Connecting The Non-Singular Origin of the Universe, The Vacuum Structure and The Cosmological Constant Problem
We consider a non-singular origin for the Universe starting from an Einstein static Universe, the so called “emergent universe” scenario, in the framework of a theory which uses two volume elements and , where is a metric independent density, used as an additional measure of integration. Also curvature, curvature square terms and for scale invariance a dilaton field are considered in the action. The first order formalism is applied. The integration of the equations of motion associated with the new measure gives rise to the spontaneous symmetry breaking (S.S.B) of scale invariance (S.I.). After S.S.B. of S.I., it is found that a non trivial potential for the dilaton is generated. In the Einstein frame we also add a cosmological term that parametrizes the zero point fluctuations. The resulting effective potential for the dilaton contains two flat regions, for relevant for the non-singular origin of the Universe, followed by an inflationary phase and , describing our present Universe. The dynamics of the scalar field becomes non linear and these non linearities produce a non trivial vacuum structure for the theory and are responsible for the stability of some of the emergent universe solutions, which exists for a parameter range of values of the vacuum energy in , which must be positive but not very big, avoiding the extreme fine tuning required to keep the vacuum energy density of the present universe small. The non trivial vacuum structure is crucial to ensure the smooth transition from the emerging phase, to an inflationary phase and finally to the slowly accelerated universe now. Zero vacuum energy density for the present universe defines the threshold for the creation of the universe.
pacs:98.80.Cq, 04.20.Cv, 95.36.+x
One of the most important and intriging issues of modern physics is the so called “Cosmological Constant Problem” CCP1 (); CCP2 (); CCP3 (); CCP4 (); CCP5 (), (CCP), most easily seen by studying the apparently uncontrolled behaviour of the zero point energies, which would lead to a corresponding equally uncontrolled vacuum energy or cosmological constant term. Even staying at the classical level, the observed very small cosmological term in the present universe is still very puzzling.
Furthermore, the Cosmological Constant Problem has evolved from the “Old Cosmological Constant Problem”, where physicist were concerned with explaining why the observed vacuum energy density of the universe is exactly zero, to different type of CCP since the evidence for the accelerating universe became evident, for reviews see AU1 (); AU2 (). We have therefore since the discovery of the accelerated universe a “New Cosmological Constant Problem” W2 (), the problem is now not to explain zero, but to explain a very small vacuum energy density.
This new situation posed by the discovery of a very small vacuum energy density of the universe means that getting a zero vacuum energy density for the present universe is definitely not the full solution of the problem, although it may be a step towards its solution.
One point of view to the CCP that has been popular has been to provide a bound based on the “anthropic principle” Anthropic (). In this approach, a too large Cosmological Constant will not provide the necessary conditions required for the existence of life, the anthropic principle provides then an upper bound on the cosmological constant.
One problem with this approach is for example that it relies on our knowledge of life as we know it and ignores the possibility that other life forms could be possible, for which other (unknown) bounds would be relevant, therefore the reasoning appears by its very nature subjective, since of course if the observed cosmological constant will be different, our universe will be different and this could include different kind of life that may be could have adjusted itself to a higher cosmological constant of the universe. But even accepting the validity of anthropic considerations, we still do not understand why the observed vacuum energy density must be positive instead of possibly a very small negative quantity. Accepting the anthropic explanation means may be also giving up on discovering important physics related to the CCP and this may be the biggest objection.
Nevertheless, the idea of associating somehow restrictions on the origin of the universe with the cosmological constant problem seems interesting. We will take on this point of view, but leave out the not understood concept of life out from our considerations. Instead, we will require, in a very specific framework, the non-singular origin of the universe. The advantage of this point of view is that it is formulated in terms of ideas of physics alone, without reference to biology, which unlike physics, has not reached the level of an exact science. Another interesting consequence is that we can learn that of a non-singularly created universe may not have a too big cosmological constant, an effect that points to a certain type of gravitational suppression of UV divergences in quantum field theory.
