Emergent Cosmology, Inflation and Dark Energy
A new class of gravity-matter models defined in terms of two independent non-Riemannian volume forms (alternative generally covariant integration measure densities) on the space-time manifold are studied in some detail. These models involve an additional (square of the scalar curvature) term as well as scalar matter field potentials of appropriate form so that the pertinent action is invariant under global Weyl-scale symmetry. Scale invariance is spontaneously broken upon integration of the equations of motion for the auxiliary volume-form degrees of freedom. After performing transition to the physical Einstein frame we obtain: (i) An effective potential for the scalar field with two flat regions which allows for a unified description of both early universe inflation as well as of present dark energy epoch; (ii) For a definite parameter range the model possesses a non-singular “emergent universe” solution which describes an initial phase of evolution that precedes the inflationary phase; (iii) For a reasonable choice of the parameters the present model conforms to the Planck Collaboration data.
Keywords:modified gravity theories, non-Riemannian volume forms, global Weyl-scale symmetry spontaneous breakdown, flat regions of scalar potential, non-singular origin of the universe
pacs:04.50.Kd, 11.30.Qc, 98.80.Bp, 95.36.+x
Modern cosmology has been formulated in an attractive framework where many aspects of the observable universe can be incorporated. In this “standard cosmological” framework, the early universe (cf. the books early-univ () and references therein) starts with a period of exponential expansion called “inflation”. In the inflationary period also primordial density perturbations are generated (Ref.primordial () and references therein). The “inflation” is followed by particle creation, where the observed matter and radiation were generated early-univ (), and finally the evolution arrives to a present phase of slowly accelerating universe accel-exp (); accel-exp-2 (). In this standard model, however, at least two fundamental questions remain unanswered:
The early inflation, although solving many cosmological puzzles, like the horizon and flatness problems, cannot address the initial singularity problem;
There is no explanation for the existence of two periods of exponential expansion with such wildly different scales – the inflationary phase and the present phase of slowly accelerated expansion of the universe.
The best known mechanism for generating a period of accelerated expansion is through the presence of some vacuum energy. In the context of a scalar field theory, vacuum energy density appears naturally when the scalar field acquires an effective potential which has flat regions so that the scalar field can “slowly roll” slow-roll (); slow-roll-param () and its kinetic energy can be neglected resulting in an energy-momentum tensor .
The possibility of continuously connecting an inflationary phase to a slowly accelerating universe through the evolution of a single scalar field – the quintessential inflation scenario – has been first studied in Ref.peebles-vilenkin (). Also, models can yield both an early time inflationary epoch and a late time de Sitter phase with vastly different values of effective vacuum energies starobinsky-2 (). For a recent proposal of a quintessential inflation mechanism based on the k-essence k-essence () framework, see Ref.saitou-nojiri (). For another recent approach to quintessential inflation based on the “variable gravity” model wetterich () and for extensive list of references to earlier work on the topic, see Ref.murzakulov-etal ().
In the present paper we will study a unified scenario where both an inflation and a slowly accelerated phase for the universe can appear naturally from the existence of two flat regions in the effective scalar field potential which we derive systematically from a Lagrangian action principle. Namely, we start with a new kind of globally Weyl-scale invariant gravity-matter action within the first-order (Palatini) approach formulated in terms of two different non-Riemannian volume forms (integration measures) quintess (). In this new theory there is a single scalar field with kinetic terms coupled to both non-Riemannian measures, and in addition to the scalar curvature term also an term is included (which is similarly allowed by global Weyl-scale invariance). Scale invariance is spontaneously broken upon solving part of the corresponding equations of motion due to the appearance of two arbitrary dimensionful integration constants. We find in the physical Einstein frame an effective k-essence k-essence () type of theory, where the effective scalar field potential has two flat regions corresponding to the two accelerating phases of the universe – the inflationary early universe and the present late universe.
In addition, within the flat region corresponding to the early universe we also obtain another phase that precedes the inflation and provides for a non-singular origin of the universe. It is of an “emergent universe” type emergent-univ (), i.e., the universe starts as a static Einstein universe, the scalar field rolls with a constant speed through a flat region and there is a domain in the parameter space of the theory where such non-singular solution exists and is stable. To this end let us recall that the concept of “emergent universe” solves one of the principal puzzles in cosmology – the problem of initial singularity singular-univ () including avoiding the singularity theorems for scalar field-driven inflationary cosmology Borde-Vilenkin-PRL ().
