Lecture Notes 4 MORE DYNAMICS OF NEWTONIAN COSMOLOGY

Transcription

1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physis Department Physis 8.286: The Early Universe Otober 1, 218 Prof. Alan Guth Leture Notes 4 MORE DYNAMICS OF NEWTONIAN COSMOLOGY THE AGE OF A FLAT UNIVERSE: We showed last time that the sale fator for a flat, matter-dominated universe behaves as a(t) = bt 2/3, (4.1) for some onstant of proportionality b. This model universe is flat in the sense that k = (or = 1), and it is matter-dominated in the sense that the mass density is dominated by the rest mass of nonrelativisti partiles. (The assumption of matterdomination entered the derivation when we assumed that the total mass M(r i ) ontained within a given omoving radius r i does not hange as the system evolves. Photons or highly relativisti partiles, by ontrast, would redshift as the universe expands, and hene they would lose energy. In Leture Notes 6 we will disuss the evolution of a universe dominated by relativisti partiles, and in Leture Notes 7 we will desribe the effets of a osmologial onstant, or vauum energy. Sine the energy loss of an expanding gas is proportional to its pressure, the matter-dominated ase an also be desribed as the ase in whih the pressure is negligible.) Using Eq. (4.1), it is easy to alulate the age of suh a universe in terms of the Hubble expansion rate. In Leture Notes 2 it was shown that the Hubble expansion rate is given by H(t) = ȧ/a, (4.2) where the over-dot has been used to denote a derivative with respet to time t. Thus, H(t) = 2 3 bt 1/3 bt 2/3 = 2 3t. (4.3) Reall that we have defined the origin of time so that the sale fator a(t) vanishes at t =, so t = is the earliest time that exists within the mathematial model. We therefore refer to t as the age of the universe, whih an then be expressed in terms of the Hubble expansion rate by t = 2 3 H 1. (4.4) We should keep in mind, however, that we would be foolish to pretend that we atually understand the origin of the universe, so the phrase age of the universe is being used

2 MORE DYNAMICS OF NEWTONIAN COSMOLOGY p. 2 loosely. It is almost ertain that our universe underwent an extremely hot dense phase at t, and we an all this phase the big bang. The variable t represents the age of the universe sine the big bang, but we an only speulate about whether the big bang atually represents the beginning of time. As we will see near the end of the ourse, the urrent understanding of inflationary osmology suggests that the big bang was very likely not the beginning of time. Consisteny requires that the age of the universe be older than the age of the oldest stars, and this requirement has turned out to be a strong onstraint. It was a very serious problem in the time shortly after Edwin Hubble s first measurement, when Hubble s bad estimate for the Hubble expansion rate led to age estimates of only several billion years. More reently, as quoted in Leture Notes 2, the Plank satellite team, using their own data ombined with other data, estimated that H = 67.66±.42 km-s 1 -Mp 1.* Using Eq. (4.4) and the relationship yr = 97.8 km-s 1 -Mp 1, we find that a flat matter-dominated universe with a Hubble expansion rate in this range (i.e., to 68.8 km-s 1 -Mp 1 ) must have an age between 9.58 and 9.7 billion years. Today the oldest stars are believed to be those in globular lusters, whih are tightly bound, nearly spherial distributions of stars that are found in the halos of galaxies. A areful study of the globular luster M4 by Hansen et al., using data from 123 orbits of the Hubble Spae Telesope, determined an age of 12.7 ±.7 billion years. Krauss and Chaboyer argue that Hansen et al. did not take into aount all the unertainties, and laim that the globular lusters in the Milky Way (inluding M4) have an age of billion years. (Both sets of authors are quoting 95% onfidene limit errors, also alled 2σ errors, meaning that the probability of the true value lying in the quoted range is estimated at 95%.) Krauss and Chaboyer estimate that the stars must take at least.8 billion years to form, implying that the universe must be at least 11.2 billion years old. * N. Aghanim et al. (Plank Collaboration), Plank 218 results, VI: Cosmologial parameters, Table 2, Column 6, arxiv: B.M.S. Hansen et al., The White Dwarf Cooling Sequene of the Globular Cluster Messier 4, Astrophysial Journal Letters, vol. 574, p. L155 (22), L.M. Krauss and B. Chaboyer, Age Estimates of Globular Clusters in the Milky Way: Constraints on Cosmology, Siene, vol. 299, pp (23), available with MIT ertifiates at 299/563/65.abstrat or for purhase at 65.abstrat

