Braneworld non-minimal inflation with induced gravity
We study cosmological inflation on a warped DGP braneworld where inflaton field is non-minimally coupled to induced gravity on the brane. We present a detailed calculation of the perturbations and inflation parameters both in Jordan and Einstein frame. We analyze the parameters space of the model fully to justify about the viability of the model in confrontation with recent observational data. We compare the results obtained in these two frames also in order to judge which frame gives more acceptable results in comparison with observational data.
- PACS numbers
98.80.Cq, 98.80.-k, 04.50.-h
- Key Words
Braneworld Inflation, Induced Gravity, Scalar-Tensor Gravity
Although the standard big bang cosmology has great successes in confrontation with observation, it suffers from some shortcomings such as the flatness, horizon and relics problems. It has been shown that an accelerating stage during the early time evolution of the universe with () has the capability to solve these problems. This is the early time inflationary stage. The inflation also provides a mechanism for production of density perturbations needed to seed the formation of structures in the universe. It has been shown that a simple scalar field (usually dubbed inflaton) whose energy dominates the universe and whose potential energy dominates over the kinetic term (the slow-roll conditions) gives the required inflation Guth (1981); Linde (1982); Albrecht (1982); Linde (1990); Liddle (2000a); Lidsey (1997); Riotto (2002); Lyth (2009). Despite the great successes of the inflation paradigm, there are several problems with no concrete solutions: natural realization of inflation in a fundamental theory, cosmological constant and dark energy problem, unexpected low power spectrum at large scales and egregious running of the spectral index are some of these problems Brandenberger (2005). Another unsolved problem in the spirit of the inflationary scenario is that we don’t know how to integrate it with ideas of the particle physics. For example, we would like to identify the inflaton, the scalar field that drives inflation, with one of the known fields of particle physics. Also, it is important that the inflaton potential emerges naturally from underlying fundamental theory Lidsey (1997).
Braneworld scenarios open new windows to address at least part of these difficulties Lidsey (2004); Buchel (2004). One of the various braneworld scenarios, is the model proposed by Dvali, Gabadadze and Porrati (DGP). This setup is based on a modification of the gravitational theory in an induced gravity perspective Dvali (2000, 2001a, 2001b); Lue (2006). This induced gravity term in the brane part of the action, leads to deviations from the standard 4-dimensional gravity over large distances. In the DGP model, the bulk is a flat Minkowski spacetime, but a reduced gravity term appears on the brane without tension. Some aspects of the braneworld inflation in the pure DGP setup are studied in Lazkoz (2004); Corradini (2008). Maeda, Mizuno and Torii have constructed a braneworld scenario which combines the Randall-Sundrum II (RS II) Randall (1999) and DGP models Maeda (2003). In this combination, an induced curvature term appears on the brane in the RS II model. This model has been called the warped DGP braneworld in literatures Cai (2004); Zhang (2006); Nozari (2007a, 2011). Some aspects of the inflation on the warped DGP setup are studied in Refs. Cai (2004); Zhang (2006); Nozari (2007a, 2011).
We note that in a braneworld setup, the induced gravity on the brane arises as a result of quantum corrections. For instance, in the Randall-Sundrum II braneworld scenario quantum corrections arise due to induced coupling between brane matter and the bulk gravitons. The induced gravity leads to the appearance of terms proportional to the 4-dimensional Ricci scalar in the brane part of the action. While the RS model gives high-energy modifications to general relativity, the DGP braneworld produces a low energy modification that leads to late-time acceleration of brane universe even in the absence of dark energy. The RS II braneworld scenario modifies certainly the high energy, ultra-violet (UV) sector of the general relativity. Also the DGP gravity is essentially a low-energy, infra-red (IR) modification of the general relativity. Since the warped DGP scenario contains both UV and IR modifications simultaneously, inflation in a warped DGP setup is physically more reasonable than the pure RS II or DGP case. An important issue we are interested in this paper, is that whether high-energy inflation is subjected to the induced gravity effect. If the induced gravity correction takes the dominant role, then there is no RS-type high-energy regime in the early universe and we recover the DGP model. From another perspective, as the energy scale of inflation grows, the induced gravity correction acts to limit the growth of amplitude relative to the 4D case Langlois (2000, 2007); Bouhamdi-Lopez (2004); Kaloper (2005). Although induced gravity is an IR modification of General Relativity and it seems that these modifications have nothing to do with inflation, however the mentioned points are important enough to be the reason for study of the warped DGP-braneworld inflation. We note also that as has been shown in Lazkoz (2004), brane assisted inflation may be equally successful beyond general relativity. It has been proved that this is the case in the RS and DGP models provided certain conditions hold. Since we considered the normal branch of solutions, as has been shown in Lazkoz (2004) the conditions for the occurrence of inflation are less restrictive.
On the other hand, considering a braneworld setup has the advantage that bulk fields such as Radions (for stability purposes) can have projection(s) on the brane that is a suitable candidate for inflaton field on the brane. The projection of the bulk inflaton on the brane behaves just like an ordinary inflaton field in four dimensions in the low energy regime. While the origin of inflaton field in standard 4D case is not so trivial, in a braneworld picture we can imagine this field as a projection of bulk field(s). This may help to reduce at least part of lacuna of standard scenario. We note also that as has been shown in Buchel (2004), inflation in warped de Sitter string theory geometries bypasses the difficulties of computing corrections to slow-roll parameter relative to the effective four dimensional perspective.
Since inflaton can interact with other fields such as the gravitational sector of the theory, in the spirit of scalar-tensor theories, we can consider a non-minimal coupling (NMC) of the inflaton field with intrinsic (Ricci) curvature on the brane. Braneworld model with scalar field minimally or non-minimally coupled to gravity have been studied extensively (see Nozari (2007b) and references therein). We note that generally the introduction of the NMC is not just a matter of taste. The NMC is instead forced upon us in many situations of physical and cosmological interest. There are compelling reasons to include an explicit non-minimal coupling in the action. For instance, non-minimal coupling arises at the quantum level when quantum corrections to the scalar field theory are considered. Even if for the classical, unperturbed theory this non-minimal coupling vanishes, it is necessary for the renormalizability of the scalar field theory in curved space. In most theories used to describe inflationary scenarios, it turns out that a non-vanishing value of the coupling constant cannot be avoided Faraoni (1996, 2000, 1999); Spokoiny (1984); Futamase (1989); Salopek (1989); Fakir (1990); Schimd (2005); Makino (1991); Fakir (1992); Libanov (1998); Hwang (1999, 1998); Tsujikawa (1999a, b, 2000a); Chiba (2000); Tsujikawa (2000b); Gunzig (2001); Koh (2005); Marco (2006); Bojowald (2006); Bauer (2008); Easson (2009); Hertzberg (2010); Pallis (2010, 2011). Nevertheless, incorporation of an explicit non-minimal coupling has disadvantage that it is harder to realize inflation even with potentials that are known to be inflationary in the minimal theory Faraoni (1996, 2000, 1999). Using the conformal equivalence between gravity theories with minimally and non-minimally coupled scalar fields, for any inflationary model based on a minimally-coupled scalar field, it is possible to construct infinitely many conformally related models with a non-minimal coupling Spokoiny (1984); Futamase (1989); Salopek (1989); Fakir (1990); Schimd (2005); Makino (1991); Fakir (1992); Libanov (1998); Hwang (1999, 1998); Tsujikawa (1999a, b, 2000a); Chiba (2000); Tsujikawa (2000b); Gunzig (2001); Koh (2005); Marco (2006); Bojowald (2006); Bauer (2008); Easson (2009); Hertzberg (2010); Pallis (2010, 2011); Kaiser (1995); Chiba (2008). However, an important question then arises: are these conformally related frames really equivalent from physics viewpoint? This issue has been considered by several authors Capozziello (1997); Nandi (1998); Flanagan (2004); Bhadra (2007); Faraoni (2007); Nozari (2009); Capozziello (2010a, b); Quiros (2011a, b) and as a part of our primary goal, we are going to address this issue from a detailed comparison of the inflationary parameters in these two (Einstein and Jordan) frames.
Based on the mentioned preliminaries, in this paper we study cosmological inflation on a warped DGP braneworld where inflaton field is non-minimally coupled to induced gravity on the brane. We present a detailed calculation of the perturbations and inflation parameters both in Jordan and Einstein frame by adopting quadratic and quartic potentials. We analyze the parameter spaces of the models with details to have a comparison between two frames and also in order to constraint these models in confrontation with recent observational data.
Ii Braneworld inflation with induced gravity in Jordan frame
The action of a warped DGP model in which a single scalar field is non-minimally coupled to induced gravity on the brane can be written in the following form
where is the five dimensional gravitational constant, is the induced Ricci scalar on the brane, is 5-dimensional Ricci scalar, is the brane tension and is the bulk cosmological constant. Also is the trace of the brane metric, . We remind that the mentioned action results in pure DGP model Dvali (2000, 2001a, 2001b) if and , and pure RSII model Randall (1999) if where is a mass scale which may correspond to the 4D Planck mass Maeda (2003). Also shows an explicit non-minimal coupling of the scalar field with induced gravity on the brane. We note that the fields and their interactions on the brane at the classical level will be determined by the bulk physics through boundary conditions on the brane. For instance, if is assumed to be a bulk scalar field, as has been shown in Himemoto (2001, 2003); Yokoyama (2001); Sasaki (2004); Nozari (2012), the effective field on the brane will be and through junction conditions on the brane. Also as we will show (see Eq. (6) below), . So, these parameters cannot be freely adjusted and are influenced by bulk physics.
The generalized cosmological dynamics in this setup is given by the following Friedmann equation
where , the energy-density corresponding to the non-minimally coupled scalar field is defined as follows
and the corresponding pressure is given by
We note that in this paper a prime represents the derivative with respect to the scalar field and a dot marks derivative with respect to the cosmic time. Now let’s to introduce the effective cosmological constant on the brane as
Since we are interested in the inflationary dynamics driven by a scalar field with a self-interacting potential, we put the effective cosmological constant equal to zero. In this way, we find
So, we can rewrite the Friedmann equation (2) as follows
Also, the second Friedmann equation is
Variation of the action (1) with respect to the scalar field gives the following equation of motion
In the slow-roll approximation, where and , energy density and equation of motion for scalar field take the following forms respectively
Also, the Friedmann equation now takes the following form
Now, we define the slow-roll parameters as follows
In the slow-roll approximation and by using equation (12) we find
where by definition
As we will show, these parameters which reflect the braneworld and non-minimal nature of our model, in the large field regime intensify the increment of the slow-roll parameters. Inflation can be attained only if ; once one of these parameters reaches unity, the inflation phase terminates. We note that and are contributions originating from braneworld nature of the setup and also the non-minimal coupling of the scalar field and induced gravity on the brane.
The number of e-folds during inflation is given by
which in the slow-roll approximation can be written as
where denotes the value of when the universe scale observed today crosses the Hubble horizon during inflation and is the value of when the universe exits the inflationary phase. For a warped DGP model with non-minimally coupled scalar field on the brane, this quantity in Jordan frame becomes
After presentation of the main equations of the setup in Jordan frame, in the next section we consider the scalar perturbation of the metric since the key test of any inflation model is the spectrum of perturbations produced due to quantum fluctuations of the fields about their homogeneous background values.
Iii Perturbations in Jordan frame
In a warped DGP braneworld model, the effective covariant equations on the brane for an arbitrary brane metric and matter distribution is given by Maartens (2010)
is the total stress-tensor on the brane and is defined as
where , the energy-momentum tensor of a scalar field non-minimally coupled to induced gravity on the brane is given by
Also we have
where is the five dimensional Weyl tensor and is the spacelike unit vector normal to the brane.
Depending on the choice of gauge (coordinates), there are many different ways of characterizing cosmological perturbations. In longitudinal gauge, the scalar metric perturbations of the FRW background are given by Bardeen (1980); Mukhanov (1992); Bertschinger (1995)
where is the scale factor on the brane, and are the metric perturbations. For the above perturbed metric, one can obtain the perturbed field equations as follows
The anisotropic stress perturbation is defined as , where is the trace of . So, is the anisotropic stress perturbation. In the Eqs. (28) and (29), and can be obtained from the standard Friedmann equation as follows
By using the continuity equation, , one can deduce
So, the perturbed effective density and pressure can be written as
can be calculated from the general definition of as
The (gauge-invariant) scalar perturbations of can be parameterized as an effective fluid with density perturbation , isotropic pressure perturbation , anisotropic stress perturbation and energy flux perturbation (see Koyama (2006); Langlois (2001)). Also and take the following forms
Equations (37) and (39) in the minimal case and within the slow-roll conditions reduce to and respectively. By perturbing the equation of motion of the scalar field (11), one obtains
Now the scalar perturbations can be decomposed to an entropy or isocurvature perturbation (the projection orthogonal to the trajectory), and adiabatic or curvature perturbations (projection parallel to the trajectory). The isocurvature perturbations are generated if inflation is driven by more than one scalar field Langlois (2000, 2007); Bassett (1999); Gordon (2001) or it interacts with other fields such as the induced gravity on the brane Bouhamdi-Lopez (2004); Kaloper (2005). The adiabatic perturbations are generated if the inflaton field is the only field in inflation period Bassett (1999); Gordon (2001); Bouhamdi-Lopez (2004); Kaloper (2005); Maartens (2000). Here, since the inflaton field is non-minimally coupled to the induced gravity on the brane, the entropy perturbations are presented in this setup Maartens (2000); Seahra (2010). A gauge-invariant primordial curvature perturbation , can be defined as follows Bardeen (1983)
This definition is valid to first order in the cosmological perturbations on scales outside the horizon. On uniform density hypersurfaces where , the above quantity reduces to the curvature perturbation, . In the warped DGP model and within the Jordan frame, we should redefine Eq. (42) as
Now, by using the energy conservation equation for linear perturbations (in an arbitrary gauge)
we can find the variation of with respect to the conformal time as
where and are given by time derivatives of equations (32) and (33) respectively.
One can split the pressure perturbation (in any gauge) into adiabatic and entropic (non-adiabatic) parts (see for instance Ref. Wands (2000))
where is the sound effective velocity. The non-adiabatic part is , where represents the displacement between hypersurfaces of uniform pressure and density. From equations (34)-(40) we can deduce
Using the equations (28)-(30) we can rewrite this relation as
where , and are defined as
respectively. Now we can rewrite the equation governing on the variation of versus the time in terms of the model’s parameters. From equations (44)-(48) we find
In the minimal case and within the standard model, the entropy perturbation vanishes for long wavelength; we have and the primordial spectrum of perturbation is due to adiabatic perturbations. But, it is obvious from equation (48) that in a DGP-inspired non-minimal setup, there is a non-vanishing contribution of the non-adiabatic perturbations, leading to non-vanishing , which affects the primordial spectrum of perturbation. We note that isocurvature perturbations are free to evolve on superhorizon scales, and the amplitude at the present day depends on the details of the entire cosmological evolution from the time that they are formed. On the other hand, because all super-Hubble radius perturbations evolve in the same way, the shape of the isocurvature perturbation spectrum is preserved during this evolution Liddle (2000b); Cid (2007).
Here we are going to obtain scalar and tensorial perturbations in our model. We take into account the slow-roll approximation at the large scales, , where we need to describe the non-decreasing modes. Then by using the relation between Ricci scalar and and , we find from equation (41)
We note that the reason for large scale assumption is that the scales of cosmological interest (e.g. for large-scale CMB anisotropies) have spent most of their time far outside the Hubble radius and have re-entered only relatively recently in the Universe history. In this respect, in the large scale the condition is an acceptable assumption. As has been shown in Refs. Amendola (2006, 2007), when this condition is satisfied, , and can be neglected. In fact, for the longitudinal post-Newtonian limit to be satisfied, we require that , and similarly for other gradient terms Amendola (2006, 2007). For a plane wave perturbation with wavelength , we see that is much smaller than when . The requirement that be also negligible implies the condition (with ), which holds if condition is satisfied for perturbation growth. This argument can be applied for and the other metric potential, too. By adopting a similar reasoning, form Eq. (30) we have
In writing the above equation we used the relation . By using equation (53) and (54), we can deduce