Introduction to Friedmann Cosmology
Andrei Lupu May 29, 2016 Contents
Introduction 3
1 Tools of the Trade 4 1.1 Metrics of Surfaces with Constant Curvature ...... 4 1.2 The Robertson-Walker Metric ...... 5 1.3 The Friedmann Equation ...... 9 1.4 The Fluid and Acceleration equations ...... 10 1.5 The Equations of State ...... 12
2 Single-Component Universes 14 2.1 Temporal Evolution of the Equations of State ...... 14 2.2 Cosmological Eras and the Benchmark Model ...... 16 2.3 Empty Universes ...... 18 2.4 Spatially Flat Universes ...... 20 2.5 Radiation Only ...... 21 2.6 Matter Only ...... 23 2.7 Lambda Only ...... 24
3 Conclusion 25
4 References 26
2 Introduction
Cosmology is a branch of astrophysics that concerns itself, not with individ- ual celestial bodies, but rather with the universe as a whole, at the largest possible scale. The cornerstone of cosmology is the set of Einstein’s ten field equations, which, under its most compact form, can be written as such: 1 8πG R − Rg + Λg = T . (1) µν 2 µν µν c4 µν These equations describe the relation between the matter and the energy permeating space-time at a given point, and the metric used at that point. In other terms, Einstein’s equations link the geometry of the space-time, as given by its curvature, to the matter and energy found in it. As a result, the solution set to these equations represents all possible universes that respect general relativity. Although tremendously powerful, Einstien’s field equations are extremely complex (the terms in the above equation are tensorial in nature) and thus difficult to solve in their full form. Consequently, decennia of cosmological research have been focused on reducing the equations using restrictive condi- tions and symmetries. One very successful and highly popular simplification has been achieved by Alexander Friedmann in the 1920’s by building on the works of Howard P. Robertson and Arthur Geoffrey Walker. The objective of this paper is therefore, in the first part, to walk through Robertson and Walker’s results in a process to understand Friedmann’s equa- tion, as well as the acceleration and fluid equations, which are required in Friedmann Cosmology. This process will also allow to define common con- cepts in the field of cosmology, including curvature, metric, homogeneity and isotropy. In the second part, the article will systematically look at specific solutions to the Friedmann’s equations that are known as ”single-component universes”. Note that this article is addressed to an audience assumed to have re- ceived formal education in differential equations, multivariable calculus and introductory astrophysics.
3 1 Tools of the Trade
1.1 Metrics of Surfaces with Constant Curvature Consider a two-dimensional surface; one of its defining property is, quite ob- viously, its curvature. The curvature could vary from point to point and, locally, the surface can either be flat, negatively curved or positively curved. For the moment, let the curvature be identical at all points on the surface. The flat and positively curved case can then easily be visualized as a plane and the surface of a sphere, respectively. The negatively curved case, how- ever, is effectively impossible to represent accurately in three dimension, but can be approximated by imagining the middlepoint of a horse saddle. Thus, a uniformly flat surface is necessarily infinite, as, by travelling constantly in any given direction, it is impossible to return at the starting point. The same holds in the negatively curved case. A surface with constant positive curvature, on the other hand, will inevitably end up closing upon itself, which means that, by travelling in a constant direction, returning to the starting point becoms inevitable. Therefore, positively curved surfaces are by definition finite. A full awareness of the curvature is necessary when attempting to de- fine distances on the surface. In general spaces, the distance is defined as the length of the geodesic, which is the shortest path between two points. On a plane, the geodesic is simply a straight line of a length determined by Pythagoras’ theorem in a cartesian coordinate system, or by its equivalent in polar coordinates. However, when the surface is either positively or neg- atively curved, the notion of a ”straight line” is meaningless and a different approach must be used. Thus, we rely on relations called ”metrics” that define the distance ds between two points. On two-dimensional surfaces, the metrics defining the distance ds between the points (r, θ) and (r + dr, θ + dθ) in a polar coordinate system take the following form : κ = 0 → ds2 = dr2 + r2dθ2 (2) r κ = +1 → ds2 = dr2 + R2sin2( )dθ2 (3) R r κ = −1 → ds2 = dr2 + R2sinh2( )dθ2, (4) R where κ is the curvature parameter, with values of 0, +1 or −1 corresponding to a flat, positive or negative curvature respectively, and R is the radius of curvature.
