Nce Again Consider a Spatially Flat Universe Containing Only Matter

EXPANSION AND GEOMETRY The equation of motion for the scale factor tin be obtained in a quasi-Newtonian fashion. Consider a sphere about some arbitrary bespeak, and permit the radius be R (t) r, where r is arbitrary. The move of a indicate at the border of the sphere will, in Newtonian gravity, be influenced merely by the interior mass. We can therefore write down immediately a differential equation Friedmann'southward equation) that expresses conservation of energy: r)² / 2 - GM / (Rr) = constant. In fact, to get this far, nosotros do require general relativity: the gravitation from mass shells at large distances is not Newtonian, so we cannot employ the usual statement about their outcome existence zippo. In fact, the result that the gravitational field inside a uniform shell is null does hold in general relativity, and is known as Birkhoff's theorem (encounter chapter 2). General relativity becomes fifty-fifty more vital in giving us the constant of integration in Friedmann's equation [problem 3.ane]:

Annotation that this equation covers all contributions to , i.e. those from matter, radiations and vacuum; it is independent of the equation of state. A common autograph for relativistic cosmological models, which are described by the Robertson-Walker metric and which obey the Friedmann equation, is to speak of FRW models.

The Friedmann equation shows that a universe that is spatially closed (with k = +1) has negative total ``free energy'': the expansion will eventually be halted by gravity, and the universe will recollapse. Conversely, an unbound model is spatially open (k = -1) and volition expand forever. This is marvelously unproblematic: the dynamics of the entire universe are the same as those of a cannonball fired vertically against the Earth's gravity. But every bit the Globe's gravity defines an escape velocity for projectiles, then a universe that expands sufficiently fast will go along to expand forever. Conversely, for a given rate of expansion there is a disquisitional density that volition bring the expansion asymptotically to a halt:

This connectedness betwixt the charge per unit of expansion of the universe and its global geometry is an astonishing and deep effect. The proof of the equation quoted in a higher place is ``only'' a question of inserting the Robertson-Walker metric into the field equations [problem three.1], but the question inevitably arises of whether there is a quasi-Newtonian manner of seeing that the issue must exist truthful; the answer is ``virtually''. Outset note that whatsoever open up model will evolve towards undecelerated expansion provided its equation of country is such that R ² is a failing role of R - the potential free energy becomes negligible by comparison with the full and tends to a constant. In this mass-gratuitous limit, at that place can be no spatial curvature and the open RW metric must be but a coordinate transformation of Minkowski spacetime. We will showroom transformation later on in this chapter and show that it implies R = ct for this model, proving the g = -ane case.

An alternative line of assault is to rewrite the Friedmann equation in terms of the Hubble parameter:

Now consider holding the local observables H and fixed merely increasing R without limit. Clearly, in the RW metric this corresponds to going to the g = 0 form: the calibration of spatial curvature goes to infinity and the comoving separation for whatsoever given proper separation goes to cypher, so that the comoving geometry becomes indistinguishable from the Euclidean form. This instance also has potential and kinetic energy much greater than total energy, so that the rhs of the Friedmann equation is finer zero. This establishes the k = 0 case, leaving the airtight universe as the only stubborn holdout against Newtonian arguments.

It is sometimes convenient to piece of work with the time derivative of the Friedmann equation, for the same reason that acceleration arguments in dynamics are sometimes more transparent than free energy ones. Differentiating with respect to time requires a knowledge of , but this can be eliminated by ways of conservation of energy: d [ c ² R ³] = -pd [R ³]. We so obtain

Both this equation and the Friedmann equation in fact arise as independent equations from unlike components of Einstein's equations for the RW metric [trouble 3.1].

DENSITY PARAMETERS ETC. The ``flat'' universe with k = 0 arises for a item critical density. Nosotros are therefore led to define a density parameter as the ratio of density to critical density:

Since and H change with fourth dimension, this defines an epoch-dependent density parameter. The current value of the parameter should strictly be denoted by ₀. Because this is such a common symbol, we shall go along the formulae uncluttered by usually dropping the subscript; the density parameter at other epochs volition be denoted by (z). The critical density therefore merely depends on the rate at which the universe is expanding. If nosotros now also define a dimensionless (current) Hubble parameter equally

A powerful gauge model for the free energy content of the universe is to divide it into pressureless matter ( R ^-3), radiation ( R ^-4) and vacuum energy ( constant). The showtime two relations just say that the number density of particles is diluted by the expansion, with photons too having their energy reduced by the redshift; the third relation applies for Einstein's cosmological constant. In terms of observables, this means that the density is written as

(introducing the normalized scale gene a = R / R ₀). For some purposes, this separation is unnecessary, since the Friedmann equation treats all contributions to the density parameter every bit:

Thus, a flat 1000 = 0 universe requires _i = i at all times, any the form of the contributions to the density, even if the equation of state cannot exist decomposed in this uncomplicated way.

which implies q = 3 _m / 2 + two _r -i for a apartment universe. One of the classical problems of cosmology is to test this relation experimentally.

