Energy and matter in the Universe

Transcription

1 Chapter 17 Energy and matter in the Universe The history and fate of the Universe ultimately turn on how much matter, energy, and pressure it contains: 1. These components of the stress energy tensor all couple to gravity. 2. This coupling determines how self-gravitation of the Universe influences the Hubble expansion. In this chapter we begin to address quantitatively the issue of the matter and energy contained in the Universe and how that determines its history. 617

2 618 CHAPTER 17. ENERGY AND MATTER IN THE UNIVERSE The Universe is mostly empty space, which might suggest that a Newtonian description of gravity (which is valid in the weak gravity limit) is adequate for describing the large-scale structure of the Universe. But whether general relativity effects are important relative to a Newtonian description may be estimated in terms of the ratio of an actual radius for a massive object compared with its radius of gravitational curvature. If we apply such a criterion to the entire Universe, reasonable estimates for the mass energy contained in the Universe indicate that the actual radius of the known Universe and the corresponding gravitational curvature radius could be comparable. Thus, a description of the large-scale structure of the Universe (cosmology) must be built on a covariant gravitational theory, rather than on Newtonian gravity. Even so, we can understand a substantial amount concerning the expanding Universe simply by using Newtonian concepts.

3 17.1. EXPANSION AND NEWTONIAN GRAVITY 619 Density = ρ Distant galaxy Earth r Homogeneous mass distribution Figure 17.1: Newtonian model of the expanding Universe Expansion and Newtonian Gravity Consider the test galaxy illustrated in Fig The gravitational potential acting on the galaxy is U = GMm, r where m is the mass of the galaxy and Total mass within sphere=m = 4 3 πr3 ρ, which is constant since ρ decreases with time and r increases but the product ρr 3 is constant. Thus U = 4 3 πgr2 ρm.

4 620 CHAPTER 17. ENERGY AND MATTER IN THE UNIVERSE Density = ρ Distant galaxy Earth r Homogeneous mass distribution If the motion of the galaxy is caused entirely by the Hubble expansion, its radial velocity relative to the Earth is v=h 0 r. This implies a kinetic energy T = 1 2 mv2 = 1 2 mh2 0 r2, where m is the inertial mass of the galaxy, assumed to be equivalent to its gravitational mass. The total energy of the galaxy is then E = T +U = 1 2 mh2 0 r2 4 3 πgr2 ρm = 1 2 mr2( H πgρ ).

5 17.2. THE CRITICAL DENSITY The Critical Density If the expansion is to halt, we must have E = 0 and thus E = 1 2 mr2( H πgρ ) H 2 0 = 8 3 πgρ. Solving for ρ, the critical density that will just halt the expansion is ρ c = 3H2 0 8πG h 2 g cm 3. The corresponding critical energy density is ε c = ρ c c 2 = h 2 MeV cm 3 = h 2 erg cm 3. The critical density corresponds to an average concentration of only six hydrogen atoms per cubic meter of space or about 140 M per cubic kiloparsec. We may distinguish three qualitative regimes for the actual density (in this simple Newtonian picture) ρ: 1. If ρ > ρ c the Universe is said to be closed and the expansion will stop in a finite amount of time. 2. If ρ < ρ c the Universe is said to be open and the expansion will never halt. 3. if ρ = ρ c the Universe is said to be flat (or euclidean) and the expansion will halt, but only asymptotically as t.

6 622 CHAPTER 17. ENERGY AND MATTER IN THE UNIVERSE Thus, in this simple Newtonian picture the ultimate fate of the Universe is determined by its present matter density. We shall see that this conclusion is modified profoundly by the apparent presence of dark energy in the actual Universe.

7 17.2. THE CRITICAL DENSITY 623 It will prove convenient to introduce the dimensionless total density parameter evaluated at the present time Ω ρ = ε = 8πGρ ρ c ε c 3H0 2. where ρ is the current total density that couples to gravity. Thus the closure condition implies that Ω=1. Note for future reference that many authors use a subscript 0 on Ω and ρ (and other cosmological parameters) to indicate explicitly that they are evaluated at the present time. Where possible we suppress these zero subscripts to avoid notational clutter. Unless otherwise noted, you should understand Ω to be Ω evaluated at the present time.