4 The mathematical concept of a metric can be extended to spaces with more than two dimensions. In the three-dimensional case, the metrics of constant curvature have infinitesimal distance ds that can be summarized as follows: 2 2 2 2 2 2 ds = dr + Sκ(r) (dθ + sin (θ)dφ ), (5) where Rsin( r ) , κ = +1 R Sκ(r) = r , κ = 0 (6) r Rsin( R ) , κ = −1. Then, by replacing the polar coordinates (r, θ, φ) by (x, θ, φ), such that 2 2 2 2 x ≡ Sκ(r), and if we substitute dΩ ≡ (dθ + sin (θ)dφ ), the metric of a three-dimensional space with constant curvature can be written under the general formula dx2 ds2 = + x2dΩ2. (7) x2 1 − κ R2
1.2 The Robertson-Walker Metric In the same train of thought, it is possible to establish a metric for space-time, which is a four-dimensional manifold composed of three spatial dimensions and one time dimension. This is precisely what Howard P. Robertson and Arthur Geoffrey Walker achieved independently for a spacetime which is homogeneous and isotropic − two properties that we will define shortly. But before studying their result, it is necessary that we clarify the notions of curvature and distance. Firstly, in the surfaces discussed in the previous section, curvature could be understood as a property emerging when considering the additional, higher dimension. The surface of a sphere, for instance, is effectively two-dimensional, as two coordinates are sufficient to describe any point on it. However, the surface could be thought of as going around the interior of the solid ball and thus enclosing a volume, which is a three-dimensional property. However, this need not be the case. Any general n-dimensional manifold can be de- fined without reference to a (n + 1)-dimensional space. Therefore, a curved four-dimensional space-time doesn’t have to be embedded in a space with five or more dimensions. For all intents and purposes, curvature can be, and will be considered as an intrinsic property of space-time itself. The same
5 goes for the expansion of the universe, which will be discussed at length this paper. Secondly, the notion of distance between two points is a purely spatial one. It was appropriate in the previous section, but it becomes limited and incomplete when treating with a universe that evolves in time. Indeed, we need a way to distinguish between two events occuring at exact same point in space, but at different moments in time. Thus, we will substitute distance for separation, and make it account for time as well as three-dimensional space. Just like a two-dimensional surface, space-time can be locally flat, posi- tively curved or negatively curved at any point in space. Then, the curvature of a region can evolve with time, especially if massive bodies such as galax- ies or planets move through it. Einstein’s Field Equations can account for the complex interactions between curvature, matter and energy at all points, which is what makes them nearly impossible to solve in the general case and what warrants the efforts of cosmologists to apply restrictive conditions. One set of such restrictions is what made H. P. Robertson and A. G. Walker famous. They focused on the restricted set of universes which satisfy two very powerful conditions of symmetry: homogeneity and isotropy. Homogeneity is the property of a space which is identically the same at all points. In terms of curvature, this means that the curvature parameter κ and the radius of curvature R is constant for every point. An isotropic space is one that looks identically the same, no matter the direction in which you look. Essentially, these are the same curvature conditions as those imposed in section 1.1. But in this case, homogeneity and isotropy must not only hold for the topography of the universe, but also for the matter density and energy density. As a result, the form of a metric describing such a universe becomes much simpler and manageable. Notice that the immediate universe around us is far from uniform: the matter density in a star is obviously different from the matter density of the intergalactic dust, and the same holds for local curvature. However, at very large scales, namely around 100 Mpc and more, the the universe can reasonably be approximated as being homogeneous and isotropic. It is important to remember however, that the equations discussed in this paper cannot be interprated at the human or even the galactic scale. For instance, if a given model describes a universe expanding as a whole, it is fallacious to conclude that Earth is also slowly moving away from the Sun.
