Starobinsky Model in Rainbow Gravity
In this paper, we study the Starobinsky model of inflation in the context of gravity’s rainbow theory. We propose that gravity rainbow functions can be written in the power-law form of the Hubble parameter. We present a detailed derivation of the spectral index of curvature perturbation and the tensor-to-scalar ratio and compare the predictions of our models with Planck 2015 data. We discover that in order to be consistent with Planck data up to C.L., the viable values of e-folds would satisfy and the rainbow parameter is nicely constrained to be .
PACS numbers: 98.80.-k, 04.50.Kd, 95.30.Sf
In the past three decades, the cosmological inflation model has been well established as the leading paradigm for the very early universe both theoretically and experimentally. It provides solutions to important problems of standard big bang cosmology such as the flatness and horizon problem and gives a mechanism for seeds of the formation of the large scale structure of the universe. Although the inflation model agrees remarkably well with the observational data, it still suffers some fundamental issues, such as the fine-tuning slow-roll potential, the initial conditions, and the trans-Planckian problem. In addition, the standard mechanism of inflation requires a scalar field (an inflaton) to drive the exponential expansion of the very early universe but physical nature and fundamental origin of an inflaton are still an open question.
Since a period of inflation may occur in the very early universe, one can, in principle, expect that inflation comes from a semi-classical theory of quantum gravity with some quantum correction of Einstein-Hilbert action at high energy. The simplest example is -gravity theories, where the modification of the action of Einstein-Hilbert action is intended to generalize the Ricci scalar to be some function of the Ricci scalar itself. (for a review see De_Felice_10 () and references therein). In the 4-dimensional action in gravity, we write the action where , is the determinant of the metric . The Ricci scalar R is defined by . One can linearize the action by introducing an auxiliary field such that where we have defined . It is obvious that by making the variation of the action with respect to , we simply get . Thus, a scalar field , which will be the inflaton in the scalar field representation, will propagate the freedom degree from modified gravity. In summary, in the case of -gravity for inflation, we can pass to its scalar-field representation to achieve some useful simplification whose mechanism has been well studied in literature, e.g. Barrow:1988xh (); Maeda:1987xf (). The well-known example is the “Starobinsky model” for inflation Starobinsky_80 (); Mukhanov_81 (); Starobinsky_83 () with a correction quadratic in the Ricci scalar in the modified gravity framework, and of an exponential potential in the scalar field framework. Starobinsky model offers the clear origin of the inflaton field stemming from gravitational theory by identifying it with the spin-0 part of spacetime metric.
Adding quantum corrections to the action is not the only possible modification to Einstein’s gravity. Almost all approaches to quantum gravity suggest that the standard energy momentum dispersion relation is deformed near the Planck scale. This feature was predicted from string field theory Kostelecky_89 (), loop quantum gravity Gambini_98 (), non-commutative geometry Carroll_01 (). The modification of the dispersion relation generally takes the form
The functions and are commonly known as the rainbow functions assuming and in the limit where is the energy scale that quantum effects of gravity become important.
One of the interesting approaches that naturally produce modified dispersion relations is called doubly special relativityAmelino-Camelia_01 (); Amelino-Camelia_02 (); Procaccini_05 (). In addition to the invariance of the speed of light, it extends special relativity by including an invariant energy scale, usually assumed to be the Planck energy. Magueijo and Smolin Magueijo_04 () generalized this idea to include curvature i.e. doubly general relativity. In their approach, the spacetime metric felt by a free particle depends on the energy or momentum of the probe particle. Thus, spacetime is represented by a one parameter family of metrics parametrized by energy of the probe , forming a rainbow of metrics, and hence this approach is called gravity’s rainbow. The rainbow modified metric can be written as
The energy dependence of the frame field can be written in terms of the energy independence frame field as and for . In the study of cosmology, the conventional Friedmann-Robertson-Walker (FRW) metric for the homogeneous and isotropic universe is replaced by a rainbow metric of the form
For simplicity, we chose and only considered the spatially flat case with . In general, the rainbow function does not depend on space-time coordinates since for any specific operation of measurement the probe’s energy can be treated as a constant. However, instead of considering any specific measurement, the author in Ling07 () suggested that this formalism can be generalized to study semi-classical effects of relativistic particles on the background metric during a longtime process. For the very early universe, we can choose massless particles which dominate the universe at that period such as gravitons and inflatons as our probes. In this case, we need to consider the evolution of the probe’s energy with cosmic time, denoted as . As a result the rainbow functions depends on time only implicitly through the energy of particles.
The rainbow universe formalism was studied in the case of Ling07 (); Ling08 () by using modified Friedmann equations and the early universe is driven by thermally fluid substance. The cosmological linear perturbations of this model was studied in Wang14 (). In this paper, we extend this formalism to more general gravity, in particular the Starobinsky model. Qualitatively, Eq. (3) implies that time and space are scaled with different scaling functions i.e. the rainbow function and the scale factor respectively. By choosing suitable form of , one can solve the horizon problem without any need of inflationary expansion Moffat_93 (); Albrecht_99 (); Barrow_99 (). In our case, however, the quadratic term in the Ricci scalar will lead to an inflationary solution for . We will show how the observational data put the constraint on the scaling function .
We organize the paper as follows. In section II, we derive the equations of motion for theory in the framework of rainbow universe where cosmological evolution of probes is taken into account and written down as an inflationary solution to these equations for the case of Starobinsky model. Then we turn to investigate cosmological perturbation in the rainbow universe in section III where the spectral index of scalar perturbation and the tensor-to-scalar ratio of the model will be calculated. In section IV, we use the Planck 2015 data set to constraint the rainbow parameter of our model. Finally, we present our conclusion in section V.
Ii theory with Rainbow gravity’s effect
In addition to the modified FRW metric (3), modifications to General Relativity are expected to be possible in the very early universe where some corrections to Einstein’s gravity may emerge at high curvature. The simplest class of modified gravity theory is gravity, where the Einstein-Hilbert term in the action is replaced by a function of the Ricci scalar. Now taking this into account we derive the field equations by following the formalism in theory described in De_Felice_10 (). We start by considering the 4-dimensional action in gravity:
where and is the matter Lagrangian. By varying the action (4) with respect to , we obtain the field equations:
where and . The energy-momentum tensor of matter fields is defined by . and both satisfy the continuity equation i.e. and respectively. In cosmology, the energy-momentum tensor of matter is given in the perfect fluid form as , where is the energy density and is the pressure. In the following, we will derive cosmological solutions to field equations (5). In so doing, let us assume the modified FRW background (3), we obtain
where the Hubble parameter and .
ii.1 Inflation with the effect of Rainbow Gravity
In the following let us focus on the Starobinsky’s model Starobinsky_80 () in which takes the following form
where has a dimension of mass and is chosen to be the same as the mass scale for quantum gravity in Eq. (1) for simplicity. This is the first inflation model related to the conformal anomaly in quantum gravity. The term is responsible for the de Sitter expansion. Because of the linear term in , the inflationary expansion ends when the term becomes smaller than the linear term . This gives the graceful exit to inflation.
In contrast to previous studies on rainbow universe Ling07 (); Ling08 (); Wang14 (), the -term plays a leading role in the very early universe instead of relativistic matter. We will set and from now on. One can also consider this assumption in a scalar field framework of theory. For the case of Starobinsky model, its classical dynamics would be equivalent to the system of one scalar (an inflaton) with an exponential potential. So, it is reasonable to assume that inflatons dominated the very early universe and neglect contributions from matter and radiation. Then the combination of equation (6) and (7) gives us
Now, we consider the evolution of modified FRW universe. For explicitness, we need to choose the form of and this can be explained in the “Einstein frame,” where the scalar degree of freedom takes a canonical form with a potential. Under the conformal transformation with , the action in Einstein frame can be written as
where the subscribe denote quantities in Einstein frame. We define the scalar field (or the inflaton) by and its potential is determined by . Note that we neglect contribution from other matters.