8 624 CHAPTER 17. ENERGY AND MATTER IN THE UNIVERSE 17.3 Cosmic Scale Factor As we have seen, the Hubble expansion makes it convenient to introduce a cosmic scale factor a(t) that sets the global distance scale for the Universe. If peculiar motion is ignored, the expansion is governed entirely by a(t) and all distances simply scale with this factor. Example: if present time is t 0 and present scale factor is a 0, a wavelength of light λ emitted at time t < t 0 is scaled to λ 0 at t = t 0 by the universal expansion: λ 0 = λ a 0 a(t). Likewise, if r 0 and ρ 0 are the present values of r and ρ, r(t) r 0 = a(t) a 0 ρ(t) ρ 0 = ( a0 a(t) ) 3. This permits us to express all dynamical equations in terms of the scale factor. Example: gravitational force acting on the galaxy F G = U r = GMm r 2 = 4 3 πgρrm, and the corresponding gravitational acceleration is r= F G m = GM r 2 Then from r(t)/r 0 = a(t)/a 0, r= r 0 ä= 4 a 0 3 πgρ 0 a 3 0 (acceleration of the scale factor). = 4 3 πgρr. a 3 r 0 a 0 a ä= 4 3 πgρ 0a 3 0 ( ) 1 a 2.

9 17.4. TIME DEPENDENCE OF THE SCALE FACTOR Time Dependence of the Scale Factor Identity: ä= 1 2 d da ȧ2 Earlier: ä= 1 2 H2 0 a3 0 Ω a 2 1 d 2 da ȧ2 = 1 2 H2 0 a3 0 Ω a 2. Solving this for dȧ 2 and integrating from the present time t 0 back to an earlier time t, t t 0 dȧ 2 = H 2 0 a3 0 Ω a da a 0 a 2 ( 1 ȧ 2 = ȧ H2 0 a3 0 Ω a 1 ), a 0 and since ȧ 0 = a 0 H 0 (Problem), where we define which must obey the condition since ȧ 2 can never be negative. ȧ 2 = a 2 0 H2 0 f(ω,t), f(ω,t)=1+ω a 0 a(t) Ω, f(ω,t) 0, We may use this condition to enumerate different possibilities for the history of the Universe. NOTE: Ω Ω 0 in these equations.

10 626 CHAPTER 17. ENERGY AND MATTER IN THE UNIVERSE 17.5 Expansion Histories for the Universe Let us consider as an example, dust-filled universes; that is, universes containing only pressureless, non-relativistic matter and negligible amounts of radiation or vacuum energy. Three qualitatively different scenarios for such a Universe, depending on the value of Ω Ω Ω < 1 (undercritical): In this case, as a(t), f(ω,t)=1+ω a 0 a(t) Ω 1 Ω>0. Thus ȧ never goes to zero (ȧ 2 f(ω,t) and we live in an open, ever-expanding universe if Ω<1. 2. Ω = 1 (critical): For this case, as a(t), f(ω,t) 0, but it only reaches 0 at t =. Hence, if Ω = 1, the universe is ever-expanding (constraint: expanding now) but the rate of expansion approaches zero asymptotically as t. 3. Ω > 1 (overcritical): Now as t increases f(ω,t) 0, but in a finite time t max. Beyond this time we still must satisfy the condition f(ω,t) 0. Thus, if Ω > 0 the expansion turns into a contraction at time t max and the universe begins to shrink.