Let us briefly recall the origin of current approach. The main idea comes from Refs.TMT-orig-1 ()-TMT-orig-3 () (see also Refs.TMT-recent-1-a ()-TMT-recent-2 ()), where some of us have proposed a new class of gravity-matter theories based on the idea that the action integral may contain a new metric-independent generally-covariant integration measure density, i.e., an alternative non-Riemannian volume form on the space-time manifold defined in terms of an auxiliary antisymmetric gauge field of maximal rank. The originally proposed modified-measure gravity-matter theories TMT-orig-1 ()-TMT-recent-2 () contained two terms in the pertinent Lagrangian action – one with a non-Riemannian integration measure and a second one with the standard Riemannian integration measure (in terms of the square-root of the determinant of the Riemannian space-time metric). An important feature was the requirement for global Weyl-scale invariance which subsequently underwent dynamical spontaneous breaking TMT-orig-1 (). The second action term with the standard Riemannian integration measure might also contain a Weyl-scale symmetry preserving -term TMT-orig-3 ().
The latter formalism yields various new interesting results in all types of known generally covariant theories:
(i) -dimensional models of gravity and matter fields containing the new measure of integration appear to be promising candidates for resolution of the dark energy and dark matter problems, the fifth force problem, and a natural mechanism for spontaneous breakdown of global Weyl-scale symmetry TMT-orig-1 ()-TMT-recent-2 ().
(ii) Study of reparametrization invariant theories of extended objects (strings and branes) based on employing of a modified non-Riemannian world-sheet/world-volume integration measure mstring () leads to dynamically induced variable string/brane tension and to string models of non-abelian confinement. Recently nishino-rajpoot () this formalism was generalized to the case of string and brane models in curved supergravity background.
(iii) Study in Refs.susy-break () of modified supergravity models with an alternative non-Riemannian volume form on the space-time manifold produces some outstanding new features: (a) This new formalism applied to minimal supergravity naturally triggers the appearance of a dynamically generated cosmological constant as an arbitrary integration constant, which signifies a new explicit mechanism of spontaneous (dynamical) breaking of supersymmetry; (b) Applying the same formalism to anti-de Sitter supergravity allows us to appropriately choose the above mentioned arbitrary integration constant so as to obtain simultaneously a very small effective observable cosmological constant as well as a very large physical gravitino mass.
The plan of the present paper is as follows. In the next Section 2 we describe in some detail the general formalism for the new class of gravity-matter systems defined in terms of two independent non-Riemannian integration measures. In Section 3 we describe the properties of the two flat regions in the Einstein-frame effective scalar potential corresponding to the evolution of the early and late universe, respectively. In Section 4 we present a numerical analysis, for a reasonable choice of the parameters, of the resulting ratio of tensor-to-scalar perturbations and show that the present model conforms to the Planck Collaboration data. In Section 5 we derive a non-singular “emergent universe” solution of the new gravity-matter system. In Section 6 a numerical study of the transition between the emergent universe phase and the slow-roll inflationary phase via a short “super-inflation” period is given in some detail. We conclude in Section 7 with some discussions.
2 Gravity-Matter Formalism With Two Independent Non-Riemannian Volume-Forms
We shall consider the following non-standard gravity-matter system with an action of the general form involving two independent non-Riemannian integration measure densities generalizing the model studied in quintess () (for simplicity we will use units where the Newton constant is taken as ):
Here the following notations are used:
and are two independent non-Riemannian volume-forms, i.e., generally covariant integration measure densities on the underlying space-time manifold:
defined in terms of field-strengths of two auxiliary 3-index antisymmetric tensor gauge fields
13. take over the role of the standard Riemannian integration measure density in terms of the space-time metric .
and are the scalar curvature and the Ricci tensor in the first-order (Palatini) formalism, where the affine connection is a priori independent of the metric . Note that in the second action term we have added a gravity term (again in the Palatini form). Let us recall that gravity within the second order formalism (which was also the first inflationary model) was originally proposed in Ref.starobinsky ().
denote two different Lagrangians of a single scalar matter field of the form (similar to the choice in Refs.TMT-orig-1 ()):
where are dimensionful positive parameters, whereas is a dimensionless one.
indicates the dual field strength of a third auxiliary 3-index antisymmetric tensor gauge field:
whose presence is crucial for non-triviality of the model.