6 In addition to imposing those two symmety requirements, Robertson and Walker allowed the universes described to evolve in time, by expanding or contracting. They subsequently derived the general form of the metric for such homogeneous, isotropic and potentially expanding universes. The result, called the Robertson-Walker metric is the following: " # dx2 ds2 = −c2dt2 + a(t)2 + x2dΩ2 . (8) x2 1 − κ 2 R0 Here, ds is the separation between events, κ is the curvature parameter and R0 is the radius of curvature at the current time. The variable t is the cos- mological proper time or cosmic time and it is defined as the time measured by an observer who has no peculiar motion and who therefore perceives the expansion around him as isotropic and homogeneous. a(t) is called the scal- ing factor and it is a dimentionless function that measures how much the universe has expanded between two moments in time. It is normalized such that a0 = a(t0) = 1 for current time t0. Also, notice that the second term on the right hand side is effectively the general form of the three-dimensional spatial metric given in eq. (7) multiplied by the scaling factor. This is expected, since space-time has a three-dimentional spatial component which, in this case, is allowed to expand in time. Using the reverse substitution as the one used for three-dimensional spaces, we can obtain the following alternate form of the metric:
2 2 2 2 2 2 2 2 2 ds = −c dt + a(t) [dr + Sκ(r) (dθ + sin θdφ )]. (9)
Whether we use the (x, θ, φ) or (r, θ, φ), the coordinate system is comoving in the Robertson-Walker metric. What this means is that, as the universe expands, the coordinates of points in space remain constant in time. The only way for an object to change its coordinates is to move with a peculiar motion, which is, by definition, independent of the expansion flow. The dynamics of test particles with peculiar motion is however beyond the scope of this paper. Albeit we have substituted the distance of section 1.1 for the notion of separation in space-time, we can still derive a useful concept called proper distance and labeled dp(t). This proper distance is effectively the length of the shortest path (the spatial geodesic) between two points at a given time, when a(t) is constant. To derive it, let us consider an observer at position
7 (0, 0, 0) and a distant galaxy at position (r, 0, 0). Since the change in angular coordinates is null when travelling along such spatial geodesics, we can take them to be equal to 0. Thus, we obtain separation
ds = a(t)dr (10) and proper distance Z r dp(t) = a(t)dr = a(t)r. (11) 0 Then, taking the temporal derivative will yield the rate of change of the proper distance, which is effectively the recession speed of the galaxy with respect to the observer: a˙ v (t) ≡ d˙ =ar ˙ = d . (12) p p a p
For a fixed current time t = t0, the recession speed of the galaxy is linearly dependant to the proper distance. Furthermore, by substituting using the definition a˙ H ≡ , (13) 0 a t=t0 we obtain the relation vp(t0) = H0dp(t0), (14) which is the famous Hubble equation. One of the reason behind the success of the Robertson-Walker metric is precisely this: despite its simplifying as- sumptions, it predicts equations which have been empirically tested. In this case, we derived the equation that Hubble observed empirically from purely theoretical premises. Similarly, it is possible to derive from the Robertson-Walker metric a relationship between the redshift of a recessing galaxy and the scale factor at the time te when the light was emitted. For a redshift z between an observed wavelength λo and an emitted wavelength λe, the relation is 1 1 + z = , (15) a(te) where λ − λ z = o e . (16) λe
8 1.3 The Friedmann Equation The Robertson-Walker metric essentially provides a specific framework in which further cosmological discussions can occur. Alexander Friedmann worked within this framework and simplified Einstein’s Field Equations us- ing the conditions of homogeneity and isotropy. The resulting Friedmann equation is the following: a˙ 2 8πG κc2 1 = 2 ε(t) − 2 2 . (17) a 3c R0 a(t)
Here, a is the scaling factor, κ is the curvature parameter, R0 is the current radius of curvature and ε(t) is the energy density permeating any point in space. The energy density has units of energy per volume and is a prop- erty of the components found in the universe, wether radiation, matter or cosmological constant; all three of which will be detailed in section 2.1. Thus, the left hand side term is clearly kinematic in nature, describing the rate of expansion of the universe. The first term on the right hand side is a measure of the gravitational effect of components inside the universe on the rate of expansion. Finally, the second term on the right hand side is a measure of curvature, which can be shown to be related to the total energy in the universe by comparing eq. (17) to the Newtonian derivation of the Friedmann equation. Using the following definition of the the Hubble parameter H(t): a˙ H ≡ , (18) a we can relate Friedmann’s equation to the observable characteristics of the universe by perfomring a simple substitution:
2 2 8πG κc 1 H(t) = 2 ε(t) − 2 2 . (19) 3c R0 a(t) At current time, this equation becomes
2 2 8πG κc H0 = 2 ε0 − 2 . (20) 3c R0 For a flat universe (κ = 0), the curvature term disappears, and we are left with 8πG H(t)2 = ε(t). (21) 3c2 9 A simple rearrangement of this equation will yield the critical density 3c2 ε (t) ≡ H(t)2, (22) c 8πG which is the exact energy densitiy required for a flat universe. If the energy density is superior to the critical one, the universe will have a positive curva- ture. If the energy density is inferior, the universe will be negatively curved. In short: +1 , ε(t) > εc(t) κ = 0 , ε(t) = εc(t) (23) −1 , ε(t) < εc(t). It is then useful to define a density parameter ε(t) Ω(t) ≡ (24) εc(t) such that +1 , Ω(t) > 1 κ = 0 , Ω(t) = 1 (25) −1 , Ω(t) < 1. Using the density parameter instead of the energy density, we can rewrite Friedmann’s equation: κc2 1 − Ω(t) = − 2 2 2 . (26) R0a(t) H(t) An important observation that can be made from this equation is that neither the left hand side nor the right hand side of the equation will ever change in sign. Thus, the sign of curvature of a universe will remain constant, independently of how it evolves in time.
