# Primordial perturbations from inflation with a hyperbolic field-space

## Abstract

We study primordial perturbations from hyperinflation, proposed recently and based on a hyperbolic field-space. In the previous work, it was shown that the field-space angular momentum supported by the negative curvature modifies the background dynamics and enhances fluctuations of the scalar fields qualitatively, assuming that the inflationary background is almost de Sitter. In this work, we confirm and extend the analysis based on the standard approach of cosmological perturbation in multi-field inflation. At the background level, to quantify the deviation from de Sitter, we introduce the slow-varying parameters and show that steep potentials, which usually can not drive inflation, can drive inflation. At the linear perturbation level, we obtain the power spectrum of primordial curvature perturbation and express the spectral tilt and running in terms of the slow-varying parameters. We show that hyperinflation with power-law type potentials has already been excluded by the recent Planck observations, while exponential-type potential with the exponent of order unity can be made consistent with observations as far as the power spectrum is concerned. We also argue that, in the context of a simple -brane inflation, the hyperinflation requires exponentially large hyperbolic extra dimensions but that masses of Kaluza-Klein gravitons can be kept relatively heavy.

^{1}

## I Introduction

Cosmic inflation is widely believed to be the most plausible explanation for the origin of temperature fluctuations of the cosmic microwave background (CMB) and large scale structure (LSS) of the Universe Guth:1982ec (); Hawking:1982cz (); Starobinsky:1982ee (); Bardeen:1983qw () (see e.g. Kodama:1985bj (); Mukhanov:1990me () for reviews). From the recent Planck observations, the primordial density perturbations generated during inflation that are almost scale-invariant and Gaussian is strongly supported by recent Planck observations Ade:2015xua (); Ade:2015ava (). This is consistent with the prediction of the simplest single-field inflation models, where the inflaton has a canonical kinetic term and a sufficiently flat potential so that it rolls slowly during inflation, and couples minimally to gravity.

Regardless of the phenomenological success, it is still nontrivial to embed the single-field slow-roll inflation into a more fundamental theory (see Baumann:2014nda (), for a review). One important concern is related with the fact that the scalar fields are ubiquitous in supergravity or string theory and in some cases, the field other than the inflaton modifies the observable predictions based on the corresponding single-field slow-roll inflation. One attempt to address this concern is spinflation Easson:2007dh (), formulated as a variant of Dirac-Born-Infeld (DBI) inflation Silverstein:2003hf (); Alishahiha:2004eh (), where inflation is driven by the motion of a D-brane. In this model, one field corresponds to the radial coordinate and the other field corresponds to the angular coordinate of the D-brane’s position in the internal space. In this model, the dynamics of the inflaton is modified so that instead of rolling straight down to the origin, it orbits around the bottom of the potential, which results in the increase of e-foldings during inflation compared with the single-field inflation considering only the radial motion. Although the idea was very interesting, quantitatively, since the original version of spinflation is based on the flat field-space, the angular momentum is diluted away soon and the increase of e-foldings is not so significant (see however discussions for the cases with an angular direction dependent potential, Gregory:2011cd (); Kidani:2014pka ()).

