Lecture Notes on General Relativity (2024)

8. COSMOLOGY

Contemporary cosmological models are based on the idea that theuniverse is pretty much the same everywhere - a stance sometimesknown as the Copernican principle. On the face of it, sucha claim seems preposterous; the center of the sun, for example, bears little resemblance to the desolate cold of interstellarspace. But we take the Copernican principle to only apply onthe very largest scales, where local variations in density areaveraged over. Its validity on such scales is manifested in a number of different observations, such as number countsof galaxies and observations of diffuse X-ray and -raybackgrounds, but is most clear in the 3° microwave backgroundradiation. Although we now know that the microwave background is notperfectly smooth (and nobody ever expected that it was), the deviations from regularity are on the order of 10^-5 or less,certainly an adequate basis for an approximate description of spacetime on large scales.

The Copernican principle is related to two more mathematicallyprecise properties that a manifold might have: isotropy and hom*ogeneity.Isotropy applies at some specific point in the space, andstates that the space looks the same no matter what direction youlook in. More formally, a manifold M is isotropic around a pointp if, for any two vectors V and W inT_pM, there is an isometry of M such that the pushforward of W under theisometry is parallel with V (not pushed forward).It is isotropy which is indicated by the observations of themicrowave background.

hom*ogeneity is the statement that the metric is the samethroughout the space. In other words, given any two points p andq in M, there is an isometry which takes p intoq. Note that there is no necessary relationship between hom*ogeneityand isotropy; a manifold can be hom*ogeneous but nowhere isotropic(such as × S² in the usual metric), or it can be isotropic around a point without being hom*ogeneous (such as a cone, which isisotropic around its vertex but certainly not hom*ogeneous). On theother hand, if a space is isotropic everywhere then it ishom*ogeneous. (Likewise if it is isotropic around one point andalso hom*ogeneous, it will be isotropic around every point.)Since there is ample observational evidence for isotropy, and the Copernican principle would have us believe that weare not the center of the universe and therefore observers elsewhereshould also observe isotropy, we will henceforth assume bothhom*ogeneity and isotropy.

There is one catch. When we look at distant galaxies, they appearto be receding from us; the universe is apparently not static, butchanging with time. Therefore we begin construction of cosmologicalmodels with the idea that the universe is hom*ogeneous and isotropicin space, but not in time. In general relativity this translates intothe statement that the universe can be foliated into spacelike slicessuch that each slice is hom*ogeneous and isotropic. We therefore consider our spacetime to be × , where represents the time direction and is a hom*ogeneous andisotropic three-manifold. The usefulness of hom*ogeneity and isotropy is that they imply that must be a maximallysymmetric space. (Think of isotropy as invariance under rotations,and hom*ogeneity as invariance under translations. Then hom*ogeneityand isotropy together imply that a space has its maximum possiblenumber of Killing vectors.) Thereforewe can take our metric to be of the form

(8.1)

Here t is the timelike coordinate, and (u¹, u², u³) are thecoordinates on ; is the maximally symmetricmetric on . This formula is a special case of (7.2), which weused to derive the Schwarzschild metric, except we have scaled tsuch that g_tt = - 1. The function a(t)is known as the scale factor, and it tells us "how big" the spacelikeslice is at the moment t. The coordinates used here,in which the metric is free of cross terms dt duⁱ and thespacelike components are proportional to a single function of t, are known as comoving coordinates, and an observer who stays at constantuⁱ is also called "comoving". Only a comovingobserver will think that the universe looks isotropic; in fact on Earth we arenot quite comoving, and as a result we see a dipole anisotropy inthe cosmic microwave background as a result of the conventionalDoppler effect.

Our interest is therefore in maximally symmetric Euclidean three-metrics. We know that maximally symmetric metrics obey

(8.2)

where k is some constant, and we put a superscript ⁽³⁾ onthe Riemann tensor to remind us that it is associated with thethree-metric , not the metric of the entire spacetime.The Ricci tensor is then

(8.3)

If the space is to be maximally symmetric, then it will certainlybe spherically symmetric. We already know something about sphericallysymmetric spaces from our exploration of the Schwarzschildsolution; the metric can be put in the form

(8.4)

The components of the Ricci tensor for such a metric can be obtainedfrom (7.16), the Ricci tensor for a spherically symmetric spacetime,by setting = 0 and = 0, which gives

(8.5)

We set these proportional to the metric using (8.3), and can solvefor (r):

(8.6)

This gives us the following metric on spacetime:

(8.7)

This is the Robertson-Walker metric. We have not yetmade use of Einstein's equations; those will determine the behaviorof the scale factor a(t).

Note that the substitutions

(8.8)

leave (8.7) invariant. Therefore the only relevant parameteris k/| k|, and there are three cases of interest: k= - 1, k = 0, and k = + 1. The k = - 1 case correspondsto constant negative curvature on , and is called open; thek = 0 case corresponds to no curvature on , and is called flat; the k = + 1 case corresponds to positive curvatureon , and is called closed.

Let us examine each of these possibilities. For the flat casek = 0 the metric on is

(8.9)

which is simply flat Euclidean space. Globally, it could describe or a more complicated manifold, such as the three-torus S¹ × S¹ × S¹. For the closed case k = + 1 we can define r = sin to write the metric on as

(8.10)

which is the metric of a three-sphere. In this case the onlypossible global structure is actually the three-sphere (except forthe non-orientable manifold P³). Finally in the open k = - 1case we can set r = sinh to obtain

(8.11)

This is the metric for a three-dimensional space of constantnegative curvature; it is hard to visualize, but think of thesaddle example we spoke of in Section Three. Globally such a space couldextend forever (which is the origin of the word "open"), but itcould also describe a non-simply-connected compact space (so "open"is really not the most accurate description).

With the metric in hand, we can set about computing the connectioncoefficients and curvature tensor. Setting da/dt,the Christoffel symbols are given by

(8.12)

The nonzero components of the Ricci tensor are

(8.13)

and the Ricci scalar is then

(8.14)

The universe is not empty, so we are not interested in vacuumsolutions to Einstein's equations. We will choose to model thematter and energy in the universe by a perfect fluid. We discussedperfect fluids in Section One, where they were defined as fluidswhich are isotropic in their rest frame. The energy-momentum tensorfor a perfect fluid can be written

(8.15)

where and p are the energy density and pressure (respectively) as measured in the rest frame, and U is the four-velocity ofthe fluid. It is clear that, if a fluid which is isotropic in someframe leads to a metric which is isotropic in some frame, the twoframes will coincide; that is, the fluid will be at rest in comovingcoordinates. The four-velocity is then

(8.16)

and the energy-momentum tensor is

(8.17)

With one index raised this takes the more convenient form

(8.18)

Note that the trace is given by

(8.19)

Before plugging in to Einstein's equations, it is educational toconsider the zero component of the conservation of energy equation:

(8.20)

To make progress it is necessary to choose an equation ofstate, a relationship between and p. Essentially all ofthe perfect fluids relevant to cosmology obey the simple equationof state

(8.21)

where w is a constant independent of time. The conservationof energy equation becomes

(8.22)

which can be integrated to obtain

(8.23)

The two most popular examples of cosmological fluids areknown as dust and radiation. Dust is collisionless,nonrelativistic matter, which obeys w = 0. Examples include ordinary stars and galaxies, for which the pressure is negligible in comparison with the energy density. Dust is alsoknown as "matter", and universes whose energy density is mostlydue to dust areknown as matter-dominated. The energy density in matterfalls off as

(8.24)

This is simply interpretedas the decrease in the number density of particles as the universeexpands. (For dust the energy density is dominated by the restenergy, which is proportional to the number density.) "Radiation"may be used to describe either actual electromagnetic radiation, ormassive particles moving at relative velocities sufficiently close to the speed of light that they become indistinguishable from photons (atleast as far as their equation of state is concerned).Although radiation is a perfect fluid and thus has an energy-momentumtensor given by (8.15), we also know that T can be expressed interms of the field strength as

(8.25)

The trace of this is given by

(8.26)

But this must also equal (8.19), so the equation of state is

(8.27)

A universe in which most of the energy density is in the form ofradiation is known as radiation-dominated. The energydensity in radiation falls off as

(8.28)

Thus, the energy density in radiation falls off slightly fasterthan that in matter; this is because the number density of photonsdecreases in the same way as the number density of nonrelativisticparticles, but individual photons also lose energy as a^-1as they redshift, as we will see later. (Likewise, massive butrelativistic particles will lose energy as they "slow down" incomoving coordinates.) We believe that today theenergy density of the universe is dominated by matter, with/ 10⁶. However, in the past the universe was much smaller, and the energy density in radiationwould have dominated at very early times.

There is one other form of energy-momentum that is sometimes considered, namely that of the vacuum itself. Introducing energyinto the vacuum is equivalent to introducing a cosmological constant.Einstein's equations with a cosmological constant are

(8.29)

which is clearly the same form as the equations with no cosmologicalconstant but an energy-momentum tensor for the vacuum,

(8.30)

This has the form of a perfect fluid with

(8.31)

We therefore have w = - 1, and the energy density is independent of a, which is whatwe would expect for the energy density of the vacuum. Since theenergy density in matter and radiation decreases as the universeexpands, if there is a nonzero vacuum energy it tends to win out overthe long term (as long as the universe doesn't start contracting).If this happens, we say that the universe becomes vacuum-dominated.

We now turn to Einstein's equations. Recall that they can bewritten in the form (4.45):

(8.32)

The = 00 equation is

(8.33)

and the = ij equations give

(8.34)

(There is only one distinct equation from = ij, due toisotropy.) We can use (8.33) to eliminate second derivatives in(8.34), and do a little cleaning up to obtain

(8.35)

and

(8.36)

Together these are known as the Friedmann equations,and metrics of the form (8.7) which obey these equations defineFriedmann-Robertson-Walker (FRW) universes.

There is a bunch of terminology which is associated with thecosmological parameters, and we will just introduce the basicshere. The rate of expansion is characterized by the Hubbleparameter,

(8.37)

The value of the Hubble parameter at the present epoch is theHubble constant, H₀. There is currently a great deal of controversy about what its actual value is, with measurementsfalling in the range of 40 to 90 km/sec/Mpc. ("Mpc" stands for"megaparsec", which is 3 × 10²⁴ cm.) Note that wehave to divide by a to get a measurable quantity, sincethe overall scale of a is irrelevant. There is also thedeceleration parameter,

(8.38)

which measures the rate of change of the rate of expansion.

Another useful quantity is the density parameter,

(8.39)

where the critical density is defined by

(8.40)

This quantity (which will generally change with time) is calledthe "critical" density because the Friedmann equation (8.36)can be written

(8.41)

The sign of k is therefore determined by whether isgreater than, equal to, or less than one. We have

The density parameter, then, tells us which of the three Robertson-Walker geometries describes our universe. Determiningit observationally is an area of intense investigation.

It is possible to solve the Friedmann equations exactly in various simple cases, but it is often more useful to knowthe qualitative behavior of various possibilities. Let us forthe moment set = 0, and consider the behavior of universesfilled with fluids of positive energy ( > 0) and nonnegativepressure (p 0). Then by (8.35) we must have < 0.Since we know from observations of distant galaxies that the universe is expanding ( > 0), this means that the universe is "decelerating." This is whatwe should expect, since the gravitational attraction of the matterin the universe works against the expansion. The fact thatthe universe can only decelerate means that it must have beenexpanding even faster in the past; if we trace the evolution backwards in time, we necessarily reach a singularity at a = 0. Notice that if were exactly zero, a(t)would be a straight line, and the age of the universe would beH₀^-1. Since is actually negative, the universemust be somewhat younger than that.

This singularity at a = 0 is the Big Bang.It represents the creation of the universe from a singular state,not explosion of matter into a pre-existing spacetime. It might behoped that the perfect symmetry of our FRW universes was responsiblefor this singularity, but in fact it's not true; the singularitytheorems predict that any universe with > 0 and p 0 musthave begun at a singularity. Of coursethe energy density becomes arbitrarily high as a 0,and we don't expect classical general relativity to be anaccurate description of nature in this regime; hopefully a consistent theory of quantum gravity will be able to fix things up.

