Newtonian Gravity and Cosmology

Transcription

1 Chapter 30 Newtonian Gravity and Cosmology 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. 949

2 950 CHAPTER 30. NEWTONIAN GRAVITY AND COSMOLOGY Density = ρ Distant galaxy Earth r Homogeneous mass distribution Figure 30.1: 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.

3 30.1. EXPANSION AND NEWTONIAN GRAVITY 951 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 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 ).

4 952 CHAPTER 30. NEWTONIAN GRAVITY AND COSMOLOGY 30.2 The Critical Density For the expansion to halt, we must have E = 0 and thus 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 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.

5 30.2. THE CRITICAL DENSITY 953 Thus, in this simple 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 vacuum energy.)

6 954 CHAPTER 30. NEWTONIAN GRAVITY AND COSMOLOGY 30.3 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. For example, neutrinos are one example of non-baryonic matter. Constraints can be placed on the baryonic matter density ρ B by comparing observed and predicted abundances of light isotopes such as 3 He and 7 Li that are formed in the early Universe. 1. One finds ρ B g cm 3, 2. This is far too small to close the Universe since ρ c g cm 3 3. However, this does not settle the issue because There is substantial evidence that most matter in the Universe is non-baryonic and in the form of dark matter that is visible only to gravitational probes. There is a substantial dark energy contribution to the current evolution of the Universe. 4. We shall discuss candidates for this non-baryonic matter and the role of vacuum energy in subsequent chapters.

7 30.3. BARYONIC AND NON-BARYONIC MATTER 955 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, we aren t even made of the right stuff!

8 956 CHAPTER 30. NEWTONIAN GRAVITY AND COSMOLOGY Observed Radial Velocity Predicted if mass traces luminosity Radius Figure 30.2: 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

9 30.3. BARYONIC AND NON-BARYONIC MATTER 957 vr (km s-1) Distance (arcminutes) Figure 30.3: Rotation curve for the Andromeda Galaxy. White points indicate measured velocities (open circles at large distance are RF observations). Observational data on the rotation curve for the Andromeda Galaxy (M31) are displayed in Fig Converting the angular size to kpc using the distance of 778 kpc to Andromeda we see that The obvious visible matter lies within about kpc of the center. RF observations suggest that the rotation curve is constant out to at least about kpc. Direct measurements suggest constant velocities out to at least 30 kpc in many spirals, and Indirect means suggest that constant velocities may extend out to 100 kpc or more in some spirals. This indicates the presence of substantial gravitating matter distributed in a halo beyond the visible matter.

10 958 CHAPTER 30. NEWTONIAN GRAVITY AND COSMOLOGY Aug 1991 Aug " Figure 30.4: Gravitational lensing: the Einstein Cross 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. A spectacular example of gravitational lensing is the Einstein Cross, shown in Fig In this image, a single object appears as four objects. A very distant quasar is thought to be positioned behind a massive galaxy. The gravitational effect of the galaxy has created multiple images through gravitational lensing on the light from the quasar. The individual stars in the foreground galaxy may also be acting as gravitational lenses, causing the images to change their relative brightness in these two images taken three years apart, as stars change position in the lensing galaxy.

11 30.3. BARYONIC AND NON-BARYONIC MATTER 959 Einstein Cross Quasar images Bar Nucleus Faint lensing galaxy Spiral arms Figure 30.5: The Einstein Cross and the lensing galaxy. The intensity has been displayed on a logarithmic scale so that the very bright quasar images and the extremely faint bar and arms of the lensing galaxy can be seen at the same time. Image courtesy W. Keel, University of Alabama. This interpretation of the Einstein Cross is bolstered by Fig. 30.5, which shows in faint outline the foreground lensing galaxy surrounding the bright central nucleus of the spiral and the four quasar images. The lensing galaxy is a relatively nearby barred spiral. Both the spiral arms and the central bar of the foreground galaxy can be seen if one looks carefully (see the annotated version of the figure in the right panel).

12 960 CHAPTER 30. NEWTONIAN GRAVITY AND COSMOLOGY 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.

13 30.4. DARK ENERGY 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 Chapter 31 that there is growing 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 two chapters.

14 962 CHAPTER 30. NEWTONIAN GRAVITY AND COSMOLOGY 30.5 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) = a(t) ( ) ρ(t) 3 a0 =, r 0 a 0 ρ 0 a(t) This permits us to express all dynamical equations in terms of the scale factor. Example: Newtonian gravitational force acting on the galaxy F G = U Mm = G r r 2 = 4 3 πgρrm, and the corresponding gravitational acceleration is r = F G m = GM r 2 = 4 3 πgρr. Then from r/r 0 = a/a 0 and ρ = (a 3 0 /a3 ) 3 a, r = r 0 ä = 4 a 0 3 πgρ 0 a 3 0 (acceleration of the scale factor). a 3 r 0 a 0 a ä = 4 3 πgρ 0a 3 0 ( ) 1 a 2.

15 30.6. DENSITY PARAMETERS Density Parameters It is convenient to introduce 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, ( ) 1 ä = 3 4πGρa3 0 a 2 Ω ρ ρ c = 8πGρ 3H0 2 = 2 ä = 1 2 H2 0 a3 0 Ω ( 4 H 3 πgρ) a 2. 0 (where ρ ρ 0 and Ω Ω 0 ). 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 (with density denoted by ρ m ) 2. Radiation (with density denoted by ρ r ) 3. Vacuum or dark energy (with density denoted by ρ v or ρ Λ ).

