2 Cosmology with a Cosmological Constant

2.1 Cosmological parameters

From the Friedmann equation (5View Equation) (where henceforth we take the effects of a cosmological constant into account by including the vacuum energy density ρΛ into the total density ρ), for any value of the Hubble parameter H there is a critical value of the energy density such that the spatial geometry is flat (k = 0):

3H2 ρcrit ≡ ----. (21 ) 8πG
It is often most convenient to measure the total energy density in terms of the critical density, by introducing the density parameter
( ) -ρ-- 8πG-- Ω ≡ ρcrit = 3H2 ρ. (22 )
One useful feature of this parameterization is a direct connection between the value of Ω and the spatial geometry:
k = sgn (Ω − 1). (23 )
[Keep in mind that some references still use “Ω” to refer strictly to the density parameter in matter, even in the presence of a cosmological constant; with this definition (23View Equation) no longer holds.]

In general, the energy density ρ will include contributions from various distinct components. From the point of view of cosmology, the relevant feature of each component is how its energy density evolves as the universe expands. Fortunately, it is often (although not always) the case that individual components i have very simple equations of state of the form

pi = wiρi, (24 )
with wi a constant. Plugging this equation of state into the energy-momentum conservation equation μν ∇ μT = 0, we find that the energy density has a power-law dependence on the scale factor,
ρ ∝ a−ni, (25 ) i
where the exponent is related to the equation of state parameter by
ni = 3(1 + wi). (26 )
The density parameter in each component is defined in the obvious way,
ρi ( 8πG ) Ωi ≡ ---- = ---2- ρi, (27 ) ρcrit 3H
which has the useful property that
Ωi-∝ a−(ni−nj). (28 ) Ωj

The simplest example of a component of this form is a set of massive particles with negligible relative velocities, known in cosmology as “dust” or simply “matter”. The energy density of such particles is given by their number density times their rest mass; as the universe expands, the number density is inversely proportional to the volume while the rest masses are constant, yielding ρM ∝ a−3. For relativistic particles, known in cosmology as “radiation” (although any relativistic species counts, not only photons or even strictly massless particles), the energy density is the number density times the particle energy, and the latter is proportional to − 1 a (redshifting as the universe expands); the radiation energy density therefore scales as ρR ∝ a−4. Vacuum energy does not change as the universe expands, so ρ Λ ∝ a0; from (26View Equation) this implies a negative pressure, or positive tension, when the vacuum energy is positive. Finally, for some purposes it is useful to pretend that the −2 −2 − ka R 0 term in (5View Equation) represents an effective “energy density in curvature”, and define 2 −2 ρk ≡ − (3k∕8πGR 0)a. We can define a corresponding density parameter

Ωk = 1 − Ω; (29 )
this relation is simply (5View Equation) divided by 2 H. Note that the contribution from Ωk is (for obvious reasons) not included in the definition of Ω. The usefulness of Ωk is that it contributes to the expansion rate analogously to the honest density parameters Ωi; we can write
( )1∕2 ∑ H (a) = H0 ( Ωi0a−ni) , (30 ) i(k)
where the notation ∑ i(k) reflects the fact that the sum includes Ωk in addition to the various components of Ω = ∑i Ωi. The most popular equations of state for cosmological energy sources can thus be summarized as follows:
| | --------------wi--ni- matter | 0 3 | | radiation |1∕3 4 (31 ) “curvature” − 1∕3 2 vacuum |− 1 0 |

The ranges of values of the Ωi’s which are allowed in principle (as opposed to constrained by observation) will depend on a complete theory of the matter fields, but lacking that we may still invoke energy conditions to get a handle on what constitutes sensible values. The most appropriate condition is the dominant energy condition (DEC), which states that μν Tμνl l ≥ 0, and μ μ T νl is non-spacelike, for any null vector lμ; this implies that energy does not flow faster than the speed of light [117]. For a perfect-fluid energy-momentum tensor of the form (4View Equation), these two requirements imply that ρ + p ≥ 0 and |ρ| ≥ |p|, respectively. Thus, either the density is positive and greater in magnitude than the pressure, or the density is negative and equal in magnitude to a compensating positive pressure; in terms of the equation-of-state parameter w, we have either positive ρ and |w | ≤ 1 or negative ρ and w = − 1. That is, a negative energy density is allowed only if it is in the form of vacuum energy. (We have actually modified the conventional DEC somewhat, by using only null vectors lμ rather than null or timelike vectors; the traditional condition would rule out a negative cosmological constant, which there is no physical reason to do.)

