No scale SUGRA SO(10) derived Starobinsky Model of Inflation

We show that a supersymmetric renormalizable theory based on gauge group SO(10) and Higgs system {\bf {10 $\oplus$ 210 $\oplus$ 126 $\oplus$ $\overline{\bf 126}$}} with no scale supergravity can lead to a Starobinsky kind of potential for inflation. Successful inflation is possible in the cases where the potential during inflation corresponds to $SU(3)_C \times SU(2)_L \times SU(2)_R \times U(1)_{B-L}$, $SU(5)\times U(1)$ and flipped $SU(5)\times U(1)$ intermediate symmetry with a suitable choice of superpotential parameters. The reheating in such a scenario can occur via non perturbative decay of inflaton i.e. through"preheating". After the end of reheating, when universe cools down, the finite temperature potential can have a minimum which corresponds to MSSM.


I. INTRODUCTION
The theory of cosmological inflation [1][2][3] not only solves the problems (flatness, horizon etc.) of standard big bang theory, but also explains the seed fluctuations which can grow via gravitational instability to form the large scale structure of the universe [4]. There are stringent constraints on inflationary theories from CMB observations [5][6][7][8] and many of the generic models like the quartic potential and quadratic potential are either ruled or disfavored by the bound on the tensor to scalar ratio which is r 0.05 < 0.12 at 95% CL from joint analysis of BICEP2/Keck array and Planck data [9].
Among the generic inflation models which survive the stringent constraint on r is R 2 inflation model of Starobinsky [1] which predicts a n s − 1 = −2/N and r = 12/N 2 ∼ 0.002 − 0.004. The theoretical motivation for the Starobinsky model is provided in [10] where it was shown that the Starobinsky can be derived from supergravity (SUGRA) with a no-scale [11][12][13] Kähler potential and a Wess Zumino superpotential with specific couplings. Supergravity models of inflation based on the Jordan frame supergravity [14][15][16] and D-term superpotential [17] also give inflation potentials which are identical to the Starobinsky potential at large field values. The natural choice for the inflaton in supergravity models are the Higgs fields of the grand unified theories. A no-scale SUGRA model of inflation based on the SU(5) GUT using the 24, 5 and 5 Higgs in the superpotential has been constructed [18]. The SU(5) symmetry breaks to MSSM with the appropriate choice of vevs for the 24 and a D-flat linear combination of H u and H d of MSSM acts as the inflaton [18].
In the present work we study inflation in a renormalizable grand unified theory based on the SO(10) gauge group with no scale SUGRA. Inflation in the context of Susy SO (10) has been studied eariler in [19][20][21][22][23] with the SO(10) invariant superpotential with minimal Kähler potential which gives polynimial potentials of inflation. However we show that a renormalizable Wess-Zumino superpotential of SO(10) GUT along with no-scale Kähler potential can give us Starobinsky kind inflation potential. The Higgs supermultiplets we consider are 10, 210, 126 (126). Among these the 210 and 126 (126) are responsible for breaking of SO(10) symmetry down to MSSM. 210 alone can give different intermediate symmetries [24] depending upon the which of its MSSM singlet field is given a vev. The minimal supersymmetric grand unified theory based on SO(10) gauge group [24][25][26][27][28] has 10(H i ), 210(Φ ijkl ) and 126(Σ ijklm )(126(Σ ijklm )) as Higgs supermultiplets. The representations H i is 1 index real, 126(Σ ijklm ) is complex (5 index, totally-antisymmetric, self dual) and 210(Φ ijkl ) is 4 index totally-antisymmetric tensor. Here i,j,k,l,m =1,2...10 run over the vector representation of SO (10). The renormalizable superpotential for the above mentioned fields is given as The No-scale form of Kähler potential we take is The 10 and 126 are required for Yukawa terms to give masses to the fermions while 126 (126) breaks the SO (10) The fields which will not break the MSSM symmetry are allowed to take vev. In this case they are The vanishing of D-terms gives the condition |σ| = |σ| [28]. The symmetry breaking path of SO (10) is given as 2. If p = 0 and a=ω=0, this results to SU (4) C × SU (2) L × SU (2) R symmetry.
The superpotential in terms of vevs of 210 is given as Here m = m Φ . Similarly no-scale Kähler potential is Here T is the single modulus field arising due to string compactification and we are taking M pl =1.
