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Physics Letters B 556 (2003) 97-104

www. elsevier. com/locate/npe

GUT scale and leptogenesis from 5D inflation

Bumseok Kyae, Qaisar Shafi

Bartol Research Institute, University of Delaware, Newark, DE 19716, USA Received 14 November 2002; received in revised form 20 January 2003; accepted 28 January 2003

Editor: M. Cvetic

Abstract

We discuss a five-dimensional inflationary scenario based on a supersymmetric SO(10) model compactified on S1 /(Z2 x Z^). Inflation is implemented through scalar potentials on four-dimensional branes, and a brane-localized Einstein-Hilbert term is essential to make both brane vacuum energies positive during inflation. The orbifold boundary conditions break the SO(10) gauge symmetry to SU(4)c x SU(2)l x SU(2)r (= H). The inflationary scenario yields 8T/T a (M/Mplanck)2, which fixes M, the symmetry breaking scale of H to be close to the SUSY GUT scale of 1016 GeV. The scalar spectral index n is 0.98-0.99, while the gravitational wave contribution to the quadrupole anisotropy is negligible (< 1%). The inflaton decay into the lightest right-handed neutrinos yields the observed baryon asymmetry via leptogenesis. © 2003 Elsevier Science B.V. All rights reserved.

There exists a class of supersymmetric models in which a close link exists between inflation and the grand unification scale [1,2]. In particular, the quadrupole microwave anisotropy is proportional to (M/MPlanck)2, where M denotes the scale of the gauge symmetry breaking associated with inflation, and MPlanck = 1.2 x 1019 GeV. Thus, M is expected to be of order 1016 GeV, to within a factor of 2 or so, depending on the details of the supersymmetric model. This is tantalizingly close to the supersym-metric grand unification scale inferred from the evolution of the minimal supersymmetric standard model (MSSM) gauge couplings, and it is therefore natural to try to realize this inflationary scenario within a grand unified framework [2]. The SO(10) model is partic-

E-mail addresses: bkyae@bartol.udel.edu (B. Kyae), shafi@bxclu.bartol.udel.edu (Q. Shafi).

ularly attractive in view of the growing confidence in the existence of neutrino oscillations [3], which require that at least two of the three known neutrinos have a non-zero mass. Because of the presence of right-handed neutrinos (MSSM singlets), non-zero masses for the known neutrinos is an automatic consequence of the see-saw mechanism [4]. Furthermore, the right-handed neutrinos play an essential role in generating the observed baryon asymmetry via leptogenesis [5], which becomes especially compelling within an inflationary framework [6]. Indeed, an inflationary scenario would be incomplete without explaining the origin of the observed baryon asymmetry, and the kind of models we are interested in here automatically achieve this via leptogenesis.

A realistic supersymmetric inflationary model along the lines we are after was presented in [7], based on the SO(10) subgroup SU(4)c x SU(2)l x SU(2)r (= H) [8]. The scalar spectral index n has a value

0370-2693/03/$ - see front matter © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0370-2693(03)00133-3

very close to unity (typically n & 0.98-0.99), while the symmetry breaking scale of H lies, as previously indicated, around 1016 GeV. The vacuum energy density during inflation is of order 1014 GeV, so that the gravitational contribution to the quadrupole anisotropy is essentially negligible. It is important to note here that the inflaton field in this scenario eventually decays into right-handed neutrinos, whose out of equilibrium decays lead to leptogenesis. An extension to the full S0(10) model is complicated by the notorious doublet-triplet splitting problem, which prevents a straightforward implementation of the inflationary scenario. Of course, the subgroup H neatly evades this problem and even allows for a rather straightforward resolution of the ' v problem'.

Our objective here is to take advantage of recent orbifold constructions of five-dimensional (5D) su-persymmetric GUTs, in which a grand unified symmetry such as S0(10) can be readily broken to its maximal subgroup H [9] (an alternative possibility is SU(5) x U(1) which we will notpursue here), with the doublet-triplet splitting problem circumvented without fine tuning of parameters. Our main challenge then is to develop a 5D framework which can be merged with the four-dimensional (4D) supersymmetric inflationary scenario based on H. Because of N = 2 SUSY (in 4D sense) in 5D bulk, the F-term inflaton potential is allowed only on the 4D orbifold fixed points (branes), where only N = 1 SUSY is preserved. We shall see how 4D inflation comes about through scalar potentials localized on the two branes by analyzing the 5D Einstein equation. A brane-localized EinsteinHilbert term is essential to make both brane vacuum energies positive definite during inflation, which is a condition required by 4D N = 1 SUSY.