On the other hand, recently, an interesting extension of spinflation, dubbed “hyperinflation”, based on the negatively-curved (hyperbolic) field space, has been proposed Brown:2017osf (). It was shown that in this model, instead of diluted away quickly, the field-space angular momentum sourced by the negative curvature modifies the dynamics of the inflaton field drastically, as we will explain in the following. Notice that internal space with negative curvature is also ubiquitous in cosmology based on high energy theories. For example, compact hyperbolic spaces have been considered in the context of the large extra-dimension scenarios in order to render the fundamental gravitational scale as low as without fine-tuning, making use of the altered spectrum of Kaluza-Klein modes Kaloper:2000jb (); Starkman:2000dy (); Starkman:2001xu (); Nasri:2002rx (); Greene:2010ch (); Kim:2010fq (). Compactification on compact hyperbolic spaces has been considered also to obtain accelerating cosmological solutions in the context of string theory setups Townsend:2003fx (); Ohta:2003pu (); Roy:2003nd (); Chen:2003ij (); Wohlfarth:2003ni (); Ohta:2003ie (); Gutperle:2003kc (); Emparan:2003gg (); Chen:2003dca (); Wohlfarth:2003kw (); Neupane:2003cs () to evade the “no-go” theorem forbidding cosmic acceleration in the standard compactifications with non-negative internal curvature Gibbons (); Maldacena:2000mw (). It can be also shown that a similar setup is derived in the context of the Higgs-dilaton cosmology GarciaBellido:2011de (); Bezrukov:2012hx (); Karananas:2016kyt (), multifield inflation with nonminimal couplings Kaiser:2012ak (); Greenwood:2012aj (); Kaiser:2013sna (); Schutz:2013fua (), and modular inflation Schimmrigk:2014ica (); Schimmrigk:2015qju (); Schimmrigk:2016bde (). Furthermore, it is known that the -attractor scenario Kallosh:2013hoa (), which is extensively studied recently, can be embedded in supergravity based on the negatively-curved Käler manifold Ferrara:2013rsa (); Carrasco:2015uma (). (See also Renaux-Petel:2015mga (); Achucarro:2016fby () for other interesting cosmological scenarios proposed recently, making use of a negatively-curved field-space.)

In Ref. Brown:2017osf (), it was shown that there exists a consistent background solution supported by the field-space angular momentum and that the model predicts scale-invariant fluctuations of inflaton field with an almost exponential enhancement factor compared with the conventional case by assuming that the inflationary background is almost de Sitter^{2}

## Ii Model and Background Dynamics

### ii.1 Model

Recently, the author of Ref. Brown:2017osf (), has proposed an interesting model whose action of the scalar fields is given by

(1) |

where we have introduced the notation, and , and correspond to the radial and angular directions, respectively, is the metric of the field-space and is related with the curvature length of the field-space. This property becomes clear if we start with the field-space whose metric is given by , change the field variable to , where is the typical mass scale related with this field, and define the new field-space metric by . In (1), the form of potential is assumed to be rotationally symmetric with a minimum at . Since the value of itself is not important from the rotational symmetry of the field-space, we discuss its dynamics in terms of . Notice that since corresponds to the radial direction, the range is limited to .

In this setup, we can define the angular momentum and the orbital kinetic energy of this field-space by

(2) |

respectively. We can see that for , where , even if is not large, can be exponentially large and it is expected that this angular motion in the field-space can give interesting phenomenology in inflation. If we consider sub-Planckian values of the inflaton field as suggested by most of stringy setups (except for axion monodromy models McAllister:2008hb ()) then is possible only if . The same condition is required also by the background dynamics that we shall consider later (see (40)).

To see that and are in principle possible, let us consider -branes in -dimensional spacetime of the form.

(3) |

where is a -dimensional metric (), is the metric of a -dimensional compact hyperbolic space (), and is a positive function often called a warp factor. Upon considering coincident -branes whose world-volume is embedded in the -dimensional spacetime as

(4) |

where represents coordinates on the brane, the induced metric of the brane world-volume is

(5) |

and thus the DBI action is

(6) |

Here, is the tension of the coincident -branes, is the string coupling, is the inverse of the string tension and we have defined

(7) |

Adding the Chern-Simons term and potentials induced by various interactions of the coincident -branes with other branes and moduli, we obtain the action

(8) |

where we have ignored terms of . Considering the radial coordinate () in the extra-dimensions so that , we now introduce the inflation field () as . This implies that and thus , where is the curvature radius of the extra-dimensions and is the string length. Obviously, the supergravity approximation is justified only if

(9) |

On the other hand, supposing that all moduli are stabilized above a certain scale sufficiently higher than the Hubble expansion rate during inflation, the -dimensional metric is described by the Einstein gravity with the Newton’s constant , where . Here, is the -dimensional Planck scale and is the volume of the -dimensional extra dimensions. Therefore,