The future evolution is different for different values of k.For the open and flat cases, k 0, (8.36) implies

(8.42)

The right hand side is strictly positive (since we areassuming > 0), so never passes through zero. Sincewe know that today > 0, it must be positive for all time.Thus, the open and flat universes expand forever - they aretemporally as well as spatially open. (Please keepin mind what assumptions go into this - namely, that thereis a nonzero positive energy density. Negative energy densityuniverses do not have to expand forever, even if they are "open".)

How fast do these universes keep expanding? Consider thequantity a³ (which is constant in matter-dominateduniverses). By the conservation of energy equation (8.20) we have

(8.43)

The right hand side is either zero or negative; therefore

(8.44)

This implies in turn that a² must go to zero in anever-expanding universe, where a . Thus (8.42)tells us that

(8.45)

(Remember that this is true for k 0.) Thus, for k = - 1the expansion approaches the limiting value 1,while for k = 0 the universe keeps expanding, but more and more slowly.

For the closed universes (k = + 1), (8.36) becomes

(8.46)

The argument that a² 0 as a still applies; but in that case (8.46) would become negative, whichcan't happen. Therefore the universe does not expand indefinitely;a possesses an upper bound a_max. As a approachesa_max, (8.35) implies

(8.47)

Thus is finite and negative at this point, so a reaches a_max and starts decreasing, whereupon (since < 0)it will inevitably continue to contract to zero - the Big Crunch.Thus, the closed universes (again, under our assumptions ofpositive and nonnegative p) are closed in time as wellas space.

We will now list some of the exact solutions corresponding to only one type of energy density.For dust-only universes (p = 0), it is convenient to definea development angle (t), rather than using t asa parameter directly. The solutions are then, for openuniverses,

(8.48)

for flat universes,

(8.49)

and for closed universes,

(8.50)

where we have defined

(8.51)

For universes filled with nothing but radiation, p = ,we have once again open universes,

(8.52)

flat universes,

(8.53)

and closed universes,

(8.54)

where this time we defined

(8.55)

You can check for yourselves that these exact solutions have the properties we argued would hold in general.

For universes which are empty save for the cosmological constant,either or p will be negative, in violation of the assumptions we used earlier to derive the general behavior ofa(t). In this case the connection between open/closed and expands forever/recollapses is lost. We begin by considering < 0. In this case is negative, and from (8.41) this can only happen if k = - 1.The solution in this case is

(8.56)

There is also an open (k = - 1) solution for > 0, given by

(8.57)

A flat vacuum-dominated universe must have > 0, and thesolution is

(8.58)

while the closed universe must also have > 0, and satisfies

(8.59)

These solutions are a little misleading. In fact the threesolutions for > 0 - (8.57), (8.58), and (8.59) -all represent the same spacetime, just in different coordinates.This spacetime, known as de Sitter space, is actuallymaximally symmetric as a spacetime. (See Hawking and Ellis fordetails.) The < 0 solution (8.56) isalso maximally symmetric, and is known as anti-de Sitter space.

It is clear that we would like to observationally determine a number of quantities to decide which of the FRW models corresponds to our universe. Obviously we would like to determineH₀, since that is related to the age of the universe.(For a purely matter-dominated, k = 0 universe, (8.49) implies that theage is 2 / (3H₀). Other possibilities would predictsimilar relations.) We would also like to know , which determinesk through (8.41). Given the definition (8.39) of ,this means we want to know both H₀ and . Unfortunatelyboth quantities are hard to measure accurately, especially .But notice that the deceleration parameter q can be relatedto using (8.35):

(8.60)

Therefore, if we think we know what w is (i.e., what kindof stuff the universe is made of), we can determine bymeasuring q. (Unfortunately we are not completely confident thatwe know w, and q is itself hard to measure. But people aretrying.)

To understand how these quantities might conceivably be measured,let's consider geodesic motion in an FRW universe. There are anumber of spacelike Killing vectors, but no timelike Killing vectorto give us a notion of conserved energy. There is, however, aKilling tensor. If U = (1, 0, 0, 0) is the four-velocity ofcomoving observers, then the tensor

(8.61)

satisfies K₎ = 0 (as you can check), and istherefore a Killing tensor. This means that if a particle hasfour-velocity V = dx/d, the quantity

(8.62)

will be a constant along geodesics. Let's think about this, firstfor massive particles. Then we will have VV = - 1, or

(8.63)

where ||² = g_ijVⁱV^j. So (8.61) implies

(8.64)

The particle therefore "slows down" with respect to the comoving coordinates as the universe expands. In fact this is anactual slowing down, in the sense that a gas of particles withinitially high relative velocities will cool down as the universeexpands.

A similar thing happens to null geodesics. In this case VV = 0, and (8.62) implies

(8.65)

But the frequency of the photon as measured by a comovingobserver is = - UV. The frequency of the photonemitted with frequency will therefore be observed witha lower frequency as the universe expands:

(8.66)

Cosmologists like to speak of this in terms of the redshiftz between the two events, defined by the fractional change inwavelength:

(8.67)

Notice that this redshift is not the same as the conventionalDoppler effect; it is the expansion of space, not the relativevelocities of the observer and emitter, which leads to theredshift.

The redshift is something we can measure; we know the rest-framewavelengths of various spectral lines in the radiation fromdistant galaxies, so we can tell how much their wavelengths havechanged along the path from time t₁ when they wereemitted to time t₀ when they were observed. We therefore know theratio of the scale factors at these two times. But we don't knowthe times themselves; the photons are not clever enough to tellus how much coordinate time has elapsed on their journey. We haveto work harder to extract this information.

Roughly speaking, since a photon moves at the speed of light itstravel time should simply be its distance. But what is the"distance" of a far away galaxy in an expanding universe?The comoving distance is not especially useful, since it is notmeasurable, and furthermore because the galaxies need not becomoving in general. Instead we can define the luminositydistance as

(8.68)

where L is the absolute luminosity of the source and F is the flux measured by the observer (the energy per unit time perunit area of some detector). The definition comes from the fact that in flat space, for a source at distance d the fluxover the luminosity is just one over the area of a sphere centeredaround the source, F/L = 1/A(d )= 1/4d². In an FRW universe,however, the flux will be diluted. Conservation of photonstells us that the total number of photons emitted bythe source will eventually pass through a sphere at comovingdistance r from the emitter. Such a sphere is at a physicaldistance d = a₀r, wherea₀ is the scale factor when the photons are observed. But the flux is diluted by two additional effects:the individual photons redshift by a factor (1 + z), and the photonsh*t the sphere less frequently, since two photons emitted a timet apart will be measured at a time (1 + z)t apart.Therefore we will have

(8.69)

or

(8.70)

The luminosity distance d_L is something we might hope to measure, since there are some astrophysical sources whoseabsolute luminosities are known ("standard candles"). But ris not observable, so we have to remove that from our equation.On a null geodesic (chosen to be radial for convenience) we have

(8.71)

or

(8.72)

For galaxies not too far away, we can expand the scale factor ina Taylor series about its present value:

(8.73)

We can then expand both sides of (8.72) to find

(8.74)

Now remembering (8.67), the expansion (8.73) is the same as

(8.75)

For small H₀(t₁ - t₀) thiscan be inverted to yield

(8.76)

Substituting this back again into (8.74) gives

(8.77)

Finally, using this in (8.70) yields Hubble's Law:

(8.78)

Therefore, measurement of the luminosity distances and redshiftsof a sufficient number of galaxies allows us to determineH₀ and q₀, and therefore takes us a long way to decidingwhat kind of FRW universe we live in. The observations themselves areextremely difficult, and the values of these parameters in thereal world are still hotly contested. Over the next decade or soa variety of new strategies and more precise application ofold strategies could very well answer these questions once and for all.