The four-dimensional inflationary model is best illustrated by considering the following superpotential which allows the breaking of some gauge symmetry G down to SU(3)c x SU(2)l x U(1)y, keeping super-symmetry (SUSY) intact [1,10]:

^infl = ks(№- M2). (1)

Here < and <f> represent superfields whose scalar components acquire non-zero vacuum expectation values (VEVs). For the particular example of G = H above, they belong to the (4,1,2) and (4,1, 2) representations of H. The <, < VEVs break H to the MSSM gauge group. The singlet superfield S provides the

scalar field that drives inflation. Note that by invoking a suitable R symmetry U(1)R, the form of W is unique at the renormalizable level, and it is gratifying to realize that R symmetries naturally occur in (higher dimensional) supersymmetric theories and can be appropriately exploited. From W, it is straightforward to show that the supersymmetric minimum corresponds to non-zero (and equal in magnitude) VEVs for < and <, while (S> = 0 [11]. (After SUSY breaking à la N = 1 supergravity (SUGRA), (S) acquires a VEV of order m3/2 (gravitino mass).)

An inflationary scenario is realized in the early universe with both <, < and S displaced from their present day minima. Thus, for S values in excess of the symmetry breaking scale M, the fields <, < both vanish, the gauge symmetry is restored, and a potential energy density proportional to M4 dominates the universe. With SUSY thus broken, there are radiative corrections from the supermultiplets that provide logarithmic corrections to the potential which drives inflation. In one loop approximation [1,12],

V « Vtree + K2M

+ (z + 1)2ln(1 + z-1)

+ (z - 1)2ln(1 - z-1)

where z = x2 = |S|2/M2, A denotes a renormaliza-tion mass scale and N denotes the dimensionality of the <, < representations. From Eq. (2) the quadrupole anisotropy is found to be [1,2]

NQ\1/2/ M y

45 J V^Planok/

x x-ly-lf{x2g)_1. (3)

The subscript 2 is there to emphasize the epoch of horizon crossing, yQ & xq(1 - 7/12xQ + •••), f(x2Q)-1 & 1/xg, for Sq sufficiently larger than M, and Nq & 45-60 denotes the e-foldings needed to resolve the horizon and flatness problems. From the expression for ST/T in Eq. (3) and comparison with the COBE result (ST/T)q & 6.6 x 10-6 [13], it follows that the gauge symmetry breaking scale M is close to 1016 GeV. Note that M is associated in our S0(10) example with the breaking scale of H (in

particular, the B-L breaking scale), which need not exactly coincide with the SUSY GUT scale. We will be more specific about M later in the text.

The relative flatness of the potential ensures that the primordial density fluctuations are essentially scale invariant. Thus, the scalar spectral index n is 0.98 for the simplest example based on W in Eq. (1). In some models n is unity to within a percent.

Several comments are in order:

• The 50-60 e-foldings required to solve the horizon and flatness problems occur when the inflaton field S is relatively close (to within a factor of order 110) to the GUT scale. Thus, Planck scale corrections can be safely ignored.

• For the case of minimal Kahler potential, the SUGRA corrections do not affect the scenario at all, which is a non-trivial result [2]. More often than not, supersymmetric inflationary scenarios fail to work in the presence of SUGRA corrections which tend to spoil the flatness of the potential needed to realize inflation.

• Turning to the subgroup H of £0(10), one needs to take into account the fact that the spontaneous breaking of H produces magnetic monopoles that carry two quanta of Dirac magnetic charge [14]. An overproduction of these monopoles at or near the end of inflation is easily avoided, say by introducing an additional (non-renormalizable) term S(<p<p)2 in W, which is permitted by the U( 1 )R symmetry. The presence of this term ensures the absence of monopoles as explained in Ref. [7]. Note that the monopole problem is also avoided by choosing a different subgroup of £0(10). In a separate publication, we will consider a scenario based on the SU(3)c x SU(2)L x U(1)Y x U(1)X subgroup of S0(10) whose breaking does not lead to monopoles. Another interesting candidate is SU(3)c x SU(2)l x SU(2)r x U(1)b-l. The salient features of the model are not affected by the monopole problem [7].