1.4 The Fluid and Acceleration equations Depending on the prefered form, the Friedmann equation has two unknowns; either a(t) and ε(t) or H(t) and Ω(t), which makes it impossible to use it alone. It is therefore necessary to find additional equations to complete the system. To derive the first equation, we must start with the first law of thermo- dynamics: dQ = dE + P dV, (27)
10 where dQ is the incremental heat transfer towards or from the system, dE is the incremental change in the internal energy of the system, P is the pressure and dV is an incremental change of volume. If the universe is homogeneous and isotropic, the heat distribution is constant, and there is no heat flow between regions. Thus, for any region, we obtain
E˙ + P V˙ = 0. (28)
Now, suppose the region is spherical (the actual shape is of no importance) with comoving radius rs and proper radius Rs(t) = a(t)rs. The volume of that sphere at any given time will be given by
4πr3a(t)3 V (t) = s (29) 3 and the rate of change of the volume will be
4πr3 a˙ V˙ = s (3a2a˙) = V 3 . (30) 3 a
Similarly, the total internal energy of the sphere is
E(t) = V (t)ε(t), (31) with rate of change E˙ (t) = V ε˙ + V˙ ε. (32) By combining equation (32) with equation (28), we obtain
a˙ a˙ V ε˙ + 3 ε + 3 P = 0. (33) a a
This is called the fluid equation and it is essentially the first law of ther- modynamics written for an isotropic, homogeneous and expanding universe. It is the second key equation that describes the expansion of the universe and can also be written, in its most compact form, as: a˙ ε˙ + 3 (ε + P ) = 0. (34) a By combining the fluid equation (34) with Friedmann’s equation (17), we can easily derive the acceleration equation which links the acceleration of the
11 universe expansion to the the energy and pressure content of the universe. The equation is of the following form: a¨ 4πG = − (ε + 3P ). (35) a 3c2 Note that both the energy density ε and the pressure P appear on the right hand side of eq. (35) and that, if they are positive, they contribute towards a negative acceleration and therefore a slowdown of the expansion of the universe. Both matter (including dark matter) and radiation have positive pressures. It is why a common interpretation of the acceleration equation is to say that the gravitational effect of the components found in a universe usually slows down its expansion. However, consider a component with pressure ε P < − . (36) 3 Such a component would have the opposite effect, and would contribute towards a positive acceleration. If that component is dominant, such that ε Puniverse < − 3 , then the expansion would effectively speed up.
1.5 The Equations of State From sections 1.3 and 1.4, we obtained three key equations describing the expansion of the universe, namely: The Friedmann equation
a˙ 2 8πG κc2 = 2 ε(t) − 2 2 (37) a 3c R0a(t) The fluid equation a˙ ε˙ + 3 (ε + P ) = 0 (38) a The acceleration equation a¨ 4πG = − (ε + 3P ). (39) a 3c2 The three unknows in this system are the scaling factor a(t), the energy density ε(t) and the pressure P (t). However, since the acceleration equation was derived from the two others, it is dependent on them. Consequently,
12 this is a situation where we have effectively two independent equations with three unknowns. It is therefore required to find an equation of the form P = P (ε). Such functions relating the pressure and the energy density are generally known as equations of state. Because the substances dealt with in cosmological contexts are analogous to dilute gases, their equation of state is a simple linear equation
P = ωε, (40) where ω is a unitless scalar depending on the substance at hand. Starting from the perfect gas law, it is possible to show that kT hv2i ω ≈ = , (41) µc2 3c2 where k is the perfect gas constant, T is the temperature, µ is the mean mass of particles and hv2i is the root mean squared velocity of the particles. 2 m If we take the air in our atmosphere, for instance, hv i ≈ 500 s , giving an ω of the order of 10−12. More generally, we will take ω ≈ 0 for non-relativistic matter whose energy is mostly contributed to by its mass, as their root mean square velocity is quite small. For simplicity, this paper will refer to such as substance as ”matter”. 1 Similarly, we will take ω = 3 for photons and for subatomic particles travelling near the speed of light, such as neutrinos. This is what we will label as ”radiation”. Looking back at the acceleration equation (39), remember that if a sub- 1 stance had ω < − 3 , it would contribute towards an accelerated expansion of the universe. Of particular interest to us is the cosmological constant ini- tially proposed by Einstein and labelled Λ. By definition, this cosmological constant will have ω = −1 and equation of state c2 P = −ε = − Λ. (42) Λ Λ 8πG Notice that, given the equation, the pressure and the energy density of the cosmological constant is constant in time, independent from the kinematics of the the universe. There are many hypotheses as to the physical interpretation of the cos- mological constant, one of which is vacuum energy originating from virtual particles. However, the exact nature of Λ is of little importance for this
13 article. Rather, we will consider it in the analysis because it allows to ac- comodate cases of accelerating expansion; one of such cases being our own universe.
2 Single-Component Universes
2.1 Temporal Evolution of the Equations of State In the first part of this article, we have successfully derived the following set of three independent equations: The Friedmann equation
a˙ 2 8πG κc2 = 2 ε − 2 2 (43) a 3c R0a The fluid equation a˙ ε˙ + 3 (ε + P ) = 0 (44) a The equation of state P = ωε. (45) Using these three equations, it is theoretically possible to obtain the values of ε(t), P (t) and a(t) at all times, and for any universe described by the Robertson-Walker metric. However, the real universe is in fact very complex, not only because it contains multiple components, each with their different equation of state, but also because there are interactions between different components in ways that make the total energy density a very difficult function to derive. The most obvious of such interactions is, for instance, the emission and absorption of photons by all atoms. The first step is therefore to assume that all interactions between compo- nents are essentially negligible. This is not as radical an assumption as it may seem. Indeed, in our universe, the number of photons originating from the Cosmic Microwave Background is overwhelmingly superior to the number of photons emitted from stellar nuclear reactions. In fact, using empirical data, it can be estimated that ε starlight ≈ 0.1. (46) εCMB
14 By neglecting the interactions between components, we also allow the total energy density and the total pressure to be a simple summation of all contributions. Thus X ε = εω (47) ω and X X P = Pω = ωεω. (48) ω ω From these summations, it is necessary that the fluid equation not only holds for the total energy density and pressure, but also for any component taken individually. We can therefore write: a˙ ε˙ + 3 (1 + ω)ε = 0. (49) ω a ω By taking ω to be constant, this becomes a separable differential equation which can be written under the form dε da ω = −3(1 + ω) (50) εω a and can be integrated to obtain
−3(1+ω) εω(a) = εω,0 · a . (51)
By simply plugging the appropriate ω for each component, we obtain, for non-relativistic matter (ω = 0): ε ε (a) = m,0 , (52) m a3 1 for radiation (ω = 3 ): ε ε (a) = r,0 , (53) r a4 and for the cosmological constant (ω = −1):
εΛ(a) = εΛ,0. (54) This is exactly what we would expect in a universe with three spatial E 3 dimensions. Indeed, the energy density is given by ε ≡ V , where V ∝ a . −3 Since we cannot create more matter, it is natural that εm ∝ a . In the
15 radiation case, we assist to the same phenomenon: the number density of photons follows an inverse cube law with respect to the scale factor. How- ever, the photons are also redshifted in the process of expansion with the proportionality relation λ ∝ a−1. Cumulated, the two effects account for equation (53). Finally, for the cosmological constant, it was already stated in section 1.5 that εΛ was defined as constant, and here we have a mathe- matical justification. Combining eq. (37) and (51), we obtain a general expression for the total energy density in the universe:
X X −3(1+ω) ε = εω = εω,0 · a , (55) ω ω
1 where (ωm = 0), (ωr = 3 ) and (ωΛ = −1). Using eq. (55), we can easily rewrite Friedmann’s equation (43) under the more explicit form
8πG X κc2 a˙ 2 = ε · a−1−3ω − . (56) 3c2 ω,0 R2 ω 0 Even with the assumption that there is no interaction between compo- nents, multi-component solutions to the Friedmann equation above can be very complex and sometimes impossible to find analytically.