In this respect, one should point out that even in the context of the inflationary scenario Inflation1 (); Inflation2 (); Inflation3 (); Inflation4 () which solves many cosmological problems, one still encounters the initial singularity problem which remains unsolved, showing that the universe necessarily had a singular beginning for generic inflationary cosmologies singularities1 (); singularities2 (); singularities3 (); singularities4 (); singularities5 ().
Here we will adopt the very attractive “Emergent Universe” scenario, where those conclusions concerning singularities can be avoided emerging1 (); emerging2 (); emerging3 (); emerging4 (); emerging5 (); emerging6 (); emerging7 (); emerging8 (); emerging9 (). The way to escape the singularity in these models is to violate the geometrical assumptions of these theorems, which assume i) that the universe has open space sections ii) the Hubble expansion is always greater than zero in the past. In emerging1 (); emerging2 () the open space section condition is violated since closed Robertson Walker universes with are considered and the Hubble expansion can become zero, so that both i) and ii) are avoided.
In emerging1 (); emerging2 () even models based on standard General Relativity, ordinary matter and minimally coupled scalar fields were considered and can provide indeed a non-singular (geodesically complete) inflationary universe, with a past eternal Einstein static Universe that eventually evolves into an inflationary Universe.
Those most simple models suffer however from instabilities, associated with the instability of the Einstein static universe. The instability is possible to cure by going away from GR, considering non perturbative corrections to the Einstein’s field equations in the context of the loop quantum gravity emerging3 (), a brane world cosmology with a time like extra dimension emerging4 (); emerging5 () considering the Starobinski model for radiative corrections (which cannot be derived from an effective action) emerging6 () or exotic matter emerging7 (). In addition to this, the consideration of a Jordan Brans Dicke model also can provide a stable initial state for the emerging universe scenario emerging8 (); emerging9 ().
In this review we study a different theoretical framework where such emerging universe scenario is realized in a natural way, where instabilities are avoided and a succesfull inflationary phase with a graceful exit can be achieved. The model we will use was studied first in SIchile (), however, in the context of this model, a few scenarios are possible. For example in the first paper on this model SIchile () a special choice of state to describe the present state of our universe was made. Then in SICCP () a different candidate for the vacuum that represents our present universe was made. The way in which we best represents the present state of the universe is crucial, since as it should be obvious, the discussion of the CCP depends on what vacuum we take. In SICCP () we expressed the stability and existence conditions for the non-singular universe in terms of the energy of the vacuum of our candidate for the present Universe. In SICCP () a few typos in SIchile () were corrected and also the discussion of some notions discussed was improved in SICCP () and more deeper studies will be done in this review.
Indeed in this review, all those topics will be further clarified, in particular the vacuum structure of this model will be extended. A very important new feature that will be presented in this review is the existence of a “kinetic vacuum”, that produces a vacuum energy state which is degenerate with the vacuum choice made in SICCP (), this degeneracy is analyzed and the dynamical role of this kinetic vacuum in the evolution of the universe and the CCP is analyzed.
We work in the context of a theory built along the lines of the two measures theory (TMT). Basic idea is developed in TMT1a (), TMT1b ()-TMT1r () TMT2 (), TMT3a ()-TMT3d (), TMT4a ()-TMT4e (), TMT5 () and more specifically in the context of the scale invariant realization of such theories TMT2 (), TMT3a ()-TMT3e (), TMT4a ()-TMT4e (), TMT5 (). These theories can provide a new approach to the cosmological constant problem and can be generalized to obtain also a theory with a dynamical spacetime dyn (), furthermore, string and brane theories, as well as brane world scenarios can be constructed using Two Measure Theories ideas G16a ()-G20 (). We should also point out that the Hodge Dual construction of TMT1d () for supergravity constitutes in fact an example of a TMT. The construction by Comelli TMT1e () where no square root of the determinant of the metric is used and instead a total divergence appears is also a very much related approach.