In this framework it is natural to choose an inflaton as our probe during the inflation. We assume that probe’s energy is proportional to the total energy density of the inflaton, . During the slow-roll inflation, we can assume further that and it can be treated as a function of the Ricci Scalar in the Jordan frame, . In the case of FRW metric (), for the Starobinsky model, it is well known that during inflation De_Felice_10 (). Hence, we can now assume that the probe energy can be expressed in terms of a geometric expansion rate and this conclusion should also be applied in the Jordan frame.
Based on these assumptions, we choose the rainbow function to be explicitly parametrized in the power-law form of the Hubble parameter i.e. with the rainbow parameter and for late time. For the inflationary period, we are interested in the limit . Then, the rainbow function can be approximated as . Equation (9) can be written as
During inflation is very slowly varying. We can keep the first two terms in Equation (12) and neglect others. This gives us
We then obtain the solution
where and are Hubble parameter and scale factor at the onset of inflation (), respectively. The slow-roll parameter is defined by
which is less than unity in the limit . One can also check that the approximate relation holds in Eq. (10) by using . The end of inflation (at time ) can be determined by the condition , i.e. . The number of -foldings from to is defined by
Since inflation ends at , we can show that
Note that for the model reverts to the Starobinsky model.
Iii Cosmological perturbation in Rainbow gravity
In this section, we focus on cosmological linear perturbation generated during inflation by following the formalism presented in De_Felice_10 (). From the point of view of a probe particle with energy , a general perturbed metric about the flat FRW universe would take the form
where , , , are scalar perturbations, , are vector perturbations, and are tensor perturbations. The rainbow function can be viewed as a smooth background function. As usual, we will focus on scalar and tensor perturbations by setting vector perturbations . Before discussing the evolution of cosmological perturbations, we construct a number of gauge-invariant quantities required to eliminate unphysical modes. Let us consider the gauge transformation and . Here and denote the infinitesimal scalar functions of the spacetime coordinates. Then we can show that the scalar metric perturbations transform as:
Note that the tensor perturbations are invariant under the gauge transformation. We can define gauge invariant quantities under the above gauge transformations as
We can choose the longitudinal gauge and . In this gauge and . Thus, without tensor perturbations, the line element takes the form
For later convenience, we define the perturbed quantity
These equations describe evolutions of the scalar perturbations. We shall solve the above equations in the context of the inflationary universe.
iii.1 Curvature perturbations
Let us consider scalar perturbations generated during inflation without taking into account the perfect fluid i.e. and . We can choose the gauge condition , so that . Since the spatial curvature on the constant-time hypersurface is related to via the relation , the quantity is often called the curvature perturbation on the uniform-field hypersurface. By setting to be zero, Eq. (31) gives us
and from the equation (31), we get
where is a comoving wave number and is defined by
To simplify Eq. (36) any further we will introduce the new variables and . The above equation can be expressed as
where a prime denotes a derivative with respect to the new time coordinates . To calculate the spectrum of curvature perturbations, let us further introduce the Hubble flow parameters (also known as slow-roll parameters) as
where . Using these definitions, can be rewritten as
Assuming that parameters are nearly constants during the inflation for and , we have . If , it follows that the term is given by
With these assumptions, the explicit solution for Eq. (38) can be found as
where , are integration constants and , are the Hankel functions of the first kind and the second kind respectively. In the asymptotic past , the solution to Eq. (38) is , this implies and . Thus we have
The power spectrum of curvature perturbations is defined by
By using Eq. (44) and the relation , we will get
Note that for we have and should be evaluated at because is fixed after the Hubble radius crossing. Now, we can define the spectral index as
During the inflationary epoch, we assume that for all , then the spectral index is reduced to
Giving that are much smaller than one, , the spectrum is nearly scale-invariant. Subsequently, the power spectrum of curvature perturbation is
iii.2 Tensor perturbations
In this section we derive appropriate expressions of the power spectrum and the spectral index of tensor perturbations. In general, the tensor perturbation can be generally written as
where and are the polarization tensors corresponding to the two polarization states of . Let be in the direction along the z-axis, then the non-vanishing components of polarization tensors are and . By neglecting the scalar and vector perturbation, the perturbed FRW metric in (20) can be written as
By using the equation of motion (5), we can show that the Fourier components satisfy the equation
where denotes and . We now follow the similar procedure to the one given in the case of curvature perturbation. Let us introduce the new variables and , then Eq. (52) can be written as
Note that the massless scalar field has dimension of mass. By choosing , we get
Hence, the solution to the equation (53) can be expressed in terms of the Hankel functions as in Eq. (44). Taking into account polarization states, the power spectrum of tensor perturbations after the Hubble radius crossing is
Note that we have used . Since all of the slow-roll parameters are very small during the inflation (), can be estimated as
In addition, the spectral index of tensor perturbations is determined by
the power spectrum can also be rewritten as
The tensor-to-scalar ratio can be obtained as
By substituting from Eq. (40), we therefore get
iii.3 The Spectra of Perturbations based on Starobinsky’s model in Gravity’s rainbow theory
Let us start by finding relations among the Hubble flow parameters. In the absence of matter field with an assumption that during the inflation, Eq. (7) gives us
For Starobinsky model, , inflation occurred in the limit and . We can approximate . By assuming that during the inflation, this leads to
The spectral index of scalar perturbations and the tensor-to-scalar ratio can be rewritten in terms of as
Let be the time at the Hubble radius crossing (). From Eq. (14), as long as the condition is satisfied, we can approximate . The number of -fold from to the end of the inflation can be estimated as .
In the following, we will express the power spectrum, the spectral index, and the tensor-to-scalar ratio in terms of -folding number . By substituting the expression for in Eq. (40) into Eq. (49) and using the relation (62), we then obtain
According to the equation (19), the quantities , , and can be written in terms of as
Note that the spectral index of scalar perturbations does not depend on the rainbow parameter.
Iv Fitting 2015 Planck data
In this examination, we confront the results predicted by our model with Planck 2015 data. In order to be consistent with Planck data, we find that the value of cannot be arbitrary. For concreteness, Fig.(1) and Fig.(2) show that to be within C.L. of Planck’15 contours value of cannot be larger than with the sizable number of e-folds. We find for and e-folds that the predictions lie well on the boundary of region of the Planck data. Moreover, we discover that the predictions are consistent with the Planck data up to C.L. if the value of satisfies . Given and e-folds, the predictions lie on the boundary of region of the Planck data.
We also consider the situation in which the values of are arbitrary but keep fixed. In this case, we obtain from Fig.(3) the window of viable values of e-folds providing our predictions consistent with the Planck contours. Surprisingly, we discover that in order to be consistent with Planck data up to C.L., the values of e-folds would satisfy with a proper choice of . However, the values of cannot be larger than .
In some realistic scenario, one can apply De_Felice_10 (). In this specific case, we find that our predictions nicely satisfy the Planck data at C.L. when we opt . Note an extreme (and highly implausible) situation where the number of e-folds can be even higher, possibly up to 100 Liddle:2003as ().
In this paper, we have studied the Starobinsky model in the framework of rainbow gravity and obtained the equations of motion where the energy of probe particles varies with the cosmological time. By assuming the rainbow function in the power-law form of the Hubble constant, we have derived the inflationary solution to the equations of motion and calculate the spectral index of curvature perturbation and the tensor-to-scalar ratio.
By comparing the results predicted by our model with Planck 2015 data, we found that the model is in agreement with the data up to if the rainbow parameter satisfies with the sizable number of e-folds associated with . We have also used the scalar power spectrum amplitude reported by Planck 2015 to determine the mass and found, for assuming , that .
Acknowledgements.We are grateful to Ahpisit Ungkitchanukit and Khamphee Karwan for discussions and helpful suggestions. P.C. thanks Peeravit Koad for technical assistance. The work of A.C. and V.Y. is supported by “CUniverse” research promotion project by Chulalongkorn University (grant reference CUAASC) and P.C. is financially supported by the Thailand Research Fund (TRF) under the project of the “TRF Grant for New Researcher” with Grant No. TRG5780143. This work is supported in part by the Special Task Force for Activating Research (STAR) Project, Ratchadaphiseksomphot Endowment Fund, Chulalongkorn University.
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