DBI Galileon inflation in background SUGRA

We introduce a model of potential driven DBI Galileon inflation in background N=1,D=4 SUGRA. Starting from D4-$\bar{D4}$ brane-antibrane in the bulk N=2,D=5 SUGRA including quadratic Gauss-Bonnet corrections, we derive an effective N=1,D=4 SUGRA by dimensional reduction, that results in a Coleman-Weinberg type Galileon potential. We employ this potential in modeling inflation and in subsequent study of primordial quantum fluctuations for scalar and tensor modes. Further, we estimate the major observable parameters in both de Sitter (DS) and beyond de Sitter (BDS) limits and confront them with recent observational data from WMAP7 by using the publicly available code CAMB.

In the very recent days a good number of theoretical physicists have devoted their attention to the development of consistent modified gravity theories which play analogous role as dark energy or the cosmological constant [1], [2]. In higher-dimensional setups as in the case of DGP model [3], [4] where self-acceleration is sourced by a scalar field (the zero helicity mode of the 5D graviton), these types of Infra-Red (IR) modification of gravity [5], [6] play a crucial role. Moreover, the DGP model replicates the general relativistic features due to non-linear interactions via the well known Vainshtein mechanism [7]- [9]. Despite its profound success it has got some serious limitations [8], [10], which are resolved by introducing a dynamical field, aka, Galileon [11]- [13] arising on the brane from the bulk in the DGP setup. The cosmological consequences of the Galilieon models have been studied to some extent in [14]- [16]. Very recently, a natural extension to the scenario has been brought forward by tagging Galileon with the good old DBI model [17], [18], resulting in so-called "DBI Galileon" [19]- [21], that has reflected a rich structure from four dimensional cosmological point of view. However, in most of the physical situations, this type of effective gravity theories are plagued with additional degrees of freedom which often results in unwanted debris like ghosts, Laplacian instabilities etc [22], [23]. In the present paper we introduce a single scalar field model described by the D3 DBI Galileon originated from D4-D4 brane anti-brane setup. This prevents the framework from having extra degrees of freedom as well as Ostrogradski instabilities [24], resulting in a higher-order derivative scalar field theory free from any such unwanted instabilities. Nevertheless, a consistent field theoretic derivation of the effective potential commonly used in the context of DBI Galileon cosmology has not been brought forth so far. On top of that, it is imperative to point out that the SUGRA origin of D3 DBI Galileon is yet to be addressed. In this article we plan to address both of these issues explicitly by deriving the inflaton potential from our proposed framework of DBI Galileon. It turns out that the derived action includes, in certain limits, the decoupling limit of DGP model as well as some consistent theories of massive gravity; and it also includes the "K-mouflage" [25], [26] and also "G/KGB" inflation [16], [27]- [29]. Moreover, in general appearance of non-vanishing superstringy frame functions in the 4D action expedites breakdown of shift symmetry. Without shift symmetry, it may happen that the theory is unstable against large renormalization. The background action chosen in our model preserves shift symmetry of the single scalar field which gives it a firm footing from phenomenological point of view as well.
The plan of the paper is as follows. First we propose a fairly general framework by taking the full DBI action in D4 brane in the background of bulk N =2, D=5 supergravity [30]- [32] including the quadratic modification in Einstein's Hilbert action via Gauss-Bonnet correction in the bulk. Hence, using dimensional reduction technique, we derive the effective action for DBI Galileon in D3 brane in the background of N =1, D=4 supergravity induced by the quadratic correction in the geometry sector and study cosmological inflationary scenario therefrom. We next engage ourselves in studying quantum fluctuations, by employing second order perturbative action for scalar and tensor modes in de Sitter (DS) and beyond de Sitter (BDS) limits, and hence calculate the primordial power spectrum of the scalar and tensor modes, their running and other observable parameters. We further confront our model with observations by using the publicly available code CAMB [33], and find them to fit well with latest observational data from WMAP7 [34] and expected to fit fair well with upcoming data from PLANCK [35].