16 964 CHAPTER 30. NEWTONIAN GRAVITY AND COSMOLOGY These densities may be used to define corresponding density parameters Ω i through ρ r (a) = ρ c Ω r ρ m (a) = ρ c Ω m ρ v (a) = ρ c Ω v, where we shall show (Chapter 31) that the total density changes with a(t) according to ( Ωr ρ(a) = ρ c a 4 + Ω ) m a 3 + Ω v (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 obtain from the observed ρ B g cm 3 that Ω = Ω m = ρ B ρ c (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 dark energy to determine the true value of Ω.

17 30.6. DENSITY PARAMETERS 965 Table 30.1: Density parameters Source Value (Ω i = ρ i /ρ c ) Total matter Ω m = 0.3 Baryonic matter Ω B = Total radiation Ω r < Total vacuum Ω v = 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 30.1 (the curvature density entry will be explained in Chapter 31).

18 966 CHAPTER 30. NEWTONIAN GRAVITY AND COSMOLOGY 30.7 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 and since ȧ 0 = a 0 H 0 (Exercise), where we define which must obey the condition since ȧ 2 can never be negative. a ( da 1 a 0 a 2 ȧ 2 = ȧ H2 0 a3 0 Ω a 1 ), a 0 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.

19 30.8. EXPANSION HISTORIES FOR THE UNIVERSE Expansion Histories for the Universe Let us consider as an example, dust-filled universes (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 0 = Ω m. 1. Ω < 1 (undercritical): In this case, as a(t), f(ω,t) = 1+Ω a 0 Ω 1 Ω > 0. a(t) Thus ȧ never goes to zero (ȧ 2 f(ω,t) and we live in an open, ever-expanding universe if Ω < 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.

20 968 CHAPTER 30. NEWTONIAN GRAVITY AND COSMOLOGY Open Ω < 1 Scale factor a(t) Flat Ω = 1 Closed Ω > 1 Now Time t Figure 30.6: Behavior of the scale factor a(t) as a function of time for a dust-filled universe. The evolution of the corresponding scale factor is sketched in Fig

21 30.9. THE DECELERATION PARAMETER 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 ä0 q 0 a 0 H0 2 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 ) }{{} Hubble 1 2 H2 0 q 0(t t 0 ) }{{} Corrections to Hubble The deviation from the Hubble law is quadratic in time to leading order.

22 970 CHAPTER 30. NEWTONIAN GRAVITY AND COSMOLOGY H0 = 72 km s -1 Mpc Scale factor relative to today Now 0.8 Ω m=0, Ω r=0 Ω v=1 q0=-1.0 Ω m=1, Ω r= Ω m=0, Ω r=0 Ω v=0 q0=1.0 1 Ω v=0.5 q0= Ω m=1, Ω r=0 Ω v=0 q0= Ω m=0, Ω r=0 5 Ω v=0 q0= Time (10 9 years) 0 Redshift Figure 30.7: 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. Quadratic deviations from the Hubble law are illustrated in Fig

23 30.9. THE DECELERATION PARAMETER Deceleration and Density Parameters Generally, the deceleration parameter q 0 is related to the density parameters Ω i through (Exercise, Ch. 31) q 0 = Ω m 2 + Ω r Ω v. The parameters of Table 30.1 suggest that the deceleration parameter for the present Universe is negative, q 0 Ω m 2 + Ω v = 1 2 (0.3) , and that the expansion is currently accelerating.

24 972 CHAPTER 30. NEWTONIAN GRAVITY AND COSMOLOGY H0 = 72 km s -1 Mpc q0 = 0 Scale factor relative to today Ω m=0, Ω r=0 Ω v=0 q0=0 Now Ω m=0, Ω r=1 Ω v=1 q0=0 0 1 Redshift Ω m=1, Ω r= Ω v=0.5 q0= Time (10 9 years) Figure 30.8: 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 30.8 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.

25 30.9. THE DECELERATION PARAMETER 973 Until very recently, the primary quest in cosmology was to determine 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 Chapter 31.

26 974 CHAPTER 30. NEWTONIAN GRAVITY AND COSMOLOGY 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 universe (Exercise) 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 = 2 ( ) (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 5 was 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.

27 LOOKBACK TIMES 975 H 0 = 72 km/s/mpc Ω=0.1 Ω=0.5 Ω=1 t L z=5 8 Figure 30.9: Geometrical interpretation of the lookback time t L for a dust Universe with three different values of the density parameter Ω = Ω m. The lookback time as a function of redshift is interpreted geometrically in Fig

28 976 CHAPTER 30. NEWTONIAN GRAVITY AND COSMOLOGY Lookback time t L (10 9 years) Ω = 0.1 Ω = 0.5 Ω = Redshift z Figure 30.10: Lookback time as a function of redshift for three different assumed values of the density parameter in a dust model with H 0 = 72 km s 1 Mpc 1. The lookback time is plotted for various assumed values of the density parameter Ω in Fig for a dust model. For small redshifts t L zτ H, as we would expect from the Hubble law. For larger redshifts the curves in Fig differ substantially from this approximation.

29 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 Chapter 31 we shall develop an understanding of the expanding Universe based of general relativity that will deal with these problems.