11 17.5. EXPANSION HISTORIES FOR THE UNIVERSE 627 Open Ω < 1 Scale factor a(t) Flat Ω = 1 Closed Ω > 1 Time t Figure 17.2: Behavior of the scale factor a(t) as a function of time for a dust-filled universe. By integrating the scale factor equation ȧ 2 = a 2 0 H2 0 f(ω,t), for a dust model (see Problems) we obtain for these three scenarios For a flat dust universe with Ω=1, ( ) 3t 2/3 a(t)=. 2t H This behavior is sketched as the Ω=1curve in Fig

12 628 CHAPTER 17. ENERGY AND MATTER IN THE UNIVERSE Open Ω < 1 Scale factor a(t) Flat Ω = 1 Closed Ω > 1 Time t For a closed dust universe with Ω>1, a(ψ)= 1 Ω 2 Ω 1 (1 cosψ) t(ψ) = 1 Ω 2H 0 (Ω 1) 3/2(ψ sinψ). where ψ 0 parameterizes the solution. This case is sketched as the Ω>1curve in the figure above.

13 17.5. EXPANSION HISTORIES FOR THE UNIVERSE 629 Open Ω < 1 Scale factor a(t) Flat Ω = 1 Closed Ω > 1 Time t For an open dust universe with Ω<1, a(ψ)= Ω (coshψ 1) 2(1 Ω) t(ψ) = 1 Ω 2H 0 (1 Ω) 3/2(sinhψ ψ). where ψ is a parameter. This behavior is sketched as the Ω<1 curve in the figure above.

14 630 CHAPTER 17. ENERGY AND MATTER IN THE UNIVERSE 17.6 Lookback Times Telescopes are time machines: Lookback time: t L how far back in time we are looking when we view an object having a redshift z, t L = t(0) t(z), where t(z = 0) is the present age of the Universe and t(z) is the age when light observed today with redshift z was emitted. Example: in a flat dust-filled universe (Problem) t(z) τ H = 2 3 (1+z) 3/2 t(0) τ H = 2 3 and the lookback time is t L = 2 τ H (1+z) 3/2 ( 1 = (1+z) 3/2 where τ H = 1/H 0 is the Hubble time. ), Thus light from an object that we observe with a redshift z 5was emitted when 1. The Universe was only 7% of its present age 2. The cosmic scale factor a(t) was six times smaller than it is today.

15 17.6. LOOKBACK TIMES 631 Scale factor relative to today (a) H 0 = 72 km/s/mpc Ω=1 0.2 Ω=0.5 3 z = 5 Ω=0.1 5 t L Time (10 9 yr) 8 Redshift Lookback time t L (10 9 yr) (b) zτ H 1 Ω = 0.1 Ω = 0.5 Ω = 1.0 H 0 = 72 km/s/mpc Redshift z 5 Figure 17.3: (a) Geometrical interpretation of the lookback time t L for z=5 in a dust universe with three different values of the density parameter Ω. (b) Lookback time as a function of redshift for different assumed density parameters in a dust model. The dashed line gives the result for Hubble s law. The lookback time as a function of redshift is interpreted graphically for a dust model in Fig. 17.3(a), and is plotted for various assumed values of the density parameter Ω in Fig. 17.3(b). For small redshift, t L zτ H, as would be expected from the Hubble law. But for larger redshifts t L differs substantially from this approximation.

16 632 CHAPTER 17. ENERGY AND MATTER IN THE UNIVERSE 17.7 The Inadequacy of Dust Models The preceding discussion has applied Newtonian gravity to a universe containing only pressureless matter (dust). Until roughly the last decade of the 20th century a covariant version of such theories was the favored model for cosmology. For example, one often-discussed model was the Einstein de Sitter universe, which was a covariant version of the Ω=1 solution with exactly a closure density of dust and zero curvature that will be described later. However, the observational evidence of that period indicated that there was not nearly enough visible matter in the Universe to constitute a closure density, suggesting an open-universe cosmology with Ω<1. We now know that the actual Universe contains additional components that influence its evolution in a highly-nontrivial way. As a result cosmology computed in a dust model is a poor approximation to the actual history of the Universe. To understand the new cosmology that emerged from these discoveries, we begin by taking inventory of these additional components. Our starting point will be the evidence that much of the matter in the Universe is not the visible matter of stars and galaxies.

17 17.8. EVIDENCE FOR DARK MATTER Evidence for Dark Matter There is strong observational evidence for large amounts of dark matter in the Universe that reveals its presence through gravity, but is not seen by any other probe. Let s review some of this evidence.