The F-term potential has the following form, Where The kinetic term is given as K j * i ∂φ i ∂φ j * . Here i runs over different fields T,p,a and ω. K i j * is the inverse of Kähler metric K j * i which is given as Where Γ = T + T * − 1 3 (|p| 2 + 3|a| 2 + 6|ω| 2 ). After simplifying the potential (7) has the following form, shown. The inflation potential is along χ 1 direction. In Fig.1a we show V(χ 1 ,χ 2 =0,χ 3 ) and in We assume that the non-perturbative Planck scale dynamics [10,18,30] fixes the vlaue of T = T * = 1 2 . After fixing the vev for T the kinetic terms of T and its mixed terms with other fields can be neglected. We then study all the possible cases of symmetry breaking mentioned eariler to study inflation in SO(10) with no-scale SUGRA. For simplicity we assume our fields to be real. The Kinetic energy term is given as To get the canonical K.E. terms we need to do the transformation to new fields χ 1 , χ 2 , χ 3 .
The potential V(χ 1 , χ 2 , χ 3 ) is flat along χ 1 direction for χ 2 =χ 3 =0 and is confining in the orthogonal (χ 2 , χ 3 ) directions as shown in Fig.1. So the potential in the limit χ 1 = 0, If we take λ= -m it gives us the Starobinsky type of potential. The potential in this specific case is, The slow roll parameters for this potential are given as Inflation ends when η ≈ 1, which corresponds to field value of χ 1 ≈ .5. To have sufficient inflation which corresponds to N e−f olds =55 gives the field value χ 1 ≈ 4.35. The power spectrum for scaler perturbation P R is given as The value of P R = (1.610±0.01)×10 −9 given by Planck [7] requires value of m = 1.311×10 −6 in Planck units. The spectral index n s =.964 and tensor to scalar perturbation ratio r=.002 for N e−f olds =55.
In this case the old to new field tranformation is given as, Then the potential in the limit χ 1 = 0, This type of potential increases exponentially with χ 1 and too steep to obey the slow roll conditions. The value of spectral index n s has negative value over a wide range of field value, so doesn't satisfy the inflationary constraints on scale invariance of scalar perturbations from observations.
• Case III: ω = 0 and p=a=0, SU In this case the old to new field tranformation is given as, Then the potential in the limit χ 1 = 0, Here α = 5λ 2 /8m 2 . In this case for α ≥ −1 potential increases exponentially with χ 1 so give similar results to case II. For α <-1 potential energy becomes negative for χ 1 > ∼ 1 and grows with large values of χ 1 .
In this case we take p=a=±ω=x, then the K.E. term and potential is given as After making the K.E. term canonical with transformation x = 3 10 tanh χ 1 √ 3 , the potential we get is for λ = − 1 3 10 3 m, which is Starobinsky kind of potential for inflation with different relation among superpotential parameters m and λ than the case I. In this case value of m=1.06 ×10 −6 is required to satisfy the constraints from CMB observations mentioned in case I.
At the end of inflation the preheating [31] can occur via non perturbative decay of inflaton χ 1 to scalar bosons which have a trilinear term with Φ in superpotential e.g. ΦH(γΣ +γΣ). Then the K Σ * Σ |W Σ | 2 and KΣ * Σ |WΣ| 2 type of terms gives us Near the origin sinh[ χ 1 The 210 inflaton produces the 10 and 126 which have Yukawa couplings with the fermions and will decay into the SM fermions and the right-handed neutrino to give a radiation dominated universe at the end of inflation. γ,γ are free parameters of the superpotential, so they can be set to have effective preheating with reheat temperature T R ∼ 10 13 GeV needed for leptogenesis [32].
During inflation Susy is broken at Planck scale and we need to restore Susy at the TeV scales to explain the light Higgs of the standard model. This can be done if the universe finds a minimum where the field values are given by [24] a = m λ Where x is the solution of following cubic equation In case of SUGRA an additional condition of W=0 is required to have MSSM symmetry. Putting these field values in the expression of superpotential and setting W = 0 gives us a fix value of x.
Since all the vevs are in units of m/λ so they can be of O(M pl ) from inflation conditions. However the main requirement of MSSM is a pair of light Higgs. In the present scenario we have a 4 × 4 mass matrix H of MSSM Higgs doublets [33]. However after making the kinetic terms canonical the fields in the mass matrix will be in new basis say χ 1 , χ 2 etc., but the form of mass matrix remains similar to given in [33], One out of the four Higgs doublets can be made light with the fine tuning condition of DetH=0.