Grand Unification and Subcritical Hybrid Inflation
Abstract
We consider hybrid inflation for small couplings of the inflaton to matter such that the critical value of the inflaton field exceeds the Planck mass. It has recently been shown that inflation then continues at subcritical inflaton field values where quantum fluctuations generate an effective inflaton mass. The effective inflaton potential interpolates between a quadratic potential at small field values and a plateau at large field values. An analysis of the allowed parameter space leads to predictions for the scalar spectral index and the tensortoscalar ratio similar to those of natural inflation. Using the ranges for and favoured by the Planck data, we find that the energy scale of the plateau is constrained to the interval , which includes the energy scale of gauge coupling unification in the supersymmetric standard model. The tensortoscalar ratio is predicted to satisfy the lower bound for folds before the end of inflation.
I I. introduction
The observations and analyses of the cosmic microwave background (CMB) by the WMAP Hinshaw:2012aka () and Planck Ade:2013uln () collaborations strongly support singlefield slowroll inflation as the paradigm of early universe cosmology. The current CMB data can be successfully described by many models of inflation. Prominent examples are the Starobinsky model Starobinsky:1980te (), chaotic inflation Linde:1983gd (), natural inflation Freese:1990rb () and hybrid inflation Linde:1993cn (), which differ significantly in their predictions for the scalar spectral index and the tensortoscalar ratio of the primordial density fluctuations. The recently released BICEP2 data Ade:2014xna (), which are presently under intense scrutiny Mortonson:2014bja (); Flauger:2014qra (); Adam:2014bub (), have renewed the interest in models with a large fraction of tensor modes.
A theoretically attractive framework is supersymmetric Dterm inflation Binetruy:1996xj (); Halyo:1996pp (); Kallosh:2003ux (). It is remarkable that it contains the rather different models listed above for different choices of the Kähler potential: For a canonical Kähler potential one obtains standard hybrid inflation, for a superconformal or noscale Kähler potential the Starobinsky model emerges Buchmuller:2012ex (), and in case of a shift symmetric Kähler potential Kawasaki:2000yn () Dterm inflation includes a “chaotic regime” with a large tensortoscalar ratio Buchmuller:2014rfa ().
In this note we study the chaotic regime of Dterm inflation in more detail. It turns out that the predictions are qualitatively similar to those of natural inflation, although the theoretical interpretation is entirely different. Moreover, there are significant quantitative differences.
The parameters of Dterm inflation are a Yukawa coupling, the FayetIliopoulos (FI) term and a gauge coupling. The last two determine the energy scale of hybrid inflation. The measured amplitude of scalar fluctuations determines as function of the Yukawa coupling. Imposing the bounds of the Planck data on and as constraints Ade:2013uln (),
(1) 
we find that has to be close to the energy scale of grand unification. Furthermore, we obtain a lower bound on the tensortoscalar ratio, for folds before the end of inflation, which is in reach of upcoming experiments.
Ii II. Subcritical Hybrid Inflation
The framework of Dterm hybrid inflation in supergravity is defined by a Kähler potential, a superpotential and a Dterm scalar potential Binetruy:1996xj (); Halyo:1996pp (); Kawasaki:2000yn (); Buchmuller:2014rfa (),
(2)  
(3)  
(4) 
The “waterfall fields” carry the U(1) charges , and the inflaton is contained in the gauge singlet . The Kähler potential is invariant under the shift where is a real constant, i.e., it is independent of the constant part of , which is identified as the inflaton field. The gauge coupling , and is a Yukawa coupling, which may be much smaller than . The only dimensionful parameter is the FI term that sets the energy scale of inflation.^{1}^{1}1Note that FI terms in supergravity are a subtle issue Binetruy:2004hh (); Komargodski:2010rb (); Dienes:2009td (); Catino:2011mu (). For recent discussions and references on fielddependent and fieldindependent FI terms, see refs. Wieck:2014xxa (); Domcke:2014zqa (). In the following we shall treat as a constant.
Standard hybrid inflation takes place at inflaton field values larger than the critical value . Here the waterfall fields have a positive mass squared and are stabilized at the origin. Classically, the potential is independent of modulus and phase of the gaugesinglet . The flatness in is lifted by quantum corrections.
For subcritical field values the complex scalar remains stabilized at the origin, whereas acquires a tachyonic instability. The sum of F and Dterms yields for the scalar potential as function of and ,
(5)  
Note that, due to the shift symmetry of the Kähler potential, the Planck suppressed terms are also only quadratic in . The scalar potential contains higher powers in , which we have neglected since they are not important for inflation.
Following ref. Buchmuller:2014rfa (), we solve the classical equations of motion for homogeneous fields, corresponding to the scalar potential (5),
(6) 
The initial conditions for the waterfall field are obtained by considering the tachyonic growth of its quantum fluctuations Felder:2000hj (); Asaka:2001ez (); Copeland:2002ku (); Dufaux:2010cf () close to the critical point ,
(7) 
Here are the momentum modes of the field operator in an exponentially expanding, spacially flat background with Hubble parameter and a timedependent inflaton field Asaka:2001ez (),
(8) 
The integration in eq. (7) extends over all soft momentum modes below where the timedependent mass operator for in the brackets of eq. (8) vanishes. At a decoherence time , where and , the waterfall field becomes classical. Matching the variance and classical field near the decoherence time, , one obtains and at . As shown in ref. Buchmuller:2014rfa (), the classical waterfall field reaches the local, inflatondependent minimum soon after the decoherence time,
(9) 
and, together with the inflaton field, it reaches the global minimum after a large number of folds.
On the inflationary trajectory, the inflaton potential takes a simple form,
(10) 
For small , the potential is quadratic, and as approaches , the potential reaches the plateau . Fig. 1 shows the potential for a certain choice of parameters. As we shall see in the following sections, in the relevant parameter range the predictions for and only depend on the potential (II). The initial conditions, in particular the initial value of and the tachyonic growth of the waterfall field only affect the total number of folds and the formation of cosmic strings.
Iii III. cosmological observables
In this section we analyse the implications of the constraints on the cosmological observables and by the Planck data on the parameters of the inflaton potential (II). Obviously, the potential only depends on two parameters, which can be chosen as
(11) 
where we have used . Then the energy density of the plateau is given by .
Scalar spectral index and tensortoscalar ratio are conveniently expressed in terms of the slowroll parameters of the inflaton potential,
(12) 
where the superscript ‘prime’ denotes the derivative with respect to , and we have set the Planck mass . Inflation ends at which is defined by . The number of folds between and can then be expressed as
(13) 
where . Solving this equation, one obtains in terms of ,
(14) 
where , , . The first 3 terms in the expansion (14) yield to sufficient accuracy. Together with the standard expressions for and ,
(15) 
where and , this yields and for a given number of folds .
Finally, a crucial observable is the amplitude of the scalar power spectrum ,
(16) 
which is determined as at % CL Ade:2013zuv () from the combined data sets of the Planck and WMAP collaborations. Imposing the central value of as constraint yields a line in the plane for a given number of folds. The result is shown in fig. 2 for and . The shaded region is consistent with the constraints (1) of the Planck data on and . The energy scale of the plateau is rather precisely determined,
(17) 
for . It is very remarkable how accurately the energy scale of the plateau agrees with the energy scale of gauge coupling unification in the supersymmetric standard model. For comparison, fig. 2 also shows the allowed region in the plane. The allowed values of , and also are superPlanckian, similar to chaotic inflation.^{2}^{2}2Note that for the treatment of the initial tachyonic growth of the waterfall field is consistent, while is small enough to allow for 60 folds below the critical point Buchmuller:2014rfa ().
Varying yields a line also in the plane for a given number of folds. In fig. 3 the result is compared with various constraints from CMB data and the prediction of natural inflation Ade:2013uln (). As one can see, subcritical hybrid inflation and natural inflation Freese:1990rb (); Freese:2014nla () yield qualitatively similar predictions. This is not surprising, given the similarity of the potential (II) to a cosinepotential.^{3}^{3}3A similar potential can be obtained in chaotic inflation with nonminimal coupling to gravity Linde:2011nh (). For a recent discussion of universality classes for models of inflation, see ref. Binetruy:2014zya (). The interpretation, however, is very different. In natural inflaton the band is obtained by varying a superPlanckian axion decay constant or, as in aligned twoaxion models Kim:2004rp (), the ratio of subPlanckian decay constants. On the contrary, in subcritical hybrid inflation different points of the band correspond to different values of the ratio of a small Yukawa coupling and a gauge coupling . The lower bound from the Planck data on the spectral index implies the lower bound on the tensortoscalar ratio for folds before the end of inflation.
Let us finally comment on the formation of cosmic strings in subcritical hybrid inflation. Cosmic strings are produced during the tachyonic growth of the waterfall field, which spontaneously breaks the U(1) symmetry. The initial average distance of the cosmic strings can be estimated as Buchmuller:2014rfa ()
(18) 
where is the scale factor and is the spinodal time at which reaches the local, inflatondependent minimum . Between and , the beginning of the last 50–60 folds, the scale factor grows by folds whereas the Hubble parameter remains almost constant, , which yields for the average string separation at ,
(19) 
The smallest value of is obtained for the largest coupling : (see figs. 1 and 2). During the final – folds the horizon at is blown up to , the size of the present universe. We thus obtain the lower bound on the average cosmic string distance
(20) 
Hence, cosmic strings are unobservable in subcritical hybrid inflation for parameters consistent with the Planck data.
Iv IV. conclusions
We have studied subcritical hybrid inflation, which occurs in supersymmetric Dterm inflation for small couplings of the inflaton to matter. The effective inflaton potential interpolates between a quadratic potential at small field values and a plateau at large field values. It is characterized by two parameters, the energy scale of the plateau, and the critical value of the inflaton field, at which the plateau is reached.
The model can accommodate the Planck data very well, and it is striking how accurately the energy scale of inflation agrees with the scale of gauge coupling unification in the supersymmetric standard model. This reopens the question on the possible connection between grand unification and inflation.
The predictions for the scalar spectral index and tensortoscalar
ratio are qualitatively similar to those from natural
inflation. Quantitatively, however, the predicted values for the
tensortoscalar ratio are larger and one obtains the lower bounds
for folds before the end of
inflation, which is in reach of upcoming experiments.
Acknowledgements
We are grateful to Valerie Domcke and Kai Schmitz for valuable
discussions and support. We also thank Rose Lerner, Alexander
Westphal and Clemens Wieck for helpful comments on the manuscript.
This work has been supported in part by the German Science Foundation
(DFG) within the Collaborative Research Center 676 “Particles,
Strings and the Early Universe”.
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