and we write Ωtot,0 = Ωm,0 + Ωr,0 + ΩΛ. In 28 1. The smooth Universe 2015 the CMB analysis with the Planck satellite determined the impressive constraint |1−Ωtot,0 | < 0.005 with high confidence. How does this constraint translate to earlier times? We need to use the Friedmann equation to answer this question, starting from the form given in Eq. (1.44). This time we use the total density parameter instead of specifying the individual ones. Using the definition of the time-dependent density parameter in Eq. (1.43), one obtains K c2 1 = Ωtot (a) − , (1.72) H (a)2 a2 which, specialised to today, turns into K c2 1 = Ωtot,0 −. (1.73) H02 Combining these two equations to eliminate the curvature K , one finds H02 a2 1 − Ωtot (a) = (1 − Ωtot,0 ) ≈ (1 − Ωtot,0 ) , (1.74) H (a)2 a2 Ωm,0 a + Ωr,0 where in the last step we inserted the Friedmann equation again, neglecting curvature (we know it is small) and dark energy (as it only becomes important at late times). We see that |1 − Ωtot (a)| grows proportional to a in the matter-dominated phase of expansion, and proportional to a2 in the radiation- dominated phase. Hence, a spatially flat universe is an unstable solution of the FLRW metric when dark matter or radiation dominate the expansion. If K = 0 exactly, it stays zero forever, but any tiny deviation from zero will become larger over time. For instance, the Planck constraint translates to |1 − Ωtot (a)| < 5 × 10−6 at z = 1000, and even smaller at earlier times. Therefore the Universe is extremely fine-tuned to have curvature very close to zero, but there is nothing in our current model that would mark K = 0 as a preferred value – this is the flatness problem. The third problem occurs when we begin to fill the Universe with particles and consider their inter- actions and evolution according to the standard model of particle physics (which we will do in detail in Chapter 2). At very early times the fundamental forces are unified, which in the case of the electromag- netic, weak, and strong nuclear forces happens at T ∼ 1028 K or t ∼ 10−36 s. As the temperature drops below this threshold, the forces separate in what is referred to as a phase transition. This can be likened to the phase transition that occurs when cooling water below 0◦ C. The microscopic structure of water is isotropic, but this symmetry is lost in the phase transition since ice crystals arrange themselves with pre- ferred orientations. A large body of water will generally not turn into a single perfect ice crystal because the growth of ice begins simultaneously, and independently, in several locations. When these different domains meet, one observes topological defects, i.e. sudden changes in the symmetry properties of the crystal. The same effect occurs in the early universe, where the size of independently developing domains is given by the particle horizon at that time. The Grand Unified Theory (GUT), which unifies the afore- mentioned three forces, predicts point-like topological defects at the GUT phase transition in the form of magnetic monopoles, a particle-like isolated magnetic north or south pole. Based on the analogy above, 1.6 How it all began 29 one expects of order one such defect per particle horizon volume. We can estimate the particle horizon size at this time with Eq. (1.66) and the time dependence of the scale factor derived in Section 1.3.4, resulting in χhor (tGUT ) = 2c tGUT. Therefore one would expect a comoving number density of magnetic monopoles of 1 nmm (tGUT ) ∼ ∼ 1082 m−3. (1.75) (2c tGUT )3 Note that in this case the comoving coordinates are normalised to the epoch tGUT , not to today as usual. Assuming that no magnetic monopoles were created or destroyed afterwards, nmm evolved proportional to a−3 , like the dark matter component. We can conservatively assume that the scale factor evolved as in the matter-dominated era over the time since the GUT era (which is the age of the Universe to very good approximation), which yields a lower limit on the relic density of monopoles of the order nmm (t0 ) ∼ 1023 pc−3. (1.76) This number is clearly discrepant with the fact that magnetic monopoles have not been observed at all to date, neither in the lab nor anywhere else in the visible Universe. Moreover, magnetic monopoles are quite massive particles wherefore they fall into the category of non-relativistic dark matter and would change the expansion history once it enters the matter-dominated phase, which is also observationally excluded. This dilemma is the monopole problem – most GUT theories predict a lot of them, but there cannot be more than one or so magnetic monopole within today’s horizon volume. The fourth problem becomes obvious once one begins to wonder how the rich structures in the Universe (that we have ignored in this chapter) originated. A strictly homogeneous and isotropic universe stays that way forever. So what did act as the seeds of structure formation? We shall refer to this puzzle as the structure problem. 