The scalar potentials have been chosen in such a way that the original action (1) is invariant under global Weyl-scale transformations:
For the same reason we have multiplied by an appropriate exponential factor the scalar kinetic term in and also and couple to the two different modified measures because of the different scalings of the latter.
Let us note that the requirement about the global Weyl-scale symmetry (6) uniquely fixes the structure of the non-Riemannian-measure gravity-matter action (1) (recall that the gravity terms and are taken in the first order (Palatini) formalism).
Let us also note that the global Weyl-scale symmetry transformations defined in (6) are not the standard Weyl-scale (or conformal) symmetry known in ordinary conformal field theory. It is straightforward to check that the dimensionful parameters present in (3)-(4) do not spoil at all the symmetry given in (6). In particular, unlike the standard form of the Weyl-scale transformation for the metric the transformation of the scalar field is not the canonical scale transformation known in standard conformal field theories. In fact, as shown in the second Ref.TMT-orig-1 () in the context of a simpler than (1) model with only one non-Riemannian measure, upon appropriate -dependent conformal rescaling of the metric together with a scalar field redefinition , one can transform the latter model into Zee’s induced gravity model zee-induced-grav (), where its pertinent scalar field transforms multiplicatively under the above scale transformations as in standard conformal field theory.
shows, following the analogous derivation in the Ref.TMT-orig-1 (), that becomes a Levi-Civita connection:
w.r.t. to the Weyl-rescaled metric :
Variation of the action (1) w.r.t. auxiliary tensor gauge fields , and yields the equations:
whose solutions read:
Here and are arbitrary dimensionful and arbitrary dimensionless integration constants.
The first integration constant in (11) preserves global Weyl-scale invariance (6), whereas the appearance of the second and third integration constants signifies dynamical spontaneous breakdown of global Weyl-scale invariance under (6) due to the scale non-invariant solutions (second and third ones) in (11).
To this end let us recall that classical solutions of the whole set of equations of motion (not only those of the scalar field(s)) correspond in the semiclassical limit to ground-state expectation values of the corresponding fields. In the present case some of the pertinent classical solutions (second and third Eqs.(11)) contain arbitrary integration constants whose appearance makes these solutions non-covariant w.r.t. the symmetry transformations (6). Thus, spontaneous symmetry breaking of (6) is not necessarily originating from some fixed extrema of the scalar potentials. In fact, as we will see in the next Section below, the (static) classical solutions for the scalar field defined through extremizing the effective Einstein-frame scalar potential (Eq.(27) below) belong to the two infinitely large flat regions of the latter (infinitely large “valleys” of “ground states”), therefore, this does not constitute a breakdown of the shift symmetry of the scalar field (6). Thus, it is the appearance of the arbitrary integration constants , which triggers the spontaneous breaking of global Weyl-scale symmetry (6).
where and are defined in (9), and are the energy-momentum tensors of the scalar field Lagrangians with the standard definitions:
with energy-momentum tensor corresponding (according to (13)):
to the following effective Einstein-frame scalar field Lagrangian:
In order to explicitly write in terms of the Einstein-frame metric (17) we use the short-hand notation for the scalar kinetic term:
and represent in the form:
whereas the effective scalar field potential reads:
Let us recall that the dimensionless integration constant is the ratio of the original second non-Riemannian integration measure to the standard Riemannian one (9).
To conclude this Section let us note that choosing the “wrong” sign of the scalar potential (Eq.(4)) in the initial non-Riemannian-measure gravity-matter action (1) is necessary to end up with the right sign in the effective scalar potential (27) in the physical Einstein-frame effective gravity-matter action (24). On the other hand, the overall sign of the other initial scalar potential (Eq.(4)) is in fact irrelevant since changing its sign does not affect the positivity of effective scalar potential (27).
Let us also remark that the effective matter Lagrangian (24) is called “Einstein-frame scalar Lagrangian” in the sense that it produces the effective energy-momentum tensor (19) entering the effective Einstein-frame form of the gravity equations of motion (18) in terms of the conformally rescaled metric (17) which have the canonical form of Einstein’s gravitational equations. On the other hand, the pertinent Einstein-frame effective scalar Lagrangian (24) arises in a non-canonical “k-essence” k-essence () type form.
