The Emergent Universe:
Inflationary cosmology with no singularity
Abstract
Observations indicate that the universe is effectively flat, but they do not rule out a closed universe. The role of positive curvature is negligible at late times, but can be crucial in the early universe. In particular, positive curvature allows for cosmologies that originate as Einstein static universes, and then inflate and later reheat to a hot big bang era. These cosmologies have no singularity, no “beginning of time”, and no horizon problem. If the initial radius is chosen to be above the Planck scale, then they also have no quantum gravity era, and are described by classical general relativity throughout their history.
I Introduction
The “standard” inflationary model is based on a flat () FriedmannRobertsonWalker (FRW) geometry, motivated by the fact that inflationary expansion rapidly wipes out any original spatial curvature. However, even though inflation drives the curvature term,
(1) 
towards zero, or equivalently, drives the total density parameter towards 1, this does not imply that . Conditions leading to the open set of values are far less finetuned than those corresponding precisely to (although closed models have other finetuning aspects). But irrespective of any arguments about finetuning, the spatial curvature of the real universe is in principle determined by observations; theory will have to give way to data if the data clearly tell us that . Recent cosmic microwave background (CMB) and other data cmb are not conclusive, but include the possibility that , with
(2) 
Future experiments such as PLANCK will reduce these error bars and give more accurate information about the curvature of the universe. (Note that the true global value of and its observed value in the local Hubble volume may differ due to cosmological perturbations seeded by inflation.) Current data mean that we should take closed universes, and therefore closed inflationary models, seriously. Inflation in a closed universe is sometimes considered to be ruled out by unrealistic finetuning linde . If this were true, then observational confirmation of would rule out inflation. But in fact there is a wide range of closed inflationary models, and it is not too difficult to construct simple and consistent models without excessive finetuning (see also Ref. ld ). It is also true that finetuned initial conditions cannot be ruled out by purely scientific arguments, as we discuss below. For the models that we consider, the value of is set by choosing one parameter, while the magnitude of largescale scalar perturbations fixes another parameter.
If is taken as 1.02, then the power spectra of CMB anisotropies and matter can show testable differences from the standard flat model ld . If is very small, then curvature is negligible as regards structure formation processes in the universe. The smaller that is, the more closely the models approximate the standard flat model. But in the early inflationary universe, positive curvature can play a significant role and lead to novel features that do not arise when , for example allowing a minimum in the scale factor which is otherwise not possible, or putting limits on the number of possible efoldings in the inflationary epoch infplus1 ; uke . Primordial history is very different when than in the case , regardless of how small is, and a variety of possibilities arise, including universes with a minimum radius.
Singularity theorems have been devised that apply in the inflationary context, showing that the universe necessarily had a beginning (according to classical and semiclassical theory) bv1 . In other words, according to these theorems the quantum gravity era cannot be avoided in the past even if inflation takes place. However, the models we present escape this conclusion, because they do not satisfy the geometrical assumptions of these theorems. Specifically, the theorems assume either (a) that the universe has open space sections, implying or , or (b) that the Hubble expansion rate is bounded away from zero in the past, . There are inflationary universes that evade these constraints and hence avoid the conclusions of the theorems (this was also noted in Ref. mv ). It is also possible to find counterexamples that are open, i.e., where assumption (a) is not violated, but this typically involves sophisticated constructions ag .
Here we consider closed models in which and can become zero, so that both assumptions (a) and (b) of the inflationary singularity theorems are violated. The models are simple, obey general relativity, and contain only ordinary matter and (minimally coupled) scalar fields. Previous examples of closed inflationary models infplus1 ; uke ; mv ; bounce are, to our knowledge, either bouncing models or models in which inflation is preceded by deceleration (so that a singularity is not avoided). The bouncing universe collapses from infinite size in the infinite past and then turns around at to expand in an inflationary phase. The canonical model for such a bounce is the de Sitter universe in the frame, with These coordinates cover the whole spacetime, which is geodesically complete sch . However, the bouncing models face serious difficulties as realistic cosmologies. The initial state is hard to motivate (collapsing from infinite size without causal interaction), and it is also difficult to avoid nonlinearities in the collapse that prevent a regular bounce.
Ii The Emergent Universe scenario
We show here that when there are closed inflationary models that do not bounce, but inflate from a static beginning, and then reheat in the usual way. The inflationary universe emerges from a small static state that has within it the seeds for the development of the macroscopic universe, and we call this the “Emergent Universe” scenario. (This can be seen as a modern version and extension of the Eddington universe.) The universe has a finite initial size, with a finite amount of inflation occurring over an infinite time in the past, and with inflation then coming to an end via reheating in the standard way. The redshift and the total number of efolds remain bounded through the expansion of the universe until the present day, because the scalefactor is bounded away from zero in the past. There is no horizon problem, since the initial state is Einstein static. Since they start as Einstein static, they avoid a singularity. The initial static state can be chosen to have a radius above the Planck scale, so that these models can even avoid a quantum gravity regime, whatever the true quantum gravity theory may be.
Because they can undergo a large amount of inflation, these models can be effectively the same as the standard flat models as regards structure formation processes. Therefore they are not vulnerable to future reductions in the observed value of below 0.02 (provided that it remains positive), and they can reproduce the successes of the standard inflationary cosmologies, but from very different primordial foundations.
We do not require exotic physics or matter. The early universe contains a standard scalar field with energy density and pressure , and possibly also ordinary matter with energy density and pressure , where . The cosmological constant is absorbed into the potential . There are no interactions between matter and the scalar field, so that they separately obey the energy conservation and KleinGordon equations,
(3)  
(4) 
The Raychaudhuri field equation
(5) 
has first integral the Friedmann equation,
(6) 
which together imply
(7) 
The Raychaudhuri equation gives the condition for inflation,
(8) 
For a positive minimum in the inflationary scale factor, ,
(9) 
where the time may be infinite. The only way to satisfy Eq. (9 ) with nonnegative energy densities is if . Closed inflationary models admit a minimum scale factor if inflation occurs for long enough, since curvature will eventually win over a slowrolling scalar field as we go back into the past (cf. infplus1 ). The inflationary singularity theorems mentioned above exclude this case, since they either only consider , or explicitly exclude the possibility .
Closed models with a minimum scale factor include both bouncing and everexpanding cases. We do not proceed further with the bouncing models because of their acausal initial state and the difficulty of achieving a stable bounce. Everexpanding models avoid these problems. A simple everinflating model is the closed model containing radiation () and cosmological constant , with scale factor h
(10) 
In the infinite past, , the model is asymptotically Einstein static, . Inflation occurs for an infinite time to the past, but at any finite time , there is a finite number of efolds,
(11) 
where the last equality holds for . The curvature parameter at is strongly suppressed by the de Sitterlike expansion:
(12) 
The exact model Eq. (10) is a simple example of Eddingtontype solutions. There are trajectories of this type in the classical phase space em satisfying Einstein’s equations. They are pastasymptotically Einstein static and everexpanding, and the fact that they exist already shows there are inflationary universes that evade the conditions of the singularity theorems presented in bv1 . However, in these models inflation does not end. Below we discuss more realistic models, based on inflationary potentials, that do exit from inflation. These universes are singularityfree, without particle horizons, and everexpanding (). Even though Emergent Universes admit closed trapped surfaces, these do not lead to a singularity, since and the weak energy condition is violated in the past ects .
The Einstein static universe is characterized by and const. Equations (3)–(7) then imply that
(13)  
(14) 
where and is the primordial vacuum energy. If the scalar field kinetic energy vanishes, i.e. if , then , so that there must be matter to keep the universe static. If the static universe has only a scalar field, i.e. if , then the field must have nonzero (but constant) kinetic energy, so that it rolls at constant speed along the flat potential. (Dynamically, this case is equivalent to a stiff fluid, , plus cosmological constant bemt .)
The radius of the initial static universe can be chosen to be above the Planck scale,
(15) 
by suitable choice of , and (with all of them ). Thus these models can in principle avoid the quantum gravity era.
A simple way to realize the scenario of the Emergent Universe is the following.
Iii A simple Emergent potential
Consider a potential that is asymptotically flat in the infinite past,
(16) 
but drops towards a minimum at a finite value . The scalar field kinetic energy density is asymptotic to the constant Einstein static value,
(17) 
where we used Eqs. (13) and (14) with . Because , no matter is needed to achieve the initial static state. The field rolls from the Einstein static state at and the potential slowly drops from its Einstein static original value. Provided that decreases more rapidly than the potential, we have , so that the universe accelerates, by Eq. (8 ). Since and , while , the KleinGordon equation (4) shows that the universe is expanding ( ).
Inflation ends at time , where . Then reheating takes place as the field oscillates about the minimum at . In the asymptotic past, , the primordial cosmological constant is given by Eq. (17) as
(18) 
so that is large for a small initial radius. At the minimum, defines the cosmological constant that dominates the late universe,
(19) 
A typical example of a potential is shown in Fig. 1, which is based on the potential presented in Ref. emerge :
(20) 
where is a constant energy scale.
The infinite time of inflation, from to (), produces a finite amount of inflation (and a finite redshift to the initial Einstein static state). The potential produces expansion that is initially qualitatively similar to the exact solution in Eq. (10), so that the total number of efolds from the initial expansion can be estimated, following Eq. (11), as
(21) 
Provided that is chosen small enough and large enough, a very large number of efolds can be produced. The parameters in the potential in Eq. (20) can be chosen so that the primordial universe is consistent with current observations emerge .
Iv Finetuning
As with standard inflationary models, finetuning is necessary to produce density perturbations at the level, and to fix so that . These issues are faced by all inflationary universe models b . The specific geometrical finetuning problem in the Emergent models is the requirement of a particular choice of the initial radius , or equivalently a specific unique choice of the primordial cosmological constant, Eq. (18). This choice must then be supplemented by a further finetuning – a choice of initial kinetic energy such that Eq. (17) holds. Both conditions are required to attain an asymptotic Einstein static state, the one ensuring asymptotic validity of the Friedmann equation, and the other, that of the Raychaudhuri equation.
The model can be criticized because of its finetuned initial state. There are two basic responses to this criticism.

The criticism does not rule out these models as valid physical models, but rather claims they are not likely to occur in reality because they are improbable. But the inflationary singularity theorems bv1 are not based on probabilities, and the Emergent models do indeed show that the geometric conditions of those theorems need not be satisfied.

The force of the finetuning criticism is based partly in philosophy not only in physics. There is no scientifically based proof that the unique physical universe has to be probable.
This points to a tension in cosmology between two viable but opposed views on how to explain the current state of the universe.

The one view is that the present state of the universe is highly probable because physical processes make it very likely to have occurred mis .

The opposite view is that Nature prefers symmetry, and the universe is likely to have originated in a highly symmetric, and necessarily finetuned, state einstat .
The key point is that there is no scientific proof that the one or the other of these approaches is the correct approach to use in cosmology. A quantum gravity theory may resolve this and other issues relating to the origin of the universe, but in the absence of such a theory, it seems worthwhile to pursue the implications of both approaches.
The underlying problem within classical and semiclassical cosmology is the uniqueness of the universe, and all the scientific and philosophical difficulties that this entails unique . One cannot straightforwardly apply statistics or probability to a unique object. There may be a scientific basis for the use of probability in the context of an ensemble of universes – a multiverse. But this is a complex and controversial proposal, and there is no evidence that it is correct; indeed it may not be scientifically testable multi . Consequently the choice between these two fundamentally different approaches to cosmological origins remains of necessity a partly philosophical one, while we lack a more complete quantum gravity theory that explains what actually underlies the existence of the real physical universe.
Furthermore, it is not simply a case of finetuning versus nonfinetuning. The standard inflationary models, which avoid a special initial state, are not free of finetuning, in particular, the finetuning entailed in the choice of . There is no mechanism to attain (as opposed to ), and no obvious way to prove observationally that to the infinite accuracy required to establish that in reality . This standard assumption would appear to involve the same kind of finetuning as occurs in the Emergent Universe, because of the exact balance required to set .
The standard inflationary models necessarily have a singularity in the past bv1 . Although it is possible to evade the singularity when ag , this requires complicated modifications of the spacetime. It also appears to invoke a finetuned initial state, not unlike the finetuning involved in the Emergent models. The choice can therefore also be seen as between a singularity and a finetuned nonsingular initial state. There are arguments for each choice. The singular inflationary models have the strength of generality of initial conditions (at some time after the classical singularity, i.e., effectively after the Planck era). They have a disadvantage of starting at a spacetime singularity where all of physics breaks down and spacetime itself comes to an end (though quantum gravity effects may be able to avoid this). The Emergent Universe has the advantage of avoiding a singularity (with or without a Planck era). There are also arguments that a highly symmetric start to the universe is necessary for the thermodynamic arrow of time to function as it does penr .
The Emergent Universe scenario gives a framework for investigating what are the implications if whatever process caused the universe to come into being, preferred the highsymmetry state of the Einstein static universe to any less ordered state. There are various arguments in support of the Einstein static model as a preferred initial state.

It is neutrally stable against inhomogeneous linear perturbations when (as in the simple Emergent model above), or when and the sound speed of matter obeys , as shown in Ref. bemt , extending the initial results of Refs. hein ; gein . It is of course unstable to homogeneous perturbations, which break the balance between curvature and energy density. This instability is crucial for producing an inflationary era.

It has no horizon problem.

It maximizes the entropy within the family of FRW radiation models gein .

It is the unique highest symmetry nonempty FRW model, being invariant under a 7dimensional group of isometries ell67 .
The Einstein static model has a welldefined vacuum, and Casimir effects (see, e.g., Ref. cas ) could play an important role in determining the primordial parameters of the Emergent Universe.
V Emergent models with a finite time of inflation
The realization of the Emergent Universe outlined above and elaborated in Ref. emerge , illustrated qualitatively in Fig. 1, has the advantage of simplicity. But the fact that the initial Einstein static state is only achieved asymptotically in the infinite past could be seen as a disadvantage. It is possible to find other realizations of the scenario in which the universe starts expanding from an Einstein static state at a finite time in the past.
For example, one can consider potentials with a critical value , where , above a stable minimum at , i.e.,
(22) 
The scalar field is initially at the critical position ; since there is no kinetic energy (), matter is necessary () to provide an initial static state. Specifically, by Eqs. (13) and (14), we have
(23) 
If the initial position is an unstable (tachyonic) maximum, i.e., , then the field rolls down to the true minimum, similar to natural inflation potentials. If the initial position is a false vacuum minimum, then the field needs to tunnel towards the true minimum.
Another alternative is an inflationary potential that depends on temperature in much the same way as in the first inflationary universe models. In the initial Einstein static state, with matter at some temperature, the field starts at a minimum, which becomes unstable after a perturbation. This leads to expansion of the universe, and consequently the temperature falls, which results in the appearance of a lower minimum.
Vi Conclusions
We have shown that inflationary cosmologies exist in which the horizon problem is solved before inflation begins, there is no singularity, no exotic physics is involved, and the quantum gravity regime can even be avoided. These Emergent Universe models can be constructed with simple potentials (illustrated schematically in Fig. 1), giving pastinfinite inflation from an asymptotically initial Einstein static state, with a bounded number of efolds and redshift, followed by reheating in the usual way. Explicit and simple forms of the potential can be found that are consistent with present cosmological observations for suitable choice of parameters in the potential emerge . Other realizations exist in which the universe starts inflating from an Einstein static state at a finite time in the past. The Emergent models illustrate the potentially strong primordial effects of positive spatial curvature, leading to a very different early universe than the standard models, while producing a late universe that can be observationally distinguished from the standard case only by highprecision observations.
If one requires, within classical and semiclassical general relativity theory, a nonsingular universe with standard fields and matter, then this is only possible if . If one further rules out bouncing models because of problems in achieving a regular bounce, then one is led to the Emergent Universe scenario, with its finetuned initial Einstein static state. This state is preferred by its appealing stability, entropy and geometric properties, and provides, because of its compactness, an interesting arena for investigating the Casimir and other effects.
The Emergent Universe scenario with provides an interesting alternative to the standard inflationary scenario with , in the case that is close to, but above, 1. This may be seen as a choice between a finetuned nonsingular initial state, and a singularity that precedes generic initial conditions. We do not claim that the Emergent scenario is superior to the standard scenario, but simply that it is worthwhile investigating, since neither it nor the standard scenario are yet ruled out by scientific arguments. In the absence of a quantum gravity theory that can explain the true nature of the origin of the universe, both approaches remain valid to investigate and test. The two approaches lead to observationally viable models, at least until there is firm observational evidence determining the curvature parameter of the universe.
If future observations turn out to provide strong evidence for positive curvature, then the standard models will be ruled out, and the Emergent models and other closed models with a singularity and deceleration preceding inflation ld , would be contenders to describe the universe. Of course if is shown to be negative, then all closed models and the standard models will be ruled out. However, if the Universe is in fact exactly flat, observations will not be able to rule out a very tiny positive curvature (and the existence of perturbations prevents a very accurate measurement). Thus if the standard is model is correct, the Emergent Universe with perturbatively small curvature cannot be falsified by observations, although it does produce a very different early universe.
Acknowledgements
We thank Anthony Aguirre, John Barrow, Bruce Bassett, David Coule, Jeff Murugan, Christos Tsagas, Tanmay Vachaspati and David Wands for helpful comments, and Emily Leeper for assistance with the figure. GE is supported by the NRF (SA) and RM by PPARC (UK). RM thanks the University of Cape Town for hospitality while part of this work was done.
References
 (1) C.L. Bennet, et al., astroph/0302207; D.N. Spergel, et al., astroph/0302209.
 (2) A. Linde, astroph/0303245.
 (3) A. Lasenby and C. Doran, astroph/0307311.
 (4) G.F.R. Ellis, W. Stoeger, P. McEwan, and P.K.S. Dunsby, Gen. Rel. Grav. 34, 1445 (2002).
 (5) J.P. Uzan, U. Kirchner, and G.F.R. Ellis, astroph/0302597.
 (6) A. Borde and A. Vilenkin, Phys. Rev. Lett. 72, 3305 (1994); ibid., Phys. Rev. D56, 717 (1997); A.H. Guth, astroph/0101507; A. Borde, A.H. Guth, and A. Vilenkin, grqc/0110012; A. Vilenkin, grqc/0204061.
 (7) C. MolinaParis and M. Visser, Phys. Lett. B455, 90 (1999).
 (8) A. Aguirre and S. Gratton, Phys. Rev. D65, 083507 (2002); ibid., grqc/0301042.
 (9) A.A. Starobinsky, Sov. Astron. Lett. 4, 82 (1978); J.D. Barrow and R.A. Matzner, Phys. Rev. D21, 336 (1980); S.W. Hawking, Nucl. Phys. B239, 257 (1984); A.V. Toporensky, Grav. Cosmol. 5, 40 (1999); N. Kanekar, V. Sahni, and Y. Shtanov, Phys. Rev. D63, 083520 (2001); C. Gordon and N. Turok, hepth/0206138; R.H. Brandenberger, S.E. Joras, and J. Martin, Phys. Rev. D66, 083514 (2002).
 (10) E. Schrödinger, Expanding Universes (Cambridge University Press, Cambridge, 1956).
 (11) E.R. Harrison, Mon. Not. R. astr. Soc. 137, 69 (1967).
 (12) M.S. Madsen and G.F.R. Ellis, Mon. Not. R. astr. Soc. 234, 67 (1988).
 (13) G.F.R. Ellis, Gen. Rel. Grav. 35, 1307 (2003).
 (14) J.D. Barrow, G.F.R. Ellis, R. Maartens, and C.G Tsagas, Class. Quantum Grav. 20, L155 (2003).
 (15) G.F.R. Ellis, J. Murugan, and C.G. Tsagas, grqc/0307112.
 (16) In a closed universe, there could be dynamical mechanisms that adjust the cosmological constant according to the size of the universe; see, J.D. Bjorken, hepth/0210202.
 (17) This approach can be seen as an extension of comments by Dingle in the 1930’s and the pioneering work of Misner in the 1960’s, e.g., C.W. Misner, Astrophys. J. Lett. 151, 431 (1968); ibid., Phys. Rev. Lett. 22, 1071 (1969).
 (18) This view was central to Einstein’s paper on the Einstein static model in 1917, and was developed into the Cosmological Principle used by McCrea, Bondi, and others to justify the RobertsonWalker metrics; see, e.g., H. Bondi, Cosmology (Cambridge University Press, Cambridge, 1960). A more informal version was used by Einstein and de Sitter to justify universe models with ; see the discussion of this often used “simplicity principle” by, R.H. Dicke and P.J.E. Peebles, in General Relativity: An Einstein Centenary Survey, ed. S. W. Hawking and W. Israel (Cambridge University Press, Cambridge, 1979), p. 504.
 (19) G.F.R. Ellis, Astrophys. Space Sci. 269, 693 (1999).
 (20) G.F.R. Ellis, U. Kirchner, and W. Stoeger, astroph/0305292.
 (21) R. Penrose, The Emperor’s New Mind (Oxford University Press, Oxford 1989), chap. 7.
 (22) E.R. Harrison, Rev. Mod. Phys. 39, 862 (1967).
 (23) G.W. Gibbons, Nucl. Phys. B292, 784 (1987); ibid., 310, 636 (1988).
 (24) G.F.R. Ellis, J. Math. Phys. 8, 1171 (1967).
 (25) I. Brevik, K.A. Milton, and S.D. Odintsov, hepth/0210286; E. Elizalde and A.C. Tort, hepth/0306049.