(10) |

This means that holds under the condition (9) if and if

(11) |

For - (or -) dimensional compact hyperbolic spaces, it is known that the ratio of the area (or volume) to the curvature length squared (or cubic) is determined by the topology of the manifold and takes values from to . We here assume that a similar statement holds for -dimensional compact hyperbolic spaces so that (11) is possible. Indeed, a simple argument Kaloper:2000jb () leads to a relation among the volume , the curvature length and the largest linear dimension in the limit as , which translates to

(12) |

where is the maximum value of . This indicates that is relatively easy to satisfy. Actually, the relation (12) implies that is possible if and only if is exponentially large. On the other hand, masses of Kaluza-Klein gravitons reflect the largest linear dimension Kaloper:2000jb () (instead of the volume ) and thus can be kept relatively heavy so that the model can pass various phenomenological tests, provided that the radion is properly stabilized Nasri:2002rx ().

### ii.2 Background dynamics

From the discussion in the previous subsection, we start with the action

(13) |

The equations of motion for the scalar fields are obtained from the variation of the action with respect to as

(14) |

where denotes the partial derivative with respect to .

Suppose that the Universe is homogeneous and isotropic with a Friedmann-Robertson-Walker (FRW) metric

(15) |

where is the scale factor whose evolution is governed by the Friedmann equations

(16) | |||

(17) |

Here, is the Hubble expansion rate, a dot denotes a derivative with respect to the cosmic time . Introducing the acceleration in the curved field-space and raising the field index, Eq. (14) becomes

(18) |

where is the Christoffel symbols associated with the field-space metric . For the quantities without the field-space indices, acts as an ordinary time derivative. For , the only nonzero and independent components of are

(19) |

In terms of and explicitly, Eqs. (18) become

(20) | |||

(21) |

where and is a constant with a dimension of mass, related with the conserved field-space angular momentum, fixed by the initial condition. We will assume that , or equivalently , which does not lose generality. Since Eq. (17) is obtained from Eqs. (16), (20) and (21), the basic equations are given by Eqs. (16) and (20) with the replacement of given by Eq. (21). Notice that since we have assumed that has a minimum at , there is a region with near . We will further assume that is kept satisfied until , where the effect of the curvature of the field-space is significant and we will concentrate on the region with .

Before analyzing the background dynamics, for later use, we will present the equations of motion of the scalar fields in a different orthonormal basis in field-space based on the adiabatic-entropic decomposition Gordon:2000hv (); GrootNibbelink:2001qt (), so that

(22) |

Here, the new basis vectors are , where is the unit vector pointing to the adiabatic direction given by

(23) |

and is the speed of the fields in the field-space, while is the entropic unit vector, which is orthogonal to ,

(24) |

The two vectors and satisfy the orthonormality condition

(25) |

In terms of the new components , the equations of motion (18) become

(26) |

Notice that are antisymmetric by definition as a consequence of the orthonormality condition (25).

The adiabatic component and the entropic component of Eq. (26) are given by

(27) |

In the following, we find inflationary solutions and as in the usual slow-roll approximations, we impose the conditions that in Eq. (16) and in Eq. (20). As we will see, since there is possibility that the motion of inflaton is dominated by the angular one and the radial field velocity is controlled by the centrifugal force, which should be distinguished from the standard slow-roll, we will call such conditions as slow-varying in the sense that and changes in very slowly. Furthermore, in order to make the effect of the curvature of the field-space significant, we also restrict ourselves to the region with , where throughout the rest of the present paper.

#### Inflationary background with constant

With the assumptions mentioned above, the basic equations become

(28) | |||

(29) |

and in order for this equation to hold at any time,

(30) |

By taking the time derivative of Eq. (30), we can relate and ,

(31) |

where the last equality holds as long as the condition

(32) |

is satisfied. We will impose the condition (32), for simplicity, from now on. Then, if the potential is steep enough to satisfy

(33) |

in terms of the parameter defined by

(34) |

we can express the time evolution of in terms of as

(35) |

which was found in Ref. Brown:2017osf (). Here, we would like to add more about the validity of this solution. Since is a constant related with the conservation of the angular momentum in the field-space, the two terms in the square root in Eq. (35), and should have the same time-dependence throughout inflation. From Eq. (28), it is possible only for the exponential type potential,

(36) |

which gives constant . Therefore, here, we will continue the discussion based on the solution characterized by Eqs. (31), (35) by assuming the exponential type potential and in the next subsubsection, we will consider more general potentials.

In terms of , the ratio between the kinetic energy of and that of is given by

(37) |

and we can express the slow-varying parameter as

(38) |

where we have used Eqs. (28) and (34) to rewrite in terms of and . Since we have considered the exponential type potential (36), is constant and . Therefore, in this case, inflation occurs when and it is obvious that the condition (32) is also satisfied during inflation. Comparing Eq. (38) with the one of the standard single-field slow-roll models , we can see that the inflationary dynamics is not controlled by , but , which comes from the fact that the radial motion is driven not by Hubble friction, but by the centrifugal force in this model. Notice that for the same potential and field value, is smaller than that would be in the standard single-field slow-roll model if

(39) |

Since this condition coincides with (33) under the slow-varying approximation, whenever this inflationary solution exists, is suppressed compared with the one would-be in standard single-field slow-roll inflation. From the discussions above, for this type of inflation occurs, should satisfy and for consistency, we must impose the condition,

(40) |

At the end of subsection II.1 we have argued that this is in principle possible.

Notice that it is well known that in the conventional single-field slow-roll inflation, we can obtain inflation solutions with the potential (36) for Lucchin:1984yf (); Kitada:1992uh (). However, in this setup, we can obtain inflationary solutions for

(41) |

which means that from Eq. (40) we can obtain
inflation from a steep potential, which can not drive single-field slow-roll inflation^{3}

#### Inflationary background with time-dependent

Although the background solution with an exponential type potential discussed in the previous subsubsection is helpful for the intuitive understanding of the effect of the field-space angular momentum, in general, is not constant during inflation. For this purpose, instead of assuming that all three terms in Eq. (29) scale, we will assume that only the last two terms scale, that is,

(42) |

so that the slow-varying of is realized solely by the effect of the angular momentum. Actually, even under this assumption, as long as the condition (32) is satisfied, we can show that Eq. (31) holds, with which the condition (42) can be rewritten as

(43) |

For the power-law type potentials , , this is a natural assumption since once the condition (33) is satisfied and inflationary starts, as goes toward , becomes larger and larger. With this assumption, we can obtain

(44) |

Although these equations seem to be same as Eqs. (35), notice that, here, we have defined in a different way as

(45) |

and it is no longer constant. Regardless of this, in the limit (43), or equivalently , two definitions (34) and (45) do not give significant difference. Since the expressions of and are unchanged in terms of , Eq. (37) holds in this case, too and from Eqs. (43) and (45), the total kinetic energy is dominated by . Then, we can express the slow-varying parameter as

(46) |

which does not change from Eq. (38). Therefore, the discussion below Eq. (38) also holds and the condition (40) is required for realizing hyperinflation in this case. As a concrete example, for the power-law type potentials, , , is given by

(47) |

Since is no longer constant, the next-order slow-varying parameter is nonzero and given by

(48) |

Since we have already imposed (32), becomes small as long as is small. For the power-law type potentials, , since , imposing Eq. (32) during inflation is reasonable and is given by

(49) |

If we adopt the slow-varying approximations together with Eq. (32), breaks down when , which means that both and must be much smaller than during inflation. Notice that we have chosen the parameter , which is related with the field-space curvature to satisfy Eq. (40), thus the other slow-roll conditions like , , never break down as long as we assume that the slow-roll background solution holds.

### ii.3 Numerical results

#### Model with

Here, we show the numerical results to confirm that the solutions describing the inflationary background discussed in the previous subsection are realized. As a first example, we consider a model with discussed in subsubsection II.2.1. Here, we first express the time evolution of the quantities in terms of and compare them with numerical results.

From Eq. (35) with Eq. (36), we obtain

(50) |

From this, we can be expressed in terms of as

(51) |

Together with Eqs. (21) and (28), we can also obtain the time evolution of , and as

(52) | |||

(53) | |||

(54) |

where in the above holds in the limit .

For the numerical calculation, we choose the values of the parameters as so that the condition (40) and and , for simplicity. The initial values are chosen as , , , , at so that the analytic solution describing the inflationary phase is realized without numerically complicated behavior. In the left panel of Fig. 1, we plot the time evolution of in terms of , which is well approximated by Eq. (51) with an appropriate choice of . In the right panel of Fig. 1, we plot the time evolution of , in terms of . Again, we confirm that it is well approximated by (54), which is constant as we have expected. Although we do not show the numerical results for , , , we can check that these quantities are also well described by the analytic solutions presented above.

#### Model with

Here, as a next example, we investigate a model with in a similar way as the model with an exponential type potential. For the setup discussed in subsubsection II.2.2 with , we obtain

(55) |

From Eq. (55), we can relate the time evolution of and by

(56) |

and this relation can be inverted with the help of the Lambert’s W function, satisfying ,

(57) |

Together with Eqs. (21) and (28), we can also obtain the time evolution of and in terms of as

(58) |

Making use of the formula of the Lambert’s W function, , we can obtain and as

(59) |

where in the above holds in the limit and Eqs. (47) and (49) are recovered in this limit.

For the numerical calculation, we choose so that the condition (40) is satisfied and , for simplicity. The initial values are chosen as , , , , at so that the approximated solution describing the inflationary phase is realized without numerically complicated behaviors. In the left panel of Fig. 2, we plot the time evolution of in terms of , which is well approximated by Eq. (57) with an appropriate choice of . In the right panel of Fig. 2, we plot the time evolution of , in terms of . Again, we confirm that it is well approximated by (59) with an appropriate choice of . Although we do not show the numerical results for , , , we can check that these quantities are also well described by the analytic solutions presented above.

## Iii Dynamics of the linear perturbations

In this section, we analyze the dynamics of the linear perturbations of the scalar fields. First, we derive the second-order action in subsection. III.1 and then, we investigate the time evolution and scale-dependence in subsection. III.2.

### iii.1 Second-order action

The modern approach to derive the second-order action was introduced by Maldacena:2002vr (); Seery:2005gb (), base on the Arnowitt-Deser-Misner (ADM) formalism Arnowitt:1962hi (). In the ADM approach, the metric is written in the form

(60) |

where is the lapse function and is the shift vector. After substituting (60) into (13), the action can be written as

(61) |

where is the scalar curvature of the hypersurface, , and are defined as

(62) |

Here, the symbol denotes the spatial covariant derivative associated with the spatial metric and is proportional to the extrinsic curvature of the hypersurface. The variation of the action with respect to N yields the energy constraint,

(63) |

while the variation of the action with respect to the shift gives the momentum constraint,

(64) |

We will study the linear perturbations about the FRW background and from now on we specify , the spatially flat slice, so that the physical degrees of freedom are fully specified. The scalar fields can be decomposed as

(65) |

where are background values of the fields and denotes the linear perturbations. In the following, we will simply write the homogeneous value as as long as this does not give confusions. We can also write the scalarly perturbed lapse and shift as

(66) |

At first-order, the momentum constraint implies

(67) |

where here is the field-space metric evaluated by the background field values. On the other hand, the energy constraint gives

(68) |

where we have extended the notation ,which is introduced in (18), to , so that

(69) |

Now that and are expressed by , the physical degrees of freedom are completely encoded in scalar field perturbations .

We now expand the action, up to quadratic order in terms of the linear perturbations. By substituting the expression (67) for , the second-order action for can be written as