1.6.3 Inflation In 1981, Alan Guth proposed a mechanism taking place in the very early Universe, around the GUT time of t ∼ 10−36 s, that was able to resolve all four major problems listed above in one go. The idea was that the Universe undergoes a short period of rapid, exponential expansion, called inflation. Although we are still waiting for conclusive proof for its existence, inflation is such an elegant solution to the four big problems (and more) that it is considered part of the standard model of cosmology. Let us assume that, at some point well in the radiation-dominated phase of expansion, the energy density of the Universe suddenly became dominated by something that behaved like the cosmological constant, i.e. had an equation of state parameter of w = −1. Then, after some time of exponential expansion, radiation took over again and the Universe resorted to ‘normal’ expansion with the same dynamics as before the onset of inflation. We shall worry about how inflation was switched on and off in the next section. 30 1. The smooth Universe We saw in Section 1.3.4 that in a universe dominated by a cosmological constant the Hubble param- eter is constant and the scale factor grows exponentially. Thus, the time dependence for the scale factor in the inflationary era is as follows,  1/2  ab (t /tb )   t ≤ tb a(t ) = ab eH (ab ) (t −tb ) tb < t ≤ tf (1.77)   a eH (ab ) (tf −tb ) (t /t )1/2  t > tf , b f where inflation began at tb and ended at tf , with corresponding labels for the scale factor. In deriving this we have assumed that ΩΛ (ab ) = 1, i.e. the cosmological constant has a value such that the cor- responding energy density matches the critical density at tb. Note that this density is much higher than the critical density today. They differ by a factor H (tb )2 /H02 ∼ 10108 , which suggests that a completely different physical process than dark energy is at work here. The length of inflation is given by tf − tb , but is usually expressed in terms of the number of e-foldings, Ne , that the scale factor has increased during inflation, af eNe = = eH (ab ) (tf −tb ). (1.78) ab If we take the inverse Hubble time at that epoch as an estimate of the Hubble parameter, H (ab )−1 ≈ tGUT , a very brief period of inflation, say 10−34 s, would have been sufficient to blow up the scale factor by e100 ∼ 1043. How does this solve the problems in the cosmological model we have considered so far? If you insert the scale factor given in Eq. (1.77) into the formula for the particle horizon, Eq. (1.66), you will find that the horizon also grew by a factor eNe during inflation (as long as Ne was large). Without inflation, χhor (tf ) = 2c tf ≈ 6 × 10−28 m, whereas χhor (tf ) ∼ 1 pc at the end of inflation with Ne = 100. As the expansion history reverted back to normal after the end of inflation, the horizon size at the time that the CMB was created was also a factor 1043 larger now, comfortably extending the region of causal contact across the whole CMB sky visible to us today. Furthermore, the densities of any relic particles whose numbers were not changed from the time that inflation began are diluted by a factor e3Ne ∼ 10126. For magnetic monopoles this implies we expect a tiny probability to see one within the Gigaparsec volumes of the visible universe today. Let us reconsider Eq. (1.74), but now between ab and af rather than between a general scale factor a and today. This yields H (af )2 af2 1 − Ωtot (ab ) = [1 − Ωtot (af )] = [1 − Ωtot (af )] e2Ne , (1.79) H (ab )2 ab2 where in the second step we have used that, during inflation, the Hubble parameter was a constant, and that the ratio of scale factors is given by Eq. (1.77). This result tells us that deviations from a flat universe that may have been present at the onset of inflation are exponentially suppressed, so that at the end of inflation curvature has to vanish to very good approximation. Note that this behaviour (which is the same in a dark energy dominated universe) is the opposite of what we saw in the matter and radiation dominated phases of expansion during which deviations from flatness grew with time. 1.6 How it all began 31 The horizon calculations above illustrate that inflation blew up volumes that were on sub-nuclear scales to astronomical scales of order one p