3 Friedmann Robertson Walker Universe

Transcription

1 3 Friedmann Robertson Walker Universe 3. Kinematics 3.. Robertson Walker metric We adopt now the cosmological principle, and discuss the homogeneous and isotropic model for the universe. This is called the Friedmann Robertson Walker (FRW) or the Friedmann Lemaître Robertson Walker model. Here the homogeneity refers to spatial homogeneity only, so that the universe will still be different at different times. Spatial homogeneity means that there exists a coordinate system whose t = const hypersurfaces are homogeneous. The time coordinate is then called the cosmic time. Thus the spatial homogeneity property selects a preferred slicing of the spacetime, reintroducing a separation between space and time. There is good evidence, that the Universe is indeed rather homogeneous (all places look the same) and isotropic (all directions look the same) at sufficiently large scales (i.e., ignoring smaller scale features), larger than Mpc. (Recall the discussion of the cosmological principle in Chapter.) Homogeneity and isotropy mean that the curvature of spacetime must be the same everywhere and into every direction, but it may change in time. It can be shown, that the metric can then be given (by a suitable choice of the coordinates) in the form [ ds 2 = dt 2 +a 2 dr 2 ] (t) Kr 2 +r2 dϑ 2 +r 2 sin 2 ϑdϕ 2, () the Robertson Walker (RW) metric in spherical coordinates. Doing a coordinate transformation we can also write it in Cartesian coordinates: ds 2 = dt 2 +a 2 (t) 2 +dy 2 +dz 2 ( + K 4 r2) 2, (2) where r 2 = x 2 +y 2 +z 2. Usually the spherical coordinates are more useful. This is thus the metric of our universe, to first approximation, and we shall work with this metric for a large part of this course. The time coordinate t is the cosmic time. Here K is a constant, related to curvature of space and a(t) is a function of time, related to expansion (or possible contraction) of the universe. We call r curv a(t)/ K (3) the curvature radius of space (at time t). The time-dependent factor a(t) is called the scale factor. When the Einstein equation is applied to the RW metric, we will get the Friedmann equations, from which we can solve a(t). This will be done in Sec For now, a(t) is an arbitrary function of the time coordinate t. However, for much of the following discussion, we will assume that a(t) grows with time; and when we refer to the age of the universe, we assume that a(t) becomes zero at some finite past, which we take as the origin of the time coordinate, t =. We define H ȧ/a. (4) That is, for the whole of Cosmology I. In Cosmology II we shall consider deviations from this homogeneity. 28

2 3 FRIEDMANN ROBERTSON WALKER UNIVERSE 29 This quantity H = H(t) gives the expansion rate of the universe, and it is called the Hubble parameter. Its present value H is the Hubble constant. The dimension of H is /time (or velocity/distance). In time dt a distance gets stretched by a factor +Hdt (a distance L grows with velocity HL). Note that although the metric describes a homogeneous universe, the metric itself is not explicitly homogeneous, because it depend also on the coordinate system in addition to the geometry. (This is a common situation, just like the spherical coordinates of a sphere do not form a homogeneous coordinate system, although the sphere itself is homogeneous.) However, any physical quantities that we calculate from the metric are homogeneous and isotropic. We notice immediately that the 2-dimensional surfaces t = r = const have the metric of a sphere with radius ar. Since the universe is homogeneous, the location of the origin (r = ) in space can be chosen freely. We naturally tend to put ourselves at the origin, but for calculations this freedom may be useful. We have the freedom to rescale the radial coordinate r. For example, we can multiply all values of r by a factor of 2, if we also divide a by a factor of 2 and K by a factor of 4. The geometry of the spacetime stays the same, just the meaning of the coordinate r has changed: the point that had a given value of r has now twice that value in the rescaled coordinate system. There are two common ways to rescale. If K, we can rescale r to make K equal to ±. In this case K is usually denoted k, and it has three possible values, k =,,+. (In this case r is dimensionless, and a(t) has the dimension of distance.) The other way is to rescale a to be one at present 2, a(t ) a =. (In this case a(t) is dimensionless, whereas r and K /2 have the dimension of distance.) To choose one of these two scalings would simplify some of our equations, but we resist the temptation, and keep the general form (). This way we avoid the possible confusion resulting from comparing equations using different scaling conventions. IfK =, thespacepart(t =const) of therobertson Walker metricisflat. The3-metric (the space part of the full metric) is that of ordinary Euclidean space written in spherical coordinates, with the radial distance given by ar. The spacetime, however, is curved, since a(t) depends on time, describing the expansion or contraction of space. In common terminology, we say the universe is flat in this case. If K >, the coordinate system is singular at r = / K. (Remember our discussion of the 2-sphere!) With the substitution (coordinate transformation) r = K /2 sinχ the metric becomes ds 2 = dt 2 +a 2 (t)k [ dχ 2 +sin 2 χdϑ 2 +sin 2 χsin 2 ϑdϕ 2]. (5) The space part has the metric of a hypersphere (a 3-sphere), a sphere with one extra dimension. χ is a new angular coordinate, whose values range over π, just like ϑ. The singularity at r = / K disappears in this coordinate transformation, showing that it was just a coordinate singularity, not a singularity of the spacetime. The original coordinates covered only half of the hypersphere, as the coordinate singularity r = / K divides the hypersphere into two halves. The case K > corresponds to a closed universe, whose (spatial) curvature is positive. 3 This is a finite universe, with circumference 2πa/ K = 2πr curv and volume 2π 2 K 3/2 a 3 = 2π 2 r 3 curv, and we can think of r curv as the radius of the hypersphere. If K <, we do not have a coordinate singularity, and r can range from to. The substitution r = K /2 sinhχ is, however, often useful in calculations. The case K < 2 In some discussions of the early universe, it may also be convenient to rescale a to be one at some particular early time. 3 Positive (negative) curvature means that the sum of angles of any triangle is greater than (less than) 8 and that the area of a sphere with radius r is less than (greater than) 4πr 2.

3 3 FRIEDMANN ROBERTSON WALKER UNIVERSE 3 Figure : The hypersphere. This figure is for K = k =. Consider the semicircle in the figure. It corresponds to χ ranging from to π. You get the (2-dimensional) sphere by rotating this semicircle off the paper around the vertical axis by an angle ϕ = 2π. You get the (3-dimensional) hypersphere by rotating it twice, in two extra dimensions, by ϑ = π and by ϕ = 2π, so that each point makes a sphere. Thus each point in the semicircle corresponds to a full sphere with coordinates ϑ and ϕ, and radius (a/ K)sinχ.

