Sub-Planckian Two-Field Inflation Consistent with the Lyth Bound
aa r X i v : . [ h e p - ph ] O c t Sub-Planckian Two-Field Inflation Consistent with the Lyth Bound
John McDonald ∗ Dept. of Physics, University of Lancaster, Lancaster LA1 4YB, UK
The BICEP2 observation of a large tensor-to-scalar ratio, r = . + . − . , implies that the inflaton f in single-field inflation models must satisfy f ∼ M Pl in order to produce sufficient inflation. This is a problem ifinteraction terms suppressed by the Planck scale impose a bound f < ∼ M Pl . Here we consider whether it ispossible to have successful sub-Planckian inflation in the case of two-field inflation. The trajectory in field spacecannot be radial if the effective single-field inflaton is to satisfy the Lyth bound. By considering a complex field F , we show that a near circular but aperiodic modulation of a | F | potential can reproduce the results of f chaotic inflation for n s and r while satisfying | F | < ∼ . M Pl throughout. More generally, for models based on a | F | potential, the simplest sub-Planckian models are equivalent to f and f / chaotic inflation. PACS numbers:
I. INTRODUCTION
The observation by BICEP2 of gravity waves from inflation [1, 2] is problematic for single-field inflation models. The Lythbound [3], following an earlier observation of Starobinsky [4] and recently generalized in [5], implies that the change of theinflaton field during inflation must be ∼ M Pl in order to generate enough inflation . This is a serious problem if there areinteraction terms in the inflaton potential suppressed by the Planck mass scale, since these will impose an upper bound f < ∼ M Pl .It may be possible to suppress such interactions via a symmetry. However, this requires a quite non-trivial symmetry, forexample a shift symmetry [6–8] which is slightly broken to allow an inflaton potential [6]. If Planck-suppressed interactions arenot suppressed by a symmetry then the only alternative is to consider multi-field inflation. One possibility is N-flation [10]. Inthis case, to achieve f i < M Pl for the i = , ..., N scalar fields in chaotic f N-flation, more than 200 scalar fields are necessary[10].Should it be that no symmetry exists to suppress Planck corrections to the inflaton potential and there is not a very largenumber of scalar fields, then we need to consider an alternative model of inflation which has sub-Planckian values of the scalarfields throughout.Here we consider whether this is possible in the case of two-field inflation. In general, this will require an unconventionalpotential. We will consider the fields to be the real and imaginary parts of a complex field F . We will show that it is possibleto construct a model in which | F | is sub-Planckian throughout and n s and r are in good agreement with Planck and BICEP2. Inorder to satisfy the Lyth bound, the minimum of the potential in field space cannot be in the radial direction, as it must describea sufficiently long path across the two-field landscape. (A similar observation was made in [5].) The path must be aperiodic inthe phase of F in order to be long enough to satisfy the Lyth bound. As an existence proof of a successful two-field inflationmodel, we will present a simple example in which the path is nearly circular but aperiodic, so that the minimum of the potentialhas a spiral-like path in the complex plane. II. INFLATION WITH AN APERIODIC MODULATED POTENTIAL
We will consider an potential of the form V ( f , q ) = f A ( f , q ) V ( | F | ) , (1)where F = ( f / √ ) e i q . f A ( f , q ) is a modulating function which creates an aperiodic local minimum of the potential, such thatthe minimum as a function of q describes a spiral-like path. In the following we will consider a sinusoidal modulating function, f A ( f , q ) = + A (cid:18) | F | m L m + q (cid:19) . (2) ∗ Electronic address: [email protected] M Pl = . × GeV. An alternative multi-field approach which may resolve the problem, M-flation, is described in [9].
The base potential V ( | F | ) is assumed to be a | F | n potential. Therefore V ( f , q ) = l | F | n M n − Pl (cid:18) + A (cid:18) | F | m L m + q (cid:19)(cid:19) , (3)where l a free parameter and we have set the dimensional mass to M Pl . In Figure 1 we show the potential as a function of f ata fixed value of q .The potential has a local minimum at f ( q ) , where (cid:18) f ( q ) √ L (cid:19) m ≈ n p − p − q (4)and n is an integer. This is a good approximation if A ( f / √ L ) m ≫
1, assuming that A is small compared to 1. In Figure 2 weshow the spiral-like minimum in the complex plane as a function of q . The distance a along the minimum in field space is relatedto q by da = s f + (cid:18) d f d q (cid:19) d q , (5)From Eq. (4), d f d q = − √ Lf ! m f m . (6)Therefore if ( √ L / f ) m ≪ da = f ( q ) d q . In this case we can consider a to be a canonicallynormalized field along the minimum of the potential. The model will behave as a single field inflation model if field orthogonalto a , f , has a mass much larger than H . We will show later that this can easily be satisfied.We can construct an effective single-field potential for the slowly-rolling field. Using da = f ( q ) d q , where f ( q ) is related to q via Eq. (4), we find Z da = Z f d q = Z f (cid:18) d q d f (cid:19) d f = Z − m (cid:18) f √ L (cid:19) m d f . (7)Therefore a = m ( m + ) (cid:16) √ L (cid:17) m (cid:0) f m + − f m + (cid:1) , (8)where we have defined a = f = f . Thus f = (cid:18)(cid:16) √ L (cid:17) m (cid:18) m + m (cid:19)(cid:19) m + (cid:18) mm + (cid:19) f m + (cid:16) √ L (cid:17) m − a m + . (9)We can then define a new slow-roll field ˆ a , where ˆ a = m ( m + ) f m + (cid:16) √ L (cid:17) m − a . (10) In [11], a model ("Dante’s Inferno") based on string axion monodromy is proposed which is dynamically similar to the model discussed here. A second axionfield q is introduced, while the potential for the monodromy axion field r is generalized to a r p potential. The dynamics of this model are essentially identicalto the present model with m =
1. The interpretation of the fields is different, with two axion fields and a potential from string dynamics in the case of [11] anda single complex field with the phase and modulus serving as the dynamical fields in the present model. In [11] the modulation is due to an additive axionterm in the potential, whereas here we consider a multiplicative modulation of the potential.
Therefore ˆ a = m ( m + ) f m + (cid:16) √ L (cid:17) m . (11)Along the minimum, V ( ˆ a ) = (cid:18) − A (cid:19) lf n ( ˆ a ) (cid:16) √ (cid:17) n M n − Pl ≈ lf n ( ˆ a ) (cid:16) √ (cid:17) n M n − Pl = l (cid:16) √ (cid:17) n M n − Pl (cid:18)(cid:16) √ L (cid:17) m (cid:18) m + m (cid:19)(cid:19) nm + ˆ a nm + . (12) FIG. 1: The modulated potential as a function of f along fixed q .FIG. 2: Aperiodic spiralling minimum of the potential. Here F = ( f + i f ) / √
2. The field can be well within the Planck scale throughoutwhile travelling a super-Planckian distance D a in field space. Therefore we can use the standard results for chaotic inflation models with power n / ( m + ) . The slow-roll parameters as afunction of the number of e-foldings N are then e = (cid:18) n ( m + ) (cid:19) N (13)and h = (cid:18) n − m − ( m + ) (cid:19) N . (14)The spectral index n s and the tensor-to-scalar ratio r are n s = + h − e = − (cid:18) n + m + ( m + ) (cid:19) N (15)and r = e = (cid:18) nm + (cid:19) N . (16)ˆ a and f are related to N by ˆ aM Pl = (cid:18) nNm + (cid:19) / (17)and (cid:18) f √ L (cid:19) m + = (cid:18) M Pl L n ( m + ) Nm (cid:19) / . (18)This model can therefore provide a good fit to the spectral index observed by Planck ( n s = . ± . r = . + . − . ) [1, 2] by choosing the parameters to reproduce the results of f chaoticinflation. In that case n s = − / N and r = / N , which gives n s = .
964 and r = .
15 for N =
55. The spectral index of ourmodel will equal that of f chaotic inflation if n = m + . (19)For example, if m = n = | F | is sub-Planckian throughout. To do this we need to fix the value of L . L is determined by the adiabaticpower spectrum, P z = V p e M Pl . (20)Therefore P z = l p (cid:18) m + n (cid:19) (cid:18) n ( m + ) m (cid:19) n ( m + ) (cid:18) L M Pl (cid:19) nmm + N m + n + ( m + ) . (21)For the case n = m =
1, this implies that L M Pl = p P z l ! / N . (22)With N =
55 and P / z = . × − , this gives L M Pl = . × − l − / . (23)From Eq. (18) we find, fL = (cid:18) NM Pl L (cid:19) / (24)therefore f M Pl = ( N ) / (cid:18) L M Pl (cid:19) / . (25)At N =
55 this gives, f M Pl = . l − / . (26)Thus inflation will be sub-Planckian as long as l is not very small compared to 1. The natural range of values for l in particlephysics models is l = . −
1, in which case | F | / M Pl = . − . D ˆ a satisfies [3, 5] D ˆ aM Pl ≥ p e min N ≈ r r N (27)where e min is the smallest value during inflation and r ≈ e min is assumed. With r = . N =
55 this requires that D ˆ a > ∼ . M Pl . In our model D ˆ a is essentially equal to ˆ a in Eq. (17). For n = m =
1, the change during 55 e-foldingsis D ˆ a ≈ M Pl , as usual for a f chaotic inflation model.We finally check the assumptions which underlie our analysis. In order to treat inflation as an effective single field model withinflaton ˆ a , we require that m f ≫ H . At the local minimum the mass squared of f is given by m f = Am l | F | n + m − M n − Pl L m . (28)Using this, the condition m f ≫ H becomes (cid:18) f √ L (cid:19) m − ≫ m A L M Pl . (29)For m = A ≫ (cid:18) L M Pl (cid:19) . (30)With L / M Pl = . × − l − / this requires that A ≫ . × − l − . This is easily satisfied so long as A is not extremelysmall.A second assumption is that ( f / √ L ) m ≫
1, in order that da = f d q is true and a can be treated as a canonically normalizedscalar field. For m = N =
55 we find that f / √ L ≈ l / . This is much larger than 1 so long as l is not extremelysmall.Finally, we require that A ( f / √ L ) m ≫ m = A ( l / ) ≫ A l / is not too small compared to 1.In summary, the model can generate inflation which is in good agreement with observations while keeping | F | sub-Planckianthroughout. For the case n = m = V ( f , q ) = l | F | (cid:18) + A (cid:18) | F | L + q (cid:19)(cid:19) , (31)where F = f e i q / √
2, and the results for n s and r are identical to the case of f chaotic inflation. With l =
1, in which case L = . × − M Pl = . × GeV, the value of | F | satisfies | F | < ∼ . M Pl throughout the final N =
55 e-foldings. Theresults are independent of the value of A so long as A is small compared to 1.A | F | potential is a natural choice for the base potential from a dimensional point of view. Assuming that m is an integer,the simplest models then correspond to m = m =
2, which behave as f and f / chaotic inflation. The former is goodagreement with present BICEP2 results but in tension with the Planck upper bound, r < .
11 (95% c.l.) [12]. The m = n s = − / N = .
970 and r = / N = .
097 when N =
55. (In this case | F | / M Pl < ∼ . l − / during inflation.)Therefore if the forthcoming Planck result for r turns out to be substantially less than the BICEP2 value, then this model couldbecome favoured. III. CONCLUSIONS
In the absence of a shift or other symmetry, the Lyth bound presents a severe problem for single-field inflation if Planck-suppressed corrections to the potential exist. N-flation would require a very large number of scalar fields to keep the inflatonsub-Planckian. The alternative is to consider multi-field inflation with a trajectory which is non-radial, so that the change inthe inflaton field can be super-Planckian while the magnitude of the field remains sub-Planckian throughout. This requires anappropriate potential landscape. We have presented an explicit example of such a model. The model is based on a complexfield with a nearly circular but aperiodic trajectory in the complex plane. This is achieved by a sinusoidal modulation of a | F | n potential. The model acts as a single-field inflation model with a spiral-like trajectory. We have shown that the simplest sinusoidalmodulation of a | F | potential exactly reproduces the values for n s and r of the f chaotic inflation model while maintaining | F | < ∼ . M Pl throughout. More generally, the simplest models based on a | F | potential favour predictions equivalent to f and f / chaotic inflation.The model therefore serves as an existence proof for two-field inflation models that are consistent with Planck and BICEP2and which satisfy the Lyth bound while remaining sub-Planckian throughout. The aperiodic modulated | F | potential we havestudied has no obvious physical explanation at present. On the other hand, the potential is a particularly simple example of atwo-field potential landscape and might therefore be realized as part of a complete theory of Planck-scale physics.We finally comment on the relation of this model to axion inflation and monodromy [11, 13–19]. In the axion inflationmonodromy model of [13] and [14], the effective theory consists of two real scalar fields with a broken shift symmetry. (Thedynamics of monodromy in the later string versions are not explicit in the four-dimensional effective theory [15].) The decayconstants of the axions are approximately aligned to produce an almost flat direction which can be used for inflation. Theeffective theory is equivalent to a single-field inflation model with a super-Planckian inflaton and Planck-suppressed correctionseliminated by the shift symmetry. The sub-Planckian nature of the model is only apparent in the full theory with two complexfields, where the two real scalar fields are seen to be two axions and the full theory is therefore sub-Planckian if their decayconstants are sufficiently sub-Planckian. In our model, the sub-Planckian nature of the model is explicit in the effective theoryitself; in the simplest n = m = f and f are no larger than O(0.01) M Pl . Our model also differs inbeing based on a single complex field, with the modulus of the field playing a role in the aperiodic potential. The axion inflationmodels of [13] and [14] make predictions which are generally different from the power-law chaotic inflation predictions of ourmodel. The string axion monodromy models of [15–17], on the other hand, are equivalent to chaotic inflation models with V ( f ) (cid:181) f and an additional sinusoidal modulation. (Recently this has been generalized to a range of power-law potentials [20].)Finally, the two-axion "Dante’s Inferno" model [11] makes predictions which are equivalent to those of the present model forthe case m = Acknowledgements
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