3 Flat Regions of the Effective Scalar Potential
Depending on the sign of the integration constant we obtain two types of shapes for the effective scalar potential (27) depicted on Fig.1 and Fig.2. Due to the vast difference in the scales of the pertinent parameters, whose estimates are given below, Fig.1 and Fig.2 represent only qualitatively the shape of .
The crucial feature of is the presence of two infinitely large flat regions – for large negative and large positive values of the scalar field . For large negative values of we have for the effective potential and the coefficient functions in the Einstein-frame scalar Lagrangian (24)-(27):
In the second flat region for large positive :
From the expression for (27) and the figures 1 and 2 we see that now we have an explicit realization of quintessential inflation scenario. The flat regions (28)-(29) and (30)-(31) correspond to the evolution of the early and the late universe, respectively, provided we choose the ratio of the coupling constants in the original scalar potentials versus the ratio of the scale-symmetry breaking integration constants to obey:
In particular, if we choose the scales of the scale symmetry breaking integration constants and , where are the electroweak and Plank scales, respectively, we are then naturally led to a very small vacuum energy density of the order:
which is the right order of magnitude for the present epoche’s vacuum energy density as already recognized in Ref.arkani-hamed (). On the other hand, if we take the order of magnitude of the coupling constants in the effective potential , then together with the above choice of order of magnitudes for the inequalities (33) will be satisfied as well and the order of magnitude of the vacuum energy density of the early universe (28) becomes:
which conforms to the BICEP2 experiment bicep2 () and Planck Collaboration data Planck1 (); Planck2 () implying the energy scale of inflation of order . However, let us remark at this point that, as shown in the next Section 4, the result for the tensor-to-scalar ratio obtained within the present model conforms to the data of the Planck Collaboration Planck1 (); Planck2 () rather than BICEP2 bicep2 ().
Let us recall that, since we are using units where , in the present case .
Before proceeding to the derivation of the non-singular “emergent universe” solution describing an initial phase of the universe evolution preceding the inflationary phase, let us briefly sketch how the present non-Riemannian-measure-modified gravity-matter theory meets the conditions for the validity of the “slow-roll” approximation slow-roll () when evolves on the flat region of the effective potential corresponding to the early universe (28)-(29).
To this end let us recall the standard Friedman-Lemaitre-Robertson-Walker space-time metric weinberg-72 ():
and the associated Friedman equations (recall the presently used units ):
describing the universe’ evolution. Here:
are the energy density and pressure of the scalar field . Henceforth the dots indicate derivatives with respect to the time .
Let us now consider the standard “slow-roll” parameters slow-roll-param ():
where measures the ratio of the scalar field kinetic energy relative to its total energy density and measures the ratio of the fields acceleration relative to the “friction” () term in the pertinent scalar field equations of motion:
with primes indicating derivatives w.r.t. .
The reason for ignoring the spatial curvature term in the second Eq.(42) is due to the fact that evolves on a flat region of and the Hubble parameter , so that grows exponentially with time making very small. Consistency of the slow-roll approximation implies for the slow-roll parameters (40), taking into account (42), the following inequalities:
which yields for the slow-roll parameter (43):
Similarly, for the second slow-roll parameter we have:
At this point let us remark that the non-canonical “k-essence” form of the effective scalar Lagrangian (24) does not affect the condition for smallness of the standard “slow-roll” parameters (40). Indeed, the definition of the first slow-roll parameter in (40) is consistent with the first Friedman equation in (37), where there is no a priori requirement for the energy density and the pressure to be defined in terms of a scalar field action with a canonical kinetic term. Similarly, the non-canonical “k-essence” form of the effective scalar Lagrangian (24) does not affect the requirement for smallness of the second “slow-roll” parameter in (40). In fact, the smallness of is explicitly displayed in Eqs.(45)-(46) because of the presence of strongly suppressing factors – exponentials of large negative values of the scalar field in the first flat region of the effective scalar potential corresponding to the early universe.