18 634 CHAPTER 17. ENERGY AND MATTER IN THE UNIVERSE Observed Radial velocity Predicted if mass traces luminosity Radius Figure 17.4: Schematic velocity curves for spiral galaxies Evidence for Dark Matter: Galaxy Rotation Curves In spiral galaxies, if we balance the centrifugal and gravitational forces at a radius R, the tangential velocity v should obey the relation GM v= R implied by Kepler s laws, with R the radius and M the enclosed mass. Well outside the main matter distribution, we expect v R 1/2. The velocities can be measured using the Doppler effect, both for visible light from the luminous matter, and from the 21 cm hydrogen line for non-luminous hydrogen. For many spirals we find not v R 1/2 but almost constant velocity well outside the bulk of the luminous matter. This is illustrated schematically in Fig

19 17.8. EVIDENCE FOR DARK MATTER 635 Figure 17.5: Rotation curve for the galaxy M33 out to a distance of about 15 kpc (50, 000 ly) from the center. Points inside about 15,000 ly are from visible starlight; points beyond that are from radio frequency (RF) observations. An example of a measured rotational curve is shown in Fig for the galaxy M33. This indicates the presence of substantial gravitating matter distributed in a halo beyond the visible matter.

20 636 CHAPTER 17. ENERGY AND MATTER IN THE UNIVERSE 1" (b) Einstein Cross (a) Cloverleaf Quasar 1" Figure 17.6: Gravitational lensing of quasars: (a) The Cloverleaf Quasar. The four images are of a single quasar lensed by foreground galaxies too faint to see in this image. (b) The Einstein Cross. The four outer images are all of a single quasar lensed by a foreground galaxy near the center of the image. Identical spectra confirm that these are images of a single object Evidence for Dark Matter: Gravitational Lensing The path of light is curved in a gravitational field. This can cause gravitational lensing, where intervening masses act as lenses to distort the image of distant objects. Spectacular examples of gravitational lensing are shown in Fig In these images, a single object appears as four objects because of lensing by a foreground galaxy. The amount of lensing depends on the total mass causing the lensing, whether visible or not.

21 17.8. EVIDENCE FOR DARK MATTER 637 The strength of a gravitational lens depends on the total mass contained within it, whether that mass is visible or not. Gravitational lenses can serve as excellent indicators of how much unseen matter is present in the region of the lens. Extensive analysis of gravitational lensing by large masses leads to conclusions similar to those suggested above by the rotation curves for spiral galaxies: More than 90% of the mass contributing to the strength of large gravitational lenses is dark.

22 638 CHAPTER 17. ENERGY AND MATTER IN THE UNIVERSE Figure 17.7: Evidence for dark matter in the Bullet Cluster. The left image shows galaxies in the cluster and total mass contours inferred from gravitational lensing. The right image shows X-ray luminosity superposed on mass contours. The simplest explanation for displacement of X-ray luminosity from mass concentrations is that the majority of the mass is dark matter now found at the two mass centers Example: The Bullet Cluster Evidence for dark matter in a galaxy cluster is displayed in Fig The double cluster of galaxies 1E (Bullet Cluster) has been studied using gravitational lensing and detection of X-rays. The double cluster represents two galaxy clusters that collided about 100 million years ago. The star distributions would have largely passed through each other but the gas would have interacted strongly through ram pressure and dark matter would have not interacted at all. The compressed gas, radiating strongly in X-rays, is displaced from the mass centers of the two clusters after the collision. The collision has separated the bulk of the dark matter (at the two local maxima of the mass contours) from the regular matter (concentrated at the sources of X-ray luminosity).

23 17.8. EVIDENCE FOR DARK MATTER kpc 10 kpc Dragonfly 44 Coma Cluster Figure 17.8: Dragonfly 44, a relatively nearby galaxy that may be almost all dark matter. Although it has a very low surface brightness, it contains of order 100 globular clusters and has a mass comparable to that of the Milky Way galaxy Dark matter in ultra-diffuse galaxies A population of ultra-diffuse galaxies (UDG) has been studied in the Coma Cluster. These are very faint but appear to have large masses. The UDG Dragonfly 44 is shown in Fig From the measured velocity dispersion the total mass of the galaxy was estimated to be comparable to that of the Milky Way, and approximately 100 globular clusters were identified. From the large mass but faint light, Dragonfly 44 was estimated to be 98% dark matter.