• At the end of inflation the scalar fields 0, (j>, and S oscillate about their respective minima. Since the </>. 4> belong, respectively, to the (4,1, 2) and (4,1,2) of SU(4)c x SU(2)l x SU(2)r, they decay exclusively into right-handed neutrinos via the superpotential couplings,

Mp^ 1 1

where the matter superfields /'",' belong to the (4,1,2) representation of //. and Mp = Mpianok/V87r = 2.44 x 1018 GeV denotes the reduced Planck mass, and Yi are dimensionless coefficients. We will have more to say about inflaton decay, the reheat temperature, as well as leptogenesis taking account of the recent neutrino oscillation data. However, we first wish to provide a five-dimensional setting for this inflationary scenario.

We consider 5D space-time (xx,y), / = 0, 1,2, 3, where the fifth dimension is compactified on an S1 /Z2 orbifold. The action is given by

j d4x j dy^-gs

-R5-AB +

Hy) (M2

S(y- yc)

where R5 (R4) is the 5-dimensional (4-dimensional) Einstein-Hilbert term,1 AB, and A2 are the bulk and brane cosmological constants, and M5 and M4 are mass parameters. The cosmological constants in the bulk and on the branes could be interpreted the vacuum expectation values of some scalar potentials from the particle physics sector. The brane curvature scalar (Ricci scalar) R4 (g/v) is defined with the induced metric of the bulk metric, g/v(x) = g/v(x, y = 0) (/,v = 0, 1,2, 3). For an inflationary solution, we take the metric ansatz,

ds2 = P2(y)(-dt2 + e2H0t dx2) + dy2, (6)

where H0 could be interpreted as the 4-dimensional Hubble constant. The non-vanishing components (/, /) and (5, 5) of the 5-dimensional Einstein equation derived from (5) gives [16]

Ap Ai A2

1 The importance of the brane-localized 4D Einstein-Hilbert term, especially for generating 4D gravity with a non-compact extra dimension was first noted in Ref. [15].

where primes denote derivatives with respect to y. The last term in the left-hand side in Eq. (7) arises from the brane scalar curvature term, and vanishes when Ho = 0.

The solutions to Eqs. (7) and (8) are given by H0

for AB< 0, (9)

I(y) = ±H0\y j + c, for Ab = 0, (10)

where k = y-AB/6M|, and c is an integration constant. Without loss of generality, we can take c positive. To avoid the existence of naked singularities within the interval -yc<y<yc, ±kyc + c > 0 and ±H0yc + c > 0 should be required. For simplicity of our discussion, let us take '+' among ± in Eqs. (9) and (10).

The introduction of the brane scalar curvature term R4 does not affect the bulk solutions, (9) and (10), but it modifies the boundary conditions. For AB < 0, the solution should satisfy the following boundary conditions at y = 0 and y = yc,

£cothc---

1 M.2 k2

2 M53 sinh2 c

6 M53'

k coth(kyc + c) =

A2 6 Mf

Hence, the integration constant c and the interval length yc are determined by A1 and A2. Similarly, the solution for AB = 0 should satisfy the boundary conditions,

1 MjHl

2 Ml c2

6 M53'

c + H0yc

so A1 and A2 determine H0/c and yc. Note that A2 must be fine-tuned to zero when A1 = 0 [17]. Hence it is natural that the scalar field which controls inflation is introduced in the bulk.

From Eqs. (11)-(14), we note that the brane cos-mological constants A1 and A2 should have opposite signs in the absence of the brane curvature scalar contribution at y = 0. However, a suitably large value of

M4/M5 can even make the sign of A1 positive. Since the introduction of the brane curvature term does not conflict with any symmetry that may be present, there is no reason why such a term with a parameter M4 that is large compared to M5 is not allowed [15]. Thus, A1 and A2 could both be positive and this fact will be exploited for implementing the inflationary scenario. We will later suggest a model for explaining how a large M4/M5 ratio may be realized.