2.2 Cosmological Eras and the Benchmark Model Although our actual universe is clearly a multiple-component one, having radiation, matter, the cosmological constant as well as a possible curvature, useful insight can be gained from studying simpler, single-component uni- verses. Indeed, the energy density of each component described in eq. (52), (53) and (54) has a different dependency on the scaling factor a. Therefore, the relative contribution of each component to the total energy density of our universe varies in time. Different periods during the evolution of the universe were dominated by different components, and we will label these periods as ”cosmological eras”. Before we proceed, it is important to understand that, in a continuously expanding universe, the scaling factor a increases monotonically with time. Therefore, it can be used interchangeably with time without any loss of specificity. For instance, if we mention an event that occured when a = 0.25,
16 we can without any doubt know that we are refering to a unique, clearly defined moment in the past. Similarly, we can use the redshift z for the same purpose, since it is linked to the scale factor by the simple relation 1 1 + z = . (57) a We can then refer to a moment a cosmological event occured by stating what redshift the light emitted at that time would have if it was observed today. To pinpoint the bounding times of the cosmological eras, it is required to work on the basis of the Benchmark Model, which is the empirically supported set of estimations for the energy density of each component at current time. It is most simply described in terms of the density parameter Ωω,0 of each component, as introduced in section 1.3. In the Benchmark Model, Ωr,0 = −5 8.4 × 10 ,Ωm,0 ≈ 0.3 and ΩΛ,0 ≈ 0.7 at current time. These numbers reveal that the evolution of the universe at current time is dominated by the cosmological constant, and, if the model is accurate, so will it be the case at any future point in time. That is because, as seen in section 2.1, the energy density of the cosmological constant is invariable with time, while both the energy density of matter and radiation decrease as the scaling factor increases. We are therefore in the final cosmological era of our universe. In the past, however, when the scaling factor was smaller, the universe −4 also went through different phases. Because εr ∝ a , radiation was the dominating component at very early cosmological times. Matter then fol- −3 lowed, with εm ∝ a . The transition between the two phases occured at the matter-radiation equivalence point given by −3 εm εm,0 × a Ωm,0 1 = = −4 = a. (58) εr εr,0 × a Ωr,0 By isolating the scale factor and plugging in the values of the Benchmark Model, we obtain 8.4 × 10−5 1 a = ≈ ≈ 2.8 × 10−4. (59) rm 0.3 3600 Similarly, we can compute the matter-Λ equivalence point, which yields
1 1 3 3 Ωm,0 0.3 amΛ = ≈ ≈ 0.75. (60) ΩΛ,0 0.7
17 Based on these results, the radiation-only model is a good approximation for our universe at times where a 2.8 × 10−4; the matter-only model is suitable for 2.8 × 10−4 a 0.75; and the Lambda-only model works for times where a 0.75, thus justifying the pertinence of studying single- component universes.
2.3 Empty Universes The first scenario we will consider is a universe devoid of matter, radiation or cosmological constant – essentially a universe with zero energy density. In this case the Friedmann equation is reduced to
2 2 κc a˙ = − 2 . (61) R0 Notice that positive curvature (κ = +1) is forbidden here. Otherwise, it would result in a complex rate of change of the scaling factor, which has no physical meaning. This leaves two possible solutions. The first one is the trivial one, where κ = 0. This reduces eq. (62) to
a˙ = 0, (62) which describes an empty, static and flat universe, also known as Minkowski space. The second solution is that with negative curvature (κ = −1). Under this condition, eq. (63) becomes c a˙ = ± , (63) R0 where the sign determines if the universe is expanding or contracting. In the expanding case, integration yields the solution t a(t) = , (64) t0
R0 where t0 = c . This empty, expanding and negatively curved universe is called the Milne universe. Here, the scaling factor increases linearly with time, because there is no energy density or gravitational effect at work. Also
18 notice that, if we combine eq. (63) and eq. (64), we obtain the age of the −1 universe equal to the Hubble time (t0 = H0 ). In such a universe, if a massless observer was to look at the light emitted at time te by a distant source, the redshift of the light would be given by: 1 t 1 + z = = 0 . (65) a(te) te By rearranging the above equation, we can find the time of emission as a function of the redshift: t H−1 t = 0 = 0 . (66) e 1 + z 1 + z Furthermore, in any universe governed by the Robertson-Walker met- ric, the proper distance between the source and the observer at the time of observation is given by Z r dp(t0) = a(t0)dr = r. (67) 0 Expressed as an integral over time between the emission and the observation of the light rather than over the comoving radial coordinate, eq. (67) becomes
Z t0 dt dp(t0) = c . (68) te a(t) In the case of the Milne universe, where a(t) = t , we obtain t0
Z t0 dt t0 dp(t0) = ct0 = ct0 ln( ). (69) te t te Then, substituting eq. (66) in the one above allows to express the proper distance as a function of the redshift: c dp(t0) = ln(1 + z). (70) H0 For small redshifts (z 1), the proper distance is reduced to a linear de- pendancy on z, which is in accordance with the observed Hubble Law.