There are good reasons to believe that the energy density in radiation today is much less than that in matter. Photons, which are readily detectable, contribute Ω γ ∼ 5 × 10−5, mostly in the 2.73 K cosmic microwave background [21187225]. If neutrinos are sufficiently low mass as to be relativistic today, conventional scenarios predict that they contribute approximately the same amount [149Jump To The Next Citation Point]. In the absence of sources which are even more exotic, it is therefore useful to parameterize the universe today by the values of ΩM and Ω Λ, with Ωk = 1 − ΩM − ΩΛ, keeping the possibility of surprises always in mind.

One way to characterize a specific Friedmann–Robertson–Walker model is by the values of the Hubble parameter and the various energy densities ρi. (Of course, reconstructing the history of such a universe also requires an understanding of the microphysical processes which can exchange energy between the different states.) It may be difficult, however, to directly measure the different contributions to ρ, and it is therefore useful to consider extracting these quantities from the behavior of the scale factor as a function of time. A traditional measure of the evolution of the expansion rate is the deceleration parameter

¨aa q ≡ − -2- ˙a = ∑ ni-−-2Ω (32 ) i 2 i 1 = -ΩM − Ω Λ, 2
where in the last line we have assumed that the universe is dominated by matter and the cosmological constant. Under the assumption that Ω Λ = 0, measuring q0 provides a direct measurement of the current density parameter ΩM0; however, once Ω Λ is admitted as a possibility there is no single parameter which characterizes various universes, and for most purposes it is more convenient to simply quote experimental results directly in terms of ΩM and ΩΛ. [Even this parameterization, of course, bears a certain theoretical bias which may not be justified; ultimately, the only unbiased method is to directly quote limits on a(t).]

Notice that positive-energy-density sources with n > 2 cause the universe to decelerate while n < 2 leads to acceleration; the more rapidly energy density redshifts away, the greater the tendency towards universal deceleration. An empty universe (Ω = 0, Ωk = 1) expands linearly with time; sometimes called the “Milne universe”, such a spacetime is really flat Minkowski space in an unusual time-slicing.

2.2 Model universes and their fates

In the remainder of this section we will explore the behavior of universes dominated by matter and vacuum energy, Ω = ΩM + Ω Λ = 1 − Ωk. According to (33View Equation), a positive cosmological constant accelerates the universal expansion, while a negative cosmological constant and/or ordinary matter tend to decelerate it. The relative contributions of these components change with time; according to (28View Equation) we have

Ω Λ ∝ a2Ωk ∝ a3ΩM. (33 )
For Ω Λ < 0, the universe will always recollapse to a Big Crunch, either because there is a sufficiently high matter density or due to the eventual domination of the negative cosmological constant. For Ω Λ > 0 the universe will expand forever unless there is sufficient matter to cause recollapse before Ω Λ becomes dynamically important. For Ω = 0 Λ we have the familiar situation in which ΩM ≤ 1 universes expand forever and ΩM > 1 universes recollapse; notice, however, that in the presence of a cosmological constant there is no necessary relationship between spatial curvature and the fate of the universe. (Furthermore, we cannot reliably determine that the universe will expand forever by any set of measurements of Ω Λ and ΩM; even if we seem to live in a parameter space that predicts eternal expansion, there is always the possibility of a future phase transition which could change the equation of state of one or more of the components.)

Given ΩM, the value of Ω Λ for which the universe will expand forever is given by

(| 0 for 0 ≤ Ω ≤ 1, { [1 ( 1 − Ω ) 4π] M Ω Λ ≥ |( 4ΩM cos3 --cos−1 ------M- + --- for ΩM > 1. (34 ) 3 ΩM 3
Conversely, if the cosmological constant is sufficiently large compared to the matter density, the universe has always been accelerating, and rather than a Big Bang its early history consisted of a period of gradually slowing contraction to a minimum radius before beginning its current expansion. The criterion for there to have been no singularity in the past is
3[1- − 1(1-−-ΩM--)] Ω Λ ≥ 4ΩMcoss 3coss Ω , (35 ) M
where “coss” represents cosh when ΩM < 1 ∕2, and cos when ΩM > 1∕2.

The dynamics of universes with Ω = ΩM + ΩΛ are summarized in Figure 1View Image, in which the arrows indicate the evolution of these parameters in an expanding universe. (In a contracting universe they would be reversed.) This is not a true phase-space plot, despite the superficial similarities. One important difference is that a universe passing through one point can pass through the same point again but moving backwards along its trajectory, by first going to infinity and then turning around (recollapse).

View Image

Figure 1: Dynamics for Ω = ΩM + ΩΛ. The arrows indicate the direction of evolution of the parameters in an expanding universe.

Figure 1View Image includes three fixed points, at (ΩM, ΩΛ ) equal to (0,0), (0,1), and (1,0). The attractor among these at (0,1 ) is known as de Sitter space – a universe with no matter density, dominated by a cosmological constant, and with scale factor growing exponentially with time. The fact that this point is an attractor on the diagram is another way of understanding the cosmological constant problem. A universe with initial conditions located at a generic point on the diagram will, after several expansion times, flow to de Sitter space if it began above the recollapse line, and flow to infinity and back to recollapse if it began below that line. Since our universe has expanded by many orders of magnitude since early times, it must have begun at a non-generic point in order not to have evolved either to de Sitter space or to a Big Crunch. The only other two fixed points on the diagram are the saddle point at (ΩM, Ω Λ) = (0,0), corresponding to an empty universe, and the repulsive fixed point at (ΩM, Ω Λ) = (1,0), known as the Einstein–de Sitter solution. Since our universe is not empty, the favored solution from this combination of theoretical and empirical arguments is the Einstein–de Sitter universe. The inflationary scenario [113Jump To The Next Citation Point159Jump To The Next Citation Point6Jump To The Next Citation Point] provides a mechanism whereby the universe can be driven to the line Ω + Ω = 1 M Λ (spatial flatness), so Einstein–de Sitter is a natural expectation if we imagine that some unknown mechanism sets Λ = 0. As discussed below, the observationally favored universe is located on this line but away from the fixed points, near (ΩM, Ω Λ) = (0.3,0.7). It is fair to conclude that naturalness arguments have a somewhat spotty track record at predicting cosmological parameters.

2.3 Surveying the universe

The lookback time from the present day to an object at redshift z∗ is given by

∫ t0 t0 − t∗ = t∗ dt ∫ 1 (36 ) = --da--, 1∕(1+z∗)aH (a)
with H (a) given by (30View Equation). The age of the universe is obtained by taking the z∗ → ∞ (t∗ → 0) limit. For Ω = ΩM = 1, this yields the familiar answer t0 = (2∕3)H −01; the age decreases as ΩM is increased, and increases as Ω Λ is increased. Figure 2View Image shows the expansion history of the universe for different values of these parameters and H0 fixed; it is clear how the acceleration caused by ΩΛ leads to an older universe. There are analytic approximation formulas which estimate (36View Equation) in various regimes [264Jump To The Next Citation Point149Jump To The Next Citation Point48Jump To The Next Citation Point], but generally the integral is straightforward to perform numerically.
View Image

Figure 2: Expansion histories for different values of ΩM and Ω Λ. From top to bottom, the curves describe (ΩM, Ω Λ) = (0.3,0.7), (0.3,0.0), (1.0, 0.0), and (4.0,0.0).

In a generic curved spacetime, there is no preferred notion of the distance between two objects. Robertson–Walker spacetimes have preferred foliations, so it is possible to define sensible notions of the distance between comoving objects – those whose worldlines are normal to the preferred slices. Placing ourselves at r = 0 in the coordinates defined by (2View Equation), the coordinate distance r to another comoving object is independent of time. It can be converted to a physical distance at any specified time t∗ by multiplying by the scale factor R0a (t∗), yielding a number which will of course change as the universe expands. However, intervals along spacelike slices are not accessible to observation, so it is typically more convenient to use distance measures which can be extracted from observable quantities. These include the luminosity distance,

∘ ----- L dL ≡ ----, (37 ) 4πF
where L is the intrinsic luminosity and F the measured flux; the proper-motion distance,
dM ≡ u, (38 ) ˙𝜃
where u is the transverse proper velocity and 𝜃˙ the observed angular velocity; and the angular-diameter distance,
d ≡ D-, (39 ) A 𝜃
where D is the proper size of the object and 𝜃 its apparent angular size. All of these definitions reduce to the usual notion of distance in a Euclidean space. In a Robertson–Walker universe, the proper-motion distance turns out to equal the physical distance along a spacelike slice at t = t0:
dM = R0r. (40 )
The three measures are related by
2 dL = (1 + z)dM = (1 + z) dA, (41 )
so any one can be converted to any other for sources of known redshift.

The proper-motion distance between sources at redshift z1 and z2 can be computed by using 2 ds = 0 along a light ray, where 2 ds is given by (2View Equation). We have

dM (z1,z2) = R0 (r2 − r1) [∫ t2 ] = R0 sinn --dt--- t1 R0a (t) (42 ) [ ∘ -----∫ 1∕(1+z2) ] = --∘-1----sinn H0 |Ωk0 | --da----, H0 |Ωk0| 1∕(1+z1) a2H (a)
where we have used (5View Equation) to solve for ∘ ----- R0 = 1∕(H0 |Ωk0 |), H (a) is again given by (30View Equation), and “sinn(x)” denotes sinh(x) when Ωk0 > 0, sin(x) when Ωk0 < 0, and x when Ωk0 = 0. An analytic approximation formula can be found in [193]. Note that, for large redshifts, the dependence of the various distance measures on z is not necessarily monotonic.

The comoving volume element in a Robertson–Walker universe is given by

3 2 √--R0r---- dV = 1 − kr2 drdΩ, (43 )
which can be integrated analytically to obtain the volume out to a distance dM:
⌊ ⌋ 1 ∘ ------------- 1 ∘ ----- V (dM) = ---3----⌈H0dM 1 + H20Ωk0d2M − ∘------sinn− 1(H0 |Ωk0|dM )⌉ , (44 ) 2H0 Ωk0 |Ωk0|
where “sinn” is defined as before (42View Equation).

2.4 Structure formation

The introduction of a cosmological constant changes the relationship between the matter density and expansion rate from what it would be in a matter-dominated universe, which in turn influences the growth of large-scale structure. The effect is similar to that of a nonzero spatial curvature, and complicated by hydrodynamic and nonlinear effects on small scales, but is potentially detectable through sufficiently careful observations.

The analysis of the evolution of structure is greatly abetted by the fact that perturbations start out very small (temperature anisotropies in the microwave background imply that the density perturbations were of order 10–5 at recombination), and linearized theory is effective. In this regime, the fate of the fluctuations is in the hands of two competing effects: the tendency of self-gravity to make overdense regions collapse, and the tendency of test particles in the background expansion to move apart. Essentially, the effect of vacuum energy is to contribute to expansion but not to the self-gravity of overdensities, thereby acting to suppress the growth of perturbations [149189].

For sub-Hubble-radius perturbations in a cold dark matter component, a Newtonian analysis suffices. (We may of course be interested in super-Hubble-radius modes, or the evolution of interacting or relativistic particles, but the simple Newtonian case serves to illustrate the relevant physical effect.) If the energy density in dynamical matter is dominated by CDM, the linearized Newtonian evolution equation is

¨δ + 2 ˙aδ˙ = 4πG ρ δ . (45 ) M a M M M
The second term represents an effective frictional force due to the expansion of the universe, characterized by a timescale −1 −1 (˙a∕a) = H, while the right hand side is a forcing term with characteristic timescale (4πG ρM )− 1∕2 ≈ Ω −1∕2H −1 M. Thus, when ΩM ≈ 1, these effects are in balance and CDM perturbations gradually grow; when Ω M dips appreciably below unity (as when curvature or vacuum energy begin to dominate), the friction term becomes more important and perturbation growth effectively ends. In fact (45View Equation) can be directly solved [119] to yield
∫ a δ (a) = 5-H2 Ω ˙a- [a′H (a′)]−3 da′, (46 ) M 2 0 M0 a 0
where H (a) is given by (30View Equation). There exist analytic approximations to this formula [48Jump To The Next Citation Point], as well as analytic expressions for flat universes [81]. Note that this analysis is consistent only in the linear regime; once perturbations on a given scale become of order unity, they break away from the Hubble flow and begin to evolve as isolated systems.

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