4 3 FRIEDMANN ROBERTSON WALKER UNIVERSE 3 corresponds to an open universe, whose (spatial) curvature is negative. The metric is then ds 2 = dt 2 +a 2 (t) K [ dχ 2 +sinh 2 χ ( dϑ 2 +sin 2 ϑdϕ 2)]. (6) This universe is infinite, just like the case K =. The RW metric (at a given time) has two associated length scales. The first is the curvature radius, 4 r curv a K /2. The second is given by the time scale of the expansion, the Hubble time or Hubble length, t H l H H = 9.778h Gyr = h Mpc. (7) In the case K = the universe is flat, so the only length scale is the Hubble length. The coordinates (t,r,ϑ,ϕ) or (t,x,y,z) of the RW metric are called comoving coordinates. This means that the coordinate system follows the expansion of space, so that the space coordinates of objects which do not move remain the same. The homogeneity of the universe fixes a special frame of reference, the cosmic rest frame given by the above coordinate system, so that, unlike in special relativity, the concept does not move has a specific meaning. The coordinate distance between two such objects stays the same, but the physical, or proper distance between them grows with time as space expands. The time coordinate t, the cosmic time, gives the time measured by such an observer at rest, at (r,ϑ,ϕ) = const. It can be shown that the expansion causes the motion of an object in free fall to slow down with respect to the comoving coordinate system. For nonrelativistic physical velocities v, v(t 2 ) = a(t ) a(t 2 ) v(t ). (8) The peculiar velocity of a galaxy is its velocity with respect to the comoving coordinate system Conformal time In the comoving coordinates of Eqs.(), (5), and (6), the space part of the coordinate system is expanding with the expansion of the universe. It is often practical to make a corresponding change in the time coordinate, so that the unit of time (i.e., separation of time coordinate surfaces) also expands with the universe. The conformal time η is defined by dη dt t a(t), or η = dt a(t ). (9) (One may also include a factor a in the definition of η to make its value independent of the normalization of a(t).) If one uses the K = k =,± normalization, the RW metric acquires the form ] ds 2 = a 2 (η) [ dη 2 + dr2 kr 2 +r2( dϑ 2 +sin 2 ϑdϕ 2), () or with the other choice of the radial coordinate, χ, [ ds 2 = a 2 (η) dη 2 +dχ 2 +Sk 2 (χ)( dϑ 2 +sin 2 ϑdϕ 2)], () 4 In the k = ± scaling a = r curv. When using this scaling, the scale factor a(t) is often denoted R(t).

5 3 FRIEDMANN ROBERTSON WALKER UNIVERSE 32 where we have defined the generalized sine sin χ (k=+) S k (χ) χ (k=) sinh χ (k=-), (2) to handle all three geometries simultaneously. The form () is especially nice for studying radial (dϑ = dϕ = ) light propagation, because the lightlike condition ds 2 = becomes dη = ±dχ. In the end of the calculation one may need to convert conformal time back to cosmic time to express the answer in terms of the latter Redshift Let us now ignore the peculiar velocities of galaxies (i.e., we assume they are = ), so that they will stay at fixed coordinate values (r,ϑ,ϕ), and find how their observed redshift z arises. We set the origin of our coordinate system at galaxy O (observer). Let the r-coordinate of galaxy A be r A. Since we assumed the peculiar velocity of galaxy A to be, the coordinate r A stays constant with time. Light leaves the galaxy at time t with wavelength λ and arrives at galaxy O at time t 2 with wavelength λ 2. It takes a time δt = λ /c = /ν to send one wavelength and a time δt 2 = λ 2 /c = /ν 2 to receive one wavelength. Follow now the two light rays sent at times t and t +δt (see figure). t and r change, ϑ and ϕ stay constant (this is clear from the symmetry of the problem). Light obeys the lightlike condition We have thus ds 2 =. (3) ds 2 = dt 2 +a 2 dr 2 (t) Kr 2 = (4) dt dr = a(t) Kr 2 (5) Integrating this, we get for the first light ray, and for the second, t2 t t2 +δt 2 t +δt dt ra a(t) = dt ra a(t) = dr Kr 2, (6) dr Kr 2. (7) The right hand sides of the two equations are the same, since the sender and the receiver have not moved (they have stayed at r = r A and r = ). Thus = t2 +δt 2 t +δt dt t2 a(t) t dt t2 +δt 2 a(t) = t 2 and the time to receive one wavelength is dt t +δt a(t) t dt a(t) = δt 2 a(t 2 ) δt a(t ), (8) δt 2 = a(t 2) a(t ) δt. (9)

6 3 FRIEDMANN ROBERTSON WALKER UNIVERSE 33 Figure 2: The two light rays to establish the redshift. As is clear from the derivation, this cosmological time dilation effect applies to observing any event taking place in galaxy A. As we observe galaxy A, we see everything happening in slow motion, slowed down by the factor a(t 2 )/a(t ), which is the factor by which the universe has expanded since the light (or any electromagnetic signal) left the galaxy. This effect can be observed, e.g., in the light curves of supernovae (their luminosity as a function of time). For the redshift we get +z λ 2 = δt 2 = a(t 2) λ δt a(t ). (2) The redshift of a galaxy directly tells us how much smaller the universe was when the light left the galaxy. The result is easy to remember; the wavelength expands with the universe. Thus the redshift z is related to the value of a(t) and thus to the time t, the age of the universe, when the light left the galaxy. We can thus use a or z as alternative time coordinates. Their relation is +z = a a or a = a +z da a = dz +z. (2) Note that while a grows with time, z decreases with time: z = at a = t = and z = at t = t Age-redshift relation If the observed redshift of a galaxy is z, what was the age of the universe when the light left the galaxy? Without knowing the function a(t) we cannot answer this and other similar questions, but it is useful to derive general expressions in terms of a(t), z, and H(t). We get H = da a dt dt = da ah = dz H(+z), (22)