24 640 CHAPTER 17. ENERGY AND MATTER IN THE UNIVERSE 17.9 Baryonic and Non-Baryonic Matter Baryonic matter is ordinary matter consisting of protons and neutrons. Non-baryonic matter consists of particles that do not undergo the strong interactions. Neutrinos are one example of non-baryonic matter. There seems to be a lot of dark matter in the Universe. How much of it is baryonic? Let us define the ratio η of baryon to photon number density, η n B n γ Ω bρ c /m B 410 cm Ω b h 2, where Ω b is the baryon density parameter (ratio of baryon density to closure density), ρ c is the closure density, m B is the average mass of a baryon, and the density of photons has been approximated by the CMB density. The abundances of light elements such as 4 He, 3 He, 2 H, and 7 Li produced by nucleosynthesis in the big bang are very sensitive to η. These indicate that η Therefore, from this value of η Ω b ( h 2 )η 0.04 and independent of how much baryonic matter has actually been observed directly strong nucleosynthesis constraints say that most the the matter in the Universe is not baryonic.

25 17.9. BARYONIC AND NON-BARYONIC MATTER 641 Extending the trend started by Copernicus: we are not the center of the Universe, and we aren t even made up of the dominant matter of the Universe. Not only are we not the center of the Universe, we aren t even made of the right stuff!

26 642 CHAPTER 17. ENERGY AND MATTER IN THE UNIVERSE Candidates for non-baryonic dark matter There are two classes of candidates for non-baryonic dark matter, each corresponding to either conjectured or known elementary particles: Cold Dark Matter (CDM), which consists of particles that decoupled very early, or that were never in thermal equilibrium. Hot Dark Matter (HDM), which consists of low-mass particles that still had relativistic velocities at the time of matter radiation decoupling.

27 CANDIDATES FOR NON-BARYONIC DARK MATTER Cold dark matter Cold dark matter had velocities well below lightspeed when galaxy formation started. Candidates for cold dark matter may be divided into Weakly Interacting Massive Particles (WIMPS), and Superlight particles with superweak interactions that were never in equilibrium. The WIMPS, because of their mass, would become nonrelativistic and decouple from the plasma earlier than would the normal leptons. Some proposed candidates for WIMPS include several exotically-named particles expected for supersymmetric theories, and a neutrino with mass greater than 45 MeV. No evidence for either presently exists. If dark matter consists of superlight particles that were never completely in equilibrium they would have been decoupled from the beginning. A prime candidate for this class of dark matter is the axion, which is a conjectured boson required in some elementary particle physics theories. There is no experimental evidence for axions at present.

28 644 CHAPTER 17. ENERGY AND MATTER IN THE UNIVERSE Hot dark matter Hot dark matter is relativistic at the time that galaxy formation begins. It could correspond to as-yet undiscovered particles but the neutrinos are obvious candidates. The present number density of neutrinos may be estimated by assuming that most neutrinos are in a uniform background of neutrinos analogous to the cosmic microwave background to be discussed later. The number density of neutrinos in this background should be related to the number density of the photons in the microwave background by a factor of Therefore, the neutrino number density for each neutrino family may be estimated as n ν 3 11 n γ 112 neutrinos cm 3. These neutrinos could close the Universe gravitationally if their mass m ν satisfies m ν n ν = Ω ν ρ c, implying that the closure mass is m ν = Ω ν ρ c n ν 95h 2 ev. The best current data indicate that neutrinos have a tiny mass and the contribution to the closure density of all neutrinos in the Universe can be no more than several percent.

29 RADIATION Radiation Astronomers classify massless and nearly massless particles such as photons, gluons, gravitons, and neutrinos as radiation. The radiation in the Universe influences its evolution The energy density of radiation in the present Universe is very small but it dominated the energy density of the very early Universe. Only a small amount of radiation density in the present Universe is found in starlight. The bulk (more than 90%) is in the cosmic microwave background (CMB) radiation, which will be discussed in later chapters.

30 646 CHAPTER 17. ENERGY AND MATTER IN THE UNIVERSE Dark Energy Dark matter may appear exotic by normal standards, since we don t know what it is and therefore do not know why it fails to couple strongly through any force other than gravity. However, we shall see in later chapters that there is strong evidence that the evolution of the present Universe is being dominated by something even more exotic: dark energy. Dark energy (also known as vacuum energy) behaves fundamentally differently from either normal matter and energy, or dark matter. It appears to cause the force of gravitation to become repulsive. To understand and to deal adequately with this remarkable notion will require a covariant formulation of gravitation. Therefore, we defer substantial discussion of the evidence for and role played by dark energy until the following chapters.

31 DENSITY PARAMETERS Density Parameters We have already introduced the total density parameter evaluated at the present time Ω ρ = 8πGρ ρ c 3H0 2. where ρ is the current total density coupled to gravity. Thus, the closure condition implies that Ω = 1 (critical density). The subscript 0 is often used on Ω and ρ to indicate explicitly that they are evaluated at the present time; we suppress that subscript to avoid notational clutter in later equations. The acceleration of the scale factor may be expressed in terms of the density parameter Ω, ä= 4 ( ) 1 3 πgρa3 0 a 2 Ω ρ ρ c = 8πGρ 3H 2 0 ä= 1 2 H2 0 a3 0 Ω a 2 (where it is understood that ρ ρ 0 and Ω Ω 0 correspond to their current values.) Anticipating the later treatment of the expansion using general relativity, we may expect that the density parameter gets contributions from three major sources in the current Universe: 1. Matter, including dark matter (with density denoted by ρ m ) 2. Radiation (with density denoted by ρ r ) 3. Vacuum or dark energy (with density denoted by ρ Λ ).

32 648 CHAPTER 17. ENERGY AND MATTER IN THE UNIVERSE These densities may be used to define corresponding partial density parameters Ω i through ρ r (a)=ρ c Ω r ρ m (a)=ρ c Ω m ρ Λ (a)=ρ c Ω Λ, where we shall show later that the total density changes with a(t) according to ( Ωr ρ(a)=ρ c a 4 + Ω ) m a 3 + Ω Λ (a(t 0 ) 1), we have assumed the standard convention of normalizing the current value of the scale parameter a(t 0 ) to unity. We shall make no explicit distinction between mass density ρ and the corresponding energy density ε = ρc 2, since they are numerically the same in c=1 units. Note that the different densities scale differently with a(t), and thus differently with time. For baryonic matter alone, we have already seen that Ω b where Ω b is a part of Ω m. (baryonic matter). This is well below the critical density (Ω=1) but, as we have previously noted, baryonic matter is not the dominant matter in the Universe and we must include the effect of non-baryonic dark matter and the effect of dark energy to determine the true value of Ω.

33 DENSITY PARAMETERS 649 Table 17.1: Density parameters Source Value (Ω i = ρ i /ρ c ) Total matter Ω m = 0.3 Baryonic matter Ω B = 0.04 Total radiation Ω r < Total vacuum Ω Λ = 0.7 Curvature Ω c 0.01 Some estimates of the current density parameters for the radiation, matter, baryonic portion of the matter, and the vacuum energy are given in Table 17.1 (the curvature density entry will be explained later).

34 650 CHAPTER 17. ENERGY AND MATTER IN THE UNIVERSE The Deceleration Parameter The density of the Universe is clearly related to the rate at which the Hubble expansion is changing with time. If we expand the cosmic scale factor to second order in time, a(t) a 0 + ȧ 0 (t t 0 )+ 1 2ä0(t t 0 ) 2 (where ȧ 0 (da/dt) t=t0, and so on), introduce the deceleration parameter at the present time q 0 q(t 0 ) through q 0 ä0 a 0 H 2 0 ä 0 = a 0 ȧ 2, 0 and utilize we obtain ȧ 0 a 0 = H 0, a(t)=a 0 1+H 0 (t t 0 ) }{{} 1 2 H2 0 q 0(t t 0 ) }{{} Hubble correction

35 THE DECELERATION PARAMETER Deceleration and Density Parameters Generally, the deceleration parameter q 0 is related to the density parameters Ω i through (Problem) q 0 = Ω m 2 + Ω r Ω Λ. The parameters of Table 17.1 suggest that the deceleration parameter for the present Universe is negative, q 0 Ω m 2 Ω Λ 0.55, and that the expansion is currently accelerating.

36 652 CHAPTER 17. ENERGY AND MATTER IN THE UNIVERSE H0 = 72 km s -1 Mpc Scale factor relative to today Ω m=0, Ω r=0 Ω Λ=1 q0=-1.0 Now Ω Λ=0 q0= Ω m=0, Ω r=0 1 Ω Λ=0.5 q0= Ω m=1, Ω r=0 Ω Λ=0 q0= Ω m=0, Ω r=0 5 Ω Λ=0 q0= Time (10 9 years) Ω m=1, Ω r=0.5 0 Redshift Figure 17.9: Quadratic deviations from the Hubble expansion. The different curves correspond to different assumed values of the density parameters and the corresponding deceleration parameter q 0. Each curve has the same linear term but a different quadratic (acceleration) term. Positive values of the deceleration parameter correspond to a slowing of the expansion and negative values to an increase in the rate of expansion with time. Some quadratic deviations from the Hubble law are illustrated in Fig

37 THE DECELERATION PARAMETER H0 = 72 km s -1 Mpc -1 q0 = 0 Scale factor relative to today Ω m=0, Ω r=0 Ω Λ=0 q0=0 Now 0.2 Ω m=1, Ω r=0 3 Ω Λ=0.5 q0= Time (10 9 years) Ω m=0, Ω r=1 Ω Λ=1 q0=0 0 1 Redshift Figure 17.10: Different choices of matter, radiation, and vacuum energy densities that give the same deceleration parameter. The curves all agree near the present time to second order, but have very different long-time behaviors Deceleration and Cosmology Figure illustrates that H 0 and q 0 determine the behavior of the Universe only near the present time. The three curves have the same H 0 and q 0 = 0, but very different mixtures of matter, radiation, and vacuum energy densities. Within the gray box the curves are essentially indistinguishable but at redshifts of 1 or larger they are very different. For example, these three curves predict ages of the Universe (intercepts with the lower axis) that differ by almost a factor of 2.

38 654 CHAPTER 17. ENERGY AND MATTER IN THE UNIVERSE Until very recently, the primary quest in cosmology was to determine with precision the Hubble constant H 0 and the deceleration parameter q 0. Acquisition of precision cosmology data through The study of high-redshift Type Ia supernovae The detailed analysis of the cosmic microwave background mean that the cosmological data now are beginning to constrain a broader range of parameters than just these two. We shall discuss this in more detail in later chapters.

39 PROBLEMS WITH NEWTONIAN COSMOLOGY Problems with Newtonian Cosmology As promised, we have been able to make considerable headway in understanding the expanding Universe simply by using Newtonian gravitational concepts. However, the purely Newtonian approach leads to some problems and inconsistencies. For example, 1. At large distances the expansion leads to recessional velocities that can exceed the speed of light. How are we to interpret this? 2. Newtonian gravitation is assumed to act instantaneously, but because light speed is the limit for signal propagation, there should be a delay in the action of gravitation. 3. In the Newtonian picture we had a uniform isotropic sphere expanding into nothing, which causes conceptual problems in interpreting the expansion. Alternatively, if the sphere is assumed to be of infinite extent, there are formal difficulties with even defining a potential. These and other difficulties suggest that we need a better theory of gravitation to adequately describe cosmologies built on expanding universes. In the following chapters we shall develop an understanding of the expanding Universe based of general relativity that will deal with these problems.