From (13) and (14), we also note that A1 and A2 in the AB = 0 case are related to the 4-dimensional Hubble constant Ho, unlike the AB < 0 case in Eqs. (11) and (12). While their non-zero values are responsible for the 3-space inflation, vanishing brane cosmologi-cal constants guarantee an effective 4-dimensional flat space-time. On the other hand, for AB < 0, the relations between the bulk and brane cosmological constants are responsible for inflation. To obtain a static solution [18], we should take H0 ^ 0 and c ^ to (or A1(2)/6M3 ^ -k(+k)) while letting the ratio H0ec/2k ^ 1.

Our main task is to embed the 4D supersymmet-ric inflationary scenario in 5D space-time [19], employing the framework and solutions discussed above. In order to extend the setup to 5D SUGRA, a grav-itino fM and a vector field BM should be appended to the graviton (funfbein) erM. Through orbifolding, only N = 1 SUSY is preserved on the branes. The brane-localized Einstein-Hilbert term in Eq. (5) is still allowed, but should be accompanied by a brane gravitino kinetic term as well as other terms, which is clear in off-shell SUGRA formalism [20]. In a higher dimensional supersymmetric theory, a F-term scalar potential is allowed only on the 4-dimensional fixed points which preserve N = 1 SUSY. We require a formalism in which inflation and the Hubble constant H0 are controlled only by the brane cosmological constants, such that during inflation the positive vacuum energy slowly decreases, and the minimum of the scalar potential corresponds to a flat 4D space-time. The solution for AB = 0 meets these requirements in the presence of the additional brane scalar curvature term at y = 0, and so we will focus only on this case.

We have tacitly assumed that the interval separating the two branes (orbifold fixed points) remains fixed during inflation. The scenario is quite different from what is often called 'D-brane inflation' [21]. The dynamics of the orbifold fixed points, unlike the

D-brane case, is governed only by the gss(x,y) component of the metric tensor. The real fields e55, B5, and the chiral fermion in the 5D gravity multiplet are assigned even parity under Z2 [20], and they compose an N = 1 chiral multiplet on the branes. The associated superfield can acquire a superheavy mass and its scalar component can develop a VEV on the brane. With superheavy brane-localized mass terms, their low-lying Kaluza-Klein (KK) mass spectrum is shifted so that even the lightest mode obtains a compactification scale mass [22]. Since this is much greater than H0, the interval distance is stable even during inflation. The stabilization of the interval distance leads to the stabilization also of the warp factor j3(y ), because the fluctuation Sft(y ) of the warp factor near the solution in Eq. (10) (also Eq. (9)) turns out to be proportional to the interval length variation Sg55 by the linearized 5D Einstein equation [23].

With AB = 0, the effective 4-dimensional reduced Planck mass squared Mp (= 1/8nGN) is given by

H0yc « 1,

Mp = M5 J dy!2 + M4212|y=0

= M|yc

':H2y2 + 2cH0yc + 2c2

+ M2c2.

For M|yc « M42 ~ M2, gravity couples universally at low energy to fields localized at y = yc and y = 0 and in the bulk, with the strength controlled by 1/M42 [24]. The 4-dimensional effective cosmological constant turns out to be

Aeff = I dy I

+ S(y)A1 + S(y - yc)A2

M\yc ( + 2cH0yc + 2c2

+ M2c2

= 3H02M2,

where the first two terms in the first line are the warp factor contributions. Hence, from Eqs. (13) and (14) Aeff vanishes when A1 = A2 = 0. Note that for

c2 A\M2 12 M\

We can directly adapt these results for the S1 /(Z2 x Z2) case.