19 2.4 Spatially Flat Universes The following sections will explore the flat single-component universes men- tioned as possible approximations to different cosmological eras in section 2.2. For the sake of clarity and efficiency, it is preferable to derive some of the properties of these universes in full generality. This is possible because their common flat topology gives them many similar attributes. In the case where there is no curvature (κ = 0) and where there is only one component permeating the universe at a time (ε = εω), the Friedmann equation becomes 8πGε (t) a˙ 2 = 0 a−(1+3ω). (71) 3c2 To solve this equation for the scaling factor, we must use the ansatz a ∝ tq, such that the left hand side ∝ t2q−2 and the right hand side ∝ t−(1+3ω)q. The resulting solution is 2 q = . (72) 3 + 3ω Note that this solution only holds for ω 6= −1, which excludes the cosmo- logical constant. This particular case will be addressed at a further point in the paper. For flat, single-component universes, the general form of the scale factor is therefore t 2/(3+3ω) a(t) = . (73) t0 The age of such a universe depends on the component at hand:
1 c2 1/2 t0 = . (74) 1 + ω 6πGε0 Then, the Hubble constant is given by
a˙ 2 H ≡ = t−1, (75) 0 a 3(1 + ω) 0 t=t0 and so we obtain the following relation between the age of the universe and the Hubble constant: 2 t = H−1. (76) 0 3(1 + ω) 0
20 In a flat, single-component universe, the light emitted at time te by a distant light source is perceived with a redshift z given by the formula
a(t ) 1 t 2/(3+3ω) 1 + z = 0 = = 0 . (77) a(te) a(te) te From eq. (78), we can compute the time at which the light was emitted:
t0 2 1 te = 3(1+ω)/2 = 3(1+ω)/2 . (78) (1 + z) 3(1 + ω)H0 (1 + z) Then, the proper distance between the light source and the observer is
c 2 −(1+3ω)/2 dp(t0) = 1 − (1 + z) . (79) H0 1 + 3ω Another emerging property of spatially flat, single-component universes is the horizon distance, which is the radius of the observable universe centered around us. Ligth emitted by objects beyond that limit has simply not had the time to reach us. From the Robertson-Walker metric, the general formula for the horizon distance is
Z t0 dt dhor(t0) = c , (80) 0 a(t) which, for spatially flat single-component universes, takes the form 3(1 + ω) c 2 dhor(t0) = ct0 = . (81) 1 + 3ω H0 1 + 3ω
1 For both the radiation (ω = 0) and the matter (ω = 3 ) cases, the horizon distance takes a finite value, limiting the extent of the visible universe.
2.5 Radiation Only Section 2.2 covered how single-component models can theoretically act as approximations for different cosmological eras. However, we must exercise caution. While it is obvious why we can rely on matter-only or Lambda-only models to study their corresponding periods, describing the early times in the history of the universe using the radiation-only model is a far less trivial simplification. Indeed, if we consider the photons of the Cosmic Microwave
21 Background, they have a current redshift zCMB = 1100. Using this value in eq. (57), we can obtain the scaling factor at the time of the photon decoupling:
1 1 −4 adecoupling = = ≈ 9.1 × 10 . (82) 1 + zCMB 1101 The photon decoupling thus occured after the matter-radiation equality, as −4 found in eq. (59) to be at a scale factor arm = 2.8 × 10 . The issue here is that the interaction between radiation and matter before the photon decoupling was effectively much more important than at current times. It is therefore necessary to justify why the major assumption made in section 2.1 – that of negligible interaction between components – still holds. The explanantion is threefold. First, the contribution of matter-radiation interactions to the energy density of matter is small when compared to the rest mass of baryons. Secondly, matter and radiation were in equilibrium before the time of recombination, meaning that the photon emission and absorption rates were equal. As a result, the interactions between the two components did not alter the number density of either photons or massive particles. Finally, and this is possibly the most convincing argument, photon- interacting baryonic matter contributes only about a sixth of the total matter energy density. The remaining energy density is due to dark matter, which naturally does not interact with radiation. For these reasons, we can accept Barbara Ryden’s statement that ”at early times – long before the time of radiation-matter equality – the universe was well described by a spatially flat, radiation-only model” (Ryden, 95). Most properties of a spatially flat, radiation-only universe can be obtained 1 by simply inserting the characteristic parameter ω = 3 into the equations derived in the previous section. Thus, the scale factor of such a universe is, from eq. (73), t 1/2 ar(t) = . (83) t0 Notice that here, the expansion of the universe will tend to slow down with time. Similarly, eq. (76) yields the age of the universe 1 t0 = , (84) 2H0 and, from eq. (81), we get the horizon distance c dhor(t0) = 2ct0 = . (85) H0
22 Here, the horizon distance is equivalent to the Hubble distance, which is an exceptional result. Should we look at a distant galaxy within a spatially flat, radiation-only universe, the proper distance at the moment of observation t0 is, from eq. (79), c 1 dp(t0) = 1 − . (86) H0 1 + z Then, the corresponding proper distance at the time of emission is c z dp(te) = 2 . (87) H0 (1 + z) In the flat, radiation-only case, the proper distance at time of emission reaches a maximum value of dp(te)max = c/(4H0), occuring at z = 1.