The two measure theories have many points of similarity with “Lagrange Multiplier Gravity (LMG)” Lim2010 (); Capozziello2010 (). In LMG there is a Lagrange multiplier field which enforces the condition that a certain function is zero. For a comparison of one of these lagrange multiplier gravity models with observations see LabranaMultiplier (). In the two measure theory this is equivalent to the constraint which requires some lagrangian to be constant. The two measure model presented here, as opposed to the LMG models of Lim2010 (); Capozziello2010 () provide us with an arbitrary constant of integration. The introduction of constraints can cause Dirac fields to contribute to dark energy FDM () or scalar fields to behave like dust like in Lim2010 () and this dust behaviour can be caused by the stabilization of a tachyonic field due to the constraint, accompanied by a floating dark energy component GSY (); AG (). TMT models naturally avoid the 5th force problem 5th ().
We will consider a slight generalization of the TMT case, where, we consider also the possible effects of zero point energy densities, thus “softly breaking” the basic structure of TMT for this purpose. We will show how the stated goals of a stable emergent universe can be achieved in the framework of the model and also how the stability of the emerging universe imposes interesting constraints on the energy density of the ground state of the theory as defined in this paper: it must be positive but not very large, thus the vacuum energy and therefore the term that softly breaks the TMT structure appears to be naturally controlled. An important ingredient of the model considered here is its softly broken conformal invariance, meaning that we allow conformal breaking terms only though potentials of the dilaton, which nevertheless preserve global scale invariance. In another models for emergent universe we have studied SICh2 (), that rule of softly broken conformal invariance was taken into account. It is also a perfectly consistent, but different approach.
The review will be organized as follows: First we review the principles of the TMT and in particular the model studied in TMT2 (), which has global scale invariance and how this can be the basis for the emerging universe. Such model gives rise, in the effective Einstein frame, to an effective potential for a dilaton field (needed to implement an interesting model with global scale invariance) which has a flat region. Following this, we look at the generalization of this model TMT5 () by adding a curvature square or simply “ term” and show that the resulting model contains now two flat regions. The existence of two flat regions for the potential is shown to be consequence of the s.s.b. of the scale symmetry. We then consider the incorporation in the model of the zero point fluctuations, parametrized by a cosmological constant in the Einstein frame. In this resulting model, there are two possible types of emerging universe solutions, for one of those, the initial Einstein Universe can be stabilized due to the non linearities of the model, provided the vacuum energy density of the ground state is positive but not very large. This is a very satisfactory results, since it means that the stability of the emerging universe prevents the vacuum energy in the present universe from being very large!. The transition from the emergent universe to the ground state goes through an intermediate inflationary phase, therefore reproducing the basic standard cosmological model as well. We end with a discussion section and present the point of view that the creation of the universe can be considered as a “threshold event” for zero present vacuum energy density, which naturally gives a positive but small vacuum energy density.
Ii Introducing a new measure
The general structure of general coordinate invariant theories is taken usually as
where . The introduction of is required since by itself is not a scalar but the product is a scalar. Inserting , which has the transformation properties of a density, produces a scalar action , as defined by Eq.(1), provided is a scalar.
In principle nothing prevents us from considering other densities instead of . One construction of such alternative “measure of integration”, is obtained as follows: given 4-scalars (a = 1,2,3,4), one can construct the density
and consider in addition to the action , as defined by Eq.(1), , defined as
is again some scalar, which may contain the curvature (i.e. the gravitational contribution) and a matter contribution, as it can be the case for , as defined by Eq.(1). For an approach that uses four-vectors instead of four-scalars see (four-vector, ).
In the action defined by Eq.(3) the measure carries degrees of freedom independent of that of the metric and that of the matter fields. The most natural and successful formulation of the theory is achieved when the connection is also treated as an independent degree of freedom. This is what is usually referred to as the first order formalism.
One can consider both contributions, and allowing therefore both geometrical objects to enter the theory and take as our action
Here and are independent.
We will study now the dynamics of a scalar field interacting with gravity as given by the following action, where except for the potential terms and we have conformal invariance, the potential terms and break down this to global scale invariance.
The suffix in is to emphasize that here the curvature appears only linearly. Here, except for the potential terms and we have conformal invariance, the potential terms and break down this to global scale invariance. Since the breaking of local conformal invariance is only through potential terms, we call this a “soft breaking”.
In the variational principle , the measure fields scalars and the “matter” - scalar field are all to be treated as independent variables although the variational principle may result in equations that allow us to solve some of these variables in terms of others.
For the case the potential terms we have local conformal invariance
and is transformed according to
where is the Jacobian of the transformation of the fields.
This will be a symmetry in the case if
Notice that can be a local function of space time, this can be arranged by performing for the fields one of the (infinite) possible diffeomorphims in the internal space.
We can still retain a global scale invariance in model for very special exponential form for the and potentials. Indeed, if we perform the global scale transformation ( = constant)
then (9) is invariant provided and are of the form TMT2 ()
and is transformed according to
We will now work out the equations of motion after introducing and and see how the integration of the equations of motion allows the spontaneous breaking of the scale invariance.
Let us begin by considering the equations which are obtained from the variation of the fields that appear in the measure, i.e. the fields. We obtain then
where . Since it is easy to check that , it follows that det if . Therefore if we obtain that , or that
where M is constant. Notice that this equation breaks spontaneously the global scale invariance of the theory, since the left hand side has a non trivial transformation under the scale transformations, while the right hand side is equal to , a constant that after we integrate the equations is fixed, cannot be changed and therefore for any we have obtained indeed, spontaneous breaking of scale invariance.
We will see what is the connection now. As we will see, the connection appears in the original frame as a non Riemannian object. However, we will see that by a simple conformal tranformation of the metric we can recover the Riemannian structure. The interpretation of the equations in the frame gives then an interesting physical picture, as we will see.
Let us begin by studying the equations obtained from the variation of the connections . We obtain then
If we define as where is the Christoffel symbol, we obtain for the equation
The general solution of Eq.(24) is
where is an arbitrary function due to the - symmetry of the curvature Lambda () ,
Z being any scalar (which means ).
If we choose the gauge , we obtain
Considering now the variation with respect to , we obtain
a constraint that allows us to solve for ,
To get the physical content of the theory, it is best consider variables that have well defined dynamical interpretation. The original metric does not has a non zero canonical momenta. The fundamental variable of the theory in the first order formalism is the connection and its canonical momenta is a function of , given by,
and given by Eq.(29). Interestingly enough, working with is the same as going to the “Einstein Conformal Frame”. In terms of the non Riemannian contribution dissappears from the equations. This is because the connection can be written as the Christoffel symbol of the metric . In terms of the equations of motion for the metric can be written then in the Einstein form (we define usual Ricci tensor in terms of the bar metric and )
In terms of the metric , the equation of motion of the Scalar field takes the standard General - Relativity form
Notice that if and also, provided is finite and there. This means the zero cosmological constant state is achieved without any sort of fine tuning. That is, independently of whether we add to a constant piece, or whether we change the value of , as long as there is still a point where , then still and ( still provided is finite and there). This is the basic feature that characterizes the TMT and allows it to solve the “old” cosmological constant problem, at least at the classical level.
In what follows we will study the effective potential (33) for the special case of global scale invariance, which as we will see displays additional very special features which makes it attractive in the context of cosmology.
Notice that in terms of the variables , , the “scale” transformation becomes only a shift in the scalar field , since is invariant (since and )
Since we can always perform the transformation we can choose by convention . We then see that as const. providing an infinite flat region as depicted in Fig. 1. Also a minimum is achieved at zero cosmological constant for the case at the point
In conclusion, the scale invariance of the original theory is responsible for the non appearance (in the physics) of a certain scale, that associated to M. However, masses do appear, since the coupling to two different measures of and allow us to introduce two independent couplings and , a situation which is unlike the standard formulation of globally scale invariant theories, where usually no stable vacuum state exists.
The constant of integration plays a very important role indeed: any non vanishing value for this constant implements, already at the classical level S.S.B. of scale invariance.
Iii Generation of two flat regions after the introduction of a term
As we have seen, it is possible to obtain a model that through a spontaneous breaking of scale invariace can give us a flat region. We want to obtain now two flat regions in our effective potential. A simple generalization of the action will fix this. The basic new feature we add is the presence is higher curvature terms in the action barrow1 ()-mijic3 (), which have been shown to be very relevant in cosmology. In particular he first inflationary model from a model with higher terms in the curvature was proposed in mijic3 ().
What one needs to do is simply consider the addition of a scale invariant term of the form
The total action being then . In the first order formalism is not only globally scale invariant but also locally scale invariant, that is conformally invariant (recall that in the first order formalism the connection is an independent degree of freedom and it does not transform under a conformal transformation of the metric).
Let us see what the equations of motion tell us, now with the addition of to the action. First of all, since the addition has been only to the part of the action that couples to , the equations of motion derived from the variation of the measure fields remains unchanged. That is Eq.(21) remains valid.
The variation of the action with respect to gives now
It is interesting to notice that if we contract this equation with , the terms do not contribute. This means that the same value for the scalar curvature is obtained as in section 2, if we express our result in terms of , its derivatives and . Solving the scalar curvature from this and inserting in the other - independent equation we get still the same solution for the ratio of the measures which was found in the case where the terms were absent, i.e. .
Following then the same steps as in the model without the curvature square terms, we can then verify that the connection is the Christoffel symbol of the metric given by
defines now the “Einstein frame”. Equations (39) can now be expressed in the “Einstein form”
Here it is satisfied that , equation that expressed in terms of becomes
. This allows us to solve for and we get,
Notice that if we express in terms of , its derivatives and , the result is the same as in the model without the curvature squared term, this is not true anymore once we express in terms of , its derivatives and .
In any case, once we insert (44) into (43), we see that the effective potential (43) will depend on the derivatives of the scalar field now. It acts as a normal scalar field potential under the conditions of slow rolling or low gradients and in the case the scalar field is near the region .
Notice that since , then if , then, as in the simpler model without the curvature squared terms, we obtain that at that point without fine tuning (here by we mean the derivative of with respect to the scalar field , as usual).
In the case of the scale invariant case, where and are given by equation (15), it is interesting to study the shape of as a function of in the case of a constant , in which case can be regarded as a real scalar field potential. Then from (44) we get , which inserted in (43) gives,
The limiting values of are:
First, for asymptotically large positive values, ie. as , we have .
Second, for asymptotically large but negative values of the scalar field, that is as , we have: .
In these two asymptotic regions ( and ) an examination of the scalar field equation reveals that a constant scalar field configuration is a solution of the equations, as is of course expected from the flatness of the effective potential in these regions.
Notice that in all the above discussion it is fundamental that . If the potential becomes just a flat one, everywhere (not only at high values of ). All the non trivial features necessary for a graceful exit, the other flat region associated to the Planck scale and the minimum at zero if are all lost . As we discussed in the model without a curvature squared term, implies the we are considering a situation with S.S.B. of scale invariance.
These kind of models with potentials giving rise to two flat potentials have been applied to produce models for bags and confinement in a very natural way bags and confinement ().
Iv A Note on the the “Einstein” metric as a canonical variable of the Theory
One could question the use of the Einstein frame metric in contrast to the original metric . In this respect, it is interesting to see the role of both the original metric and that of the Einstein frame metric in a canonical approach to the first order formalism. Here we see that the original metric does not have a canonically conjugated momentum (this turns out to be zero), in contrast, the canonically conjugated momentum to the connection turns out to be a function exclusively of , this Einstein metric is therefore a genuine dynamical canonical variable, as opposed to the original metric. There is also a lagrangian formulation of the theory which uses , as we will see in the next section, what we can call the action in the Einstein frame. In this frame we can quantize the theory for example and consider contributions without reference to the original frame, thus possibly considering breaking the TMT structure of the theory through quantum effects, but such breaking will be done “softly” through the introduction of a cosmological term only. Surprisingly, the remaining structure of the theory, reminiscent from the original TMT structure will be enough to control the strength of this additional cosmological term once we demand that the universe originated from a non-singular and stable emergent state.
V Generalizing the model to include effects of zero point fluctuations
The effective energy-momentum tensor can be represented in a form like that of a perfect fluid
here . This defines a pressure functional and an energy density functional. The system of equations obtained after solving for , working in the Einstein frame with the metric can be obtained from a “k-essence” type effective action, as it is standard in treatments of theories with non linear kinetic terms or k-essence modelsk-essence1 ()-k-essence4 (). The action from which the classical equations follow is,
where it is understood that,
We have two possible formulations concerning : Notice first that and are different objects, the is the Riemannian curvature scalar in the Einstein frame, while is a different object. This will be treated in two different ways:
1. First order formalism for . Here is a lagrangian variable, determined as follows, that appear in the expression above for can be obtained from the variation of the pressure functional action above with respect to , this gives exactly the expression for that has been solved already in terms of , etc, see Eq. (44).
2. Second order formalism for . that appear in the action above is exactly the expression for that has been solved already in terms of , etc. The second order formalism can be obtained from the first order formalism by solving algebraically R from the Eq. (44) obtained by variation of , and inserting back into the action.
One may also use the method outlined in (R2Bulg, ) to find the effective action in the Einstein frame, in (R2Bulg, ) the problem of a curvature squared theory with standard measure was studied. The methods outlined there can be also applied in the modified measure case (mahary, ), thus providing another derivation of the effective action explained above.
The problem that we have to solve to find the effective lagrangian is basically finding that lagrangian tat will produce the effective energy momentum tensor in the Einstein frame by the variation of the metric
In contrast to the simplified models studied in literaturek-essence1 (); k-essence2 (); k-essence3 (); k-essence4 (), it is impossible here to represent in a factorizable form like . The scalar field effective Lagrangian can be taken as a starting point for many considerations.
In particular, the quantization of the model can proceed from (47) and additional terms could be generated by radiative corrections. We will focus only on a possible cosmological term in the Einstein frame added (due to zero point fluctuations) to (47), which leads then to the new action
This addition to the effective action leaves the equations of motion of the scalar field unaffected, but the gravitational equations aquire a cosmological constant. Adding the term can be regarded as a redefinition of
As we will see the stability of the emerging Universe imposes interesting constraints on
After introducing the term, we get from the variation of the same value of , unaffected by the new term, but as one can easily see then does not have the interpretation of a curvature scalar in the original frame since it is unaffected by the new source of energy density (the term), this is why the term theory does not have a formulation in the original frame, but is a perfectly legitimate generalization of the theory, probably obtained by considering zero point fluctuations, notice that quantum theory is possible only in the Einstein frame. Notice that even in the original frame the bar metric (not the original metric) appears automatically in the canonically conjugate momenta to the connection, so we can expect from this that the bar metric and not the original metric be the relevant one for the quantum theory.
Vi Analysis of the Emergent Universe solutions
We now want to consider the detailed analysis of The Emerging Universe solutions and in the next section their stability in the TMT scale invariant theory. We start considering the cosmological solutions of the form
in this case, we obtain for the energy density and the pressure, the following expressions. We will consider a scenario where the scalar field is moving in the extreme right region , in this case the expressions for the energy density and pressure are given by,
It is interesting to notice that all terms proportional to behave like “radiation”, since is satisfied. here the constants and are given by,
It will be convenient to “decompose” the constant into two pieces,
since as , . Therefore has the interesting interpretation of the vacuum energy density in the vacuum. As we will see, it is remarkable that the stability and existence of non-singular emergent universe implies that , and it is bounded from above as well.
The equation that determines such static universe , , gives rise to a restriction for that have to satisfy the following equation in order to guarantee that the universe be static, because is proportional to , we must require that , which leads to
This equation leads to two roots, the first being
The second root is:
It is also interesting to see that if the discriminant is positive, the first solution has automatically positive energy density, if we only consider cases where , which is required if we want the emerging solution to be able to turn into an inflationary solution eventually. One can see that the condition for the first solution reduces to the inequality , where , since we must have , otherwise we get a negative kinetic term during the inflationary period, and as we will see in the next section, we must have that from the stability of the solution, and as long as , it is always true that this inequality is satisfied.
Before going into the subject of the small perturbations and stability of these solutions, we would like to notice the “entropy like” conservation laws that may be useful in a non perturbative analysis of the theory.
In fact in the region, we have the exact symmetry . and considering that the effective matter action here is , we have the conserved quantity
It is very interesting to notice that
where assumes the “entropy density” form
provided we identify the “Temperature” T with .
Vii Stability of the static solution
We will now consider the perturbation equations. Considering small deviations of the from the static emerging solution value and also considering the perturbations of the scale factor , we obtain, from Eq. (55)
at the same time can be obtained from the perturbation of the Friedmann equation
and since we are perturbing a solution which is static, i.e., has , we obtain then
we also have the second order Friedmann equation
For the static emerging solution, we have , , so
where we have chosen to express our result in terms of , defined by , which for the emerging solution has the value . Using this in 69, we obtain
we now consider the perturbation of the eq. (70). In the right hand side of this equation we consider that , with
and therefore, the perturbation of the Eq. (70) leads to,
notice that the sum of the last two terms in the expression for , that is vanish since , for the same reason, we have that , which brings us to the simplified expression
to avoid negative kinetic terms during the slow roll phase that takes place following the emergent phase, we must consider , so, we see that the second solution is unstable and will not be considered further.
Now in the case of the first solution, E. (62) (), then becomes
so the condition of stability becomes , or , squaring both sides and since , we get , which means , and therefore , multiplying by , we obtain, , replacing the values of , given by 57 we obtain the condition
Now there is the condition that the discriminant be positive
since , , meaning that , we see that we obtain a positive upper bound for the energy density of the vacuum as , which must be positive, but not very big.
Viii Inflation and its Graceful Exit
The emerging phase owes its existence to a strictly constant vacuum energy (which here is represented by the value of ) at very large values of the field . In fact, while for the effective potential of the scalar field is perfectly flat, for any the effective potential acquires a non trivial shape. This causes the transition from the emergent phase to a slow roll inflationary phase which will be the subject of this section.
Following SIchile (), we consider now then the relevant equations for the model in the slow roll regime, i.e. for small and when the scalar field is large, but finite and we consider the first corrections to the flatness to the effective potential. Dropping higher powers of in the contributions for the kinetic energy and in the scalar curvature , we obtain
Here, as usual . In the slow roll approximation, we can drop the second derivative term of and the second power of in the equation for and we get
where . The relevant expression for will be that given by (45), i.e., where all higher derivatives are ignored in the potential, consistent with the slow roll approximation.
We now display the relevant expressions for the region of very large, but not infinite , these are:
The relevant constants that will affect our results are, , as given by (59) and and given by
Using Eq. (91) we can calculates the key landmarks of the inflationary history: first, the value of the scalar field where inflation ends, and a value for the scalar field bigger than this () and which happens earlier, which represents the “horizon crossing point”. We must demand then that a typical number of e-foldings, like , takes place between , until the end of inflation at .
To determine the end of inflation, we consider the quantity and consider the point in the evolution of the Universe where , only when , we have an accelerating Universe, so the point represents indeed the end of inflation. Calculating the derivative with respect to cosmic time of the Hubble expansion using (90) and (89), we obtain that the condition gives