7 3 FRIEDMANN ROBERTSON WALKER UNIVERSE 34 so that the age of the universe at redshift z is t(z) = and the present age of the universe is t dt = t t(z = ) = z dz H(+z ). (23) dz H(+z ). (24) The difference gives the light travel time, i.e., how far in the past we see the galaxy, 3..5 Distance t t(z) = z dz H(+z ). (25) In cosmology, the typical velocities of observers (with respect to the comoving coordinates) are small, v < km/s, so that we do not have to worry about Lorentz contraction (or about the velocity-related time dilation) and in the FRW model we can use the cosmic rest frame. The expansion of the universe brings, however, other complications to the concept of distance. Do we mean by the distance of a galaxy how far it is now (longer), how far it was when the observed light left the galaxy (shorter), or the distance the light has traveled (intermediate)? By proper distance d(t) between two objects 5 is defined as their physical distance measured along the hypersurface of constant cosmic time t. By comoving distance we mean the proper distance scaled to the present value of the scale factor (or sometimes to some other special time we choose as the reference time). If the objects have no peculiar velocity their comoving distance at any time is the same as their proper distance today. To calculate the proper distance d(t) between galaxies (one at r =, another at r = r A ) at time t, we need the metric, d(t) = r A ds. We integrate along the path t,ϑ,ϕ = const, or dt = dϑ = dϕ =, so ds 2 = a 2 (t) dr 2 2, and get or ra d(t) = a(t) = Kr dr Kr 2 K /2 a(t)arcsin(k /2 r A ) = K /2 a(t)χ = r curv χ, (K > ) a(t)r A, (K = ) K /2 a(t)arsinh( K /2 r A ) = K /2 a(t)χ = r curv χ. (K < ) a(t)s K (r A) (26) Using the generalized sine S k (x) of Eq. (2) we define a slightly more generalized sine S K (x) S K (x) K /2 S k ( K /2 x), (27) K /2 sin(k /2 x), (K > ) x, (K = ) K /2 sinh( K /2 x), (K < ) 5 or more generally between two points (r,ϑ,ϕ) on the t = const hypersurface. In relativity, proper length of an object refers to the length of an object in its rest frame, so the use of the word proper in proper distance is perhaps proper only when the objects are at rest in the FRW coordinate system. Nevertheless, we define it now this way. (28)

8 3 FRIEDMANN ROBERTSON WALKER UNIVERSE 35 and write S k and S K for their inverse functions. The functions S K and S K convert between the two natural unscaled (i.e., you still need to multiply this distance by the scale factor a) radial distance definitions for the RW metric: d a = χ, (29) K /2 the proper distance measured along the radial line, and r which is related to the length of the circle and area of the sphere at this distance with the familiar 2πr and 4πr 2 : K /2 arcsin(k /2 r) = K /2 χ, (K > ) S K (r) = r, (K = ) (3) K /2 arsinh( K /2 r) = K /2 χ. (K < ) Thus so that d(t) = a(t)s K (r) = a(t) K /2 S k ( K /2 r) (3) r = S K (d/a), (32) where d/a = d(t)/a(t) is independent of time, for the relation between the proper distance from the origin and the r coordinate. As the universe expands, this distance grows, d(t) = a(t)s K (r) = a +z S K (r) = dc +z, (33) where d c d(t ) is the present proper distance to r, or the comoving distance to r. Neither the proper distance d, nor the coordinate r of a galaxy are directly observable. Observable quantities are, e.g., the redshift z, location on the sky (ϑ,ϕ) when the observer is at r = ), the angular diameter, and the apparent luminosity. We want to use the FRW model to relate these observable quantities to the coordinates and actual distances. Let us first derive the distance-redshift relation. We see a galaxy with redshift z; how far is it? (We assume z is entirely due to the Hubble expansion, +z = a/a, i.e., we ignore the contribution from the peculiar velocity of the galaxy or the observer). Since for light, ds 2 = dt 2 +a 2 dr 2 (t) =, (34) Kr2 we have dr dt = a(t) Kr 2 t where d/a = d(t)/a(t) is independent of t. The comoving distance to redshift z is thus d c (z) = a t t t dt r a(t) = dr = d Kr 2 a = dc, (35) a dt dt a(t) = x = +z x /dt, (36) where x a/a = /(+z) H ȧ a = x dt and x = da a = dz +z. (37)

9 3 FRIEDMANN ROBERTSON WALKER UNIVERSE 36 Figure 3: Calculation of the distance-redshift relation. (For the a = scaling, x is simply a.) In terms of redshift and Hubble parameter, this becomes d c (z) = z The proper distance at the time the light left the galaxy is d(z) = +z dz H(z ). (38) z dz H(z ). (39) The distance light has traveled (i.e., adding up the infinitesimal distances measured by a sequence of observers at rest along the light path) is equal to the light travel time, Eq. (25). In a monotonously expanding (or contracting) universe it is intermediate between d and d c. We encounter the beginning of time, t =, at a = or z =. Thus the comoving distance light has traveled during the entire age of the universe is d hor = dz H(z ). (4) This distance (or the sphere with radius d hor, centered on the observer) is called the horizon, since it represent the maximum distance we can see, or receive any information from. There are actually several different concepts in cosmology called the horizon. To be exact, the one defined above is the particle horizon. Another horizon concept is the event horizon, which is related to how far the light can travel in the future. The Hubble distance H is also often referred to as the horizon (especially when one talks about subhorizon and superhorizon distance scales).

10 3 FRIEDMANN ROBERTSON WALKER UNIVERSE 37 Figure 4: Defining the angular diameter distance Volume Theobjects we observe lie on our past light cone, and the observed quantities are z,ϑ,ϕ, so these are the observer s coordinates for the light cone. What is the volume of space corresponding to a range z ϑ ϕ? Note that the light cone is a lightlike surface, so its volume is zero. Here we mean instead the volume that we get when we project a section of it onto the t = const hypersurface crossing this section at a particular z (which is unique when z = dz is infinitesimal). From Eq. (39), the proper distance corresponding to dz is dz/( + z)h(z). Directly from the RW metric, the area corresponding to dϑdϕ is ardϑ arsinϑdϕ, so that the volume element becomes dv = a2 r 2 sinϑ (+z)h(z) dzdϑdϕ = a 2 r 2 dzdω (4) (+z)h(z) and the comoving volume is dv c = (+z) 3 dv = a2 r2 dzdω. (42) H(z) These are the volume elements for counting the number density or comoving number density of galaxies from observations. If the number of galaxies is conserved, in a homogeneous universe their comoving number density should be independent of z. Thus, in principle, from such observations one should be able to determine H(z). In practice this is made difficult by evolution of galaxies with time and the fact that it is more difficult to observe galaxies at larger z Angular diameter distance The distance-redshift relation (36 or 38) obtained above would be nice if we already knew the function a(t). We can turn the situation around and use an observed distance-redshift relation, to obtain information about a(t), or equivalently, about H(z). But for that we need a different distance-redshift relation, one where the distance is replaced by some directly observable quantity. Astronomers employ various auxiliary distance concepts, like the angular diameter distance or the luminosity distance. These would be equal to the true distance in Euclidean non-expanding space. To answer the question: what is the physical size of an object, whom we see at redshift z, subtending an angle ϑ on the sky? we need the concept of angular diameter distance d A. In Euclidean geometry, s = ϑd or d = s ϑ. (43) Accordingly, we define d A s ϑ, (44)