2.6 Matter Only This is the case of a spacially flat universe containing only non-relativistic matter (ω = 0), also known as an Einstein-de Sitter universe. Just like in the radiation-only case, we can easily use the equations of section 2.4 to obtain the properties of the universe. Thus, the scale factor becomes t 2/3 am(t) = , (88) t0 with the age of the universe being 2 t0 = , (89) 3H0 and the horizon distance 2c dhor(t0) = 3ct0 = . (90) H0 The proper distance between an observer and a distant light source would be 2c 1 dp(t0) = 1 − √ (91) H0 1 + z at time of observation and 2c 1 dp(te) = 1 − √ (92) H0(1 + z) 1 + z at time of emission. The maximum proper distance at time of emission is dp(te)max = 8c/(27H0), obtained when z = 5/4.
23 2.7 Lambda Only In section 2.4, we derived the general properties of spatially flat universes, excluding the case where ω = −1. Here, we will address this exception, which corresponds to a flat universe permeated only by the cosmological constant, Λ. This universe is also known as a de Sitter universe. In this universe, Friedmann’s equation takes the form 8πGε a˙ 2 = Λ a2. (93) 3c2 By making the substitution
a˙ 8πGε 1/2 H ≡ = Λ , (94) 0 a 3c2 t=t0 we can rewrite eq. (93) simply as
a˙ = H0a. (95)
The solution to the above equation in the context of an expanding universe is a = eH0(t−t0), (96) which describes an infinitiely old universe that is expanding exponentially in time. A physical interpretation of this result is once again to associate the cosmological constant to vacuum energy provided by the spontaneous generation and anihilation of virtual particle pairs. As the universe expands but the energy density of Λ stays constant, the total energy in the universe increases, thus accelerating the expansion. In this spatially flat, Λ-only universe, we can consider an observer receiv- ing the light of a distant galaxy. Then, the proper distance between them at the time of observation is
Z t0 c c H0(t0−t) H0(t0−te) dp(t0) = c e dt = e − 1 = z. (97) te H0 H0 Similarly, at the time of emission, we obtain c z dp(te) = . (98) H0 1 + z
24 This model is of particular interest because our own universe has, as dis- cussed in section 2.2, recently entered its final, Λ-dominated phase. Although reducing the Benchmark Model to the one discussed here for current times would be an oversimplification – matter still plays a non-negligible role – the approximate long-term behaviours of both models is similar, as discovered by two teams, the Supernova Cosmology Project (Perlmutter et al. 1999, ApJ, 517, 565) and the High-z Supernova Search Team (Riess et al. 1998, AJ, 116, 1009). Indeed, these research projects studied the redshift and the distance of several Type Ia supernovae and concluded that our universe is in fact undergoing an accelerating expansion. Furthermore, the temporal evolution of the equations of state tells us that, if the universe keeps expanding, the cosmological constant will become overwhelmingly dominant. It is therefore possible to emit the reasonable hypothesis that the fate of our universe will be identical to that of a Λ-only universe. That is, in the limit t → ∞, εm → 0 and εr → 0, which is referred to as the heat death of the universe or the Big Chill.
3 Conclusion
The objective of this paper was to provide an introduction to the frame- work underlying Friedmannian cosmology, as well as to solve the Friedmann equation for simple single-component universes. This work is effectively a summary of selected chapters of Barbara Ry- den’s book titled ”Introduction to Cosmology”. Some of the discussions, however, feature a different emphasis and are complemented by additional details and observations that have arisen during my personal reading and during quasi-weekly meetings with my professor and project supervisor, Ivan T. Ivanov, without whom this project would never have been completed.
25 4 References
• Ryden, Barbara. Introduction to Cosmology. Department of Astronomy. The Ohio State University. January 13, 2006. 295 p.
• International Summer School for Young Physicists, The Perimeter Institute for Theoretical Physics, Waterloo, 2015. Lecture series: ”Using Gravity to Solve all of our Problems”. Class notes. Teacher: Todd Sierens.
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