Modular inflation observables and j-inflation phenomenology

Journal of High Energy Physics, Sep 2017

Modular inflation is the restriction to two fields of automorphic inflation, a general group based framework for multifield scalar field theories with curved target spaces, which can be parametrized by the comoving curvature perturbation ℛ and the isocurvature perturbation tensor S IJ . This paper describes the dynamics and observables of these per-turbations and considers in some detail the special case of modular inflation as an extensive class of two-field inflation theories with a conformally flat target space. It is shown that the nonmodular nature of derivatives of modular forms leads to CMB observables in modular invariant inflation theories that are in general constructed from almost holomorphic modular forms. The phenomenology of the model of j-inflation is compared to the recent observational constraints from the Planck satellite and the BICEP2/Keck Array data.

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Modular inflation observables and j-inflation phenomenology

Received: June Modular in ation observables and j -in ation Rolf Schimmrigk 0 1 2 Indiana University South Bend 0 1 2 0 at target space. It is shown that 1 1700 Mishawaka Ave. , South Bend, IN 46634 , U.S.A 2 gg @ is the d'Alembert operator Modular in ation is the restriction to two elds of automorphic in ation, a general group based framework for multi eld scalar eld theories with curved target spaces, which can be parametrized by the comoving curvature perturbation R and the isocurvature perturbation tensor SIJ . This paper describes the dynamics and observables of these perturbations and considers in some detail the special case of modular in ation as an extensive class of two- eld in ation theories with a conformally the nonmodular nature of derivatives of modular forms leads to CMB observables in modular invariant in ation theories that are in general constructed from almost holomorphic modular forms. The phenomenology of the model of j-in ation is compared to the recent observational constraints from the Planck satellite and the BICEP2/Keck Array data. Cosmology of Theories beyond the SM; Discrete Symmetries - 3 Perturbed multi eld in ation 4 Modular in ation Scalar eld dynamics Curved target background dynamics Slow-roll dynamics Dynamics of perturbations The slow-roll form of the power spectrum The tensor-to-scalar ratio Kinetic term Modular potentials and symmetry breaking Modular Eisenstein series 1 Introduction 2 Multi eld dynamics 2.1 2.2 2.3 3.1 3.2 3.3 4.1 4.2 4.3 5.1 5.2 5.3 6 8 9 7 j-in ation 5 Observables in modular in ation Modular in ation parameters Modular building blocks of physical observables The almost holomorphic modularity of CMB observables Dynamics via transfer functions Observables of j-in ation 8.1 Modular form of observables 8.2 j-in ation observables and the Planck probe Conclusion 1 Introduction In ation is a framework de ned on the space F (M; X) of scalar eld theories de ned by eld multiplets I , I = 1; : : : ; n on the spacetime manifold M taking values in the target space X. The eld space F (M; X) is in general assumed to be an unstructured set on which dynamical variables and observables are de ned, providing the underlying theory spaces of in ation. The dynamics of the background is determined by the metric GIJ ( K ) on the { 1 { J . Early references that admit a nontrivial geometry include [3{6]. Depending on the origin of the model, the action can either be taken to be given by E and V in polynomial form, or more generally can be a non-rational function, such as in the case of brane-induced in ation. There are several reasons why it is of interest to consider a symmetry based approach to in ation. One motivation arises from the shift-symmetry, an ad-hoc operation that is often invoked because of concerns about higher order corrections to observable parameters of in ationary models. This is reminiscent of duality considerations, which have led in gauge and string theory to the embedding of analogous dualities into larger discrete and continuous groups. A group-theoretic approach is furthermore useful because it endows the in aton target space with structure that allows the exploration of the in ationary theory space in a more systematic way by using the resulting geometry as an organizing principle. Symmetries appear as well in in ation theories based on moduli, sometimes also called modular in ation, in which in ation is driven by some of the moduli that appear in many fundamental theories [7]. Such moduli based models can provide special cases of the framework developed here. The idea of automorphic in ation is to consider the space of the in ationary multi eld space de ned by theories that are invariant under in nite discrete symmetry groups that contain the in aton shift symmetry. This leads to the notion of automorphic in ation as a structured framework of multi eld in ation [8, 9]. An immediate consequence of the existence of such symmetries is that the space of automorphic eld theories acquires a foliation, with leaves that are speci ed by numerical characteristics, de ned in terms of the group theoretic and automorphic structure, speci cally the underlying continuous group G, the discrete group in G that extends the shift symmetry, and the types of the automorphic forms that de ne the building blocks of the in aton potential. The simplest case of automorphic in ation is obtained when the discrete group is the modular group, leading to the special case of modular in ation, in which case the continuous group G is xed to be the Mobius group SL(2; R) and the discrete group is a subgroup SL(2; Z).1 The theory of modular forms has been developed over the past century in greatest detail for groups that are of congruence type with some level N , but more general groups of Fuchsian type are also possible. The potential V ( I ) is constructed in terms of modular forms fi of weight wi relative to (N ). The resulting theory space of modular in ation is thus determined by the numerical characteristics (N; wi). The kinetic term of all modular in ation models is determined by the Poincare metric of the upper halfplane, leading to a conformally at target space for the in aton. The invariance of this metric under the continuous group of Mobius transformations leads via the introduction of the in aton potential to a breaking of the group PSL(2; R) to the discrete group of modular transformations PSL(2; Z), or subgroups thereof. The energy scale of this breaking is determined by CMB constraints, leading to a weakly broken symmetry in the case considered here. 1In the case of in ation driven by two or more moduli elds the presence of appropriate symmetries places such models in the framework of modular, and more generally, automorphic in ation. { 2 { HJEP09(217)43 Automorphic in ation was brie y outlined in [8], and the general framework was described in more detail in [9], with emphasis on the automorphic side of the theory. The purpose of the present paper is to focus in more detail on the class of modular in ation and the particular model of j-in ation. The phenomenological analysis of general modular in ation involves derivatives of the potential, hence derivatives of modular forms. However, derivatives of modular forms are not modular forms, raising the issue of the modular structure of observables in this framework. It is shown that the basic CMB observables are in general determined not by modular forms per se but by almost holomorphic modular forms. The phenomenological analysis of the speci c model of j-in ation, introduced in [8], is compared to observational constraints from the microwave anisotropy. Sections 2 and 3 brie y introduce the general multi eld in ationary background dynamics and perturbation theory, with emphasis on a closed system of equations for the isocurvature perturbation described by an antisymmetric rank two tensor SIJ . This generalizes an earlier result obtained in the special case of at two- eld in ation to an arbitrary number of elds with a general curved target space. Sections 4 and 5 consider modular in ation, with focus on the general structure of physical observables in this framework, showing in particular that they are modular invariant in a generalized sense. Section 6 brie y describes the integration of the in ationary evolution in terms of the transfer function, and sections 7 and 8 analyze the speci c model of j-in ation. The nal section presents some conclusions. 2 Multi eld dynamics Automorphic in ation as a group theoretic framework for multi eld in ation involves eld spaces that are obtained as coset spaces of continuous groups, hence are curved. The specialization to modular in ation leads to an extensive class of two- eld models with conformally at target spaces. In this section some essential features of these theories are described. There are di erent ways to encode the dynamics of the adiabatic and isocurvature perturbations. In the following the comoving curvature perturbation R will be adopted as the adiabatic mode, while the isocurvature perturbations are encoded in an antisymmetric tensor denoted by SIJ . The focus on the latter is suggested by the dynamics of R, leading to an isocurvature dynamics di erent from the usual dynamics based on projections of the Sasaki-Mukhanov variables. While the geometry in automorphic in ation is derived from the structure of the underlying Lie group G and certain subgroups, it is best to leave the metric of the in aton eld space arbitrary and the number of elds of the in aton multiplet I unconstrained, so as to indicate the general features of the framework. Modular in ation is then recovered by setting G = SL(2; R) and considering the eld space G=K with the compact subgroup K = SO(2; R), which is isomorphic to the complex upper halfplane. 2.1 Scalar eld dynamics The scalar eld action is considered to be of the type A[ I ; GIJ ; g ] = Z d x 4 p g MP2l R 2 1 2 I (2.1) { 3 { where the spacetime metric has the signature ( ; +; +; +) and MPl = 1=p8 GN is the reduced Planck mass associated to Newton's constant GN . The target space spanned by the in aton elds I , I = 1; : : : ; N , in general has a non-trivial geometry determined by the metric GIJ that is assumed to be Riemannian. The dynamics of the system (g ; GIJ ; I ) involves the geometry of the target space as well as that of spacetime via the Einstein equations and the Klein-Gordon equation. Assuming that the covariant derivative on the in aton space is of Levi-Civita type, the Euler-Lagrange form of the latter takes the form J (2.2) HJEP09(217)43 where g = p g I JK are the target space Christo el symbols. This Klein-Gordon equation can be written in terms of a covariant derivative D , which can be viewed as a combination of the spacetime Koszul connection and a contribution of the curved target space as I I ) GIJ V;J = 0: (2.3) (2.4) (2.5) (2.6) (2.7) (2.8) (2.9) 2.2 Curved target background dynamics The dynamics of the background is assumed to belong to the class of theories characterized by the action (2.1). The Klein-Gordon equation is given by Dt _I + 3H _I + GIJ V;J = 0; I _J W K : JK where the dot indicates the derivative with respect to t and Dt is the covariant derivative on the target space, de ned for vector elds W I in terms of the connection coe cients IKJ as Here the IKJ are universally assumed to be the Christo el symbols determined by the The background equations constraining the Hubble parameter H = a_=a are given in the Newtonian gauge with metric ds2 = dt2 + a2(t) ij dxidxj ; { 4 { and are given by where the density and the pressure p of the isotropic uid tensor where in the present paper the spatial metric ij is chosen to be at for simplicity, by the two Friedman-Lemaitre equations It is useful to note that the variation of the Hubble parameter is given by 2MP2l (GIJ _I _J ): := := _ H H2 = 1 d H dt ; V , and the acceleration of the in aton is assumed to be small as well. More precisely, it is conventional to introduce the parameters _I = H2 = GIJ V;J 3H V 3MP2l : { 5 { where the background dynamics has been used and _ v = background in aton speed, in the notation of [ 10, 11 ], respectively. The parameter can q GIJ _I _J denotes the be written as [ 12 ] projection of the acceleration vector onto the in aton velocity. The slow-roll form of the background equation can then be used to eliminate the background eld velocity and the slow-roll form of the rst Friedman-Lemaitre equation reduces to (2.10) (2.11) (2.12) (2.13) (2.14) (2.15) (2.16) (2.17) (2.18) HJEP09(217)43 While j-in ation is an example of two- eld in ation, it is conceptually more transparent to leave the number of elds in the following brief discussion of in ationary perturbation theory unrestricted. For scalar eld theory with at target spaces this has been considered in many references, including [ 1, 2, 10, 13 ], and reviews can be found in [ 14, 15 ]. A comprehensive review for curved target space in ation has not yet been written, but the references [11, 16{26] contain brief descriptions of some aspects of multi eld in ation with a non-trivial eld space geometry, and [12, 19{22, 27, 28] are concerned with covariant formulations that extend the construction of [29]. The metric perturbations are conventionally parametrized as ds2 = (1 + 2')dt2 + 2aB;idxidt + a2 ((1 2 ) ij + 2E;ij ) dxidxj ; (3.1) where ' = in the absence of anisotropic stresses, and di erent gauges are de ned via the vanishing of some of these perturbations. The notation adopted in (3.1) is close to that of the reviews [30, 31]. The in ationary dynamics has been constrained over the past two decades by the CMB satellite probes COBE, WMAP and Planck, providing experimental results for some observational variables associated to the gravitational and in aton perturbations at the percent level, and non-trivially bounding others. 3.1 Dynamics of perturbations A commonly adopted perturbation is the comoving curvature perturbation, de ned for general uids in the Newton gauge as [16, 32, 33] QI = Q I + _I H ; R = H I _ I Q ; { 6 { where is the spatial metric perturbation and u is obtained from the divergence part of the energy momentum tensor perturbation T0i = The above de nition is often written in a di erent form by introducing q = ( + p) u and writing R in terms of q. In multi eld scalar eld theory with curved target space geometry the perturbation can be expressed in terms of the Sasaki-Mukhanov variables [34, 35] as where I is the normalized in aton velocity I = _I = _ and _ is the in aton speed de ned above. Adopting the notation of [22], the variables QI denote the covariant form of the eld (3.2) (3.3) (3.4) (3.5) perturbations I (t; ~x) := I (t; ~x) I (t) de ned in terms of the geodesic path between the perturbed eld I (t; ~x) and the background eld I (t) given by [29] For the time evolution of R we nd here in terms of rank two tensors the equation where and the gradient tensor is de ned as I = Q I IJ := MP2l VV;IJ : This suggests identifying the dimensionless variables SIJ as the fundamental perturbations that source the large scale time evolution of R. They will be referred to as the rank 2 tensor of the isocurvature perturbations. In terms of the isocurvature variables SIJ adopted here as the basic nonadiabatic perturbations, the slow-roll approximation of the R-dynamics on large scales is given by where the slow-roll parameter IJ is de ned as The time evolution of the isocurvature perturbation tensor in this approximation is given by DtSIJ = 2H( KL K L )SIJ + H KLGK[I SJ]L 3 MP2l [I RKJ]LM K SLM ; (3.12) where the brackets U [I V J] indicate antisymmetrization without the conventional factor of 1/2, and Dt is the covariant derivative acting on the contravariant tensor SIJ . The differential equations (3.10) and (3.12) form a closed system of evolution equations for the comoving curvature perturbation R and the isocurvature perturbations SIJ which shows how in the general multi eld case the latter mix during the evolution and how they couple to the curvature tensor. Equation (3.10) shows that the adiabatic perturbation remains constant on large scales if the vector de ned by the slow-roll contraction IJ J of the normalized in aton velocity I is orthogonal to the isocurvature contraction SIJ J of I . Alternatively one can view the r.h.s. as a quadratic form DIJ := GIK SKL LJ (3.13) de ned on the tangent space of the target manifold. The system (3.10), (3.12) generalizes the at target two- eld dynamics of ref. [13] to curved eld spaces of arbitrary dimension. { 7 { The power spectrum associated to a dimensionless perturbation, in the following generically denoted by O(t; ~x), is de ned in terms of the correlator as hO(t; ~k)O0(t; ~k0)i =: (2 )3 (3)(~k ~k0)POO0 (k): with the associated dimensionless power spectrum de ned as POO := k 3 2 2 POO0 : In the present case the variables O; O0 are given by the perturbations (R; SIJ ). The dynamics of R identi es the isocurvature tensor contraction as the essential source of the large scale behavior of the adiabatic perturbation. We associate to the tensor SIJ a dimensionless isocurvature scalar such that its power spectrum at horizon crossing is identical to that of R by introducing is the absolute value of the normalized acceleration vector construction the power spectrum PSS of S as de ned in (3.14) is identical to that of PRR, I := Dt I . Per given in its dimensionless form by while the cross-correlation power vanishes at horizon crossing (3.14) (3.15) (3.16) (3.17) (3.18) (3.19) (3.20) (3.21) (3.22) 3.2 The slow-roll form of the power spectrum The slow-roll approximation of the adiabatic power spectrum PRR can be expressed either in terms of the potential or in terms of the slow-roll parameters IJ introduced above and which resolves the parameter de ned above. Using the resulting _ 2 = (V =3)GIJ I J leads to the power spectrum at horizon crossing as I = MPl V V;I ; PRR = 1 V 12 2MP4l GIJ I J : nOO0 = 1 + d ln POO : d ln k As noted above, PRS = 0 and PSS = PRR at horizon crossing. The spectral indices nOO0 are obtained from the power spectrum POO0 as The shift by unity is conventional for the adiabatic perturbation, but is not always adopted for isocurvature perturbations in the literature. The slow-roll form of the power spectrum above then leads to the spectral index The constraints on nOO0 obtained by WMAP [36] and Planck [37, 38] therefore restrict the shape of the potentials. Gravitational waves play an important role in constraining the viable part of the in ationary theory space. While no primordial signal has been detected, satellite probes like WMAP and Planck have led to upper bounds that models have to satisfy. Conventionally, these bounds are formulated in terms of the tensor-to-scalar ratio r, which is constructed from the tensor power spectrum [39] and the scalar power spectrum. In multi eld in ation di erent forms for r have been considered. A convenient way to de ne r is as the ratio of the tensor-to-adiabatic scalar amplitude HJEP09(217)43 PT = 2 2 H MPl 2 r := PT ; PRR r = 8GIJ I J : which leads to the slow-roll parameter form The tensor spectral index, de ned via the ansatz can be written in terms of the slow-roll parameters as leading with eq. (3.25) to the (r; nT )-relation PT (k) = PT (kp) k kp nT ; nT = GIJ I J ; r = 8nT in the slow-roll approximation. This relation is a ected by the transfer functions, as discussed further below. 4 Modular in ation In this section the framework of automorphic in ation is specialized to the case of modular in ation, a particular class of two- eld scalar eld theories coupled to gravity with a nontrivial target space geometry, which is of coset type G=K, where G is a Lie group and K G is a maximal compact subgroup. The general framework was introduced in [8] and its structure described in more detail in [9] in the higher rank case. Classical modular forms [40, 41] were introduced in the second half of the 19th century as functions on the complex upper halfplane because this space is mapped to itself by the modular group SL(2; Z). The general concept was introduced by Klein [42] in the context of various discrete subgroups SL(2; Z). Thinking about forms in this way is computationally useful, but conceptually not the most illuminating approach, and it is { 9 { (3.23) (3.24) (3.25) (3.26) (3.27) (3.28) most advantageous to have both the domain theoretic and the group theoretic formulations available. Such a framework is described in refs. [8, 9]. The result is that in the modular context the group theoretic set-up is given by the pairs of groups (G; ), where G = SL(2; R) is semisimple. The domain theoretic structure is obtained by considering a maximal compact subgroup K G, which in this case is the rotation group K = SO(2; R), both of which act via the Mobius transformation. More details can be found in [9]. The discrete groups can be of Fuchsian type but the most well-developed theory is that of di erent types of congruence groups N SL(2; Z), where the level N determines the de ning congruence constraint. For Hecke groups the matrices HJEP09(217)43 = a b c d ! 2 0(N ) SL(2; Z) K = 1= 2. Thus the space (H; ds2) with the metric (4.3) has constant negative Gaussian curvature (4.1) (4.2) (4.3) (4.4) (4.5) (4.6) (4.7) satisfy the constraint c 0(mod N ). Other possibilities include groups usually denoted by 1(N ) and (N ). These will collectively be denoted as N in the following. 4.1 Kinetic term The bounded domain H is two-dimensional, which implies via the Bianchi identity that the Riemann curvature tensor takes the form RIJKL = K(GIK GJL GILGJK ); where K = R=2 is the Gaussian curvature expressed in terms of the Ricci scalar R. The metric on H induced by the Cartan-Killing form B on the Lie algebra sl(2; R) is the Poincare metric The eld theoretic form of the Poincare metric leads to the non-vanishing Christo el symbols ds2 = d d (Im )2 = dx2 + dy2 y2 ds2 = GIJ d I d J = 2 ( 2)2 IJ d I d J The curvature tensor RIJKL has only one independent component leading to the curvature scalar is invariant under the group de ned by the linear fractional transformations g = a b c d ! ; a + b c + d ; g = (4.8) (4.9) (4.11) (4.12) (4.13) (4.14) are matrices with real entries. These Mobius transformations map the upper halfplane to itself and de ne the continuous symmetry group of the free modular theory. Since the center of SL(2; R) acts trivially the continuous invariance group is given by PSL(2; R). The metric (4.4) has a structure that is reminiscent of the metrics encountered in single eld pole in ation considered in refs. [43, 44] in that it has a double pole for vanishing 2 . In modular in ation the boundary Im( ) = 0 however is not part of the domain of the in aton eld and neither the metric nor the potential are de ned on the real axis of the complex plane. 4.2 Modular potentials and symmetry breaking The action of modular in ation can be written as Amod = Z d x 4 p g MP2l R 2 1 2 where G is the complex form of the Poincare metric. The potentials of modular two- eld theory are de ned in terms of modular forms that descend from the group to the upper halfplane. The general construction of the in aton target space X in terms of the Lie group G has been described in detail in [9]. SL(2; R) ! C via the 1-cocycle Modular forms on the upper halfplane H are induced by the group function : where = gi. They are de ned with respect to discrete subgroups N of the modular group G(Z) = SL(2; Z) and are characterized by the level N of the subgroups N , their weights w, and by a character N via their transformation behavior, which for 2 N and the action given by the discrete Mobius transformation = (a + b)=c + d) is de ned to be f ( ) = N (d)(c + d)wf ( ): The potential V ( I ) is de ned in terms of modular functions F ( ) on the upper halfplane, as well as some function (F ) as J (g; ) = (c + d); g = f ( ) = (c + d)w (g); a b c d ! V = 4 (F; F ); where 4 is an energy scale and F and is given by powers of the norm function, leading to are dimensionless. A simple class of functions Vp := 4jF ( )j2p: In the present paper the focus will be on the p = 1 case. The modular functions F can in general be viewed as a discrete subgroup of the modular group SL(2; Z), but the most detailed theory of forms has been formulated for congruence groups N of various types with level N . The introduction of the potential thus breaks the continuous Mobius group of the previous subsection to the discrete subgroups, which can be written schematically as In j-in ation, the example considered further below, the Mobius symmetry is weakly broken since the constraints from the CMB determine the energy scale to be much lower than the Planck scale. 4.3 Modular Eisenstein series As in the automorphic case, Eisenstein series play a key role for modular forms of arbitrary weight because for the full modular group they span the subspace complementary to the cusp forms. Holomorphic Eisenstein series are obtained by following the general construction brie y outlined in the general case in the previous subsection. The details of how to obtain from the group theoretic Eisenstein series on G = SL(2; R) the classical Eisenstein functions Ew( ) on the upper halfplane, given in terms of the divisor function the Bernoulli numbers Bw, and q = e2 i with Ew( ) = 1 w(n) := X dw; d n j 2 H, as 2w X Bw n w 1(n)qn; can be found in ref. [9]. There are di erent ways to obtain the values for Bw, for example via the generating function x=(ex 1) = Pm1=0 Bmxm=m!, or in terms of the Riemann zeta function via Euler's formula as Bw = 2w! (w)=(2 i)w. For w > 2 these functions are modular. In the case of j-in ation the forms of weight 2; 4; 6 are relevant and with Euler's results for (2); (4) and (6) [45] the zeta function relation leads to B2 = 1=6; B4 = 1=30; B6 = 1=42. These ingredients will be used below to de ne j-in ation. 5 Observables in modular in ation It is shown in this section that for general modular invariant in ation the physical observables are determined by modular forms that are almost holomorphic, but in general (4.15) (4.16) (4.17) (4.18) not holomorphic. The explicit form of the spectral index and the tensor-to-scalar ratio is determined in terms of the de ning modular function F of the in ationary potential. For two- eld in ation the dynamical system introduced above simpli es considerably because there is only one independent isocurvature perturbation. The general adiabatic equation (3.10) takes the form R_ = 2H K which in the case of modular in ation simpli es further because the metric is conformally at G11 = G22 = G, resulting in with The isocurvature equation (3.12) reduces to where the abbreviations = IJ I J and DtS12 = H ss + 2 1 3 MP2lK 1 S12; are covariant objects obtained from (3.11). This specialization of the general dynamics of the system (R; SIJ ) derived above extends the discussion of ref. [13] for two- eld in ation with a at eld space metric to curved target spaces. 5.1 Modular in ation parameters The in ationary analysis considered in section 3 involves the geometry of the potential. General expressions for the observables in modular in ation thus involve derivatives of modular forms. For a potential of modular functions for a modular function F the slow-roll parameters I de ned in eq. (3.19) take the form V = 4jF j2 I = iI 1 MPl F V := 1 GIJ 2 I J = 2 M2P2l (Im ) and the acceleration of the scale parameter is directly determined by the behavior of These parameters determine the tensor-to-adiabatic scalar ratio and part of the spectral index. The remaining ingredient of nRR is the parameter matrix IJ de ned in (3.11). Decomposing the covariant derivative into its at and Christo el contributions leads to the at part IJ = iI+J MP2l 2 F IJ = IJ + IJ ( 1)I + ( 1)J F + ( 1)I+J 1 F 0 : F For modular in ation models in which the curvature contribution is small, like in j-in ation, the at limit provides a very good approximation to the full result. 5.2 Modular building blocks of physical observables Modular functions F can be written as quotients of modular forms, hence the computation of F 0 reduces to the computation of f 0 for modular forms of some arbitrary weight w. However, the derivative of a modular form is not a modular form. This raises the issue of what precisely the modular structure is of the physical observables in modular invariant in ation. The general structure of the derivative takes the form df d where f~ is a modular form of weight (w + 2). The problem to specify the rst term in f 0 is nontrivial and was described in some detail in [9]. Very brie y, in the case of the full modular group it is determined by functions H (z) considered in [ 46 ] where q = e2 iz and the functions jn are constructed iteratively via the normalized weight zero Hecke operators T0(m) as jn(z) = j1(z) , where j1(z) = j(z) 744. With the modular form f~ can be written as H (z) := 1 X jn( )qn; n=0 T0(n) e := < 8> 1=2 if > : 1=3 1 = i = 3 if otherwise ;> 9 > = f~(z) = f X e ord (f )H (z); 2F (5.9) (5.10) (5.11) (5.12) (5.13) (5.14) (5.15) where ord (f ) is the vanishing order of f , which is constrained by the valence formula 1 2 where 3 = e2 i=3 [ 47, 48 ]. 1 3 ord =i(f ) + ord = 3 (f ) + ord1(f ) + ord (f ) = (5.16) X 2F i; 3 w 12 ; The nonmodularity of f 0 arises from the second term because the Eisenstein series E2, de ned as in (4.18), transforms under the modular group SL(2; Z) as E2( ) = (c + d)2E2( ) 6i c(c + d); and therefore is not a modular form. This shows that in the in ationary context the relevant space is not just generated by E4 and E6, but must also include the Eisenstein series of weight two. E2 is the paradigmatic example of a quasimodular form, a notion for which various de nitions have been introduced. From a physical point of view it is best to focus on the transformation behavior of these functions under the modular group. The example of E2 indicates a structure that is reminiscent of the case of Christo el symbols, objects with an inhomogeneous transformation behavior, where the tensor behavior is modi ed by an additional term. Similarly, a quasimodular form transforms like an ordinary modular form, but with additional terms. For a quasimodular form f q of weight w this can be written as (5.17) (5.18) (5.19) (5.20) (5.21) f q( ) = (cz + d)wf ( ) + (cz + d)w X where the sum is nite, and fm are holomorphic functions. This de nition is more direct, but not less general, than Nahm's de nition as given in [49]. The original de nition of quasimodular forms was based on the \constant" term of a nearly holomorphic, or almost holomorphic, form as de ned by Shimura [50, 51]. The nonmodularity of f 0 a priori induces nonmodular terms in the observables of modular in ation, hence this raises the question what exactly the modular nature of these objects is in modular invariant in ation. For modular functions F = f =g with modular forms f; g of equal weight the derivative turns out to be simple because of a cancellation and we obtain where f~; g~ are as in (5.12). Thus the rst derivative of modular functions are modular forms of weight two. The second derivative F 00 is no longer modular and by using the above formula for the derivative iteratively we nd F 00 = 4 iF 0 where f~~ and g~~ are obtained from the derivatives of the modular forms f~ and g~. Important for the IJ -induced terms in the spectral observables is the combination F 00=F 0. The nonmodular part of this quotient is therefore given by F 00 F 0 nmod i 3 = E2: The main result of this and the following discussion is that there are further terms in the modular in ation observables that are induced by the nontrivial geometry of the target space. These terms combine with the nonmodular Eisenstein series E2 into a new function that is modular, but not holomorphic, de ned as The modularity of this function follows from the transformation behavior Eb2( ) = E2( ) 3 (Im ) The Eisenstein series Eb2 is an example of a nearly holomorphic, or almost holomorphic, modular form. Such a form of weight w for the modular group SL(2; Z) is a function f on the upper halfplane that is a polynomial in 1=(Im ) with coe cients that are holomorphic functions. The almost holomorphic modularity of CMB observables The results above can now be used to address the question raised earlier about the modular structure of the CMB observables. The tensor-to-scalar ration r is determined by the modular form F 0=F as hence is modular invariant in terms of holomorphic modular forms (and their complex conjugates). The spectral indices involve the second derivatives F 00, hence are not modular in the same sense. From the expressions obtained above for the parameters I and IJ we obtain for modular in ation the spectral index nRR as M2P2l (Im )Im F 0 F ; (5.25) where the nal term is induced by the target space metric. The nonmodular contribution to the second term in the square brackets combines with the last term in this equation so that the modularity of the spectral index can be made manifest by writing mod + 3 Im Eb2 F This completes the derivation of the most important observables for the general framework of modular in ation based on arbitrary modular invariant functions F . The weight structure of the in ationary variables can be resolved into their holomorphic and antiholomorphic factors (w; w). It follows from the above that the slow-roll parameters I are determined by forms of weight (2,0) and (0,2), while the parameter V , which is proportional to r, is of weight (0,0). The two- eld dynamics at the beginning of this section can now be made explicit in terms of the de ning modular invariant function F as mod mod mod + 3 3 Im Im Eb2 F Eb2 F F 0 # F 0 # + 3 Re Eb2 F : (5.27) These parameters complete the speci cation of the two- eld dynamics for general modular in ation and determine the transfer functions considered in the next section. 6 Dynamics via transfer functions It is useful to note that a system of equations of the form PRR(t) = (1 + TR2 S )PRR(t ) PRS (t) = TRS TSS PRR(t ) PSS (t) = TS2S PRR(t ) can be integrated formally in terms of transfer functions as where R_ = AHS S_ = BHS R(t) = R(t ) + TRS S(t ) S(t) = TSS S(t ); Z t t Z t t TSS (t; t ) = exp dt0B(t0)H(t0) TRS (t; t ) = dt0A(t0)H(t0)TSS (t0; t ): The details of the coe cient functions A; B depend on whether the target space is at or curved, and on the details of the model. For at targets they can be expressed in terms of the contractions of the slow-roll parameters I and IJ , as illustrated in the case of two- eld in ation in [13]. For curved targets the dynamics involves the curvature of the eld manifold, as indicated by eq. (3.12). For two- eld in ation this specializes to eq. (5.4), with parameters given in eq. (5.27). The resulting power spectra of the adiabatic and isocurvature perturbations (6.1) (6.2) (6.3) (6.4) lead to evolving spectral indices nRR(t) = nRR(t ) nRS (t) = nRR(t ) nSS (t) = nRR(t ) (A + B TRS ) 1 + T 2 2TRS RS A TRS 2B ; 2B where the on A; B indicates evaluation at horizon crossing. These indices can alternatively be expressed in terms of the correlation fraction corr = PRS = PRRPSS = TRS = for which constraints have been determined by the Planck collaboration. The results of eq. (5.27) complete the speci cation of the evolution of these spectral indices for general p As mentioned earlier, the gravitational tensor power spectrum can be quanti ed in a variety of ways when isocurvature perturbations are present. Using the tensor-to-adiabatic scalar power ratio (3.24) leads via the evolution of PRR to the evolution of r as r = r 1 + T 2 ; RS i.e. to a suppression of r for post-horizon crossing times, hence changing the relation between r and nT . Alternatively, this can again be expressed in terms of the correlation fraction corr. For more than two elds the adiabatic dynamics (3.10) shows that the isocurvature correlators hSIJ (~k)SKL(~k0)i lead to a further suppression of the tensor ratio, turning the above relation again into an inequality, a fact that was anticipated in [13] on dim Mw(SL(2; Z)) = ( w 12 w 1 + 12 for w 2(mod 12) for w 6= 2(mod 12) ) : (6.5) q 1 + T 2 RS (6.6) (7.1) (7.2) (7.3) the basis of a two- eld discussion. 7 j-in ation There are a number of prominent modular forms that can be used within the framework of modular in ation. The model considered in [8] is based on a function that is basic to all modular functions, the j function, which up to scaling is the Klein invariant J ( ). The fundamental nature of this function is indicated by its origin in the Eisenstein series E4 and E6, which provide a basis of the space of all modular forms of the full modular group G(Z) = SL(2; Z) SL(2; R). Denoting the space of such modular forms by M (SL(2; Z)) = Mw(SL(2; Z)); M w M (SL(2; Z)) = hE4; E6i; where Mw(SL(2; Z)) denotes the spaces of weight w forms, gives where the Eisenstein series Ew are de ned in (4.18). Modular invariant functions with respect to subgroups N SL(2; Z) can be constructed by considering quotients of modular forms of equal weight. For the full modular group the dimension of the space Mw(SL(2; Z)) for even w is given by [ 47 ] This shows that for even w < 12 these spaces are at most one-dimensional. At weight w = 12 one encounters the rst cusp form, given by the Ramanujan form, which can be written in terms of the Dedekind eta function which is closely related to the partition function, hence the harmonic oscillator, as 1 n=1 1 n=1 ( ) = q1=24 Y (1 qn); ( ) = ( )24 = q Y (1 qn)24; ( ) = E3( ) 4 E2( ) 6 1728 : or alternatively in terms of the Eisenstein series (4.18) as This form does not vanish on H, hence one can obtain modular functions without poles on H by using as the denominator. An important modular invariant function obtained in this way is the j-function j( ) := E3( ) 4 ( ) : (7.4) (7.5) Up to a factor j( ) is the Klein invariant J ( ) = j( )=1728 [52]. The di erent normalizations are motivated by the fact that the Fourier expansion of j( ) has integral coe cients 1 q j(q) = + 744 + 196884q + 21493760q2 + 864299970q3 + 20245856256q4 + while the Klein invariant has nice values at the parabolic and elliptic points of the fundamental domain of the modular group. Di erent de nitions and normalizations of exist in the literature, but all de nitions of the j-functions are such that they lead to the same q-expansion. The valence formula (5.16) implies that the j-function is holomorphic on H, with a simple pole at in nity and a triple zero at 3 = e2 i=3. This follows from the fact that the Ramanujan form is holomorphic on H with a simple zero at in nity, hence is non-vanishing on the upper halfplane. The Eisenstein series E4 is homolorphic with a simple zero at 3 = e2 i=3, hence is non-vanishing on H and at in nity. The j-invariant is an ubiquitous function which plays an important role in completely di erent contexts. After having been discussed earlier by Kronecker in 1857 [53] in the context of complex multiplication, and Hermite in 1858 [54] in the context of solving the quintic equation, it was interpreted by Dedekind in 1877 [55] as the function that maps the fundamental domain one-to-one onto C. The j-function arises also in the geometry of elliptic curves and in the representation theory of the Fischer-Griess monster group. The connection to the latter arises because its Fourier coe cients encode the dimensions of the representations of this nite group, thereby linking j to the largest of the nite simple groups via the theory of vertex algebras [56]. Part of the importance of the j-function centered around (Re ) = 0. derives from the fact that every modular function on the upper halfplane can be expressed as a rational function of j [ 47 ]. A brief history of this tantalizing object can be found in [57]. The Ramanujan form does not appear to be less versatile, with applications that range from a geometric interpretation as a motivic modular form, to partition functions of the bosonic string, as well as the entropy of certain types of black holes. In ationary potentials can be constructed by considering dimensionless functions F (j; j). An immediate class of examples that can be considered is given by More general classes of models that could be considered are based on potentials of the form Vp( 1; 2) = 4jj( )j2p: Vp;q( 1; 2) = 4jjq( )j2p; V ( 1; 2) = 4jj( )j2: where jq( ) for q prime are modular functions at level q derived from j. In the remainder of this paper the focus will be on the model with potential Figure 1 shows a graph of the absolute value of the j-function close to the boundary (Im ) = 0 of a domain X that is centered around (Re ) = 0. 8 Observables of j-in ation The satellite probes WMAP [36] and Planck [37, 38] have constrained the scalar power spectrum, in particular its amplitude and spectral index, and put bounds on the tensor amplitude. These results thus provide a set of observables (ARR; nRR; r) for early universe models. (7.9) (7.10) (7.11) In order to express the observables in terms of the basic functions of j-in ation, it is useful to write the slow-roll parameters I in terms of the Eisenstein series. The construction of the derivatives outlined above leads in the special case of the Eisenstein series to E40 = E60 = 2 i 3 (E4E2 i E6E2 E6) E42 ; forms. of the j-functions, which can be obtained as relations that were originally obtained by Ramanujan [ 58 ]. The appearance of the quasimodular Eisenstein series E2 changes the transformation behavior of derivatives of modular The observables of j-in ation can then be obtained by using in addition the derivatives # # (8.1) (8.2) (8.3) (8.4) (8.5) (8.6) This leads to the Eisenstein form of the slow-roll parameters j0 j j00 j 2 i E6 E4 and the parameter that determines the slow-roll acceleration of the scale parameter a(t) is given by V = I J = 8 2(Im ) 2 MP2l E6 2 E4 2 : This implies that a > 0 for values of that are close to the zero of the weight six Eisenstein series, which is given by = i. At the point = i the Eisenstein series E4 does not have a zero or a pole. The spectral index expressed in terms of the Eisenstein series then takes the form = . The last term on the r.h.s. is induced by the curved target space metric. It combines with the E2-term in the square bracket to an expression that contains the almost holomorphic modular form Eb2 as a factor In the lowest order in the slow-roll approximation this determines the tensor spectral index via (3.28). The evaluation of the observables is here at t , the time at which the pivot scale crosses the horizon during in ation. The rst constraint on the parameters of the model can be obtained from the requirement that the number of e-folds Z te t N = dt H(t) This can also be derived as a specialization of the general modular in ation result for nRR derived earlier in the paper. The amplitude A R of the scalar power spectrum can be expressed in terms of the j-function and the Eisenstein series Ew as A RR = 1 where the r.h.s. is to be evaluated with the in aton values I = I = such that nRR and the tensor-to-scalar ratio r, considered below, are within the experimental range. Once nRR (and r) have been used to determine I at the pivot scale one can use the experimental result for A RR to determine the energy scale of j-in ation. The isocurvature power at horizon crossing is the same as that of the adiabatic perturbation PSS = PRR and the cross correlation vanishes PRS = 0. The tensor-to-scalar ratio r of multi eld in ation with curved targets (3.25) takes for modular in ation the form r = 8(Im )2 IJ I J ; and with the j-in ation expression for the parameters I one obtains the Eisenstein form for r as r = 128 2 (Im ) 2 E6 E4 2 : (8.7) (8.8) (8.9) (8.10) (8.11) should fall into the standard range N 2 [60; 70]. This interval is not sharp, and values within a wider range have been considered in the literature. The input for the computation of the number of e-folds is the in ationary dynamics which in the slow-roll approximation takes for j-in ation the form _I = 2 iI 1 p E4 E6 + ( 1)I E6 E4 jjj; where G = ( = 2)2 is the conformal factor of the modular in ation target space metric. Integrating the j-function then leads to the number of e-folds N = p 1 2 Z te 3 MPl t dt jj( )j: 8.2 j-in ation observables and the Planck probe The parameter space of j-in ation is strati ed by the energy scale that enters all the observables. Given a speci c choice of and the pivot value I of the in aton, the spectral = i. The curved path indicates an N = 60 trajectory. from the valence formula (5.16) as nite at this point [ 47 ] indices nRR and nT , as well as the tensor ratio r, can be computed from the Eisenstein formulae above. Furthermore, the scale can be obtained via eq. (8.7) from the Planck scalar amplitude [37, 38], which in turn allows to determine the number of e-folds N . A detailed scan of the parameter space given by ( ; ) can be performed, leading to di erent neighborhoods U ( ) in the upper halfplane that can be tested against the satellite probe constraints. The form of the slow-roll parameters I in eq. (8.3) and the scalar spectral index in eq. (8.5) show that the horizon crossing value of the in aton should be chosen in a neighborhood of the zero of the Eisenstein series E6, which can be obtained = i = p 1. The Eisenstein series E2 and E4 are E2(i) = 3 E4(i) = 3 (1=4)8 (2 )6 ; (8.12) hence I , nRR and r are regular functions in this neighborhood. A zoom of the potential close to this point with a particular in aton trajectory is shown in gure 2. It illustrates in more detail the ridge along the (Re ) = 0 line that is suppressed in the large scale view of the potential of gure 1. After xing the scale at a super-Planckian value, the in aton values at horizon crossing can be chosen such that after integrating the j-in ation dynamics (8.10) the orbit I (t) leads to a number N of e-folds between horizon crossing and the end of in ation that falls within the standard range N = [50; 70]: The eld ( I ) therefore traverses a super-Planckian distance in eld space during in ation. Figure 3 gives an illustration of the behavior for a few trajectories in the target space X associated to di erent scales that lead to the central value N = 60 and are consistent with CMB phenomenology. These models in particular all lead to spectral indices nRR = 0:96 , compatible with the Planck result [37], and the tensor-to-scalar ratio takes values in the range r 2 [10 8; 0:08], compatible with the result of the Planck Collaboration [59], which reports for the tensor ratio r at the pivot scale kp = 0:002Mpc the bound r0:002 0:11, while the BICEP2/Keck/Planck Collaboration reports r0:05 0:12 [ 60 ]. The energy scale determining the height of the potential includes the range 2 [10 6; 10 4]MPl for the realizations discussed here. More generally, the regions U ( = i) around the slow-roll point = i, identi ed above via the Eisenstein series, contain for varying many orbits that are consistent with Planck probe results. Moving too far from the slow-roll point violates the slow-roll condition, hence leads to tensor ratios r that are too large, as expected. On the in ationary time scales during which large scale perturbations cross the horizon the e ect on the observables of the transfer function TOO0 , which can be expressed in terms of the Eisenstein series Ew using the results above and earlier in the paper, is small. 9 Modular in ation is a class of two- eld models obtained as a specialization of multi eld automorphic eld theories by restricting the automorphic group G(Z) to be given by the modular group SL(2; Z), or subgroups thereof. The framework considered here di ers from in ation theories based on moduli, sometimes also called modular in ation. Moduli in ation posits that some of the moduli that arise in string theory, in particular in Calabi-Yau compacti cations, are involved in the in ationary process. Cases of string theory induced potentials exhibiting automorphic symmetries provide special examples that t into the more general framework of automorphic and modular in ation considered here and in [8, 9]. Modular in ation models present the simplest class of theories that allow to embed the shift symmetry into a group, in the process leading to a strati ed theory space, in which the individual leaves that provide the building blocks of the resulting foliation are characterized by the weights and levels of the de ning modular forms. The eld theory space of automorphic in ation in general, and modular in ation in particular, has a nontrivial geometry that is encoded in the Riemannian metric GIJ derived in a canonical way from the underlying group structure. In this paper a detailed description has been given of the general multi eld curved target space dynamics and its specializations to two- eld in ation and modular in ation, including a formulation of some of the variables that enter the phenomenological analysis for general modular potentials. An important problem arises from the fact that derivatives of modular forms are not modular, raising the issue of the modular nature of physical observables in modular invariant in ation. It was shown that the nonmodular contributions of the derivatives of the in aton potential combine with the nonmodular terms induced by the curved target space into almost holomorphic modular forms that in turn lead to CMB observables that are almost holomorphic modular invariant. An example of a modular function is given by the j-function, de ned as a quotient of modular forms of weight twelve forms relative to the full modular group, i.e. level one. The in ationary two- eld model that results from the simplest potential based on this function leads to a slow-roll phenomenology that is consistent with the observational results from the Planck satellite probe. 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Rolf Schimmrigk. Modular inflation observables and j-inflation phenomenology, Journal of High Energy Physics, 2017, 43, DOI: 10.1007/JHEP09(2017)043