11 3 FRIEDMANN ROBERTSON WALKER UNIVERSE 38 where s is the actual diameter of the object (the diameter it had when the light we see left it), and ϑ is the observed angle. For large-scale structures, which expand with the universe, we use the comoving angular diameter distance d c A sc /ϑ, where s c = ( + z)s is the comoving diameter of the structure. Thus d c A = (+z)d A. From the RW metric, the physical length s corresponding to an angle ϑ is, from ds 2 = a 2 (t)r 2 dϑ 2 s = a(t)rϑ. Thus d A (z) = a(t)r = a +z r = a +z S K(d c /a ) = = a +z S K a +z S K ( a +z z ( a x dz H(z ) ) /dt ) (45) The comoving angular diameter distance is then ( d c A = a r = a S K x a +z /dt ) = a S K ( a z dz ) H(z. (46) ) For the flat (K = ) FRW model these relations simplify to d A (z) = dc +z = d(z) = +z +z x /dt = z dz +z H(z ) (47) and d c A = +z x /dt = z dz H(z ), (48) so that the angular diameter distance is equal to the proper distance when the light left the object and the comoving angular diameter distance is equal to the comoving distance. For large distances (redshifts) the angular diameter distance may have the counterintuitive property that after some z it begins to decrease as a function of z. Thus objects with are behind other objects as seen from here will nevertheless have a smaller angular diameter distance. There are two reasons for such behavior: In a closed (K > ) universe objects which are on the other side of the universe (the 3-sphere), i.e., with χ > π/2, will cover a larger angle as seen from here because of the spherical geometry (if we can see this far away). This effect comes from the S K in Eq. (45). An object at exactly opposite end (χ = π would cover the entire sky as light from it would reach us from every direction after traveling half-way around the 3-sphere. In our universe these situations do not occur in practice, because lower limits to the size of the 3-sphere 6 are larger than the distance light will travel during the age of the universe. The second reason, which does apply to the observed universe, and applies only to d A, not to d c A, is the expansion of the universe. An object, which does not expand with the universe, occupied a much larger comoving volume in the smaller universe of the past. This effect is the /(+z) factor in Eq. (45), which for large z decreases faster than the other part grows. In other words, the physical size of the 2-sphere corresponding to a given redshift z has a maximum at some finite redshift (of the order z ), and for larger redshifts it is again smaller. (The same behavior applies to the proper distance d(z).) 6 Observations tell us that the curvature of the universe is very small, so that the we cannot determine which of the three geometries applies to our universe.

12 3 FRIEDMANN ROBERTSON WALKER UNIVERSE 39 Suppose we have a set of standard rulers, objects that we know are all the same size s, observed at different redshifts. Their observed angular sizes ϑ(z) then give us the observed angular diameter distance as d A (z) = s/ϑ(z). This observed function can be used to determine the expansion history a(t), or H(z) Luminosity distance In transparent Euclidean space, an object whose distance is d and whose absolute luminosity (radiated power) is L would have an apparent luminosity l = L/4πd 2. Thus we define the luminosity distance of an object as L d L 4πl. (49) Consider the situation in the FRW universe. The absolute luminosity can be expressed as: L = number of photons emitted time their average energy = N γe em t em. (5) Iftheobserverisat acoordinatedistancer fromthesource, thephotonshaveat that distance spread over an area A = 4πa 2 r 2. (5) The apparent luminosity can be expressed as: l = number of photons observed time area their average energy = N γe obs t obs A. (52) The number of photons N γ is conserved, but their energy is redshifted, E obs = E em /( + z). Also, if the source is at redshift z, it takes a factor +z longer to receive the photons t obs = (+z)t em. Thus, l = N γe obs t obs A = N γe em t em (+z) 2 4πa 2. (53) r2 From Eq. (49), d L L 4πl = (+z)a r = (+z) 2 d A (z) = (+z)d c A (z). (54) Compared to the comoving angular diameter distance, d c A (z), we have a factor (+z), which causes d L to increase faster with z than d c A (z). There is one factor of ( + z)/2 from photon redshift and another factor of (+z) /2 from cosmological time dilation, both contributing to making large-redshift objects dimmer. Compared to d A (z), thereis another factor of (+z) from the expansion of the universe, which we discussed in Sec. 3..7, which causes distant objects to appear larger on the sky, but does not contribute to their apparent luminosity. Thus the surface brightness (flux density per solid angle) of objects decreases with redshift as d 2 A/d 2 L = (+z) 4 (55) (flux density l d 2 L, solid angle Ω d 2 A.) Suppose that we have a set of standard candles, objects that we know all have the same L. From their observed redshifts and apparent luminosities we get an observed luminosity-distanceredshift relation d L (z) = L/4πl, which can be used to determine a(t), or H(z).

13 3 FRIEDMANN ROBERTSON WALKER UNIVERSE Hubble law In Sec. we introduced the Hubble law z = H d d = H z, (56) which was based on observations (at small redshifts). Now that we have introduced the different distance concepts, d, d c, d A, d c A, d L, in an expanding universe, and derived exact formulas for them in the RW metric, we can see that for z (when we can approximate H(z) = H ) all of them give the Hubble law as an approximation, but all of them deviate from it, in a different manner, for z and larger.