3 MORE DYNAMICS OF NEWTONIAN COSMOLOGY p. 3 (That is, the stars are at least = 1.4 billion years old, and ould not have formed until the universe was already.8 billion years old.) With either of these estimates for the age of the oldest stars, however, the age of the universe is inonsistent with the measured value of the Hubble expansion rate and the assumption of a flat, matter-dominated universe. We will see later in these leture notes that the age alulation gives a larger answer if we assume an open universe ( ρ/ρ < 1), but inflationary osmology predits that should be very lose to 1, and starting with the first BOOMERANG long duration balloon experiment of 1997,* measurements of the flutuations in the CMB have indiated a nearly flat universe. The CMB measurements now imply that = 1 to high auray. Combining their own data with results of other experiments, the Plank team onluded that =.9993 ±.37. We will see in Leture Notes 7, however, that the age disrepany problem is ompletely resolved by the inlusion of dark energy. With about 7% of the mass density in the form of dark energy, H = 67.7 km-s 1 -Mp 1 is onsistent with an age of ±.2 billion years, whih is our urrent best estimate for the age of the universe. Sine the age estimates are not onsistent if we assume a matter-dominated flat universe, the age alulations help to support the proposition that our universe is dominated by dark energy, whih we will disuss in detail in Leture Notes 7. THE BIG BANG SINGULARITY: This mathematial model of the universe starts from a onfiguration with a(t) =, whih orresponds to infinite density. From Eq. (4.3) we see that the initial value of the Hubble expansion rate is also infinite. We will see shortly that these infinities are not peuliarities of the flat universe model, but our also in the models for either a losed or open universe. This instant of infinite density is alled a singularity. One should realize, however, that there is no reason to believe that the equations whih we have used are valid in the viinity of this singularity. Thus, although our mathematial models of the universe ertainly begin with a singularity, it is an open question whether the universe atually began with a singularity. If we use our equations to follow the history of the universe further and further into the past, the universe beomes denser and denser without limit. At some point one enounters densities that are so far beyond our experiene that the equations are no longer to be trusted. We will disuss later in the ourse where this point may our, but for now I just want to make it lear that the singularity should not be onsidered a reliable onsequene of the theory. * P. D. Mauskopf et al., Measurement of a peak in the osmi mirowave bakground power spetrum from the North Amerian test flight of BOOMERANG, Astrophysial Journal Letters, vol. 536, pp. L59 L62 (2), N. Aghanim et al. (Plank Collaboration), ited above, Table 4, Column 4. N. Aghanim et al. (Plank Collaboration), ited above, Table 2, Column 6.

4 MORE DYNAMICS OF NEWTONIAN COSMOLOGY p. 4 THE HORIZON DISTANCE: Sine the age of the universe in the big bang model is finite, it follows that there is a theoretial upper limit to how far we an see. Sine light travels at a finite speed, there will be partiles in the universe that are so far away that light emitted from these partiles will not yet have reahed us. The present distane of the furthest partiles from whih light has had time to reah us is alled the horizon distane. If the universe were stati and had age t, then the horizon distane would be simply t. In the real universe, however, everything is onstantly in motion, and the value of the horizon distane has to be alulated. The alulation of the horizon distane an be done most easily by using omoving oordinates. The speed of light rays in a omoving oordinate system was given in Eq. (2.8) as dx dt = a(t). (4.5) (Reall that this formula is based on the statement that the speed of light in meters per seond is onstant, but the sale fator a(t) is needed to onvert from meters per seond to nothes per seond.) One an then alulate the oordinate horizon distane (i.e., the horizon distane in nothes ) by alulating the distane whih light rays ould travel between time zero and some arbitrary final time t. By integrating Eq. (4.5), one sees that l,horizon (t) = t The physial horizon distane at time t is then given by l p,horizon (t) = a(t) a(t ) dt. (4.6) t a(t ) dt. (4.7) For the speial ase of a flat, matter-dominated universe, one an use Eq. (4.1) to obtain l p,horizon (t) = bt 2/3 t bt 2/3 dt. (4.8) Carrying out the integration, the physial horizon distane for a flat matter-dominated universe is found to be l p,horizon (t) = 3t. (4.9) The horizon distane an also be expressed in terms of the Hubble expansion rate, by using Eq. (4.4): l p,horizon (t) = 2H 1. (4.1)

5 MORE DYNAMICS OF NEWTONIAN COSMOLOGY p. 5 Taking H = 67.7 km-s 1 -Mp 1, the horizon distane today, under the assumption of a flat matter-dominated universe, is found to be about 29 billion light-years. The fator of 3 on the right-hand side of Eq. (4.9) implies that the horizon distane is three times larger than it would be in a stati universe, and hene it is signifiantly larger than most of us would probably guess. The reason, of ourse, is that the horizon distane refers to the present distane of the most distant matter that an in priniple be seen. However, the light that we reeive from distant soures was emitted long ago, and the present distane is large beause the matter has been moving away from us ever sine. In the limiting ase of matter exatly at the horizon, the light that we reeive today left the objet at t =, at the instant of the big bang. You might wonder why light from every objet did not reah us immediately, sine the sale fator a(t) was zero at t =, so the initial distane between any two objets was zero. To be honest, you are pretty safe in believing whatever you like about what happens at the initial singularity. The lassial desription ertainly breaks down as the mass density approahes infinity, and there does not yet exist a satisfatory quantum desription. So, if you want to believe that everything ould ommuniate with everything else at the instant of the singularity, nobody whom I know ould prove that you are wrong. But nobody whom I know ould prove that you are right, either. If one ignores the possibility of ommuniation at the singularity, however, it is then a well-defined question to ask how far light signals an travel one the lassial desription beomes valid. As t the sale fator a(t) approahes zero, but its time derivative ȧ(t) approahes infinity. If we think about some objet at oordinate distane l from us, at early times its physial distane a(t)l was arbitrarily small, but its veloity of reession ȧ(t)l was arbitrarily large. If suh an objet emitted a light pulse in our diretion at some very early time t = ɛ >, even though the light pulse would have traveled toward us at the speed of light, the expansion of the universe was so fast that the distane between us and the light pulse would have initially inreased with time. The oordinate distane that the light pulse ould travel between then and now is at most equal to l,horizon, as given Eq. (4.6), so the pulse ould reah us by now only if the present distane to the objet is less than the horizon distane as given by Eq. (4.7). EVOLUTION OF A CLOSED UNIVERSE: The time evolution equations are easiest to solve for the ase of a flat (k = ) universe, but the equations for a losed or open universe are also soluble. We will first onsider the losed universe, for whih k >, E <, and > 1. From Leture Notes 3, we write the Friedmann equation as (ȧ a ) 2 = 8π k2 Gρ 3 a 2, (4.11)

6 MORE DYNAMICS OF NEWTONIAN COSMOLOGY p. 6 where the mass density ρ an be written as [ ] 3 a(ti ) ρ(t) = ρ(t i ). (4.12) a(t) (The above equation is a slight generalization of Eq. (3.23), ρ(t) = ρ i /a 3 (t). Eq. (3.23) was derived under the assumption that a(t i ) = 1, while Eq. (4.12) makes no suh requirement.) Here we will use Eq. (4.12) to onlude that the quantity ρ(t)a 3 (t) is independent of time. To obtain the desired solution in an eonomial way, it is useful to identify from the beginning the quantities of physial interest. Note that a is measured in units of, for example, meters per noth, while k is measured in units of noth 2. Thus the quantity a/ k has units of physial length (meters, for example), and is therefore independent of the definition of the noth. We will therefore hoose ã(t) a(t) k (4.13) as the variable to use in our solution of the differential equation. Similarly we will use t t, (4.14) rather than t, so that all the spaetime variables have units of length. Multiplying the Friedmann equation (4.11) by a 2 /(k 2 ), it an be rewritten as 1 k 2 ( ) 2 da = 8π Gρa 2 dt 3 = 8π 3 k 2 1 Gρa 3 k 3/2 2 k a 1. In the seond line the fators have been arranged so that we an rewrite it as (4.15) ( ) 2 dã = 2α d t ã 1, (4.16) where α 4π Gρã (4.17) Note that the parameter α also has the units of length. While the above expression for α ontains the produt ρ(t)ã 3 (t), Eq. (4.12) guarantees that this quantity is independent of time, so α is a onstant. Eq. (4.16) an be solved formally by rewriting it as d t = dã ã dã = 2α ã 1 2αã ã 2 (4.18)

7 MORE DYNAMICS OF NEWTONIAN COSMOLOGY p. 7 and then integrating both sides. One ould use indefinite integrals, as we did for the k = ase in Eq. (3.37). Using indefinite integrals, the arbitrary onstant of integration would beome an arbitrary onstant in the solution to the equation. We found, however, that the arbitrary onstant ould be eliminated by hoosing the zero of time so that a(t) = at t =. Here, mainly for the purpose of demonstrating an alternative method, I will use definite integrals. In this method the arbitrary onstant in the solution will be fixed by the speifiation of the limits of integration. Following the standard onvention, I will again hoose the zero of time to be the time at whih the sale fator is equal to zero. Using a subsript f to denote an arbitrary final time, one has t f = t f d t = ãf ã dã 2αã ã 2, (4.19) where ã f ã( t f ). The subsripts f will be dropped when the problem is finished, but for now it is onvenient to use them to distinguish the limits of integration from the variables of integration. The only remaining step is to arry out the integration shown in Eq. (4.19). Completing the square in the denominator, and then replaing the variable of integration by x ã α, (4.2) one finds t f = = ãf ãf α α ã dã α 2 (ã α) 2 (x + α) dx α2 x 2. The integral an now be simplified by the trigonometri substitution whih leads to t f = α θf (4.21) x = α os θ, (4.22) (1 os θ)dθ = α(θ f sin θ f ). (4.23) Sine θ f denotes the final value of θ, we an ombine Eqs. (4.2) and (4.22) to find x f = ã f α = α os θ f, so ã f = α(1 os θ f ). (4.24) Dropping the subsript f and realling the definitions (4.13) and (4.14), the two equations above provide a solution to our problem:

8 MORE DYNAMICS OF NEWTONIAN COSMOLOGY p. 8 t = α(θ sin θ), a k = α(1 os θ). (4.25a) (4.25b) These two equations provide a parametri desription of the funtion a(t). That is, Eq. (4.25a) determines in priniple the funtion θ(t), and then Eq. (4.25b) defines the funtion a ( θ(t) ). (There is, however, no expliit expression for θ(t), so the funtion a(t) annot be onstruted expliitly.) Some of you may reognize these equations as the equations for a yloid. The urve whih they trae an be generated by imagining a graph of a/ k vs. t, with a disk of radius α that rolls along the t-axis, as shown below. As the disk rolls, the point P traes the graph of a/ k vs. t: Figure 4.1: Evolution of a losed universe. The figure shows a graph of a/ k vs. t for a losed matter-dominated universe. If one imagines a irle rolling on the t axis, a point on the irle traes out a yloid, whih is exatly the equation for a/ k for a losed universe. The insert at the upper right inludes labels for various distanes, showing the onnetion with Eq. (4.25). The relation between Eqs. (4.25) and the rolling disk an be seen in the insert at the top right: after the disk has rolled through an angle θ, the horizontal omponent of P is given by αθ α sin θ, and the vertial omponent is given by α α os θ.

9 MORE DYNAMICS OF NEWTONIAN COSMOLOGY p. 9 The model for the losed universe reahes a maximum sale fator when θ = π, whih orresponds to a time t = πα. The orresponding value of a is given by a max k = 2α. (4.26) By that point the pull of gravity has overome the inertia of the expansion, and the universe begins to ontrat. The sale fator during the ontrating phase reverses the behavior it had during the expansion phase, and the universe ends in a big runh. The total lifetime of this universe is then t total = 2πα = πa max k. (4.27) The angle θ is sometimes alled the development angle, beause it desribes the stage of development of the universe. The universe begins at θ =, reahes its maximum expansion at θ = π, and then is terminated by a big runh at θ = 2π. Eqs. (4.25) ontain the two parameters α and k, whih might lead one to believe that there is a two-parameter family of losed universes. We must remember, however, that these equations still allow us the freedom to define the noth, so the numerial value of k is physially irrelevant. Many books, in fat, define k to always have the value +1 for a losed universe. The parameter α, on the other hand, is physially meaningful, and is related to the total lifetime of the losed universe. (When general relativity effets are desribed in Leture Notes 5, we will learn that a losed universe atually has a finite size, and that the maximum size is determined by α.) Thus there is really only a one-parameter lass of solutions. THE AGE OF A CLOSED UNIVERSE: The formula for the age of a losed universe an be obtained from the formulas in the previous setion, but we have to do a little work. Eqs. (4.25) tell how to express the age in terms of α and θ, but this is not the result we want. We would prefer to relate α and θ to other quantities that are in priniple measurable. Sine we need to determine two variables, α and θ, we will have to imagine measuring two physial quantities. These two measurable quantities an be taken to be the Hubble expansion rate H and the mass density parameter ρ/ρ. Our goal, then, is to express all the quantities related to the losed universe model in terms of H and. To start, the mass density ρ an be rewritten as ρ, where ρ = 3H 2 /8πG (see Eq. (3.33)). So ρ = 3H2 8πG, (4.28)

10 MORE DYNAMICS OF NEWTONIAN COSMOLOGY p. 1 or equivalently 8π 3 Gρ = H2. (4.29) Realling that H = ȧ/a (see Eq. (2.7)), the above formula an be substituted into the Friedmann equation (4.11), yielding whih an then be solved for a 2 to give H 2 = H 2 k2 a 2, (4.3) ã 2 = a2 k = 2 H 2 ( 1). (4.31) If we want to get our signs right for the entire evolution of the losed universe, we need to be areful. Note that Eq. (4.31) does not determine the sign of ã, sine it only speifies the value of ã 2. The sale fator is by definition positive, however, and for the ase under onsideration k >. We adopt the standard onvention that the square root of a positive number is positive, so k >. Thus, the definition ã a/ k implies that ã >, so Eq. (4.31) implies that ã = a k = H 1. (4.32) We write Eq. (4.32) with absolute value indiators around H, beause during the ontrating phase H is negative, while we know that only the positive square root of Eq. (4.31) is physially relevant. We are now ready to evaluate α, using the definition (4.17). Using Eqs. (4.28) and (4.32) to replae ρ and ã, we find α = 2 H. (4.33) ( 1) 3/2 Reall that α has diret physial meaning if our universe is losed, then the total lifetime of the universe (from big bang to big runh) would be given by 2πα/. The value of θ an now be found from Eq. (4.25b), using Eq. (4.32) to replae a/ k on the left-hand side, and Eq. (4.33) to replae α on the right-hand side. After these substitutions, Eq. (4.25b) beomes H 1 = 2 H whih an be solved for either os θ or for : (1 os θ), (4.34) ( 1) 3/2 os θ = 2, (4.35)

11 MORE DYNAMICS OF NEWTONIAN COSMOLOGY p. 11 Using = os θ. (4.36) sin θ = ± 1 os 2 θ = ± 2 1, (4.37) Eq. (4.25a) an now be rewritten to obtain the desired formula for the age of the universe: t = { ( arsin ± 2 ) 1 2 } 1 2 H ( 1) 3/2 (for a losed universe), (4.38) where the sign hoies orrespond to Eq. (4.37) (i.e., if the upper sign is used in Eq. (4.37), then both upper signs should be used in Eq. (4.38).) Both the square root and the inverse sine funtion are multibranhed funtions, so the evaluation of Eq. (4.38) requires some additional information. It is easy to see whih branh to use, however, if one remembers that Eq. (4.38) is a rewriting of Eq. (4.25a), and that in Eq. (4.25a) the variable θ runs monotonially from to 2π over the lifespan of the losed universe. To desribe the branhes in detail, it is neessary to divide the full yle, with θ varying from to 2π, into quadrants. The first quadrant is from θ = to θ = π/2, where we see from Eq. (4.36) that θ = π/2 orresponds to = 2. Thus, the first quadrant orresponds to the beginning of the expanding phase, with 1 2. For this quadrant sin θ >, so we use the upper signs in Eq. (4.38), and the inverse sine is evaluated in the range to π/2. The other quadrants are understood in the same way, produing the following table of rules: Quadrant Phase Sign Choie sin 1 () 1 Expanding 1 to 2 Upper to π 2 2 Expanding 2 to Upper π 2 to π 3 Contrating to 2 Lower π to 3π 2 4 Contrating 2 to 1 Lower 3π 2 to 2π Our universe is not urrently matter-dominated, but it ould be just barely losed. If we onsidered the hypothesis that our universe was matter-dominated, it would ertainly be in the expanding phase, with < 2, and so it would be in the first quadrant. That means that the age would be given by Eq. (4.38), using the upper signs, and evaluating the inverse sine funtion in the range of to π/2.