II. THE BACKGROUND ACTION
Let us demonstrate briefly the construction of DBI Galileon starting from N =2,D=5 SUGRA along with Gauss Bonnet correction in D4 brane set up. The full five dimensional model is described by the following action The D4 brane action decomposed into two parts as where the DBI action and the Wess-Zumino action are given by respectively where T (4) is the D4 brane tension, α ′ is the Regge Slope, exp(−Φ) is the closed string dilaton and C 0 is the Axion.
The gauge invariant combination of rank 2 field strength tensor, appearing in D4 brane, is F AB = B AB + 2πα ′ F AB and {F 2 , B 2 } represents 2-form U (1) gauge fields. In eqn(2.1) N = 2, D = 5 bulk supergravity action can be written as [30] S (5) In this context the 5-dimensional coordinates X A = (x α , y), where y parameterizes the extra dimension compactified on the closed interval [−πR, +πR] and Z 2 symmetry is imposed. We consider 5D Yang Mills SUGRA model which is described by the field content {em µ , Ψ ĩ µ , A Ĩ µ , λ ia , Φ x } whereμ = (µ, 5) are curved andm = m,5 are flat 5D indices with µ, m their corresponding 4D indices. The remaining indices are I = 0, 1, ....., n, a = 1, 2, ....., n and x = 1, 2, ...., n. The SUGRA multiplet consists of the fünfbien em µ , two graviphoton A 0 µ and two gravitini Ψ ĩ µ , where i = 1, 2 is the simplectic SU (2) R index. Moreover, there exists n vector multiplets, counting the Yang Mills fields (A ã µ ). The spinor and the scalar fields included in the vector multiplets are collectively denoted by λ ia , Φ x respectively. The indices a and x are flat and curved indices respectively of the 5D manifold M. It is important to mention here that the Chern-Simons terms can be gauged away assuming cubic constraints and Z 2 symmetry. Now we consider full particle spectrum , the Z 2 even fields {e m µ , e5 5 , For computational purpose it is useful to define the five dimensional generating function(G) of supergravity in this setup as where the supergravity Kähler moduli fields are given by T = which is assumed to be stabilized under first approximation and K(Φ, Φ † ) represents generalized Kähler function.
Including the kinetic term of the five dimensional field Φ and rearranging into a perfect square, the 5D bulk supergravity action can be expressed as where the 5D potential The field equations in presence of Gauss-Bonnet term can be expressed as AB , (2.11) where the covariantly conserved Gauss-Bonnet tensor AB R ABCD(5) R (2.12) which acts as a source term. It is useful to introduce the 5D metric in conformal form h(y) ds 2 4 + h(y)G(y)dy 2 = exp(2A(y)) ds 2 4 + R 2 β 2 dy 2 , (2.13) with warp factor exp(2A(y)) = 1 (2.14) and ds 2 4 = g αβ dx α dx β is FLRW counterpart. In order to write down explicitly the expression for D4 brane action, the induced metric can be shown as The 5D energy momentum tensor for the set up reads (2.17) On the other hand, the Klein-Gordon equation of motion in 5D can be expressed as Now using the scaling relations Φ A = T (4)Φ A , G AB = exp(−Φ)g AB and b AB = √ h(y) T (4) B AB the 5D action for D4 brane can be expressed in more convenient form as Here we use the fact that no spatial direction along which the scalar fields are only time dependent lead to B µ ν = 0 and F µν = 0 in the background. Consequently Maxwell's field equations are unaffected in 4D after dimensional reduction. In this context the 5D D4 brane potential is given by where Klebanov Strassler warped geometry motivated frame function f (Φ) = exp(Φ)h(y) is originated from Coulomb interaction between D4-D4 brane and the implicit brane function G(Φ, X) =

III. GENERATION OF D3 DBI GALILEON
In this section we employ dimensional reduction technique to derive a N =1, D=4 SUGRA and the inflaton potential therefrom that results in DBI Galileon on the D3 brane. For convenience we deal with different contributions to the action (2.1) separately.

A. The Einstein-Hilbert Action
After integrating out the contribution from the five dimension, the Einstein Hilbert action in four dimension can be written as 1) where the explicit expression for I(1) is mentioned in appendix. It is important to mention here that the 5D Planck mass and 4D Planck mass are related through T (4) .

C. The D3 Brane Action
To reduce the D4 brane action we employ the method of separation of variable Φ(X A ) = Φ(x µ , y) =φ(x µ )χ(y) where χ(y) = exp( 2πiy R ). Consequently the D3 brane action turns out to be The effective Klebanov Strassler and Coulomb frame function are hereby expressed asf (φ) ≃ 1 (f0+f2φ 2 +f4φ 4 ) and ν (4) . The outcome of dimensional reduction is reflected through the constants mentioned in the appendix. The scaled D3 brane potential turns out to beṼ where the D3 brane tension is given by Further, imposing Z 2 symmetry toφ via Φ(0) = Φ(πR) = 0 and compactifying around a circle (S 1 ) Now using the above mentioned ansatz for method of separation of variable we get for superpotential. Hence, the N = 1, D = 4 supergravity F-term potential turns out to be (3.11) Now using the ansatz for the Kähler potential K(φ exp( 2πiy , exp(− 2πiy R )) = 1 eqn(3.11) reduces to the following form: with the general Kähler metric K αβ In most of the simple situations, we are interested in the Canonical metric structure defined by K αβ 1 = δ αβ . Consequently the N = 1, D = 4 SUGRA action turns out to be where the canonical F-term potential can be recast as To derive the expression for inflaton potential we start with a specific superpotential [36] introduced as an inflaton with n ≥ 2. Here g(∼ O (1)) is the positive and real coupling constant and v is the VEV of the fieldφ. In this model U (1 ) R symmetry is dynamically broken to a discrete Z 2n R symmetry at the scale v << 1. Consequently the inflaton transforms asφ −→ exp 2αi This leads to the following form of the bulk contribution to the potential It has a minimum at |φ| min ≃ . However, in the context of SUGRA, we may interpret that such negative potential energy is almost canceled by positive contribution due to the local supersymmetry breaking, Λ 4 SUGRA , and that the residual positive energy density is responsible for the present dark energy. Then, we can relate the energy scale of this model to the gravitino mass as m 3/2 ≃ n n+1 v 2 g 1 n v 2 . Identifying the real part of φ with the inflaton φ ≡ √ 2Reφ, the dynamics of the inflaton is governed by the following potential, Imposing renormalization condition, here we restrict ourselves to n = 2 leading to effective N = 1, D = 4 SUGRA potential (3.17)
The collective effect of equation(3.8) and equation (3.17) gives the total D3 DBI Galileon potential as: are tree level constants. Now including the contribution from one-loop radiative correction, the renormalizable potential is as under where D 0 = 0. It is the Coleman Weinberg potential [37], provided the coupling constant satisfies the Gellmann-Low equation [38] in the context of Renormalization group. Here the first term in the eqn(4.4) physically represents the energy scale of inflation ( 4 √ C 0 ).
The number of e-foldings for D3 DBI Galileon can be expressed as where φ i and φ f are the corresponding values of the inflaton field at the beginning and end of inflation.

V. QUANTUM FLUCTUATIONS AND OBSERVABLE PARAMETERS
Let us now engage ourselves in analyzing quantum fluctuation in our model and its observational imprints via primordial spectra generated from cosmological perturbation. To serve this purpose, following the well known ADM formalism, the action can be expanded in the second order as: where ζ comes from the ADM metric at the linear level and carries the signature of curvature perturbation. In equation(5.1) where c 2 s takes into account the nontrivial modification due to Galileon. Similarly for tensor modes, equation(5.1) can be recast as: where Following the same prescription we can establish equation(5.3) for tensor modes provided c s is replaced by c T . The Bunch-Davies mode function turns out to be (Throughout the paper we have used DS for de-Sitter results and BDS for beyond de-Sitter results.) and in the super-Hubble limit H  . Further we have introduced five new parameters during the analysis of primordial quantum fluctuation defined as: Now using eqn(5.6) the two-point correlation function for scalar modes can be expressed as: where the dimensionless Power spectrum for scalar modes P ζ (k) at the horizon crossing turns out to be: ⋆ corresponds to the horizon crossing. Similarly the dimensionless Power spectrum for tensor modes reads: (5.14) Consequently the ratio of tensor to scalar power spectrum can be expressed as: Further, the scale dependence of the perturbations, described by the scalar and tensor spectral indices, as follows: Consistently, the consistency relation is also modified to: The expressions for the running of the scalar and tensor spectral index in this specific model with respect to the logarithmic pivot scale at the horizon crossing are given by: (5.20) Here we have used a shorthand notation ab = a ′ b − ab ′ where ′ = d dφ . Figure(3(a)) represents the scale dependent power spectrum ( P ζ ) with respect to the scalar spectral index(n ζ ). It directly shows that both the DS and BDS analysis follow the same characteristics but the estimated windows for the observational parameters (P ζ , n ζ ) are slightly different, but both of them are within the observational bound. Figure(3(b)) shows the characteristic differences between the behavior of DS and BDS scale dependent power spectrum with respect to the momentum scale (k). Here DS behavior is quasi-statically flat, but BDS characteristics is rapidly increasing with respect to the scale. In figure(3(c)) we have plotted the the scale dependent tensor to scalar ratio for DS and BDS limit. Most significantly they show complementary characteristics with the scale and intersects at a point where both the analysis will be equivalent. In the next section, we will estimate these parameters by confronting the results directly to WMAP7 results.

VI. PARAMETER ESTIMATION AND CONFRONTATION WITH WMAP7
Using the parameter space for the model parameters (C i , D i ) we have estimated the window of the cosmological parameters from our model which confronts observational data well in 56 < N < 70. In Table(I) we have tabulated the relevant observational parameters estimated from our model for both DS and BDS limit.   Further, we use the publicly available code CAMB [33] to verify our results directly with observation. To operate CAMB at the pivot scale k 0 = 0.002M pc −1 the values of the initial parameter space are taken for lower bound of C ′ i s and N = 70. Additionally WMAP7 years dataset for ΛCDM background has been used in CAMB to obtain CMB angular power spectrum. In Table(II) we have given all the input parameters for CAMB. Table(III) shows the CAMB output, which is in good agreement with WMAP7 [34] data. In figure(4)

VII. SUMMARY AND OUTLOOK
In this article we have proposed a model of single field inflation in the context of DBI Galileon cosmology in D3 brane. We have demonstrated the technical details of construction mechanism of an one-loop 4D inflationary potential via dimensional reduction starting from D4 brane in N =2,D=5 SUGRA including the quadratic Gauss-Bonnet correction term that leads to an effective N =1,D=4 SUGRA in the D3 brane, which is precisely the DBI Galileon in our framework. Hence we have studied inflation using the one loop effective potential by estimating the observable parameters originated from primordial quantum fluctuation for scalar and tensor modes, in the de-Sitter An interesting open issue in this context is to study the primordial non-Gaussian features of DBI Galileon introduced in the present article. As has been pointed out recently [39]- [41] there is a tension between bispectrum (f N L ) and tensor-to-scalar ratio (r) in DBI inflation, which is a generic sensitivity problem. It will be interesting to investigate whether our proposed framework of DBI Galileon can resolve this issue. We are already in progress in this direction and have obtained some interesting results. A detailed report on this issue will be brought forth shortly.