14 3 FRIEDMANN ROBERTSON WALKER UNIVERSE Dynamics 3.2. Friedmann equations Note: If you do not understand the next two paragraphs, don t worry. They are properly explained in the General Relativity course. The point of them is just to give a hint where equations (68) and (69) come from. But all that is needed for this course from here is to accept these two equations. We shall now apply the Einstein equation to the homogeneous and isotropic case, which leads to Friedmann Robertson Walker (FRW) cosmology. The metric is now the RW metric, g µν = a 2 Kr 2 a 2 r 2 a 2 r 2 sin 2 ϑ Calculating the Einstein tensor from this metric gives. (57) Gˆˆ Gˆˆ = 3 a 2(ȧ2 +K) (58) = a 2(2äa+ȧ2 +K) = Gˆ2ˆ2 = Gˆ3ˆ3. (59) We use here the orthonormal basis (signified by theˆover the index). We assume the perfect fluid form for the energy tensor T µν = (ρ+p)u µ u ν +pg µν. (6) Isotropy implies that the fluid is at rest in the RW coordinates, so that uˆµ = (,,,) and (remember, gˆµˆν = η µν = diag(,,,)) ρ T ˆµˆν = p p. (6) p Homogeneity implies that ρ = ρ(t), p = p(t). The Einstein equation Gˆµˆν = 8πGT ˆµˆν becomes now Let us rearrange this pair of equations to 7 3 a 2(ȧ2 +K) = 8πGρ (62) (ȧ ) 2 2ä a K a a 2 = 8πGp. (63) 7 Including the cosmological constant Λ these equations take the form 3 +K) Λ = 8πGρ (64) a (ȧ 2(ȧ2 ) 2 2ä a K +Λ = 8πGp. (65) a a2

15 3 FRIEDMANN ROBERTSON WALKER UNIVERSE 42 (ȧ a ) 2 + K a 2 = 8πG 3 ρ (68) ä a = 4πG (ρ+3p). 3 (69) These are the Friedmann equations. ( Friedmann equation in singular refers to Eq. (68).) Using the Hubble parameter H ȧ/a Ḣ = ä a ȧ2 a 2 we can write the Friedmann equations also as ä a = Ḣ +H2 (7) H 2 = 8πG 3 ρ K a 2 (7) Ḣ = 4πG(ρ+p)+ K a 2. (72) The general relativity version of energy and momentum conservation, energy-momentum continuity, follows from the Einstein equation. In the present case this becomes the energy continuity equation ρ = 3(ρ+p)ȧ a. (73) Since the fluid is at rest, there is no equation for the momentum. (Exercise: Derive this from the Friedmann equations!) The Friedmann equation(68) connects the three quantities, the density ρ, the space curvature K/a 2, and the expansion rate H of the universe, ρ = 3 8πG (H 2 + Ka 2 ) = ρ c + 3K 8πGa 2. (74) (Note that the curvature quantity K/a 2 is invariant under the r coordinate scaling we discussed earlier.) Here we have defined the critical density ρ c 3H2 8πG, (75) correspondingto a given value of the Hubbleparameter. 8 The critical density changes in time as the Hubble parameter evolves. The present value of the critical density is given by the Hubble constant as ρ c ρ c (t ) 3H2 8πG = h 2 kg/m 3 =.54h 2 GeV/m 3 = 2.77 h 2 m /Mpc 3. (76) or, in the rearranged form, (ȧ ) 2 + K a a Λ = 8πG ρ (66) ä a Λ = 4πG (ρ+3p). (67) 3 3 We shall not include Λ in these equations. Instead, we allow for the presence of vacuum energy ρ vac, which has the same effect. 8 We could also define likewise a critical Hubble parameter H c corresponding to a given density ρ, but since, of the above three quantities, the Hubble constant has usually been the best determined observationally, it has been better to refer other quantities to it.

16 3 FRIEDMANN ROBERTSON WALKER UNIVERSE 43 The nature of the curvature then depends on the density ρ: ρ < ρ c K < (77) ρ = ρ c K = (78) ρ > ρ c K >. (79) (8) The density parameter Ω is defined Ω ρ ρ c (8) (where all three quantities are functions of time). Thus Ω = implies a flat universe, Ω < an open universe, and Ω > a closed universe. The Friedmann equation can now be written as Ω = + K H 2 a 2, (82) a very useful relation. Here K is a constant, and the other quantities are functions of time Ω(t), H(t), and a(t). Note that if Ω < (or > ), it will stay that way. And if Ω =, it will stay constant, Ω = Ω =. Observations suggest that the density of the universe today is close to critical, Ω. To solve the Friedmann equations, we need the equation of state that relates p and ρ. In general, the pressure p of matter may depend also on other thermodynamic variables than the energy density ρ. The equation of state is called barotropic is p is uniquely determimed by ρ, i.e., p = p(ρ). Regardless of the nature of matter, in a homogeneous universe we have p = p(ρ) in practice if the energy density decreases monotonously with time, since p = p(t), ρ = ρ(t) and we can invert the latter to get t(ρ), so that we can write p = p(t) = p(t(ρ)) p(ρ). We define the equation-of-state parameter w p ρ (83) so that we can formally write the equation of state as and the energy continuity equation as ρ ρ = 3(+w)ȧ a a p = wρ, (84) dlnρ = 3(+w)dlna, (85) where, in general, w = w(t). Equation (85) can be formally integrated to { ρ a } { z } = exp 3[+w(a )] da ρ a = exp dz 3[+w(z )] +z. (86) The simplest case is the one where p ρ, so that w = const: in which case the solution of (85) is p = wρ, w = const, (87) There are three such cases: ρ a 3(+w) ρ = ρ ( a a ) 3(+w). (88)

17 3 FRIEDMANN ROBERTSON WALKER UNIVERSE 44 Matter (w = ) (called matter in cosmology, but dust in general relativity), meaningnonrelativistic matter (particle velocities v ), forwhichp ρ, sothat wecanforget the pressure, and approximate p =. From Eq. (73), d(ρa 3 )/dt =, or ρ a 3. Radiation (w = /3), meaning ultrarelativistic matter (where particle energies are their rest masses, which is always true for massless particles like photons), for which p = ρ/3. From Eq. (73), d(ρa 4 )/dt =, or ρ a 4. Vacuum energy (w = ) (or the cosmological constant), for which ρ = const. From Eq. (73) follows the equation of state for vacuum energy: p = ρ. Thus a positive vacuum energy 9 corresponds to a negative vacuum pressure. You may be used to pressure being positive, but there is nothing unphysical about negative pressure. In other contexts it is often called (positive) tension instead of (negative) pressure. We know that the universe contains ordinary, nonrelativistic matter. We also know that there is radiation, especially the cosmic microwave background. In Chapters 4 and 5 we shall discuss how the different known particle species behave as radiation in the early universe when it is very hot, but as the universe cools, the massive particles change form ultrarelativistic (radiation) to nonrelativistic (matter). During the transition period the pressure due to that particle species falls from p = ρ/3 to p. We shall discuss these transition periods in Chapter 5. In this chapter we focus on the later evolution of the universe(after big bang nucleosynthesis). Then the known forms of matter and energy in the universe can be divided into these two classes: matter (p ) and radiation (p ρ/3). Except that we do not know the small masses of neutrinos. Depending on the values of these masses, neutrinos may make this radiation-to-matter transition sometime during this later evolution. We already revealed in Chapter that the present observational data cannot be explained in terms of known forms of particles and energy using known laws of physics, and therefore we believe that there are other, unknown forms of energy in the universe, called dark matter and dark energy. Dark matter has by definition negligible pressure, so that we can ignore its pressure in the Friedmann equations. However, to explain the observed expansion history of the universe, an energy component with negative pressure is needed. This we call dark energy. We do not know its equation of state. The simplest possibility for dark energy is just the cosmological constant (vacuum energy), which fits the data perfectly. Therefore we shall carry on our discussion assuming three energy components: matter, radiation, and vacuum energy. We shall later comment on how much current observations actually constrain the equation of state for dark energy. 9 In the quantumfield theory view, vacuum is the minimum energy density state of the system. Therefore any other contribution to energy density is necessarily positive, but whether the vacuum energy density itself needs to be nonnegative is less clear. Other physics except general relativity is sensitive only to energy differencies and thus does not care about the value of vacuum energy density. In general relativity it is a source of gravity, but cannot be distinguished from a cosmological constant, which is a modification of the law of gravity by an arbitrary constant that could be negative just as well as positive. For simplicity we will here include the possible cosmological constant in the concept of vacuum energy, and thus we should allow for negative vacuum energy density also. In Chapter 4 we derive formulae for the pressures of different particle species in thermal equilibrium. These always give a positive pressure. The point is that there we ignore interparticle forces. To make the pressure from particles negative, would require strong attractive forces between particles. But the vacuum pressure is not from particles, its from the vacuum. If the dark energy is not just vacuum energy, it is usually thought to be some kind of field. For fields, a negative pressure comes out more naturally than for particles.

18 3 FRIEDMANN ROBERTSON WALKER UNIVERSE 45 If the universe contains several energy components ρ = i ρ i with p i = w i ρ i (89) without significant energy transfer between them, then each component satisfies the energy continuity equation separately, ρ i = 3(+w i )ȧ ρ i a. (9) and, if w i = const, ρ i a 3(+w i) ( ) a 3(+wi ) ρ i = ρ i. (9) a In the early universe there were times where such energy transfer was important, but after BBN it was negligible, so then we have the above case with ρ = ρ r +ρ m +ρ vac with w r = /3, w m =, w vac =,. (92) We can then arrange Eq. (68) into the form (ȧ a ) 2 = α 2 a 4 +β 2 a 3 Ka 2 + Λ, (93) 3 where α, β, K, and Λ = 8πGρ vac are constants. The four terms on the right hand side are due to radiation, matter, curvature, and vacuum energy, in that order. As the universe expands (a grows), different components on the right hand side become important at different times. Early on, when a was very small, the universe was radiation-dominated. If the universe keeps expanding without limit, eventually the vacuum energy will become dominant(already it appears to be the largest term). In the middle we may have matter-dominated and curvature-dominated eras. In practice it seems the curvature of the universe is quite small and therefore there never was a curvature-dominated era, but there was a long matter-dominated era. We know that the radiation component is insignificant at present, and we can forget it in Eq. (93), if we exclude the first few million years of the universe from discussion. In the inflationary scenario, there was something resembling a very large vacuum energy density in the very early universe (during the first small fraction of the first second), which then disappeared. So there may have been a very early vacuum-dominated era (inflation). Let us now solve the Friedmann equation for the case where one of the four terms dominates. The equation has the form (ȧ a ) 2 = α 2 a n or a n 2 da = αdt. (94) Integration gives 2 n an 2 = αt, (95) where we chose t =, so that a(t = ) =. We get the three cases: n = 4 radiation dominated a t /2 n = 3 matter dominated a t 2/3 n = 2 curvature dominated (K < ) a t The cases K > and vacuum energy have to be treated differently (exercise).

19 3 FRIEDMANN ROBERTSON WALKER UNIVERSE 46 Example: Einstein de Sitter model. Consider the simplest case, Ω = (K = ) and Λ =. The first couple of million years when radiation can not be ignored, makes an insignificant contribution to the present age of the universe, so we can ignore radiation also. We have now the matter-dominated case. For the density we have ( a ρ = ρ a The Friedmann equation is now (ȧ a2 a ) 3 = Ω ρ c ( a a ) 2 = 8πG ( 3 ρ a ) 3 c }{{} a a /2 da = H a 3/2 dt H 2 a /2 da = H a 3/2 a t2 t dt ) 3 ( a ) 3 = ρc. (96) a 2 3 (a3/2 2 a 3/2 ) = H a 3/2 (t 2 t ). Thus we get t 2 t = 2 3 H [( )3 a2 a 2 ( a a )3 2 ] = 2 3 H ] (+z 2 ) 3/2 (+z ) 3/2 [ (97) where z is the redshift. Let t 2 = t be the present time (z = ). The time elapsed since t = t corresponding to redshift z is [ t t = 2 ( ] )3 a 2 3 H = 2 [ ] 3 H. (98) (+z) 3/2 a Let t = and t 2 = t(z) be the time corresponding to redshift z. The age of the universe corresponding to z is t = 2 ( )3 a H = 3 H = t(z). (99) (+z) 3/2 a This is the age-redshift relation. For the present (z = ) age of the universe we get t = 2 3 H. () The Hubble constant is H h km/s/mpc = h/( yr), or H = h yr. Thus { t = h yr h=.7 yr = 3. 9 () yr h=.5 The ages of the oldest stars appear to be at least about 2 9 years. Considering the HST value for the Hubble constant (h =.72±.8), this model has an age problem Cosmological parameters We divide the density into its matter, radiation, and vacuum components ρ = ρ m +ρ r +ρ vac, and likewise for the density parameter, Ω = Ω m + Ω r + Ω Λ, where Ω m ρ m /ρ c, Ω r ρ r /ρ c, and Ω Λ ρ vac /ρ c Λ/3H 2. Ω m, Ω r, and Ω Λ are functions of time (although ρ vac is constant, ρ c (t) is not). In standard notation, Ω m, Ω r, and Ω Λ denote the present values of these density parameters, and we write Ω m (t), Ω r (t), and Ω Λ (t), if we want to refer to their values at other times. Thus we write Ω Ω m +Ω r +Ω Λ. (2)