The value of at the end of the slow-roll regime is determined from the condition which through (45) yields:
The number of e-foldings (see, e.g. second Ref.primordial ()) between two values of cosmological times and or analogously between two different values and becomes:
In what follows the subscript is used to indicate the epoch where the cosmological scale exits the horizon.
In the following we will describe the scalar and tensor perturbations for our model. Following Refs.Garriga (); p1 () the power spectrum of the scalar perturbation for a non-canonical kinetic term in the slow-roll approximation is given by:
where denotes the “speed of sound” and is defined as , and . Here is a function of the scalar field and is the scalar kinetic term as in (21). The constant , and denotes the derivative with respect . In particular, in the present case (24) where .
The scalar spectral index is given by:
On the other hand, it is well known that the generation of tensor perturbations during inflation would generate gravitational waves. The spectrum of the tensor perturbations was calculated in Ref.Garriga () and is given by:
and the tensor spectral index can be expressed in terms of the parameter as . An important observational quantity is the tensor-to-scalar ratio satisfying a generalized consistency relation in which . These observational quantities should be evaluated at (see Eq.(49)).
Considering the slow-roll approximation the power spectrum of the scalar perturbation (to leading order) from Eq.(50) becomes:
where the constant is given by
From Eq.(51) the scalar spectral index , becomes:
Here we took into account the relation between and the number of e-foldings (49), which can be written as:
where the constant is given by:
In this form, we can obtain the value of the parameter for given values , ,, and parameters when the number of e-folds and the power spectrum are given.
From Eq.(55) and considering that , the relation between the tensor-to-scalar ratio and the spectral index , i.e., the consistency relation, , is given by:
Here we note that working to leading order the consistency relation , becomes independent of the integration constants and (11).
In Fig.3 we show the evolution of the tensor-to-scalar ratio w.r.t. the scalar spectral index for three different values of the parameter . Here we show the two-dimensional marginalized constraints, at 68 and 95 levels of confidence, for the tensor-to-scalar ratio and the spectral index from BICEP2 experiment in connection with Planck + WP + highL bicep2 (). In order to write down values that relate the ratio and the spectral index we considered the consistency relation given by Eq.(59). Also, we have used the values , , , and .
From the plot in Fig.3 we note that the tensor-to-scalar ratio , and our model is disproved from BICEP2, since according to the latter the ratio with the ratio disproved at 7.0. Nevertheless, the result for tensor-to-scalar ratio has become less clear when serious criticisms of BICEP2 appeared in the literature. In particular, the Planck Collaboration has issued the data about the polarized dust emission through an analysis of the polarized thermal emissions from diffuse Galactic dust, which suggest that BICEP2 data of the gravitational wave result could be due to the dust contamination Planck1 (). Thereby, a detailed analysis of Planck and BICEP2 data would be required for a definitive answer. In this form, previous CMB observations from the Planck satellite and other CMB experiments obtained only an upper limit for the tensor-to-scalar ratio, in which 0.11 (at 95 confidence level) Planck2 (). Therefore, we find that the value is well supported by the confidence levels from Planck data. In particular, the value corresponds to . Also, we note that when we increase the value of the parameter , the value of the tensor-to-scalar ratio .
Besides, in particular for the values and (recall the subscript indicating the epoch where the cosmological scale exits the horizon) we obtained for the parameter from Eq.(58) that , which corresponds to the value of , and , which corresponds to the parameter . In this form the constraint for is given by . Here, we have used the same values of , , and from Fig.3.
Numerically, from Eq.(55) we find a constraint for the parameter given by for the values and the number , which corresponds to the value of , and , which corresponds to . In this way, the range of the parameter is . As before, we have considered the same values of , and from Fig.3.
5 Non-Singular Emergent Universe Solution
We will now show that under appropriate restrictions on the parameters there exist an epoch preceding the inflationary phase. Namely, we derive an explicit cosmological solution of the Einstein-frame system with effective scalar field Lagrangian (24)-(27) describing a non-singular “emergent universe” emergent-univ () when the scalar field evolves on the first flat region for large negative (28). For previous studies of “emergent universe” scenarios within the context of the less general modified-measure gravity-matter theories with one non-Riemannian and one standard Riemannian integration measures, see Ref.TMT-recent-1-a ()-TMT-recent-1-c ().
The emergent universe condition (60) implies that the -velocity is time-independent and satisfies the bi-quadratic algebraic equation: