Multifield Dynamics in Higgs-otic Inflation

In Higgs-otic inflation a complex neutral scalar combination of the $h^0$ and $H^0$ MSSM Higgs fields plays the role of inflaton in a chaotic fashion. The potential is protected from large trans-Planckian corrections at large inflaton if the system is embedded in string theory so that the Higgs fields parametrize a D-brane position. The inflaton potential is then given by a DBI+CS D-brane action yielding an approximate linear behaviour at large field. The inflaton scalar potential is a 2-field model with specific non-canonical kinetic terms. Previous computations of the cosmological parameters (i.e. scalar and tensor perturbations) did not take into account the full 2-field character of the model, ignoring in particular the presence of isocurvature perturbations and their coupling to the adiabatic modes. It is well known that for generic 2-field potentials such effects may significantly alter the observational signatures of a given model. We perform a full analysis of adiabatic and isocurvature perturbations in the Higgs-otic 2-field model. We show that the predictivity of the model is increased compared to the adiabatic approximation. Isocurvature perturbations moderately feed back into adiabatic fluctuations. However, the isocurvature component is exponentially damped by the end of inflation. The tensor to scalar ratio varies in a region $r=0.08-0.12$, consistent with combined Planck/BICEP results.


Introduction
It is very hard to protect fundamental scalars in field theory from large quantum corrections.
It is well known that these quantum corrections easily drive the scalar masses to the size of the largest ultraviolet cut-off in the theory. Even so, light scalars seem to play an important role in our understanding of the observed properties of particle physics and cosmology. On the one hand the recent observation of the Higgs particle at the LHC [1] is consistent with a fundamental SU (2) × U (1) complex doublet remaining light well below any ultraviolet cutoff. On the cosmology side, precision data increasingly favours the existence of an inflaton scalar, with mass well below the Planck scale, leading to a period of fast expansion of the universe during which quantum perturbations generate CMB anisotropies and plant the seeds for galaxy formation.
Low-energy supersymmetry provides a rationale for the stability of the Higgs mass against quantum fluctuations. However, the fact that no trace of SUSY has been found yet at the LHC and the measured value of the Higgs mass m h 125 GeV cast some doubts on the idea of low-energy SUSY. Indeed, such a Higgs mass is quite high compared to the expectations from the MSSM and can only be reached for very heavy SUSY spectra. One can think of disposing altogether with the idea of SUSY and admit that the Higgs field is fine-tuned to be light, perhaps on the basis of anthropic arguments. Still this is probably not enough, since if we insist on the validity of a non-SUSY SM structure at higher energies, the renormalisation group evolution of the Higgs self-coupling is such that the potential becomes unstable (or metastable) for scales larger than ∼ 10 10 GeV, possibly indicating the presence of new physics at these scales, well below the Planck scale (see e.g. [2]).
A most elegant way to stabilise the Higgs potential for scales above ∼ 10 10 GeV is again SUSY. Indeed, the scalar potential of e.g. the MSSM is positive definite for scales above SUSY breaking and hence the potential is automatically stable. It was shown in [3,4] (see also [5][6][7]) that under very general assumptions, if SUSY is broken at scales 10 10−13 GeV, the resulting Higgs mass at low-energies is consistent with its measured mass ∼ 125 GeV.
Of course, with such a high SUSY breaking scale the Higgs mass is again unprotected and one should admit fine-tuning as the ultimate reason for its lightness. This fine-tuning could come from an underlying multiverse within string theory, which would also be at the root of the understanding of the smallness of the cosmological constant.
On the cosmology side, a fundamental scalar degree of freedom, the inflaton, seems to provide the simplest explanation for a variety of cosmological observations. That scalar again will have to be protected from acquiring a large mass and interactions that spoil slow-roll.
On the other hand, as we already remarked, light scalars usually do not stay light in a field theory. Intuitively, one may expect having two fundamental scalars light to be even more unlikely than a single (Higgs) one. Thus if the Higgs field itself could act as the inflaton the combination Higgs/inflaton would look more likely within a landscape of theories. Of course, independently from any landscape argument, it is a natural question whether the Higgs and inflaton scalar could be one and the same. This has been considered often in the past, see e.g. [8][9][10][11][12][13][14][15], see [16] for a review and references.
In ref. [17] it was proposed that the neutral Higgs system of the MSSM with SUSY broken at a large scale ∼ 10 13 GeV could be in charge of cosmic inflation. This goes under the name of Higgs-otic inflation [18], since it is in some sense a MSSM Higgs version of Linde's chaotic inflation [19]. This proposal is quite economical since it addresses several issues simultaneously. It provides stability for the Higgs scalar potential at the right scale, is consistent with the observed value of the Higgs mass and a neutral Higgs component acts as a complex inflaton. The inflaton has a trans-Planckian field range and leads to a certain variety of (non-canonical) 2-field chaotic like inflation. In order for this to be sensible we need a theory of quantum gravity, which is naturally identified with string theory.
In general, a substantial contribution of tensor perturbations is generated.
The Higgs-otic model is a 2-field inflaton system. In [18] a study of the cosmological observables was made for the model focusing only on the adiabatic perturbations and ignoring possible 2-field specific effects like the generation of isocurvature perturbations. This was an important pending issue since 2-field effects can, in principle, substantially modify the cosmological observables and furthermore the Planck satellite has provided stringent bounds on isocurvature perturbations. In the present paper we perform a systematic analysis of the observables in the Higgs-otic 2-field inflation system. We find that, as expected, adiabatic and isocurvature perturbations form a coupled system and there is super-horizon evolution of the curvature perturbations. This leads in general to a relative increase of adiabatic perturbations and consequently to a reduction of the tensor to scalar ratio r compared to the computation in ref. [18]. The range of variation of n s gets smaller and is centered around the region alowed by Planck data with a tensor to scalar ratio in a range r = 0.08 − 0.12.
Moreover, the isocurvature component is always very suppressed at the end of inflation, consistent with upper Planck bounds.
The structure of this paper is as follows. We review the main points of Higgs-otic inflation in the next section, in which the relevant definitions and the inflaton potential are described. In section 3 we review the main issues of 2-field inflation as applied to the case of Higgs-otic inflation. Section 4 presents the results for Higgs-otic inflation for three representative points in the parameter space of the induced soft-terms for the Higgs-inflaton system. The latter are determined by a real positive parameter 0 , H, h being the neutral Higgs scalars driving inflation [18]. The first case (A = 0.83) correspond to the canonical Higgs-otic model in which the lightest scalar field at the minimum of the potential (at scale M SS ) can be identified with the SM Higgs field. A second case (A = 0.7) analyses how those results are changed if there is some effect (like modified RG running) slightly modifying the Higgs-otic setting. For completeness we finally present a third case with A = 0.2 in which the inflaton cannot be identified with the MSSM Higgs fields but could be relevant in some extensions of the MSSM. At the end of this section we show the expectations for the r − n s plot in the Higgs-otic model for a variety of initial conditions and mass scales. Finally, section 5 is left for the conclusions.
-3 -In this section we briefly review and extend the results presented in [17,18] on Higgs-otic inflation. We thus motivate the form of the non-minimal 2-field inflaton potential to be analysed in the following sections. Readers interested only in the inflationary analysis may safely jump to the next section.
Higgs-otic inflation refers to theories in which the inflaton is a complex scalar giving rise to gauge symmetry breaking, while attaining large field inflation. The most obvious and natural candidate for that is the SM Higgs field itself, as described in [18] . Nevertheless the same idea may be applied to other BSM fields introduced for other purposes, as we briefly discuss below. The essential ingredient is the identification of a complex inflaton with the position moduli of some Dp-brane system in string compactifications. 1 The motion of the brane corresponds to the gauge symmetry breaking through a scalar vev. For concreteness we will consider the position moduli of a D7-brane describing cycles in a T 2 torus inside a Type IIB orientifold compactification, although the setting may be easily generalized to other string configurations 2 . As we emphasised above, the reason to go to a string setting in order to implement this idea is twofold. First, we will be interested in producing a large field model in which inflation implies trans-Planckian excursions of the inflaton. In order to do so in a sensible manner we need a theory of quantum gravity. Indeed within string theory the vevs of scalar fields may be trans-Planckian and one can still maintain an effective potential Lagrangian which makes sense. Second, string theory has the required properties/symmetries in order to keep under control Planck-supressed corrections which become obviously important at trans-Planckian field values. There are modular and shift symmetries which force these potential corrections to be subleading. So let us consider, for definiteness, a Type IIB orientifold yielding a theory with N = 1 supersymmetry in D = 4. Gauge interactions appear from stacks of D7-branes wrapping 4-cycles in the CY compact manifold. Those will include the SM gauge interactions as well as the SM fields. They come from zero modes of D = 8 complex scalars Φ and A upon reduction to D = 4. In particular, the position of the D7-branes is parametrized by the vevs of Φ zero modes. On a stack of N D7 the gauge group is U (N ) and the scalars transform as adjoint chiral multiplets. The gauge symmetry may be reduced to that of the SM (or some extension) by e.g. an orbifold twist in the compactification. Generically only some fields of the adjoint scalars Φ survive, like those parametrising the motion of those D7-branes which 1 The same structure appears in terms of continuous complex Wilson lines in models with extra dimensions.
From the string theory point of view they correspond to equivalent T-dual compactifications. 2 For an introduction to orientifold constructions see [42] and references therein.
can move along flat directions. This is the case of D7-branes associated to SM Higgs vevs in ref. [18].
The generic presence of closed string fluxes in Type IIB compactifications will give rise to a non-trivial potential for these fields, see ref. [43][44][45][46][47][48][49][50]. We thus assume, as is customary, that there are imaginary self-dual (ISD) fluxes G 3 acting as a background. Such backgrounds are known to be solutions of Type IIB D = 10 equations of motion in warped CY backgrounds. In these backgrounds there are two classes of ISD fluxes, with tensor structure G (0,3) and G (2,1) , respectively. The first correspond locally to components G123 and the second to G12 3 . From now on we will denote G ≡ G123 and S ≡ 3jk G 3jk . The first class of flux, G, breaks SUSY and gives rise to SUSY-breaking soft terms, scalar and gaugino masses. The second class, S, preserves SUSY and gives rise to supersymmetric F-terms, i.e., µ-terms. These induced masses will eventually be identified with the mass scale of the inflaton. The cosmological bounds on the density scalar perturbations applied to our model fix this scale to be of order 10 12 − 10 13 GeV [51]. In order to identify the inflaton with an MSSM Higgs boson we will see that the simultaneous presence of both classes of fluxes, G and S, is required. This implies that the supersymmetry breaking scale M ss (ie. the scale of the soft terms) in this large field inflation setting will also be of order 10 12 − 10 13 GeV. This is consistent with the scenario of Intermediate SUSY breaking in [3,[5][6][7] which was shown to be consistent with a Higgs mass of 126 GeV [4]. The structure of mass scales in the Higgs-otic setting is summarised in where there is a non-canonical kinetic factor f (Φ) which depends on the scalar (F-term) potential V F as and Str denotes the symmetrised trace over gauge indices. Here g s is the string coupling, V 4 the volume wrapped by the D7-brane and Z −2 = 1 in the absence of warping, which we will assume in what follows. In addition there is the D7 tension µ 7 = M 8 s /(2π) 7 , with M s = (α ) −1/2 the string scale. To obtain Eq. (2.1) we have kept all the terms involving higher order powers of the position modulus Φ, as required in a consistent large field inflation model. This is doable because the DBI action contains all perturbative corrections in α , whose effect is encoded in the non-canonical kinetic factor f (Φ). For small field f ∼ 1 and we recover the result prescribed by the 4d supergravity effective action [18].
The above action concerns the U (N ) adjoint in the world volume of D7-branes. However, it still applies after we make an orbifold projection that converts the adjoint into a set of bifundamentals, some of which are identified with the Higgs field. For example [17,18] one can start from a set of 6 D7-branes yielding an initial U (6) gauge and project out yielding a U (3) × U (2) × U (1) gauge group. One linear combination of the U (1)'s can be identified with hypercharge, whereas the other two are anomalous and become massive in the usual way. The adjoint Φ contains doublets surviving projection, i.e.
The same scalar potential may be obtained in an N = 1 supergravity formulation not resting on the DBI+CS action. However, such a SUGRA treatment will miss the important effect of the non-canonical kinetic term, which gives rise to the flattening of the potential, see [18].
where one defines with γ = π − Arg(GS) given by the relative phase of both fluxes.
The model is essentially double chaotic inflation with the key difference of non-canonical kinetic terms. To see different limits of this expression it is useful to define the real variable A which controls the relative size of SUSY (S) versus non-SUSY (G) fluxes, i.e. (2.7) Note that 0 ≤ A ≤ 1. This parameter may also written in terms of the masses of the above There are two interesting limits in which this 2-field scalar potential becomes effectively a single field potential: • A = 1. In this case |G| = |S| and the field h becomes massless. If we want to eventually fine-tune a Higgs doublet to remain at low-energies as a SM Higgs, much below the inflaton mass scale, we would need to be close to that situation. The Higgs field H will be the one producing inflation in this limit.
• A = 0. This happens, in particular, in the SUSY case in which G = 0. In this case the inflaton mass would be supersymmetric and σ = 1 √ 2 |h| 2 + |H| 2 acts as an effective single inflaton. In this case one can hardly identify any of these scalars with (neutral) SM Higgs fields, since they will have large masses, of order the inflaton mass 10 13 GeV. On the other hand such fields could be identified with other BSM fields which were not required to survive at low energies and could act as inflatons. Examples of these could be SU (5)  It is useful to write down the scalar potential in Eq. (2.5) in angular variables. Defining the D-term flat direction as one rewrites the potential as where we define Note that M SS measures the size of the soft terms and hence the inflaton mass. One can estimate the size of these parameters from purely stringy arguments by assuming an isotropic compactification, Z g s 1 and standard GUT values for the gauge couplings [18]. One finds where M s is the string scale, assumed to be of order 10 16 − 10 17 GeV. In this way one has an inflaton mass of order M SS 10 13 GeV, consistent with the cosmological bounds.
In angular variables the relevant piece of the action may be written as Eq.(2.10) will be our inflaton potential, although one should not forget that the kinetic terms are not canonical and this fact should be taken into account. This will lead to a general flattening effect, as we discuss below. Note that the shape of the scalar potential (before flattening) depends essentially on the value of the A-parameter defined above. In fig. 2 we show the structure of the scalar potential for three characteristic values A = 0.1, 0.5, 0.95.
In the one-field limits, A = 0, 1, one can always get canonical kinetic terms by making the following field redefinition (2.15) In those limits one has a potential The kinetic factor is then From Eq. (2.15) one finds the canonically normalised scalar ϕ There is not a simple analytic expression for the inverse function φ(ϕ), which is what one would want in order to get the final form of the canonically normalised potential. However, for large fields one can check that the second piece in Eq. (2.18) is small and one can invert analytically ϕ(φ) and obtain the scalar potential for the new variable Note that, for large field, the potential becomes linear in the inflaton ϕ. This behaviour for large field is also present in the results of the complete 2-field analysis below. However, finding such an analytic expression is not always possible. In general, for multiple fields there does not exist a transformation that makes the metric flat on the moduli space parametrized by φ a . Therefore, to make a complete analysis we will have to use the general Lagrangian of multiple fields 20) where in our case the metric will be given by and use the generalized expressions of the cosmological observables for non-canonical kinetic terms and multiple fields introduced in the next chapter.

Basics of 2-field inflation
We are interested in studying cosmological perturbations of a system of two scalar fields coupled to Einstein gravity. The action for such a system, if one allows for a curved field space, takes the form (8πG = M 2 P l = 1): where g µν is the spacetime metric with determinant g and G ab the field-space metric. The scalar fields φ 1 and φ 2 span a 2-dimensional manifold with all relevant information contained in G ab . The Christoffel symbols, Riemann tensor and Riemann scalar of the scalar manifold are defined in the usual way. Assuming that the scalar fields depend only on (cosmic) time, φ a = φ a 0 (t) and that the spacetime metric is the flat FRW metric: the equations of motion for the scalar fields are given by: This can expression can be written more economically using the covariant derivative defined The function a is the scale factor and the Hubble parameter, H, is defined as: H = ∂ t ln(a). The equation of motion for the scale factor is: Physically, measures the change of the Hubble sphere of the local universe. Hence, 0 < < 1 corresponds to a shrinking Hubble sphere, which is a good definition of an inflationairy period.
In order for inflation to last a sufficient amount of time the derivative of has to be small as well, so η 1. It is possible to use the equations of motion (3.3) and (3.4) to relate tȯ 0 , which implies that η is related to the tangential acceleration of the background trajectory.
In order to make connection with observations it is necessary to consider quantum perturbations around classical background solutions. This involves expanding the scalar and metric degrees of freedom in terms of the background quantities (φ a 0 and g 0 ab ) and perturbations, finding the equations of motion for the gauge invariant perturbations and solving them [52]. The spectral index and tensor-to-scalar ratio are defined in terms of the power spectra of the quantum fluctuations of scalar and tensor modes as we will shortly review.
It is convenient to define a local frame on the trajectory in field space given by [53][54][55]: These vectors form an orthonormal basis of the tangent space of the scalar manifold. Hence, they can be used to decompose the physically relevant quantities along the normal and tangential directions with respect to the background trajectory. In particular, the derivatives of the scalar potential can be written in this basis as V φ = T a ∂ a V and V N = N a ∂ a V (a labels both components of the basis vectors as well as fields). The total acceleration of the fields can be derived by taking the covariant derivative of T a , using the equations of motion and projecting with N a . One obtains:φ The parameter η needs to be generalised to capture the full dynamics in field space. Recalling that in single field settings η measures the acceleration of the scalar field, one finds that its convenient multifield generalisation is which measures the acceleration for the field φ a .
In the local (T, N ) basis, η a is projected on the following parallel and perpendicular components: such that: where a α and a † α are the usual creation and annihilation operators. We have switched to conformal time defined by: dτ = 1/a(t)dt. The Greek index α labels the quantum modes of the perturbations, consistency requires that α = 1, 2 in the 2-field case.
The equations of motion for the Mukhanov-Sasaki variables in this basis take the form [55]: We see that the 2-field perturbation system consists of a set of pairwise coupled harmonic oscillators. The coupling between curvature (v T α ) and isocurvature (v N α ) modes is controlled by η ⊥ which is inversely proportional to the curvature radius of the background trajectory. It follows that the coupling between curvature and isocurvature will be strong whenever there is a sharp turn in the background trajectory. The symmetric mass matrix Ω of Eq. (3.18) and (3.19) has the following elements: 20) where R is the Ricci scalar of the scalar manifold and the third slow-roll parameters are defined as: i.e. the initial conditions are simply given by the Bunch-Davies vacuum. Note that in Eq.
(3.25) it is understood that δ T 1 = 1 etc. and derivatives define the initial condition for d dτ v T,N α . It is important to note that there are two sets of equations of motion for the Muhkanov-Sasaki variables (in total 4 equations) and corresponding initial conditions, one for each value of α and that both should be taken into account when computing the inflationary observables.
The power spectra are defined in terms of the scalar 2-point functions as where P ζ (k, τ ) and P S (k, τ ) denote the dimensionless power spectra for the curvature and isocurvature modes respectively. Given that in multi-field models there can be superhorizon evolution of the perturbations, these are to be evaluated at the end of inflation. This is to be contrasted to single field models, where the freezing of curvature perturbations on superhorizon scales means the power spectra can be evaluated at horizon exit. From Eq.
(3.26) one can compute the spectral index for the curvature perturbations as well as the amplitude at the pivot scale k * A s = P ζ (k * , τ end ), (3.29) which in the absence of an analytical solution to Eqs. (3.18), (3.19) must be computed numerically. Noting that the tensor modes' evolution is unaffected by the number of dynamical fields driving the background expansion, the amplitude of the tensor power spectrum is given as in the single field case by which implies the following definition of the tensor to scalar ratio Besides probing the scalar and tensor power spectra, observations also put bounds on the total fraction of primordial isocurvature, defined as From the theoretical point of view the isocurvature fraction depends on the mass of the isocurvature modes (Ω N N ) and on the strength of their coupling to the adiabatic perturbations. The observational bounds on β iso can vary by many orders of magnitude, depending on how primordial isocurvature is transferred to the post-inflationary Universe. From [58] we find that the less constraining bound is of the order β iso ≤ 10 −3 (3.33) at the end of inflation.
In addition to putting constraints on the fraction of primordial isocurvature, observations also put constraints on the non-linear non-Gaussianity parameters f N L . Producing large non-Gaussianities would, in principle, spoil the validility of the model. However, for 2-field models the non-linear f N L are of the order of the slow-roll parameters and hence they are surpressed in our model, see [59], [60]. We will not consider non-Gaussianities beyond this point.
In order to understand how to relate the observables defined above with those of single field inflation let us once more take the decoupling limit η ⊥ → 0 and reduce the equations of motion to up to overall unimportant phases, with v T 2 = v N 1 = 0. Note that ν T = 3/2 and ν N < 3/2 to zeroth order in the slow-roll expansion.
On superhorizon scales, in the decoupling limit, one can show that the curvature perturbations are frozen as in the pure single field case, since Q T = v T /a ∝ a 0 while the isocurvature perturbations decay as Q N = v N /a ∝ a ν N −3/2 ∼ a − 2M 2 9H 2 to zeroth order in slow roll.
In order to make contact with observations one can compute the dimensionless power spectrum of curvature perturbations in this limit, finding on superhorizon scales, which implies the following definitions for the spectral index and the amplitude Given that in the η ⊥ = 0 limit the curvature perturbations are frozen once they leave the horizon, these observables can be evaluated at horizon exit, when k = aH.
By performing a similar computation for the isocurvature modes one can show that the amplitude of the isocurvature power spectrum at horizon crossing is the same as for the adiabatic modes. As noted above, due to the fact that ν N < 3/2, the isocurvature perturbations decay on superhorizon scales with a rate controlled by the ratio M 2 /H 2 . This implies that the isocurvature fraction at the end of inflation scales as and is therefore suppressed if at some stage during observable inflation M ≥ H.
The tensor to scalar ratio is given in the decoupling limit by its single field expression where * denotes evaluation at horizon crossing since both adiabatic and tensor perturbations are frozen outside the horizon.

Results
Having fraction, β iso , at the end of inflation on the k 50 and k 60 scales. We observe that β iso varies by many orders of magnitude, being larger for late turning trajectories (large θ 0 ) where power transfer between isocurvature and curvature perturbations is less efficient and the attenuation of isocurvature power is mostly driven by its decay on superhorizon scales.
In the same way as the multi-field effects can lead to an underestimate of the scalar amplitude, they will also impact other inflationary observables, in particular the tightly constrained spectral index and the tensor to scalar ratio. The single and multi-field estimates for these quantities are plotted as functions of the initial condition θ 0 in figure 5.
Starting with the tensor to scalar ratio, we observe that the effect of the multi-field dynamics is to flatten the peak and therefore to bring the results more in line with the PLANCK 2015 constraint of r < 0.114. This effect is partially due to the tensor modes beings unaffected by the multifield effects and in part due to the fact that the single field estimate for the amplitude of the scalar perturbations is a bad approximation for trajectories that turn early. Therefore by underestimating the amplitude of the scalar fluctuations, the single field formula overestimates the tensor to scalar ratio by In what concerns the spectral index, we also see that the multi-field estimate is con- siderably sharper than what one would expect by applying single field results. We observe that the peak at low θ 0 is absent and that the trough is shallower. We recall that the single field estimate for n s , Eq. (3.39), is obtained by taking the decoupling limit η ⊥ → 0. This condition is clearly violated by the early turning trajectories, for which η ⊥ peaks as the scalar modes leave the horizon. We therefore conclude that the peak in the low θ 0 region is spurious 4 . To understand the change in n s for larger θ 0 trajectories it is useful to rewrite eq. (3.18) . 4 Comparing the results for the single field estimates of figure 5 and those of [18] we see that they differ in the low θ0 range, where [18] has no peak. This difference can be traced back to how one generalises η = V V for multifield cases. If one takes η to be the smallest eigenvalue of then indeed there is no peak . However we use here a different, and more accurate, prescription which can be indeed derived from the decoupling limit, as we argued in section 3.
We see that the equation of motion for the curvature perturbation is equivalent to a frictionless harmonic oscillator with a "time" dependent proper frequency subject to an external force whose magnitude is set by the isocurvature perturbation. The effects of a turn in the background trajectory, which gives rise to the external force, will be more pronounced on k-modes for which the ratio between the external force and the elastic force is larger. This ratio is well approximated by the simpler relation between .

(4.2)
Since the amplitude of different k-modes around the pivot scale will be affected differently by a turn in the background trajectory, there will be superhorizon evolution of the spectral index for the curvature perturbations. From the solutions in the decoupling limit (η ⊥ = 0) one finds that on superhorizon scales and before the turn where n 0 s denotes the curvature spectral index before the turn in the trajectory and is assumed to be n 0 s < 1. As for the isocurvature perturbations, one may expand which leads to To understand how this changes the spectral index consider a pair of k-modes around the pivot scale k * : k − < k * < k + . It follows that since the k + mode power is more enhanced than the k − mode power resulting in a spectral index closer to unity This behaviour can be clearly observed in figure 6, where the k − power is less enhanced than the k + , resulting in a more even distribution of power and in an increase in the spectral index. Note that, before the turn (N e < 50), the single field estimate is actually a good approximation to the full result and that it only fails due to the sharp turn in the background trajectory that causes conversion of isocurvature into curvature power.    In such a case, very small values for the inflaton parameter A would be consistent.
While the search for specific MSSM extensions with this structure is quite interesting, the inflationary dynamics may be studied in a model independent manner assuming that such new non-Higgs degrees of freedom correspond again to the position moduli of D-branes.
We study here for comparison the case with A = 0.2 (corresponding to m H /m h = 1.22).
In figure 9 we present the background evolution for sample trajectories in such a regime.
In this case we see that the trajectories are essentially straight along the σ direction with only slight turning in the last 10 efoldings of expansion. This is in accordance with the fact that θ becomes massless in the limit of vanishing A. The straight trajectories imply that η ⊥ , being inversely proportional to the curvature radius, vanishes everywhere except at the very end of inflation (where the mild turning takes place) as can be seen in figure 9. As mentioned above η ⊥ controls the coupling between the curvature and isocurvature modes and so this limit corresponds to a decoupling regime of Eqs. (3.18) and (3.19). We therefore expect the single field estimates of Eqs. (3.39) and (3.41) for the inflationary observables to provide a good approximation to the full result. In fact, if one employs the multi-field formalism in the computation of the observables and compares it with the single field estimates, one finds that there is agreement at the level of a few percent. This is displayed in figure 10 where the naive single field bands track the exact results to the point of being almost indistinguishable.
Since isocurvature is practically decoupled from curvature throughout the observable inflationary range, the isocurvature fraction at the end of inflation as estimated by β iso is larger that in the previous cases, as can be seen in  Table 3. Ratio between the amplitude of the curvature perturbations at the end of inflation and the single field estimate of eq.

VaryingĜ
Finally let us discuss the effects of varyingĜ over the results. As we commented in section 2,Ĝ parametrises the total amount of flux quanta in Planck units, which determines in turn the ratio between the supersymmetry breaking scale and the string scale. In particular, with n being the flux quanta. We will take the string scale of order M s ∼ 10 16 − 10 17  . Tensor to scalar ratio and spectral index for Higgs-otic inflation with A = 0.83 and arbitraryĜ. The data is superimposed over the recent Planck exclusion limits [58]. The color pattern (from red to blue) corresponds to (higher or lower) density of initial condition points . There is a lower cutoff on the density required to be plotted (or equivalently in the level of fine-tuning allowed) missing around 10 % of the points.
Notice thatĜ not only enters in the absolute value of the potential, but also in the field redefinition to get canonical kinetic terms. The biggerĜ is, the stronger is the flattening of the potential and the results are closer to those of linear inflation. This is the reason why our results actually interpolate between quadratic (smallĜ, negligible flattening) and linear (bigĜ, strong flattening). However, as we already said, those closer to linear inflation are in better agreement with both density scalar perturbations and tensor-to-scalar ratio constraints from Planck. In figure 11 the color pattern from red to blue refers to the density of points, being the red regions the most populated. This could have been anticipated from

Conclusions
The current lack of experimental evidence for low-energy supersymmetry coupled with the possibility of observing primordial gravitational waves in the near future drives us to analyse a model in which these two seemingly unconnected phenomena have their roots in the same sector of the theory: Higgs-otic inflation.
In Higgs-otic inflation the scale of SUSY breaking, M SS ∼ 10 13 GeV, is identified with the inflation's mass and the Universe's expansion is driven by the MSSM Higgs scalars. Such a high inflationary scale gives rise to a sizeable tensor fraction and implies super-Planckian field excursions for the inflatons. The model is embedded into a local string construction, in which the Higgses/inflatons are D-brane position moduli. Such a stringy embedding allows for control over higher order corrections to the inflationary potential that are usually problematic for large field inflation models.
In this work we have analysed in detail the observational signatures of this model, taking into account the effects of isocurvature perturbations that are unavoidable whenever there are multiple dynamical fields during inflation. We found that the effects of 2-field dynamics on the observables are more pronounced whenever there is a turn in the background trajectory.
In Higgs-otic inflation the presence of such a turn during the last 60 efoldings of expansion is controlled by the choice of background fluxes. For flux choices leading to almost straight trajectories, the single field estimates for the inflationary observables derived in [18] are a good approximation to the full result as we have seen in Sec. 4.3. On the other hand for those flux choices consistent with the identification of the inflaton with the MSSM Higgses, we find that the trajectories will necessarily present some degree of turning. At the turn isocurvature and curvature perturbations are coupled, with isocurvature sourcing the curvature power. This implies superhorizon evolution of the curvature perturbations and translates into an increase of the adiabatic perturbations' amplitude and the corresponding decrease in the tensor to scalar ratio. Since different k-modes are affected differently by a turn in the trajectory, there will be superhorizon evolution of the spectral index, which we have shown to lead to a sharper prediction for n s when compared to the single field approximation employed in [18]. In what concerns the isocurvature power at the end of inflation, we have shown that it is always suppressed enough to be compatible with the less stringent upper bounds from PLANCK 2015. We have shown in 4.1 and 4.2 that the observational footprint of Higgs-otic inflation is narrower than what one would expect by simply performing an adiabatic projection: the range over which n s varies is smaller and centred around the Planck best fit value and the tensor to scalar ratio is reduced significantly yielding a region r = 0.08 − 0.12 depending on the model initial conditions.
In summary, our analysis demonstrates that the observable signatures of Higgs-otic inflation are in line with the results of the joint Planck/BICEP analysis [61]. It would be interesting to address additional questions like moduli stabilisation and reheating in this model. We leave those aspects to future research.