20 3 FRIEDMANN ROBERTSON WALKER UNIVERSE 47 The present radiation density is relatively small, Ω r 4 (we shall calculate it in Chapter 5). So we usually write just Ω = Ω m +Ω Λ. (3) The radiation density is also known very accurately from the temperature of the cosmic microwave background, and therefore Ω r is not usually considered as a cosmological parameter (in the sense of an inaccurately known number that we try to fit with observations). The FRW cosmological model is thus defined by giving the present values of the three cosmological parameters, H, Ω m, and Ω Λ. Observations favor the values h.7, Ω m.3, and Ω Λ.7. (We have already discussed the observational determination of H. We shall discuss the observational determination of Ω m and Ω Λ both in this Chapter and later.) Since the critical density is h 2, it is often useful to use instead the physical or reduced density parameters, ω m Ω m h 2, ω r Ω r h 2, which are directly proportional to the actual densities in kg/m 3. (An ω Λ turns out not be so useful and is not used.) Age of the FRW universe Considernow the general case with arbitraryvalues of Ω m and Ω Λ. Now thefriedmann equation has four terms on the right hand side, (ȧ ) 2 = 8πG ( a 3 Ω a ) 4 8πG ( rρ c + }{{} a 3 Ω a ) 3 mρ c +ΩΛ H 2 }{{} a K a 2 Ω rh 2 Ω mh 2 da dt = H a Ω r a 2 a 2 +Ω m a a +Ω Λ a 2 a 2 KH 2 a 2. Defining we get x a a = +z, (4) da a dt dt = H Ωr x 2 +Ω m x +Ω Λ x 2 +( Ω ), (5) where we have used Eq. (82). We shall later have much use for this convenient form of the Friedmann equation. Now we integrate from it the time it takes for the universe to expand from a to a 2, or from redshift z to z 2, t2 t dt = x2 x /dt = H +z 2 (6) ( Ω )+Ω +z r x 2 +Ω m x +Ω Λ x 2 This is integrable to an elementary function if two of the four terms under the root sign are absent. From this we get the age-redshift relation t(z) = t dt = H +z (7) ( Ω )+Ω r x 2 +Ω m x +Ω Λ x2. (This gives t(z), that is, t(a). Inverting this function gives us a(t), the scale factor as a function of time. Now a(t) is not necessarily an elementary function, even if t(a) is. Sometimes one can get a parameter representation a(ψ), t(ψ) in terms of elementary functions.)

21 3 FRIEDMANN ROBERTSON WALKER UNIVERSE Matter only 2. Flat universe a/a. a/a time (Hubble units) time (Hubble units) Figure 5: The expansion of the universe a(t) for a) the matter-only universe Ω Λ =, Ω m =,.2,...,.8 (from top to bottom) b) the flat universe Ω = (Ω Λ = Ω m ), Ω m =,.5,.2,.4,.6,.8,.,.5 (from top to bottom). The time axis gives H (t t ), i.e,. corresponds to the present time. In Fig. 5 we have integrated Eq. (5) from the initial conditions a = a, ȧ = H a, both backwards and forwards from the present time t = t to find a(t) as a function of time. For the present age of the universe we get t = t dt = H ( Ω )+Ω r x 2 +Ω m x +Ω Λ x 2. (8) The simplest cases, where only one of the terms under the square root is nonzero, give: radiation dominated (Ω r = Ω = ): t = 2 H matter dominated (Ω m = Ω = ): t = 2 3 H curvature dominated (Ω = ): t = H vacuum dominated (Ω Λ = Ω = ): t =. These results can be applied also at other times, e.g., during the radiation-dominated epoch the age of the universe was related to the Hubble parameter by t = 2 H and during the matterdominated epoch by t = 2 3 H (assuming that we can ignore the effect of the earlier epochs on the age). Returning to the present time, we know that Ω r is so small that ignoring that term causes negligible error. Example: Age of the open universe. Consider now the case of the open universe (K < or Ω < ), but without vacuum energy (Ω Λ = ), and approximating Ω r. Integrating Eq. (8) (e.g., with substitution x = Ωm Ω m sinh 2 ψ 2 ) gives for the age of the open universe t = H = H [ Ωm +Ω m x Ω m Ω m arcosh 2( Ω m ) 3/2 ( )] 2. (9) Ω m A special case of the open universe is the empty, or curvature-dominated, universe (Ω m = and Ω Λ = ). Now the Friedmann equation says /dt = H, or a = a H t, and t = H.

22 3 FRIEDMANN ROBERTSON WALKER UNIVERSE 49 From the cases considered so far we get the following table for the age of the universe: Ω m Ω Λ t H..9H.3.8H.5.75H (2/3)H Therearemanywaysofestimatingthe matterdensity Ω m ofthe universe, someofwhicharediscussed in Chapter 8. These estimates give Ω m.3. With Ω m =.3, Ω Λ = (no dark energy), and h =.72, we get t = years. This is about the same as the lowest estimates for the ages of the oldest stars. Since it should take hundreds of millions of years for the first stars to form, the open universe (or in general, a no-dark-energy universe, Ω Λ = ) seems also to have an age problem. The cases (Ω m >, Ω Λ = ) and (Ω = Ω m +Ω Λ =, Ω Λ > ) are left as exercises. The more general case (Ω, Ω Λ ) leads to elliptic functions Distance-redshift relation From Eq. (36), the comoving distance to redshift z is d c (z) = a t t dt dt a(t) = x = x /dt, () where x a/a = /(+z). We have /dt from Eq. (5), giving d c (z) = x H Ωm x +Ω r x 2 +Ω Λ x 2 +( Ω ) = H H = H +z +z +z ΩΛ x 4 +( Ω )x 2 +Ω m x+ω r ΩΛ x 4 +( Ω )x 2 +Ω m x Ω (x x 2 ) Ω Λ (x x 4 )+x 2, () wherewehavedroppedtheω r term, whichhasnegligibleeffect, andutilized Ω m = Ω Ω Λ toget the last form. This is the distance-redshift relation. We see that it depends on three independent cosmological parameters, for which we have taken H, Ω, and Ω Λ. In this parametrization, the distance at a given redshift is proportional to the Hubble distance, H. If we give the distance in units of H, then it depends only on the two remaining parameters, Ω and Ω Λ. If we increase Ω keeping Ω Λ constant (meaning that we increase Ω m ), the distance corresponding to a given redshift decreases. This is because the universe has expanded faster in the past (see Fig. 5), so that there is less time between a given a = a /(+z) and the present. The distance to the galaxy with redshift z is shorter, because photons have had less time to travel. Whereas if we increase Ω Λ with a fixed Ω (meaning that we decrease Ω m ), we have the opposite situation and the distance increases. (Note that (x x 2 ) and (x x 4 ) are always positive in Eq. ). If the galaxy has stayed at the same coordinate value r, i.e., it has no peculiar velocity, then the comoving distance is equal to its present distance. The actual distance to the galaxy at the