12 MORE DYNAMICS OF NEWTONIAN COSMOLOGY p. 12 EVOLUTION OF AN OPEN UNIVERSE: The evolution of an open universe an be alulated in a very similar way, exept that one must use hyperboli trigonometri substitutions in order to arry out the ruial integral. Fortunately, none of the ompliations with multibranhed funtions will our in this ase. For the open universe k <, E >, and < 1. We will define κ k to avoid the inonveniene of working with a negative quantity, and we will define ã(t) a(t) κ (4.39) instead of ã = a(t)/ k. Eq. (4.16) is then replaed by while Eq. (4.17), ( ) 2 dã = 2α d t ã + 1, (4.4) α 4π 3 Gρã 3 2, ontinues to be valid, although the meaning of ã has hanged. Eqs. (4.19) and (4.21) are then replaed by ãf ã dã t f = 2αã + ã 2 In this ase one uses the substitution and one obtains the result = = ãf ãf +α α ã dã (ã + α) 2 α 2 (x α) dx x2 α 2. (4.41) x = α osh θ, (4.42) t = α(sinh θ θ), a κ = α(osh θ 1). (4.43a) (4.43b)

13 MORE DYNAMICS OF NEWTONIAN COSMOLOGY p. 13 THE AGE OF AN OPEN UNIVERSE: The methods are again idential to the ones used previously, so I will just state the results. Eqs. (4.35), (4.36) and (4.37) are replaed by and = sinh θ = osh θ, osh θ = 2 osh 2 θ 1 = 2 1 and the final result for the age of the universe beomes (4.44), (4.45) t = { ( )} sinh 1 2H(1 ) 3/2 (for an open universe). (4.46) Below is a graph of Ht versus, using Eqs. (4.38) and (4.46). The graph shows that the urve is atually ontinuous at = 1, even though the expressions (4.38) and (4.46) look rather different. In fat, these two expressions are really not so different. Although it is not obvious, the two expressions are different ways of writing the same analyti funtion. You an verify this by using the usual tehniques to evaluate square roots of negative numbers, and the trigonometri and hypertrigonometri funtions of imaginary arguments. To summarize, the age the universe an be expressed as a funtion of H and as [ ( )] sinh 1 if < 1 2(1 ) 3/2 H t = 2/3 if = 1 ( [sin 1 ± 2 ) 1 2 ] 1 if > 1 2( 1) 3/2 (4.47) Graphially,

14 MORE DYNAMICS OF NEWTONIAN COSMOLOGY p. 14 Figure 4.2: The age of a matter-dominated universe, expressed as Ht (where t is the age and H is the Hubble expansion rate), as a funtion of. The urve desribes all three ases of an open ( < 1), flat ( = 1), and losed ( > 1) universe. THE EVOLUTION OF A MATTER-DOMINATED UNIVERSE: The following graph shows the evolution of the sale fator for all three ases: open, losed, or flat. The graphs were onstruted using Eqs. (4.1) for the flat ase, (4.43) for the open ase, and (4.25) for the losed ase. The parametri equations are plotted by hoosing a finely spaed grid of values for θ, using the formulas to determine t and a for eah value of θ. Sine the only parameter, α, appears merely as an overall fator, a graph that is valid for all values of α an be obtained by plotting the dimensionless ratios a/(α k ) and t/α:

15 MORE DYNAMICS OF NEWTONIAN COSMOLOGY p. 15 Figure 4.3: The evolution of a matter-dominated universe. Closed and open universes an be haraterized by a single parameter α, defined by Eq. (4.17). With the salings shown on the axis labels, the evolution of a matter-dominated universe is desribed in all ases by the urves shown in this graph. Although the graph shows all three ases, it must be remembered that it is still restrited by the assumption that the universe is matter-dominated that is, the mass density is dominated by nonrelativisti matter, for whih pressure fores are negligible.