To see how inflation is realized in this 5D setting, let us consider the 4D SU(4)c x SU(2)l x SU(2)r(= H) supersymmetric inflationary model [7]. An effective 4D theory with the gauge group H is readily obtained from a 5D S0(10) gauge theory if the fifth dimension is compactified on theorbifold S1 /(Z2 x z2 ) [9], where Z2 reflects y ^ -y, and Z2 reflects y' ^ -y' with y' = y + yc/2. There are two independent orbifold fixed points (branes) at y = 0 and y = yc/2, with N = 1 SUSYs and gauge symmetries S0(10) and H, respectively, [9]. The S0(10) gauge multiplet (AM,X1 ,X2decomposes under H as

V45 ^ V(15,1,1) + V(1,3,1) + V(1,1,3) + V(6,2,2)

+ £(15,1,1) + £(1,3,1) + £(1,1,3) + ^(6,2,2),

where V and £ denote the vector multiplet (AV,Xl) and the chiral multiplet ((& + iA5)/y/2, X2), respectively, and their (Z2,Z^) parity assignments and KK masses are shown in Table 1.

The parities of the chiral multiplets £'s are opposite to those of the vector multiplets V's in Table 1 and hence, N = 2 SUSY explicitly breaks to N = 1 below the compactification scale n/yc. As shown in Table 1, only the vector multiplets, V(1511), V(131), and V(1,13) contain massless modes, which means that the low energy effective 4D theory reduces to N = 1 supersymmetric SU(4)c x SU(2)l x SU(2)r . The parity assignments in Table 1 also show that the wave function of the vector multiplet V(6 2 2) vanishes at the brane located at y = yc/2 (B2) because it is assigned an odd parity under Z2, while the wave functions of all the vector multiplets should be the same at the y = 0 brane (B1). Therefore, while the gauge symmetry at B1 is S0(10), only SU(4)c x SU(2)l x SU(2)R is preserved at B2 [25].

The 5D inflationary solution requires positive vacuum energies on both branes B1 and B2. While the scalar potential in Eq. (2) would be suitable for B2, an appropriate scalar potential on B1 is also required.

Table 1

(Z2, Z2) parity assignments and Kaluza-Klein masses (n = 0,1,2,...) for the vector multiplet in S0(10)

Vector ^(15,1,1) ^(1,3,1) v(1,1,3) v(6,2,2)

(Z2,Z2) (+,+) (+,+) (+,+) (+,-)

Masses 2nn/yc 2nn/yc 2nn/yc (2n + 1)n/yc

Chiral s(15,1,1) E(l,l,3) s(6,2,2)

(Z2,Z^) Masses (-,-) (2n + 2)n/yc (-,-) (2n + 2)n/yc (-,-) (2n + 2)n/yc (-,+) (2n + 1)n/yc

Since the boundary conditions inEqs. (13) and (14) require A1 and A2 to simultaneously vanish, it is natural to require S to be a bulk field. Then, the VEVs of S on the two branes can be adjusted such that the boundary conditions are satisfied. As an example, consider the following superpotential on B1,

WB1 = k1s(zz- M2), (19)

where Z and Z are S0(10) singlet superfields on the B1 brane with opposite U(1)R charges. The condition for a positive brane cosmological constant on B1 is found from (13) to be (H0/c)(M2/M53) > 2. For k — 10-3, say, and c — 1, we have H0 — 1010 GeV and M5 — 1015 GeV (so that M4 — Mp). Thus, there exists a hierarchy of order 103 between the 5D bulk scale M5 and the four-dimensional brane mass scale M4. To see how this hierarchy could arise, consider the case where the brane-localized gravity kinetic term has the canonical form but not the bulk term. Thus,

C = MPc-fW)

Z4 - V(\$\)

where $ is some scalar field, V(\$\) its associated potential, we take M4 = MP. Let us assume that like e5, $ acquires a Planck scale mass and VEV on the brane. Then, at the minimum of V(\$\), the 5D Einstein equation determining the background geometry is effectively given by Eqs. (7) and (8), with M5 = e-{f(\$\))/3MP. Taking f(\$\) = 2\$\/MP, for instance, and ($) & 10MP would lead to M5 — 1015 GeV as required.

After inflation is over, the oscillating system consists of the complex scalar fields @ = (8$ + 8$), where 8<j> = <j> - M (8<f> = <f> - M), and S, both with masses equal to w„,n = \J~1kM . Through the superpotential couplings in Eq. (4), these fields decay into a

pair of right-handed neutrinos and sneutrinos, respectively, with an approximate decay width [7]

nimfi 8;r

where Mi denotes the mass of the heaviest right-handed neutrino with 2Mi < mmfl, so that the inflaton decay is possible. Assuming an MSSM spectrum below the GUT scale, the reheat temperature is given by [26]

1 .- 1 /55\1/4 __

Tr^-JrMp^— ( — ) (22)

12\Nq)

For yQ — unity (see below), and Tr < 109 5 GeV from the gravitino constraint [27], we require Mi < 10101010 5 GeV

In order to decide on which Mi is involved in the decay [28], let us start with atmospheric neutrino (vM-vT) oscillations and assume that the light neutrinos exhibit an hierarchical mass pattern with m.T,^>m2^>

m\. Then ^A«^ «a »13 «a m2m/M3, where mo3 (= mt (M)) denotes the third family Dirac mass which equals the asymptotic top quark mass due to SU(4)c. We also assume a mass hierarchy in the right-handed sector, M3 > M2 > M1. The mass M3 arises from the superpotential coupling Eq. (4) and is given by M3 = 2y3M2/MP — 1014 GeV, for M — 1016 GeV and Y3 ~ unity. This value of M3 is in the right ball park to

generate an m3

'Am^Awith mt(M)'

110 GeV [26]. It follows from (22) that Mi in (21) cannot be identified with the third family right-handed neutrino mass M3. It should also not correspond to the second family neutrino mass M2 if we make the plausible assumption that the second generation Dirac mass should lie in the few GeV scale. The large mixing angle MSW solution of the solar neutrino problem

requires that J Am2 , «a m2 ~ GeV2/M2 ~ eV,

so that M2 > 1011-1012 GeV. Thus, we are led to conclude [28] that the inflaton decays into the lightest (first family) right-handed neutrino with mass

M1 - 1010-1010 5 GeV,

such that 2M1 < minfl.

The constraint 2M2 > minfl yields yQ < 3.34y2, where M2 = 2y2M2/MP. We will not provide here a comprehensive analysis of the allowed parameter space but will be content to present a specific example, namely,

M « 8 x 1015 GeV,

minfl - 1013 GeV (- M2),

with yQ « 0.4 (corresponding to xq near unity, so that the inflaton S is quite close to M during the last 50-60 e--foldings).

Note that typically k is of order 10-2- few x 10-4 [7], so that the vacuum energy density during inflation is ~ 10-4-10-8MGUT. Thus, in this class of models the gravitational wave contribution to the quadru-pole anisotropy (ST/T)q is essentially negligible (< 10-8). With k - few x 10-4 (10-3), the scalar spectral index n « 0.99 (0.98).

The decay of the (lightest) right-handed neutrinos generates a lepton asymmetry which is given by [29]

10 16тг

Tr \/Mi

mnü)\M2

c^s^sin2S(mD2 - m^1)2

|(h>|2(m22s2 + m2lC2) ' (25)

where the VEV |(h>| « 174 GeV (for large tanp), mD1,2 are the neutrino Dirac masses (in a basis in which they are diagonal and positive), and ce = cos 0, se = sin 0, with 0 and 8 being the rotation angle and phase which diagonalize the Majorana mass matrix of the right-handed neutrinos. Assuming ce and se of comparable magnitude, taking mD2 > mD1, and using (23) and (24), Eq. (25) reduces to

nL — « 10

-8.5 „2

cesin25l 109.s QeVA2 x 1010-5 GeV

1013GeVy mD2 ^

~M2 ) V^lOGeV/ ' (26)

which can be in the correct ball park to account for the observedbaryonasymmetry nB/s (« -28/79nL/s).

In conclusion, our goal in this Letter has been to demonstrate the existence of realistic models which

nicely blend together four particularly attractive ideas, namely, supersymmetric grand unification, extra dimension^), inflation and leptogenesis. The doublet-triplet problem is circumvented by utilizing orbifold breaking of S0(10), which may also help in suppressing dimension five proton decay. There are two predictions concerning inflation that are particularly significant. Namely, the scalar spectral index n lies very close to unity (« 0.98-0.99), and the gravitational wave contribution to (ST/T)q is highly suppressed (~ 10-8-10-9). Finally, the inflaton decay produces heavy right-handed Majorana neutrinos (in our case the lightest one), whose subsequent out of equilibrium decay leads to the baryon asymmetry via leptogenesis. We expect to generalize this approach to other symmetry breaking patterns of £0(10) in a future publication.

Acknowledgements

Q.S. thanks Gia Dvali and George Lazarides for useful discussions. We also acknowledge useful discussion with Jim Liu. The work is partially supported by DOE under contract number DE-FG02-91ER40626.

References

[1] G. Dvali, Q. Shafi, R. Schaefer, Phys. Rev. Lett. 73 (1994) 1886, hep-ph/9406319.

[2] For a comprehensive review and additional references see, G. Lazarides, hep-ph/0111328.

[3] S. Fukuda, etal., Super-Kamiokande Collaboration, Phys. Rev. Lett. 85 (2000) 3999;

S. Fukuda, et al., Phys. Lett. B 539 (2002) 179.

[4] M. Gell-Mann, P. Ramond, R. Slansky, in: P. van Nieuwen-huizen, D.Z. Freedman (Eds.), Supergravity: Proceedings, North-Holland, Amsterdam, 1979, p. 315, Print-80-0576 (CERN);

T. Yanagida, in: O. Sawada, A. Sugamoto (Eds.), Workshop on the Unified Theory and Baryon Number in the Universe, KEK, Tsukuba, 1979, p. 95.

[5] M. Fukugita, T. Yanagida, Phys. Lett. B 174 (1986) 45.

[6] G. Lazarides, Q. Shafi, Phys. Lett. B 258 (1991) 305.

[7] R. Jeannerot, S. Khalil, G. Lazarides, Q. Shafi, JHEP 0010

(2000) 012, hep-ph/0002151.

[8] J.C. Pati, A. Salam, Phys. Rev. D 10 (1974) 275.

[9] R. Dermisek, A. Mafi, Phys. Rev. D 65 (2002) 055002, hep-ph/0108139;

See also, T. Asaka, W. Buchmüller, L. Covi, Phys. Lett. B 523

(2001) 199, hep-ph/0108021;

L. Hall, Y. Nomura, T. Okui, D. Smith, Phys. Rev. D 65 (2002) 035008, hep-ph/0108071.

[10] E.J. Copeland, A.R. Liddle, D.H. Lyth, E.D. Stewart,

D. Wands, Phys. Rev. D 49 (1994) 6410.

[11] S.F. King, Q. Shafi, Phys. Lett. B 422 (1998) 135, hep-ph/9711288.

[12] S. Coleman, E. Weinberg, Phys. Rev. D 7 (1973) 1888.

[13] D.H. Lyth, A. Riotto, Phys. Rep. 314 (1999) 1, hep-ph/9807278;

E.F. Bunn, A.R. Liddle, M. White, Phys. Rev. D 54 (1996) 5917.

[14] G. Lazarides, M. Magg, Q. Shafi, Phys. Lett. B 97 (1980) 87.

[15] G. Dvali, G. Gabadadze, M. Porrati, Phys. Lett. B 485 (2000) 208, hep-th/0005016;

G. Dvali, G. Gabadadze, M. Kolanovic, F. Nitti, Phys. Rev. D 64 (2001) 084004, hep-ph/0102216.

[16] T. Nihei, Phys. Lett. B 465 (1999) 81, hep-ph/9905487;

N. Kaloper, Phys. Rev. D 60 (1999) 123506, hep-th/9905210;

H.B. Kim, H.D. Kim, Phys. Rev. D 61 (2000) 064003, hep-ph/9909053;

See also, J.E. Kim, B. Kyae, H.M. Lee, Nucl. Phys. B 582 (2000) 296, hep-th/0004005;

J.E. Kim, B. Kyae, H.M. Lee, Nucl. Phys. B 591 (2000) 587, Erratum;

J.E. Kim, B. Kyae, Phys. Lett. B 486 (2000) 165, hep-th/0005139;

P. Binetruy, C. Deffayet, D. Langlois, Nucl. Phys. B 565 (2000) 269, hep-th/9905012;

S. Nojiri, S.D. Odintsov, Phys. Lett. B 484 (2000) 119, hep-th/0004097;

C. Csaki, M. Graesser, C. Kolda, J. Terning, Phys. Lett. B 462 (1999) 34, hep-ph/9906513;

J.M. Cline, C. Grojean, G. Servant, Phys. Rev. Lett. 83 (1999) 4245, hep-ph/9906523;

H.B. Kim, Phys. Lett. B 478 (2000) 285, hep-th/0001209.

[17] See also, J.E. Kim, hep-th/0210117, in this paper inflation is discussed based on a self-tuning mechanism for the cosmolog-ical constant.

[18] L. Randall, R. Sundrum, Phys. Rev. Lett. 83 (1999) 3370, hep-ph/9905221.

[19] Higher dimensional cosmology has a long history. See, for example, Q. Shafi, C. Wetterich, Phys. Lett. B 129 (1983) 387; Q. Shafi, C. Wetterich, Phys. Lett. B 152 (1985) 51;

Q. Shafi, C. Wetterich, Nucl. Phys. B 289 (1987) 787;

For a recent discussion and additional references see,

M. Bastero-Gil, V. Di Clemente, S.F. King, hep-ph/0211012.

[20] M. Zucker, Phys. Rev. D 64 (2001) 024024, hep-th/0009083;

B. Kyae, Q. Shafi, Phys. Rev. D 66 (2002) 095009, hep-ph/0204041.

[21] G. Dvali, S.-H.H. Tye, Phys. Lett. B 450 (1999) 72, hep-ph/9812483;

G. Dvali, Q. Shafi, S. Solganik, hep-th/0105203;

C.P. Burgess, M. Majumdar, D. Nolte, F. Quevedo, G. Rajesh, R.-J. Zhang, JHEP 0107 (2001) 047, hep-th/0105204;

G. Shiu, S.-H.H. Tye, Phys. Lett. B 516 (2001) 421, hep-th/0106274;

C. Herdeiro, S. Hirano, R. Kallosh, JHEP 0112 (2001) 027, hep-th/0110271;

B. Kyae, Q. Shafi, Phys. Lett. B 526 (2002) 379, hep-ph/0111101;

J. Garcia-Bellido, R. Rabadan, F. Zamora, JHEP 0201 (2002) 036, hep-th/0112147;

R. Blumenhagen, B. Körs, D. Lüst, T. Ott, Nucl. Phys. B 641 (2002) 235, hep-th/0202124.

[22] Y. Nomura, D. Smith, N. Weiner, Nucl. Phys. B 613 (2001) 147, hep-ph/0104041;

N. Arkani-Hamed, L. Hall, Y. Nomura, D. Smith, N. Weiner, Nucl. Phys. B 605 (2001) 81, hep-ph/0102090; Z. Chacko, M.A. Luty, E. Ponton, JHEP 0007 (2000) 036, hep-ph/9909248.

[23] Z. Chacko, P.J. Fox, Phys. Rev. D 64 (2001) 024015, hep-th/0102023;

C. Csaki, M.L. Graesser, G.D. Kribs, Phys. Rev. D 63 (2001) 065002, hep-th/0008151;

J.E. Kim, B. Kyae, H.M. Lee, Phys. Rev. D 66 (2002) 106004, hep-th/0110103.

[24] B. Kyae, hep-th/0207272.

[25] A. Hebecker, J. March-Russell, Nucl. Phys. B 625 (2002) 128, hep-ph/0107039.

[26] G. Lazarides, R.K. Schaefer, Q. Shafi, Phys. Rev. D 56 (1997) 1324, hep-ph/9608256.

[27] J. Ellis, J.E. Kim, D. Nanopoulos, Phys. Lett. B 145 (1984) 181;

M.Yu. Khlopov, A.D. Linde, Phys. Lett. B 138 (1984) 265; For a review and additional references see, W. Buchmüller, in: G.C. Branco, Q. Shafi, J.I. Silva-Marcos (Eds.), Nato Science Series II, Vol. 34, 2001.

[28] See also, J.C. Pati, hep-ph/0209160, in this paper it is also discussed how large neutrino mixings compatible with observations can arise from S0(10) and SU(4)C x SU(2)L x SU(2)R.

[29] G. Lazarides, Q. Shafi, N.D. Vlachos, Phys. Lett. B 427 (1998) 53, and references therein.