23 3 FRIEDMANN ROBERTSON WALKER UNIVERSE 5 2 Age of the universe / H - no big bang accelerating --- decelerating Ω Λ.5 open --- closed recollapses eventually Ω m Fig. by E. Sihvola Figure 6: The age of the universe as a function of Ω m and Ω Λ.

24 3 FRIEDMANN ROBERTSON WALKER UNIVERSE 5 2. Matter only 2. Flat universe distance (H - ).2..8 distance (H - ) redshift redshift Figure 7: The distance-redshift relation, Eq. (), for a) the matter-only universe Ω Λ =, Ω m =,.2,...,.8 (from top to bottom) b) the flat universe Ω = (Ω Λ = Ω m ), Ω m =,.5,.2,.4,.6,.8,.,.5 (from top to bottom). The thick line in both cases is the Ω m =, Ω Λ = model. time t the light left the galaxy is d (z) = dc (z) +z. (2) We encounter the beginning of time, t =, at a = or z =. Thus the comoving distance light has traveled during the entire age of the universe, the horizon distance, is d hor = H ΩΛ x 4 +( Ω )x 2 +Ω m x+ω r. (3) The simplest cases, where only one of the terms under the square root is nonzero, give: radiation dominated (Ω r = Ω = ): d hor = H = 2t matter dominated (Ω m = Ω = ): d hor = 2H = 3t curvature dominated (Ω = ): d hor = vacuum dominated (Ω Λ = Ω = ): d hor =. These results can be applied also at other times, e.g., during the radiation-dominated epoch the horizon distance was related to the Hubble parameter and age by d hor = H = 2t and during the matter-dominated epoch by d hor = 2H = 3t (assuming that epoch had already Of these cases, the strict forms of the two last ones, pure curvature (Ω = ) and pure vacuum (Ω Λ = Ω = ) do not actually fit in the FRW framework, where the starting assumption was spatial homogeneity that formed the basis of separation between time and space. This separation requires a physical quantity that evolves in time, in practice the energy density ρ(t), so that the t = const slices can be defined as the ρ = const hypersurfaces. Now in these two cases, ρ = const (either or the vacuum value), and does not provide this separation. These cases are called the Milne universe and the de Sitter space (or anti-de Sitter space for ρ vac < ) and are discussed in the General Relativity course. For our purposes, we should instead consider these as limiting cases where there is also another density component that is just very small (a nonzero Ω m or Ω r that is ). Then this other component necessarily becomes important in the early universe, as a. This means that d hor is not, just very large. The same applies to the infinite age of the vacuum-dominated universe.

25 3 FRIEDMANN ROBERTSON WALKER UNIVERSE 52 lasted long enough so that we can ignore the effect of the earlier epochs on the age and horizon distance). Returning to the present time, we know that Ω r is so small that ignoring that term causes negligible error. Example: Distance and redshift in the flat matter-dominated universe. Let us look at the simplest case, (Ω m,ω Λ ) = (,) (with Ω r ), in more detail. Now Eq. () is just d c (z) = H +z ( = 2H x/2 Expanding / +z = 2 z z2 5 6 z3 we get ). (4) +z d c (z) = H (z 3 4 z z3 ) (5) so that for small redshifts, z we get the Hubble law, z = H d. At the time when the light we see left the galaxy, its distance was ( ) d (z) = +z dc (z) = a(t)r = 2H +z (6) (+z) 3/2 = H (z 7 4 z z3 ) (7) so the Hubble law is valid for small z independent of our definition of distance. The distance d(t) = a(t)r to the galaxy grows with the velocity d = rȧ = rah, so that today d = ra H = d c H = 2( / +z). This equals (the speed of light) at z = 3. We note that d (z) has a maximum d (z) = 8 27 H at z = 5 4 (+z = 9 4 ). This corresponds to the comoving distance d c (z) = 2 3 H. See Fig. 9. Galaxies that are further out were thus closer when the light left, since the universe was then so much smaller. The distance to the horizon in this simplest case is d hor d c (z = ) = 2H = 3t. (8) Just like any planar map of the surface of the Earth must be distorted, so is that of the curved spacetime. Even in the flat-universe case, the spacetime is curved due to the expansion. Thus any spacetime diagram is a distortion of the true situation. In Figs. 9 and there are three different ways of drawing the same spacetime diagram. In the first one the vertical distance is proportional to the cosmic time t, the horizontal distance to the proper distance at that time, d(t). The second one is in the comoving coordinates (t,r), so that the horizontal distance is proportional to the comoving distance d (Note that for Ω =, i.e., K =, we have d c = a r, see Eq. (33)). The third one is in the conformal coordinates (η,r), with normalization a =. The last one has the advantage that light cones are always at a 45 angle. This is thus the Mercator projection 2 spacetime. Angular diameter distance: From Eq. (32), the coordinate r is related to the comoving distance d c by r = S K (d c /a ), (9) and using the distance-redshift relation, Eq. (), we have d A (z) = a [ +z S K a H +z ], (2) Ω (x x 2 ) Ω Λ (x x 4 )+x 2 2 The Mercator projection is a way of drawing the map of the Earth so that the points of compass correspond to the same direction everywhere on the map, e.g., northeast and northwest are always 45 from the north direction.