Lastly, it is oftentimes necessary to know the present value of the scale cistron, which may exist read directly from the Friedmann equation:

The Hubble constant thus sets the curvature length, which becomes infinitely big as approaches unity from either direction. Only in the limit of goose egg density does this length get equal to the other common measure of the size of the universe - the Hubble length, c / H ₀.

SOLUTIONS TO THE FRIEDMANN EQUATION The Friedmann equation is so named because Friedmann was the starting time to appreciate, in 1922, that Einstein's equations admitted cosmological solutions containing affair merely (although information technology was Lemaître who in 1927 both obtained the solution and appreciated that it led to a linear distance-redshift relation). The term Friedmann model is therefore often used to indicate a matter-merely cosmology, even though his equation includes contributions from all equations of country.

The Friedmann equation may be solved most simply in ``parametric'' form, by recasting it in terms of the conformal fourth dimension d = c dt / R (cogent derivatives with respect to by primes):

which is straightforward to integrate provided _v = 0. Solving the Friedmann equation for R (t) in this mode is important for determining global quantities such as the present historic period of the universe, and explicit solutions for particular cases are considered below. However, from the betoken of view of observations, and in particular the distance-redshift relation, it is non necessary to keep by the direct route of determining R (t).

To the observer, the development of the calibration factor is virtually straight characterised by the change with redshift of the Hubble parameter and the density parameter; the evolution of H (z) and (z) is given immediately by the Friedmann equation in the grade H ² = 8 Chiliad / 3 - kc ² / R ². Inserting the higher up dependence of on a gives

This is a crucial equation, which tin exist used to obtain the relation between redshift and comoving distance. The radial equation of motility for a photon is R dr = c dt = c dR / = c dR / (RH). With R = R ₀ / (1 + z), this gives

This relation is arguably the single well-nigh of import equation in cosmology, since it shows how to chronicle comoving distance to the observables of redshift, Hubble constant and density parameters. The comoving altitude determines the credible brightness of distant objects, and the comoving volume element determines the numbers of objects that are observed. These attribute of observational cosmology are discussed in more than detail below in section 3.4.

Lastly, using the expression for H (z) with (a) - ane = kc ² / (H ² R ²) gives the redshift dependence of the full density parameter:

This last equation is very important. Information technology tells us that, at high redshift, all model universes apart from those with only vacuum energy will tend to look like the = i model. This is not surprising given the form of the Friedmann equation: provided R ² -> every bit R -> 0, the -kc ² curvature term will become negligible at early on times. If i, and so in the distant past (z) must have differed from unity by a tiny amount: the density and rate of expansion needed to have been finely balanced for the universe to aggrandize to the present. This tuning of the initial atmospheric condition is called the flatness problem and is one of the motivations for the applications of quantum theory to the early universe that are discussed in later chapters.

Thing-DOMINATED UNIVERSE From the observed temperature of the microwave background (2.73 K) and the supposition of three species of neutrino at a slightly lower temperature (see later chapters), nosotros deduce that the total relativistic density parameter is _r h ² 4.ii x 10^-v, then at nowadays it should exist a adept approximation to ignore radiation. Still, the different redshift dependences of matter and radiation densities mean that this assumption fails at early on times: _m / _r (1 + z)^-1. One of the critical epochs in cosmology is therefore the point at which these contributions were equal: the redshift of matter-radiations equality

At redshifts college than this, the universal dynamics were dominated past the relativistic-particle content. By a coincidence discussed beneath, this epoch is shut to some other of import event in cosmological history: recombination. Once the temperature falls below 10⁴ Chiliad, ionized fabric can form neutral hydrogen. Observational astronomy is simply possible from this betoken on, since Thomson scattering from electrons in ionized material prevents photon propagation. In practise, this limits the maximum redshift of observational interest to virtually one thousand (as discussed in detail in chapter nine); unless is very depression or vacuum energy is important, a thing-dominated model is therefore a expert approximation to reality.

By conserving matter, we can introduce a characteristic mass M _*, and from this a characteristic radius R _*:

where we have used the expression for R ₀ in the kickoff step. When just matter is present, the conformal-time version of the Friedmann equation is unproblematic to integrate for R (), and integration of dt = d / R gives t ():

This cycloid solution is a special example of the general solution for the evolution of a spherical mass distribution: R = A [1 - C_chiliad ()], t = B [ - S_k ()], where A ^three = GMB ² and the mass Grand need non be the mass of the universe. In the general example, the variable is known every bit the development angle; it is only equal to the conformal time in the special instance of the solution to the Friedmann equation. We will later use this solution to study the evolution of density inhomogeneities. The evolution of R (t) in this solution is plotted in figure iii.4. A particular point to notation is that the behaviour at early times is always the same: potential and kinetic energies greatly exceed full energy and we always have the k = 0 form R t ^2/3.

At this point, we accept reproduced one of the great conclusions of relativistic cosmology: the universe is of finite historic period, and had its origin in a mathematical singularity at which the scale factor went to cypher, leading to a divergent spacetime curvature. Since zero scale cistron also implies infinite density (and temperature), the inferred moving picture of the early universe is one of unimaginable violence. The term large bang was coined by Fred Hoyle to describe this kickoff, although it was intended somewhat critically. The trouble with the singularity is that it marks the breakdown of the laws of physics; we cannot extrapolate the solution for R (t) to t < 0, and and so the origin of the expansion becomes an unexplained purlieus condition. It was simply after about 1980 that a consistent ready of ideas became available for ways of avoiding this barrier, every bit discussed in chapter eleven.

The parametric solution cannot be rearranged to give R (t), merely information technology is conspicuously possible to solve for t (R). This is most simply expressed in terms of the density parameter and the age of the universe at a given phase of its development:

(three.44)

When we insert the redshift dependences of H (z) and (z),

(3.45)

this gives the states the time-redshift relation. An alternative route to this issue would have been to use the general differential expression for comoving distance dr / dz; since c dt = [R ₀ / (ane + z)] dr, this gives the age of the universe as an integral over z.

An authentic and very useful approximation to the above verbal issue is

(iii.46)

which interpolates betwixt the exact ages of H ^-one for an empty universe and 2/3 H ^-i for a critical-density = 1 model.

Thing PLUS RADIATION Background The parametric solution tin can be extended in an elegant way for a universe containing a mixture of matter and radiation. Suppose we write the mass inside R equally

(3.47)

reflecting the R ^-3 and R ^-4 dependencies of thing and radiations densities respectively. At present define dimensionless masses of the grade y GM / (c ² R ₀), which reduce to y _{g, r} = k _{m, r} / [2( - 1)]. The parametric solutions then become

(3.48)

MODELS WITH VACUUM ENERGY The solution of the Friedmann equation becomes more than complicated if we allow a significant contribution from vacuum energy - i.e. a not-zero cosmological abiding. Detailed discussions of the problem are given by Felten & Isaacman (1986) and Carroll, Press & Turner (1992); the most of import features are outlined below.

The Friedmann equation itself is independent of the equation of land, and just says H ² R ² = kc ^two / ( - 1), whatever the course of the contributions to . In terms of the cosmological constant itself, we have

(3.49)

STATIC UNIVERSE The reason that the cosmological constant was showtime introduced by Einstein was non simply because there was no general reason to expect empty space to exist of zero density, just because it allows a non-expanding cosmology to exist constructed. This is perhaps not so obvious from some forms of the Friedmann equation, since now H = 0 and = ; if nosotros cast the equation in its original form without defining these parameters, then zero expansion implies

(3.50)

Since tin can have either sign, this appears not to constrain one thousand. However, we also want to have nada acceleration for this model, and so need the fourth dimension derivative of the Friedmann equation: = -4 GR ( + 3p) / three. A further condition for a static model is therefore that

(three.51)

Since = -p for vacuum free energy, and this is the only source of pressure if we ignore radiation, this tells us that = iii _vac and hence that the mass density is twice the vacuum density. The total density is hence positive and k = 1; we have a airtight model.

Notice that what this says is that a positive vacuum energy acts in a repulsive way, balancing the attraction of normal matter. This is related to the idea of + threep every bit the effective source density for gravity. This insight alone should make one capeesh that the static model cannot exist stable: if we perturb the scale factor by a minor positive amount, the vacuum repulsion is unchanged whereas the ``normal'' gravitational attraction is reduced, so that the model volition tend to expand further (or contract, if the initial perturbation was negative). Thinking along these lines, a tidy history of science would have required Einstein to predict the expanding universe in advance of its observation. However, it is perhaps not then surprising that this prediction was never clearly made, despite the fact that expanding models were studied by Lemaître and by Friedmann in the years prior to Hubble'south piece of work. In those days, the thought of a quasi-Newtonian approach to cosmology was non developed; the mutual difficulty of obtaining a clear concrete estimation of solutions to Einstein's equations obscured the meaning of the expanding universe even for its creators.

DE SITTER Space Earlier going on to the full general case, it is worth looking at the endpoint of an outwards perturbation of Einstein's static model, offset studied by de Sitter and named after him. This universe is completely dominated past vacuum energy, and is clearly the limit of the unstable expansion, since the density of affair redshifts to cipher while the vacuum energy remains constant. Consider over again the Friedmann equation in its general form ² - 8 K R ² / 3 = -kc ²: since the density is constant and R volition increase without limit, the two terms on the lhs must eventually become nearly exactly equal and the curvature term on the rhs will be negligible. Thus, even if 1000 0, the universe will have a density that differs but infinitesimally from the critical, and so that we can solve the equation by setting k = 0, in which example

(3.52)

An interesting estimation of this behaviour was promoted in the early days of cosmology by Eddington: the cosmological constant is what caused the expansion. In models without , the expansion is just an initial status: anyone who asks why the universe expands at a given epoch is given the unsatisfactory reply that information technology does so considering it was expanding at some earlier time, and this concatenation of reasoning comes upwardly against a barrier at t = 0. It would exist more satisfying to have some mechanism that set up the expansion into motion, and this is what is provided by vacuum repulsion. This trend of models with positive to end up undergoing an exponential phase of expansion (and moreover one with = 1) is exactly what is used in inflationary cosmology to generate the initial conditions for the big bang.

THE STEADY-STATE MODEL The behaviour of de Sitter infinite is in some ways reminiscent of the steady-state universe, which was popular in the 1960s. This theory drew its motivation from the philosophical problems of big-blindside models - which begin in a singularity at t = 0, and for which earlier times have no meaning. Instead, Hoyle, Bondi and Gold suggested the perfect cosmological principle in which the universe is homogeneous not but in space, but too in time: apart from local fluctuations, the universe appears the same to all observers at all times. This tells u.s. that the Hubble constant really is constant, and and so the model necessarily has exponential expansion, R exp(Ht), exactly as for de Sitter infinite. Furthermore, it is necessary that thou = 0, every bit may exist seen by considering the transverse office of the Robertson-Walker metric: d ² = [R (t) S₁₀₀₀ (r) d ]². This has the convention that r is a dimensionless comoving coordinate; if nosotros dissever past R ₀ and change to physical radius r', the metric becomes d ² = [a (t) R ₀ S_k (r' / R ₀) d ]². The electric current scale factor R ₀ now plays the part of a curvature length, determining the distance over which the model is spatially Euclidean. However, any such curvature radius must be abiding in the steady-state model, so the only possibility is that it is space and that grand = 0. We thus see that de Sitter infinite is a steady-state universe: information technology contains a constant vacuum energy density, and has an infinite age, lacking whatever big-bang singularity. In this sense, some aspects of the steady-land model have been resurrected in inflationary cosmology. All the same, de Sitter space is a rather uninteresting model because information technology contains no matter. Introducing matter into a steady-state universe violates energy conservation, since thing does not accept the p = - c ² equation of state that allows the density to remain constant. This is the most radical attribute of steady-land models: they require continuous creation of matter. The energy to accomplish this has to come from somewhere, and Einstein'due south equations are modified past calculation some ``cosmos'' or ``C-field'' term to the free energy-momentum tensor:

(3.53)

The consequence of this extra term must be to cancel the matter density and force per unit area, leaving only the overall effective form of the vacuum tensor, which is required to produce de Sitter infinite and the exponential expansion. This ad hoc field and the lack of whatever physical motivation for it beyond the cosmological trouble information technology was designed to solve was always the most unsatisfactory feature of the steady-state model, and may business relationship for the strong reactions generated by the theory. Certainly, the debate between steady-land supporters and protagonists of the big blindside produced some memorable displays of vitriol in the 1960s. At the start of the decade, the point at effect was whether the proper density of active galaxies was abiding every bit predicted by the steady-country model. Since the radio-source count data were in a somewhat primitive state at that fourth dimension, the fence remained inconclusive until the detection of the microwave background in 1965. For many, this spelled the end of the steady-country universe, but doubts lingered on about whether the radiations might originate in interstellar grit. These were perhaps simply finally laid to rest in 1990, with the demonstration that the radiation was near exactly Planckian in grade (come across affiliate 9).

BOUNCING AND LOITERING MODELS Returning to the general example of models with a mixture of energy in the vacuum and normal components, we accept to distinguish three cases. For models that beginning from a large bang (in which case radiation dominates completely at the primeval times), the universe will either recollapse or expand forever. The latter outcome becomes more than likely for low densities of matter and radiation, but high vacuum density. It is however also possible to have models in which there is no big bang: the universe was collapsing in the afar past, merely was slowed by the repulsion of a positive term and underwent a ``bounce'' to reach its present land of expansion. Working out the weather condition for these different events is a matter of integrating the Friedmann equation. For the addition of , this can only in full general exist done numerically. Withal, we tin can find the conditions for the dissimilar behaviours described above analytically, at least if nosotros simplify things past ignoring radiation. The equation in the form of the time-dependent Hubble parameter looks like

(three.54)

and we are interested in the conditions under which the lhs vanishes, defining a turning signal in the expansion. Setting the rhs to zero yields a cubic equation, and it is possible to requite the weather under which this has a solution (run across Felten & Isaacman 1986), which are as follows.

e'er implies recollapse, which is intuitively reasonable (either the mass causes recollapse before dominates, or the density is low enough that comes to dominate, which cannot atomic number 82 to infinite expansion unless is positive.

(two)	If is positive and one thousand < 1, the model always expands to infinity.
(three)	If m > 1, recollapse is only avoided if v exceeds a critical value (3.55)
(4)	If is big enough, the stationary point of the expansion is at a < 1 and we have a bounce cosmology. This critical value is (3.56) where the function f is similar in spirit to Cgrand : cosh if m < 0.5, otherwise cos. If the universe lies exactly on the disquisitional line, the bounciness is at infinitely early times and we have a solution that is the result of a perturbation of the Einstein static model. Models that about satisfy the critical 5 ( m) relation are known as loitering models, since they spend a long time close to constant scale factor. Such models were briefly pop in the early on 1970s, when in that location seemed to be a sharp tiptop in the quasar redshift distribution at z two. However, this no longer seems a reasonable explanation for what is undoubtedly a mixture of evolution and observational option.

In fact, bounce models can be ruled out quite strongly. The same cubic equations that ascertain the critical conditions for a bounce also give an inequality for the maximum redshift possible (that of the bounciness):

(three.57)

A reasonable lower limit for _k of 0.1 then rules out a bounciness in one case objects are seen at z > two.

The main results of this section are summed upwardly in figure iii.five.. Since the radiation density is very modest today, the principal chore of relativistic cosmology is to work out where on the _thing - _vacuum plane the existent universe lies. The existence of loftier-redshift objects rules out the bounce models, and then that the idea of a hot big blindside cannot be evaded. As subsequent chapters volition show, the data favour a position somewhere near the point (1,0), which is the worst possible situation: it means that the bug of recollapse and closure are very difficult to resolve.

Apartment UNIVERSE The nearly of import model in cosmological research is that with chiliad = 0 -> _total = 1; when dominated by matter, this is oft termed the Einstein-de Sitter model. Paradoxically, this importance arises because information technology is an unstable land: as nosotros accept seen earlier, the universe will evolve abroad from = 1, given a slight perturbation. For the universe to have expanded past so many e-foldings (factors of eastward expansion) and however still take ~ ane implies that it was very close to being spatially flat at early times. Many workers accept conjectured that it would be contrived if this flatness was other than perfect - a prejudice raised to the status of a prediction in about models of inflation.

Although it is a mathematically distinct example, in practice the backdrop of a flat model can unremarkably exist obtained by taking the limit -> 1 for either open or closed universes with one thousand = ± 1. Nevertheless, it is usually easier to start again from the k = 0 Friedmann equation, ⁱⁱ = 8 Chiliad img src="../GIFS/rho2.gif" alt="rho" align=middle> R ⁱⁱ / (3c ²). Since both sides are quadratic in R, this makes it clear that the value of R ₀ is arbitrary, unlike models with one: the comoving geometry is Euclidean, and there is no natural curvature scale.

It at present makes more sense to work throughout in terms of the normalized scale factor a (t), then that the Friedmann equation for a thing-radiation mix is

(3.58)

which may exist integrated to give the time as a function of calibration factor:

(iii.59)

this goes to 2/3 a ^3/2 for a thing-only model, and to 1/two a ² for radiation only.

One further mode of presenting the model's dependence on time is via the density. Following the above, it is like shooting fish in a barrel to show that

(3.60)

The whole universe thus always obeys the rule-of-thumb for the collapse from rest of a gravitating body: the collapse time i / sqrt(G ).

Because _r is and so small, the deviations from a matter-simply model are unimportant for z 1000, and so the altitude-redshift relation for the grand = 0 affair plus radiation model is effectively just that of the _m = 1 Einstein-de Sitter model. An alternative 1000 = 0 model of greater observational interest has a significant cosmological abiding, so that _grand + _v = 1 (radiation being neglected for simplicity). This may seem contrived, just in one case k = 0 has been established, it cannot modify: individual contributions to must adjust to keep in balance. The advantage of this model is that information technology is the merely way of retaining the theoretical attractiveness of m = 0 while changing the age of the universe from the relation H ₀ t ₀ = 2/iii, which characterises the Einstein-de Sitter model. Since much observational evidence indicates that H ₀ t ₀ 1 (meet chapter 5), this model has received a good deal of involvement in recent years. To keep things unproblematic we shall neglect radiation, so that the Friedmann equation is

(3.61)

and the t (a) relation is

(3.62)

The x ^four on the bottom looks similar trouble, but it can be rendered tractable by the substitution y = sqrt(x ⁱⁱⁱ | _{one thousand} - 1| / _m), which turns the integral into

(3.63)

where we include a simple approximation that is authentic to a few % over the region of interest ( _k 0.1). In the general case of significant merely k 0, this expression all the same gives a very expert approximation to the exact result, provided _m is replaced by 0.7 _k - 0.3 _v + 0.3 (Carroll, Press & Turner 1992).

HORIZONS For photons, the radial equation of motion is just c dt = R dr. How far tin can a photon go far a given fourth dimension? The answer is clearly

(3.65)

i.e. just the interval of conformal fourth dimension. What happens as t ₀ -> 0 in this expression? We tin replace dt past dR / , which the Friedmann equation says is dR / sqrt( R ^two) at early times. Thus, this integral converges if R ² -> as t ₀ -> 0, otherwise information technology diverges. Provided the equation of state is such that changes faster than R ^-2, lite signals can only propagate a finite altitude between the large bang and the present; there is so said to be a particle horizon. Such a horizon therefore exists in conventional large bang models, which are dominated past radiation at early times.

A particle horizon is not at all the same thing every bit an event horizon: for the latter, we ask whether r diverges as t -> . If it does, then seeing a given event is but a question of waiting long plenty. Clearly, an upshot horizon requires R (t) to increase more quickly than t, then that distant parts of the universe recede ``faster than light''. This does not occur unless the universe is dominated by vacuum free energy at late times, as discussed higher up. Despite this distinction, cosmologists usually say the horizon when they mean the particle horizon.

There are some unique aspects to the idea of a horizon in a closed universe, where you can in principle return to your starting point by continuing for long enough in the same direction. However, the related possibility of viewing the back of your caput (admitting with some fourth dimension filibuster) turns out to be more than difficult once dynamics are taken into business relationship. For a matter-only model, it is like shooting fish in a barrel to prove that the horizon only merely reaches the stage of allowing a photon to circumnavigate the universe at the point of recollapse - the ``big crisis''. A photon that starts at r = 0 at t = 0 will return to its initial position when r = 2 , at which betoken the conformal fourth dimension = ii also (from above) and the model has recollapsed. Since we alive in an expanding universe, it is not even possible to see the same object in 2 different directions, at radii r and ii - r. This requires a horizon size larger than ; but conformal time = is attained only at maximum expansion, so converse pairs of high-redshift objects are visible only in the collapse phase. These constraints practice not utilize if the universe has a significant cosmological constant; loitering models should permit 1 to see converse pairs at approximately the aforementioned redshift. This effect has been sought, but without success.

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