Can Effective Field Theory of inflation generate large tensor-to-scalar ratio within Randall Sundrum single braneworld?
In this paper my prime objective is to explain the generation of large tensor-to-scalar ratio from the single field sub-Planckian inflationary paradigm within Randall Sundrum (RS) single braneworld scenario in a model independent fashion. By explicit computation I have shown that the effective field theory prescription of brane inflation within RS single brane setup is consistent with sub-Planckian excursion of the inflaton field, which will further generate large value of tensor-to-scalar ratio, provided the energy density for inflaton degrees of freedom is high enough compared to the brane tension in high energy regime. Finally, I have mentioned the stringent theoretical constraint on positive brane tension, cut-off of the quantum gravity scale and bulk cosmological constant to get sub-Planckian field excursion along with large tensor-to-scalar ratio as recently observed by BICEP2 or at least generates the tensor-to-scalar ratio consistent with the upper bound of Planck (2013 and 2015) data and Planck+BICEP2+Keck Array joint constraint.
It is a very good-old assumption from superstring theory Green1 (); Green2 (); Polchinski2 () that we are living in 11 dimensions and different string field theoretic setups are connected with each other via stringy duality conditions. Among varieties of string theories, the 10-dimensional heterotic string theory is a strong candidate for our real world as the theory may contain the standard model of particle physics and is related to an 11-dimensional theory written on the orbifold . Within this field theoretic setup, the standard model particle species are confined to the 4-dimensional space-time which is the sub-manifold of . On the contrary, the graviton degrees of freedom propagate in the total space-time. In a most simplified situation, one can think about a 5-dimensional problem where the matter fields are confined to the 4-dimensional spacetime while gravity acts in 5 dimensional bulk spacetime Maartens:2010ar (); Brax:2004xh (). Amongst very successful propositions for extra dimensional models, Randall & Sundrum (RS) one brane Randall:1999vf () and two brane Randall:1999ee () models are very famous theoretical prescription in which our observable universe in embedded on 3-brane which is exactly identical to a domain wall in the context of 5-dimensional anti-de Sitter () space-time. Various cosmo-phenomenological consequences along with inflation have been studied from RS setup in refs. Choudhury:2011sq (); Choudhury:2011rz (); Choudhury:2012ib (); Choudhury:2013yg (); Choudhury:2013aqa (); Choudhury:2013eoa (); Choudhury:2014hna (); Das:2013lqa (); Banerjee:2014wea (); Minamitsuji:2005xs (); Himemoto:2000nd (); Mukhopadhyaya:2002jn (); Ghosh:2008vc (); Koley:2005nv ().
The primordial inflation has two key predictions - creating the scalar density perturbations and the tensor perturbations during the accelerated phase of expansion Mukhanov:1981xt (); Mukhanov:1990me (). Very recently, BICEP2 333BICEP2 result was quite recently put into question by several works Liu:2014mpa (); Mortonson:2014bja (); Flauger:2014qra (); Adam:2014gaa (). Also accounting for the contribution of foreground dust will shift the value of tensor-to-scalar ratio downward by an amount which will be better constrained by the joint analysis performed by Planck and BICEP2/Keck Array team Ade:2015tva (). The final result is expressed as a likelihood curve for r, and yields an upper limit at confidence. Marginalizing over dust and , lensing B-modes are detected at significance. Very recently in Ade:2015lrj () the Planck team also fixed the upper bound on the tensor-to-scalar ratio is at C.L. and perfectly consistent with the joint analysis performed by Planck and BICEP2/Keck Array team. Ade:2014xna () team reported the detection of the primordial tensor perturbations through the B-mode polarization as:
where is the tensor-scalar ratio. Explaining this large tensor-to-scalar ratio is a challenging issue for particle cosmologist because of the Lyth bound Lyth:1996im (), one would expect a super-Planckian excursion 444Field excursion of the inflation filed is defined as: , where represent the field value of the inflaton at the momentum scale which satisfies the equality, , where represent the scale factor , Hubble parameter, the conformal time and pivot momentum scale respectively. Also is the field value of the inflaton defined at the end of inflation. Here the super-Planckian excursion is described by, , which is applicable for large filed models of inflation Baumann:2014nda (); Baumann:2009ds (); Linde:2014nna (); Kallosh:2014xwa (); Kallosh:2014rga () and sub-Planckian excursion is characterized by, , which hold good in case of small field models of inflation Lyth:1998xn (); Mazumdar:2010sa (); Martin:2014vha (); Martin:2013nzq (). of the inflaton field in order to generate large tensor-to-scalar ratio. It is important to mention here that super-Planckian field excursion computed from the inflationary paradigm is necessarily required to embed the setup with effective field theory description 555In case of super-Planckian field excursion it is necessarily required to introduce the higher order quantum corrections including the effect of higher derivative interactions appearing through the local modifications to GR plays significant role in this context Choudhury:2013yg (). For an example, within 4D Effective Field Theory picture incorporating the local corrections in GR one can write the action as, In this case the appropriate choice of the co-efficients of the correction factors would modify the UV behaviour of gravity. But such local modification of the renormalizable version of GR typically contain debris like massive ghosts which cannot be regularized or avoided using any field theoretic prescriptions. If the quantum correction to the usual classical theory of gravity represented via Einstein-Hilbert term is dominated by higher derivative nonlocal corrections Chialva:2014rla (); Biswas:2011ar (); Biswas:2013cha () then one can avoid such ghost degrees of freedom, as the role of these corrections are significant in super-Planckian (or trans-Planckian) scale to make the theory UV complete Chialva:2014rla (). For an example, within 4D Effective Field Theory picture incorporating the non-local corrections in the gravity sector one can write the action as Biswas:2011ar (): where are analytic entire functions containing higher derivatives up to infinite order, where is the 4D d’Alembertian operator. On the other hand in the matter sector incorporating the effects of quantum correction through the interaction between heavy and light (inflaton) field sector and finally integrating out the heavy degrees of freedom from the 4D Effective Field Theory picture the matter action, which admits a systematic expansion within the light inflaton sector can be written as Baumann:2014nda (); Assassi:2013gxa (): where are dimensionless Wilson coefficients that depend on the couplings g of the UV theory, and are local operators of dimension . This procedure typically generates all possible effective operators consistent with the symmetries of the UV theory. Also and describe the part of total Lagrangian density involving only the light and heavy fields, and includes all possible interactions involving both sets of fields within Effective Field Theory prescription. After removal of heavy degrees of freedom the effective action is splitted into a renormalizable part: and a sum of non-renormalizable corrections appearing through the operators . Such operators of dimensions less than four are called “relevant operators”. They dominate in the IR and become small in the UV. In 4D Effective Field Theory the operators of dimensions greater than four are called irrelevant operators. These operators become small in the IR regime, but dominate in the UV end. However such corrections are extremely hard to compute and at the same time the theoretical origin of all such corrections is not at all clear till now as it completely belongs to the hidden sector of the theory Assassi:2013gxa (). One of the possibilities of the origin of such hidden sector heavy field is higher dimensional Superstring Theory or its low energy supergravity version. Such a higher dimension setups dimensionally reduced to the 4D Effective Field Theory version via various compactifications. In such a case the corrections arising from graviton loops will always be weighted by the UV cut-off scale which is fixed at Planck scale , while those coming from heavy sector fields will be suppressed by the background scale of heavy physics relevant for those fields , where . Present observational status suggests that the scale of such hidden scale is constrained around the GUT scale ( GeV) Choudhury:2014kma (); Choudhury:2014wsa (). In this connection Randall-Sundrum (RS) model is one of possible remedies to solve the trans-Planckian problem of field excursion as the 5D cut-off scale of such theory (see section II for details) is one order smaller than the 4D cut-off scale of the Effective Field Theory i.e. the Planck scale to explain the latest ATLAS bound on the lightest graviton mass and the Higgs mass within the estimated125 GeV against large radiative correction upto the cut-off of the Model Das:2013lqa () in the phenomenological ground. In this work using model independent semi-analytical analysis within inflationary setup we have explicitly shown that 5D cut-off of RS model is also one order smaller than the 4D cut-off scale (see section III for details). This also suggests that within RS setup the higher order quantum corrections appearing in the gravity as well in the matter sector of the theory is very small in the 4D Effective Field Theory version. During our analysis we have further taking an ansatz where the non-renormalizable 4D Planck scale suppressed effective operators only modify the effective potential. Consequently with the renormalizable part of the potential such corrections will add and finally give rise to the total potential as stated in Eq (13). . At present it is a very challenging task for the theoretical physicists to propose a new mechanism or technique through which it is possible to accommodate sub-Planckian inflation to generate large tensor-to-scalar ratio. The first possibility of addressing this issue is to incorporate the features of spectral tilt, running and running of the running by modifying the scale invariant power spectrum. Obviously, the current data can also be explained by the sub-Planckian excursion of the inflaton field in the context of single field inflation as discussed in Choudhury:2013iaa (); Choudhury:2014kma (); Choudhury:2014wsa (); Choudhury:2013woa (); Choudhury:2014hua (), where in these class of models sufficient amount of running and running of the running in tensor-to-scalar ratio has been taken care of. A small class of potentials inspired from particle physics phenomenology i.e. high scale models of inflation in the context of MSSM, MSSM etc Choudhury:2013jya (); Choudhury:2014sxa (); Choudhury:2014uxa (); Choudhury:2011jt () will serve this purpose. The next possibility is modified gravity or beyond General Relativistic (GR) framework through which it is possible to address this crucial issue within single field inflationary scenario where the effective field theory description holds perfectly. The prime motivation of this work to show explicitly how one can address this issue in beyond GR prescription. In this work I investigate the possibility for RS single brane setup in which one can generate large tensor-to-scalar ratio along with sub-Planckian field excursion from a large class of models of inflation within effective field theory prescription Baumann:2014nda (); Baumann:2009ds (); Cheung:2007st (); Weinberg:2008hq (); Tsujikawa:2014mba (); Senatore:2010wk (); LopezNacir:2011kk (); Agarwal:2013rva (); Assassi:2013gxa (); Baumann:2011nm (); Creminelli:2013xfa (); Khosravi:2012qg (), and within this setup it is feasible to describe a system through the lowest dimension operators compatible with the underlying symmetries 666Assisted inflation Liddle:1998jc (); Copeland:1999cs (); Kanti:1999ie (); Kanti:1999vt (); Mazumdar:2001mm (); Green:1999vv (); Malik:1998gy () and N-flation Dimopoulos:2005ac (); Cicoli:2014sva (); Easther:2005zr () within multi-field inflationary description, asymptotically free gravity Chialva:2014rla (); Biswas:2011ar (); Biswas:2013cha (); Stelle:1976gc (); Stelle:1977ry (); Nunez:2004ts (), shift symmetry Brax:2005jv (); Choudhury:2013zna () are the various possibilities in which it is possible to achieve sub-Planckian field excursion along with large tensor-to-scalar ratio and finally the trans-Planckian field excursion issue can be resolved within Effective Field Theory prescription..
In this paper, I derive the direct connection between field excursion and tensor-so-scalar ratio in the context of effective theory inflation within Randall-Sundrum (RS) braneworld scenario in a model independent fashion. For clarity in the present context the bulk space-time is assumed to have 5 dimensions. By explicit computation I have shown that the effective field theory of brane inflation within RS setup is consistent with sub-Planckian VEV and field excursion, which will further generate large value of tensor-to-scalar ratio when the energy density for inflaton degrees of freedom is high enough as compared to the visible and hidden brane tensions in high energy regime. Last but not the least, I have mentioned the stringent constraint condition on positive brane tension as well as on the cut-off of the quantum gravity scale to get sub-Planckian field excursion along with large tensor-to-scalar ratio.
Ii Brane inflation within Radall-Sundrum single brane setup
Let me start the discussion with a very brief introduction to RS single brane setup. The RS single brane setup and its generalized version from a Minkowski brane to a Friedmann- Robertson-Walker (FRW) brane were derived as solutions in specific choice of coordinates of the 5D Einstein equations in the bulk, along with the junction conditions which are applied at the -symmetric single brane. A broader perspective, with non-compact dimensions, can be obtained via the well known covariant Shiromizu-Maeda-Sasaki approach Shiromizu:1999wj (), in which the brane and bulk metrics take its generalized structure. The key point is to use the Gauss-Codazzi equations to project the 5D bulk curvature along the brane using the covariant formalism. Here I start with the well known 5D Rundall Sundrum (RS) single brane model action given by Randall:1999vf ():
where the extra dimension “y” is non-compact for which the covariant formalism is applicable. Here be the 5D quantum gravity cut-off scale, be the 5D bulk cosmological constant, be the bulk field Lagrangian density, signifies the Lagrangian density for the brane field contents. It is important to mention that the the scalar inflaton degrees of freedom is embedded on the 3 brane which has a positive brane tension and it is localized at the position of orbifold point in case of single brane. The 5D field equations in the bulk, including explicitly the contribution of the RS single brane is given by Shiromizu:1999wj (); Maartens:2010ar ():
where characterizes any 5D energy-momentum tensor of the gravitational sector within bulk specetime. On the other hand the total energy-momentum tensor on the brane is given by: where is the energy-momentum tensor of particles and fields confined to the single brane. Further applying the well known Israel–Darmois junction conditions at the brane Shiromizu:1999wj (); Maartens:2010ar () finally one can arrive at the -dimensional Einstein induced field equations on the single brane given by Shiromizu:1999wj (); Maartens:2010ar (); Brax:2004xh ():
where represents the energy-momentum on the single brane, is a rank-2 tensor that contains contributions that are quadratic in the energy momentum tensor Shiromizu:1999wj (); Maartens:2010ar () and characterizes the projection of the 5-dimensional Weyl tensor on the 3-brane and physically equivalent to the non-local contributions to the pressure and energy flux for a perfect fluid Shiromizu:1999wj (); Maartens:2010ar (); Brax:2004xh ().
In a cosmological framework, where the 3-brane resembles our universe and the metric projected onto the brane is an homogeneous and isotropic flat Friedmann-Robertson-Walker (FRW) metric, the Friedmann equation becomes Shiromizu:1999wj (); Maartens:2010ar (); Brax:2004xh ():
where is the 3-brane tension. Within RS setup the quantum gravity cut-off scale i.e. the 5D Planck mass and effective 4D Planck mass are connected through the visible brane tension as:
Assuming that, as required by observations, the 4D cosmological constant is negligible in the early universe the localized visible brane tension is given by:
where be the scaled 5D bulk cosmological constant defined as:
where be the positive brane tension, signifies the energy density of the inflaton field and be the reduced 4D Planck mass. Using Eq (8) in Eq (7), the 5D quantum gravity cut-off scale can be expressed in terms of 5D cosmological constant as:
In the low energy limit in which standard GR framework can be retrieved. On the other hand, in the high energy regime as the effect of braneworld correction factor is dominant which is my present focus in this paper. Consequently in high energy limit , Eq (10) is written using the slow-roll approximation as:
where be the inflaton single field potential which is expanded in a Taylor series around an intermediate field value 777Here and represent the inflaton field value at the starting point of inflation and at the end of inflation. as:
where denotes the height of the potential, and the coefficients: , determine the shape of the potential in terms of the model parameters. The prime denotes the derivative w.r.t. . Here as a special case one can consider a situation where the intermediate field value is identified with the VEV of the inflaton field field i.e.
where be the Bunch-Davies vacuum state using which the VEV is computed in curved space-time. In a most simplest case the numerical value of the VEV is computed from the flatness condition:
provided . In a more advanced situation where inflation is driven by saddle point and inflection point, one can impose the flatness constraint on the potential as:
for inflection point Choudhury:2013jya (); Choudhury:2014sxa (); Choudhury:2014uxa (); Allahverdi:2006we () 888The present observational data from Planck and BICEP2 prefers the inflection point models of inflation compared to the saddle point, as the predicted value for the scalar spectral tilt obtained from saddle point inflationary models is low.. Moreover here it is important mention that the inflaton field belongs to the the visible sector of RS setup in which effective field theory prescription perfectly holds good. Even for zero VEV of the inflaton, , Eq (13) also holds good. One can further simplify the expression for the potential by applying symmetry in the inflaton field as:
where the expansion co-efficients are defined as:
where corresponds to the field value at the end of inflation which can be obtained from the following equation:
In terms of the momentum, the number of e-foldings, , can be expressed as Burgess:2005sb ():
where is the energy density at the end of inflation, is an energy scale during reheating, is the present Hubble scale, corresponds to the potential energy when the relevant modes left the Hubble patch during inflation corresponding to the momentum scale , and characterises the effective equation of state parameter between the end of inflation, and the energy scale during reheating. Within the momentum interval, , the corresponding number of e-foldings is given by, , as
where and represent the scale factor and the Hubble parameter at the CMB scale and end of inflation. One can estimate the contribution of the last term of the right hand side by using Eq (13) as:
where and represent two series sum given by:
where the field excursion is defined as, , where and signify the inflaton field value at the at the last scattering surface (LSS) of CMB or more precisely at the horizon crossing 999Here horizon crossing stands for the physical situation where the corresponding momentum scale satisfies the equality , where be the associated wavelength of the scalar and tensor modes whose snapshot are observed at the LSS of CMB. After crossing the horizon all such modes goes to the super-Hubble region in which the momentum scale i.e. , which implies the corresponding wavelengths of the scalar snd tensor modes are too small to be detected. On the other hand, before the horizon crossing there will be region in a smooth patch within sub-Hubble region where the corresponding momentum scale i.e. , which can be detected via various observational probes. and at the end of inflation respectively. Now I explicitly show that both of the series sum are convergent in the present context. To hold the effective field theory prescription one need to satisfy the following sets of criteria:
This implies that, both and are convergent and from Eq (36) we get:
which perfectly holds good for zero VEV inflaton case. Let us investigate the symmetric case in which one can write:
where and represent two series sum given by:
Here also the similar criteria hold good to apply the effective field theory prescription which make the series sum and convergent. Consequently, for all the physical situations described in this paper Eq (31) reduces to:
Iii Field excursion within effective theory description
In the high energy limit of RS braneworld the tensor-to-scalar ratio satisfies the following consistency condition at the leading order of the effective field theory:
where and are the scalar and tensor power spectrum at any scale . It is important to note that the following operator relationship holds good in the high energy limit of RS braneworld:
In Eq (40) the tensor-to-scalar ratio can be parametrized at any arbitrary momentum scale as:
where be the pivot scale of momentum. In Eq (42) the subscript signifies the tensor and scalar modes obtained from cosmological perturbation in RS braneworld. Here , and represent the tensor and scalar spectral tilt, running and running of the running in RS braneworld respectively. See appendix where all these definitions are explicitly given. Also in Eq (42) I mention four possibilities as given by:
Case I stands for a situation where the spectrum is scale invariant,
Case II stands for a situation where spectrum follows power law feature through the spectral tilt ,
Case III signifies a situation where the spectrum shows deviation from power low in presence of running of the spectral tilt along with logarithmic correction in the momentum scale (as appearing in the exponent) and
Case IV characterizes a physical situation in which the spectrum is further modified compared to the Case III, by allowing running of the running of spectral tilt along with square of the momentum dependent logarithmic correction.
Here all the observables appearing in the left side of Eq (44) can also be expressed in terms of the slow-roll parameters in RS single braneworld. See the appendix for details. Further using the the limiting results on I get:
Most importantly Eq (99) and Eq (100) fix the value of within the desired range demanded by the observational probes. This can be easily done by putting constraint on the brane tension of the single brane and the Taylor expansion co-efficients of the effective potential within RS setup. Also this makes the analysis consistent presented in this paper. Further from Eq (99) and Eq (100) one can write the field excursion for the both the physical situations as: