New Shapes of Primordial Non-Gaussianity from Quasi-Single Field Inflation with Multiple Isocurvatons
Michael McAneny Alexander K. Ridgway






CALT-TH-2019-010










Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125


We study a simple extension of quasi-single field inflation in which the inflaton interacts with multiple extra massive scalars known as isocurvatons.  Due to the breaking of time translational invariance by the inflaton background, the theory includes kinetic mixings among the inflaton and isocurvatons.  These mixings give rise to novel new features in the primordial non-Gaussianities of the scalar curvature perturbation.  A noteworthy feature is the amplitude of the squeezed bispectrum can grow nearly as (k_s/k_l)^-3 while oscillating as cosγlog(k_s/k_l), where k_s/k_l is the ratio of the lengths of the short and long wavevectors.  Observation of such a shape would provide evidence for the existence of multiple isocurvatons during inflation.  In addition, we consider the effects of these non-Gaussianities on large-scale structure.     


New Shapes of Primordial Non-Gaussianity from Quasi-Single Field Inflation with Multiple Isocurvatons
    Michael McAneny and Alexander K. Ridgway
    December 30, 2023
=====================================================================================================






§ INTRODUCTION


The inflationary paradigm <cit.> posits a period of time shortly after the big bang during which the universe's energy density was dominated by vacuum energy and the size of the universe grew exponentially.  Such an era would explain the near isotropy of the CMB and the near flatness of the universe.  At the same time, inflation provides a quantum mechanical origin for the energy density perturbations which have an almost scale-invariant Harrison-Zel'dovich power spectrum <cit.> (see <cit.> for a review).

The simplest inflationary theories are known as single-field inflation models and involve a single scalar field called the “inflaton."  In slow-roll models, the inflaton vacuum expectation value ϕ_0 is initially displaced from the minima of the inflaton potential V_ sr(ϕ).  The potential is chosen such that there is a period of time (the inflationary era) during which ϕ_0's potential energy dominates over its kinetic energy and drives the exponential expansion of the universe.  After more than 50-60 e-folds of expansion, ϕ_0's kinetic energy becomes large and inflation ends.

Since ϕ̇_0 ≠ 0, time-translational invariance is spontaneously broken.  This gives rise to a Goldstone boson π that sources scalar curvature perturbations <cit.>.  In the simplest single-field models, the curvature perturbations are approximately Gaussian <cit.>.  To produce large primordial non-Gaussianities (PNG), one can add extra fields to the inflationary field content and include interactions between π and the new fields[Large non-Gaussianities can also be achieved in more complicated single field models, see for example <cit.>.].  This leads to interesting new shapes and features of  primordial non-Gaussianity that could be observed in the CMB and large-scale structure (LSS) and used to constrain the inflationary theory.

Quasi-single field inflation (QSFI) is a well-studied extension of single-field inflation models <cit.> that could potentially give rise to significant PNG.  In QSFI, one adds a single extra scalar field σ of mass m, known as the isocurvaton, and includes a kinetic mixing between σ and π of the form[Kinetic mixing terms such as μπ̇σ can appear in the Lagrangian because the inflaton vev spontaneously breaks Lorentz invariance] μπ̇σ and a potential for the isocurvaton V(σ).  In QSFI the isocurvaton never fulfills the roll of the inflaton, rather, its purpose is to generate PNG through its interactions with π.  

The connected three- (bispectrum) and four- (trispectrum) point functions of π in QSFI have been studied extensively <cit.>.  The squeezed limit of the bispectrum, which occurs when the lengths of two of the three wavevectors are roughly equal and much larger than the length of the third wavevector, has been shown to exhibit particularly interesting behavior.  Let k_l denote the length of the larger wavevectors and k_s the length of the shorter one.  It has been shown that if m and μ are both much smaller than the Hubble constant during inflation H, then the magnitude of the squeezed bispectrum in QSFI grows approximately as (k_s/k_l)^-3.  In the opposite limit, m,μ≫ H, the magnitude of the squeezed bispectrum was found to oscillate logarithmically in k_s/k_l but grow only as (k_s/k_l)^-3/2 <cit.>.

Since single-isocurvaton QSFI gives rise to distinct PNG, it is worthwhile to study theories that include multiple isocurvatons σ_I (we will call these theories multi-isocurvaton QSFI).  It turns out that interactions such as ρ (σ̇_1σ_2 - σ̇_2 σ_1) and μ_1π̇σ_1 give rise to novel features in the PNG.  Specifically, for certain choices of ρ and μ_1, the squeezed bispectrum undergoes logarithmic oscillations in k_s/k_l and grows approximately as (k_s/k_l)^-3.  This is very different from the behavior of the squeezed bispectrum in single-isocurvaton QSFI, which can only exhibit oscillations or nearly cubic power law growth in different limits of μ and m.  

Several previously studied inflationary models predict log-oscillating or oscillating shapes for the bispectrum <cit.>. Using shape templates, the Planck collaboration has begun constraining these oscillating/log-oscillating bispectra <cit.>. However, none of these shapes simultaneously exhibit log-oscillations and nearly cubic power law growth in the squeezed limit.  In this paper, we provide a bispectrum shape template for multi-isocurvaton QSFI that could be searched for experimentally.  

Previous work has demonstrated that non-Gaussianities that grow like (k_s/k_l)^-3 in the squeezed limit can have significant effects on large-scale structure <cit.>.  For example, in single-isocurvaton QSFI models with m,μ≪ H, non-Gaussian contributions to the dark matter halo-halo power spectrum P_hh(k_s) become much larger than the Gaussian contribution as k_s→ 0.  The halo-halo power spectrum P_hh(q) will also be sensitive to the extra fields and interactions present in multi-isocurvaton QSFI.  We will show that for certain choices of the parameters ρ and μ_1, the bispectrum and trispectrum contributions to P_hh(k_s) will dominate over the Gaussian contribution while oscillating logarithmically as k_s→ 0.    

The purpose of this work is to determine the shapes of the bispectrum and trispectrum in multi-isocurvaton QSFI and explore their effects on LSS.  In section <ref> we write down a general quadratic Lagrangian for multi-isocurvaton QSFI and determine the mode functions of the π and σ_I fields.  In section <ref> we compute the primordial bispectrum and trispectrum due to a cubic interaction of the form σ_1^3, and provide a template for the bispectrum shape.  Finally, in section <ref> we calculate the contribution of the PNG to the halo-halo power spectrum P_hh(k_s) and halo-matter power spectrum P_hm(k_s).





§ THE MODEL AND MODE FUNCTIONS


We consider multi-isocurvaton QSFI, which include an inflaton field and N extra scalar fields known as “isocurvatons."  The inflaton develops a time-dependent vacuum expectation value ϕ_0 that sources a background de-Sitter metric

    ds^2 = 1/(Hτ)^2
        (dτ^2 - (dx^i)^2)

where H is the Hubble parameter during inflation and τ is proper time.  We assume that ϕ_0 exhibits a slow-roll trajectory, which means ϕ̈_0 ≃ 0 and H is approximately constant throughout the inflationary era (see <cit.> for a review of slow-roll inflation).  The quantum fluctuations of the inflaton and isocurvaton fields perturb the metric about (<ref>) and source scalar and tensor curvature fluctuations, ζ(τ,x⃗) and γ_ij(τ,x⃗).

To describe the field fluctuations, we use the effective field theory of inflation formalism <cit.>.  Following <cit.>, we choose uniform inflaton gauge in which the inflaton fluctuations are set to zero.  We then observe that the time-dependent inflaton vev spontaneously breaks time diffeomorphism invariance, giving rise to a Goldstone boson π that transforms as π(x) →π(x) - ξ(x) under time-diffeomorphisms t → t + ξ(x).  The degrees of freedom in the effective theory are π, the metric fluctuations, and the fluctuations of the N isocurvaton fields σ_I.  To construct the effective theory, one writes all possible terms invariant under the full set of space-time diffeomorphisms involving these fields (see <cit.> for a complete derivation in the context of single-isocurvaton QSFI).  

We will be interested in computing the momentum space correlation functions of π at the time when the modes exit the horizon.  This implies we can work in the “decoupling limit" of the effective theory <cit.> and set the metric perturbations to zero.  
Correlation functions of ζ in the gauge where π is zero can be related to correlation functions of π in the decoupling limit by <cit.>

    ζ = -(H/ϕ̇) π.

The leading quadratic Lagrangian simplifies to any term quadratic in π and σ_I that is consistent with a shift symmetry in π and a background de-Sitter space-time:

    L_2 = 1/2(H τ)^2   ((∂_τπ)^2-c̃_π^2(∂_i π)^2+Z̃_IJ∂_τσ_I ∂_τσ_J-c̃^2_σ IJ∂_i σ_I ∂_i σ_J+2β̃_I∂_τπ∂_τσ_I + δ̃_I∂_i π∂_i σ_I.   .-(Hτ)^-2m̃^2_IJσ_I σ_J -2(H τ)^-1ρ̃_IJσ_I∂_τσ_J  - 2(H τ)^-1μ̃_I σ_I∂_τπ).

Note, repeated indices are summed over and we have included the √(-g) factor from the action in the Lagrangian.  We have dropped terms quadratic in the fields that have more than two derivatives because they are suppressed by powers of the cutoff Λ of the effective theory.  

Several terms in (<ref>) can be eliminated by field redefinitions.  For example, the δ̃_I∂_iπ∂_i σ_I interaction can be absorbed into other couplings by performing the time diffeomorphism that induces the shift π→π + δ̃_I σ_I/(2 c^2_π).  Moreover, we can rotate and re-scale σ_I to diagonalize c̃^2_σ IJ and set Z̃_I J  = 1.  Equation (<ref>) then simplifies to 

    L_2 = 1/2(Hτ)^2(    (∂_τπ)^2-c_π^2(∂_i π)^2+(∂_τσ_I)^2-c_σ I^2(∂_i σ_I)^2+2β_I ∂_τπ∂_τσ_I      -(H τ)^-2 m_IJ^2σ_I σ_J-2(Hτ)^-1ρ_IJσ_I ∂_τσ_J -2(H τ)^-1μ_I σ_I∂_τπ).

The matrix m^2_IJ is symmetric while ρ_IJ is anti-symmetric[Any symmetric part of ρ_IJ can be absorbed into m^2_IJ through an integration by parts.].  The interactions μ_Iπ̇σ_I and ρ_IJσ_I σ̇_J could result from a UV theory containing terms such as (σ_I/Λ) g^μν∂_μϕ∂_νϕ and (σ_I/Λ)g^μν∂_μσ_J ∂_νϕ.  The kinetic mixings in (<ref>) arise because the inflaton vev spontaneously breaks Lorentz invariance.  If ϕ̇_0 = 0, Lorentz invariance is unbroken and the kinetic mixings must vanish.  By dimensional analysis, this means μ_I and ρ_IJ are proportional to ϕ̇_0/Λ.  Using (H^2/ϕ̇_0)^2 ∼ 2π^2Δ_ζ^2, where 

    Δ_ζ^2=k^3/2π^2P_ζ(k)≃ 2.11× 10^-9

is the dimensionless power spectrum <cit.>, the cutoff can be expressed in terms of |μ_I| as

    Λ/H∼H/|μ_I|× 10^4.

For μ_I ∼ O(H), this implies that Λ≫ H and higher derivative terms are suppressed.


We can recover previously studied single-isocurvaton QSFI models by taking limits of (<ref>).  If we take N=1, β_1 = 0 and c^2_π=c^2_σ1=1, we recover the quadratic part of the QSFI Lagrangian originally considered by Chen and Wang <cit.>.  The resulting Lagrangian only has two parameters, μ_1 and m_11.  Single-isocurvaton QSFI with generic speeds of sound and nonzero β_1 was studied in <cit.>.  The presence of a nontrivial ρ_IJ matrix is the main new aspect of theories with N>1 isocurvatons. 


One way to treat the kinetic mixings parameterized by μ_I, β_I and ρ_IJ is to write the Fourier transforms of the π and σ_I fields in terms of a common set of raising and lowering operators

    π̂(τ, x)    = ∫d^3k/(2π)^3H/k^3/2∑_i=1^N+1(â_k^(i)π^(i)(η)e^-ik·x + c.c)σ̂_I(τ,x)    = ∫d^3k/(2π)^3H/k^3/2∑_i=1^N+1(â_k^(i)σ_I^(i)(η)e^-ik·x + c.c)

where η = kτ.  The mode functions π^(i)(η) and σ^(i)(η) obey the Euler-Lagrange equations obtained from (<ref>),

    σ^(i)_I”-2/ησ^(i)_I'+c^2_σ Iσ^(i)_I+m^2_IJ/η^2σ^(i)_J +μ_I/ηπ^(i)'+β_I(π^(i)”-2/ηπ^(i)')+ρ_IJ/η(2σ^(i)_J'-3/ησ^(i)_J)= 0   π^(i)”-2/ηπ^(i)'+c^2_ππ^(i)-μ_I/η(σ^(i)_I' - 3/ησ^(i)_I)+β_I(σ^(i)_I”-2/ησ^(i)_I') = 0

where primes denote derivatives with respect to η.  The mode functions asymptotically obey the Bunch-Davies vacuum condition.  Since equations (<ref>) are coupled, they are difficult to solve analytically for general parameters[Analytic progress can be made in the context of single-isocurvaton QSFI in the regimes where μ_1, m_11≪ H <cit.>, or μ_1, m_11≫ H <cit.>.].  Instead, we use the numerical solutions of (<ref>) to perform most of the calculations in our analysis.  However, one can derive the small η behavior of the mode functions analytically, which turns out to fix the wavevector dependence of the squeezed and collapsed limits of the bispectrum and trispectrum.

In the limit[We have chosen the convention in which inflation occurs between -∞ < τ≤ 0] -η << 1, we can neglect the terms in (<ref>) proportional to the speeds of sound.  The leading late time behavior for the mode functions can be written as

    π^(i)(η)    = a^(i)_s (-η)^s    σ^(i)_I(η) = b^(i)_I,s (-η)^s.

For example, specializing to N = 2 and inserting (<ref>) into (<ref>) yields the following equation for s,

    (s-3)s            (s-3)(sβ_1-μ_1)      (s-3)(sβ_2-μ_2)
    
        s((s-3)β_1 + μ_1)            m_11^2+(s-3)s      m_12^2+(3-2 s)ρ
    
        s((s-3)β_2 + μ_2)            m_12^2+(2s - 3)ρ     m_22^2 + (s-3)s
     = 0,

which can be solved for six different roots:

    s = 0,s_-,s_-^*,3-s_-,3-s_-^*,3.

In general, s_- is a complex number satisfying[For general model parameters Re[s_-] can be less than 0, however the modes would then be tachyonic and grow rather than decay as η→ 0.] 0<Re[s_-]≤ 3/2.  The s=0 solution arises from the shift symmetry in π and can only exist in the π mode functions.  The leading η behavior of the mode functions is 

    π^(i)(η)    = a^(i)_0+a^(i)_s_-(-η)^s_- + a^(i)_s_-^*(-η)^s_-^*+…σ^(i)_I(η)    = b^(i)_I,s_-(-η)^s_- + b^(i)_I,s_-^*(-η)^s_-^*+….

Equations (<ref>) imply that as η→ 0, π^(i) approaches a constant while σ^(i)_I decays to 0. 

The late time behavior of σ_I in multi-isocurvaton QSFI can be very different from its late time behavior in single-isocurvaton QSFI <cit.>.  If we write s_- ≡α + iγ, equation (<ref>) becomes

    σ^(i)_I(η) = b^(i)_I,s_-(-η)^α e^iγlog(-η) + b^(i)_I,s_-^*(-η)^α e^-iγlog(-η).

Observe that the modes oscillate logarithmically in η with frequency γ.  Moreover, α dictates how quickly the modes decay at late times.  In the original Chen and Wang theory <cit.> γ can only be nonzero when α = 3/2 (see e.g. <cit.>).  This means that while the isocurvaton's mode functions exhibit oscillatory behavior at late times, they decay quickly as η→ 0.  On the other hand, in a multi-isocurvaton QSFI theory with ρ_IJ≠ 0, one can obtain γ≠ 0 with α < 3/2, which means that σ^(i)_I can oscillate while decaying slowly.


To illustrate this, we specialize to the case of two isocurvatons and focus on two sets of parameters which will serve as our benchmark models.  The first set is

    μ_1 = m_12 = m_21 = β_I =0, c_π= c_σ_1 = c_σ_2 = 1,   μ_2 = 0.6H, m_11^2 = m_22^2 =-ρ_12^2 = -(5H)^2

which yields s_-≃ 0.06 - 5.00 i, while the second is

    μ_1 = m_12 = m_21 = β_I =0, c_π= c_σ_1 = c_σ_2 = 1,   μ_2 = 0.4H, m_11 = m_22 = 0.3 H, ρ_12 = H

which yields s_-≃ 0.46 - 1.00i.  Notice, the masses squared in (<ref>) are negative, which is usually a signal of tachyonic modes whose mode functions diverge as η→ 0.  However, due to the kinetic mixing, the mass squared parameters that appear in the Lagrangian do not equal the physical masses squared.  Indeed, α > 0 for this set of parameters, which implies σ_I → 0 as η→ 0.

For α<3/2 and γ≠ 0, α and γ are typically the same order of magnitude.  Some tuning is required in order to produce α≪ 1 with γ∼ O(1).  This means, rapid oscillations that decay slowly cannot be produced without some degree of tuning between model parameters[For example, in eq. (<ref>), we tuned m_ii^2=-ρ^2.].




§ PRIMORDIAL NON-GAUSSIANITY


In the previous section, we showed that theories with multiple-isocurvaton  have a  kinetic mixing term parameterized by the matrix ρ_IJ that cannot exist in single-isocurvaton models.  If this term is present, the mode functions of π and σ_I can exhibit oscillatory behavior that decays slowly at late times.  We now study the effects of this behavior on the non-Gaussianities of the scalar curvature perturbations of the metric, ζ.  

We are interested in computing the “in-in" correlation functions of ζ at τ = 0, which are related to those of π through (<ref>).  
The in-in correlator of an operator O can be expressed in the so-called “commutator form" (see for example  <cit.>) as

    ⟨ O(0)⟩ = ∑_N=0^∞ i^N∫_-∞^0 d τ_N∫_-∞^τ_N dτ_N-1…∫_-∞^τ_2 dτ_1⟨ [H_int(τ_1),[H_int(τ_2),… [H_int(τ_N), O(0)]…]]⟩_I

where the fields on the right hand side of (<ref>) evolve according to (<ref>) and (<ref>).  In general, the interaction Hamiltonian consists of an isocurvaton potential V(σ_I) as well as interactions involving combinations of π and σ_I.  For simplicity, we assume the potential consists of a cubic interaction involving only the σ_1 field and that the interaction Hamiltonian is dominated by this interaction[The potential V(σ_I) is not related to the other operators in the effective theory by any symmetry and can, in principle, be the largest term in the interaction Hamiltonian.]:

    H_int(τ) = 1/(H τ)^4∫ d^3xV”'/3!σ_1(x)^3 + ….

In the next two subsections, we compute the bispectrum and trispectrum of ζ due to this interaction.


 §.§ Bispectrum





We first consider the three-point function of ζ.  Let ζ_ k denote the Fourier transform of ζ( x,0).  The bispectrum B_ζ is defined as

    ⟨ζ_ k_1ζ_ k_2ζ_ k_3⟩ = B_ζ( k_1, k_2, k_3)(2π)^3δ^3( k_1+ k_2+ k_3).

Using equations (<ref>), (<ref>), (<ref>) and (<ref>), it is straightforward to show that to leading order in V”'

    B_ζ(k_1,k_2,k_3)    = -2(H^2/ϕ̇_0)^3V”'/H1/k_1^3k_2^3k_3^3   ×∫_-∞^0dτ/τ^4 Im(π^(i)(0)σ^(i)*_1(k_1τ)π^(j)(0)σ^(j)*_1(k_2τ)π^(k)(0)σ^(k)*_1(k_3τ))

where repeated mode labels are summed.  Note, the sum π^(i)(0)σ_1^(i)*(k_iτ) is nonzero because of the kinetic mixings.   



The τ integral in (<ref>) is potentially IR divergent due to the factor of 1/τ^4 in the integrand.  Even though we do not have explicit expressions for the mode functions, it can be shown using the canonical commutation relations (see Appendix <ref>) that the integral is indeed finite in the IR.  One can then evaluate the bispectrum using the numerical solutions of (<ref>). 

Consider the squeezed limit of (<ref>), which occurs when k_l≡ k_1∼ k_2 and k_s ≡ k_3 ≪ k_l, i.e. there are two long sides and one short side of the triangle traced out by the k_i.  We can factor out the momentum dependence from the time integral by changing integration variables to η = k_lτ and expanding to leading order in k_s/k_l using (<ref>).  We find

    B_ζ^sq(k_l,k_s)   = -4(H^2/ϕ̇_0)^3(V”'/H)1/k_l^6(k_s/k_l)^-3+α(   cos(γlog k_s/k_l)Re[a^(i)_0b^(i)*_1,s_-y^*(α,γ)]
        +   sin(γlog k_s/k_l)Im[a^(i)_0b^(i)*_1,s_-y^*(α,γ)] )

where

    y(α,γ)   =∫_-∞^0dη/(-η)^4-α-i γIm[(π^(i)(0)σ^(i)*_1(η))^2].

The squeezed bispectrum oscillates logarithmically in k_s/k_l with angular frequency γ and the amplitude grows as (k_s/k_l)^-3+α.  As mentioned earlier, in single-isocurvaton QSFI, γ can only be nonzero when α = 3/2, which means the amplitude of an oscillating squeezed bispectrum can only grow as (k_s/k_l)^-3/2 in these models.  This is not the case for the multi-isocurvaton models given by (<ref>) or (<ref>), whose oscillating bispectrum grow approximately as (k_s/k_l)^-2.94 and (k_s/k_l)^-2.54 respectively.  







 To illustrate the momentum dependence of the full bispectrum, it is useful to define the shape function

    S(k_1,k_2,k_3)=κ(k_1k_2k_3)^2B_ζ(k_1,k_2,k_3).

  The normalization factor κ is chosen so that S(k,k,k) = 1.





In Fig. <ref>, we plot the shape function of multi-isocurvaton QSFI for the model parameters (<ref>) and (<ref>).  For comparison, we also include the shape function of local non-Gaussianity

    S^ loc(k_1,k_2,k_3)=1/3(k_1k_2k_3)^2[1/(k_1k_2)^3+1/(k_2k_3)^3+1/(k_3k_1)^3]

which is close to the shape function of the single-isocurvaton QSFI originally considered by Chen and Wang in the limit μ_1,m_11≪ H <cit.>. 

Fig. <ref> is consistent with the analytic results for the squeezed bispectrum of multi-isocurvaton QSFI.  In the limit k_3/k_1 → 0, the shape function oscillates logarithmically in k_3/k_1 and its amplitude has power law growth.  On the other hand, in single-isocurvaton QSFI, the shape function for an oscillating bispectrum decays to 0 as k_3/k_1→ 0.  Note, to get very rapid oscillations and nearly local power law growth for the multi-isocurvaton shape, one needs to tune parameters as in (<ref>).  However, even the shape for the untuned parameters (<ref>) displays a visible turn due to nonzero γ.




For single-isocurvaton QSFI with γ=0, the shape function has been approximated as <cit.>:

    S_α^ QSFI(k_1,k_2,k_3)=3^3α-2(k_1^2+k_2^2+k_3^2)(k_1+k_2+k_3)^1-3α/(k_1k_2k_3)^1-α

A good phenomenological fit to the multi-isocurvaton QSFI shape is

    S_α,γ,ϕ^ OQSFI(k_1,k_2,k_3)=C(cos(γlog(k_2 k_3/k_1(k_1+k_2+k_3))+ϕ)+2 perms.)S_α^ QSFI(k_1,k_2,k_3)

where the normalization C=(3cos(γlog(1/3)+ϕ))^-1 enforces S^ OQSFI=1 in the equilateral limit.  Note that the shape function is parameterized by three numbers α, γ, and a phase ϕ.  In Fig. <ref>, we plot the shape functions evaluated numerically against the shape functions computed with (<ref>).  

We can also define the parameter f_ NL^ OQSFI, corresponding to the magnitude of this shape.  In keeping with convention[Conventionally, f_ NL is defined via B_Φ(k_1,k_2,k_3)=6A^2f_ NL1/k_1^2k_2^2k_3^2S(k_1,k_2,k_3) where A is given by the power spectrum's normalization P_Φ(k)=A/k^3 (see, e.g. <cit.>).], we define:

    B_ζ(k_1,k_2,k_3)=18/5(k^3P_ζ(k))^2 f_ NL^ OQSFI1/k_1^2k_2^2k_3^2S_α,γ,ϕ^ OQSFI(k_1,k_2,k_3).

By matching the squeezed limit of (<ref>) onto (<ref>), we can obtain V”'/H as a function of f_ NL^ OQSFI:

    V”'/H=-9/10(|a^(i)_0|^2)^3/2√(2π^2 Δ_ζ^2)f_ NL^ OQSFIC/√(Im[a^(i)_0b^(i)*_1,s_-y^*(α,γ)]^2+1/9Re[a^(i)_0b^(i)*_1,s_-y^*(α,γ)]^2 ).






Even for γ∼ O(0.1), the oscillating QSFI bispectrum shape still displays qualitatively distinct features. Specifically, the shape can get large and negative in the squeezed limit.  This is different from many other bispectrum shapes that grow in the squeezed limit because they typically remain positive rather than become negative (see <cit.>).

 







 §.§ Collapsed Trispectrum


We now consider the four-point function of ζ, which is defined as

    ⟨ζ_ k_1ζ_ k_2ζ_ k_3ζ_ k_4⟩ = T_ζ( k_1, k_2, k_3, k_4)(2π)^3δ^3( k_1+ k_2+ k_3+ k_4).

We focus on the collapsed limit, which occurs when k_l1≡ k_1∼ k_2, k_l2≡ k_3∼ k_4, and k_s≡ | k_1+ k_2|=| k_3+ k_4|≪ k_li.  The integrand will be symmetric in k_l1 and k_l2, which means we can untangle the limits of the time integrals.  The collapsed four-point then simplifies to

    T_ζ^coll(k_l1,k_l2,k_s)=4   (H^2/ϕ̇_0)^4(V”'/H)^21/k_l1^9/2k_l2^9/2(k_s^2/k_l1k_l2)^-3/2+α[   cos(γlogk_l2/k_l1)(|b^(i)_1,s_-|^2+|b^(i)_1,s_-^*|^2)|y(α,γ)|^2
        +2   cos(γlogk_s^2/k_l1k_l2)Re[b^(i)_1,s_- b^(i)*_1,s_-^*y(α,γ)^2]
        -2   sin(γlogk_s^2/k_l1k_l2)Im[b^(i)_1,s_- b^(i)*_1,s_-^*y(α,γ)^2]].

Note, not every term oscillates logarithmically in k_s/k_l.  Instead, there are two terms which oscillate with angular frequency 2γ as well as one that does not oscillate at all in k_s/k_l.  

It turns out that a loop contribution to the trispectrum can produce terms that oscillate with frequency 4γ.  We explore this in Appendix <ref>.





§ LARGE-SCALE STRUCTURE


In this section, we determine the effects of PNG on large-scale structure, specifically the halo-halo power spectrum P_hh and the matter-halo power spectrum P_hm.  It is well known that in certain inflationary theories, such as single-isocurvaton QSFI,  the contributions of PNG to P_hh and P_hm can become much larger than the Gaussian contributions at wavevectors of order (10^2  Mpc/h)^-1 <cit.>.  We now consider this in multi-isocurvaton QSFI.  

The primordial curvature perturbations are related to the linearly evolved smoothed matter density perturbations today, δ_R(k), by <cit.>

    δ_R( k)=2k^2/5Ω_m H_0^2T(k)W_R(k)ζ_ k

where Ω_m is the ratio of the matter density and the critical density today, H_0 is the Hubble constant today, T(k) is the BBKS transfer function <cit.> and W_R(k) is a window function smoothing over radius R.  We use the top-hat window function 

    W_R(k) = 3 ( sinkR - kR  coskR)/(kR)^3.
.  



Since we are interested in scales of order 10^2 Mpc/h, δ_h can be related to δ_R through a bias expansion.  For simplicity, we use a local-in-matter-density bias expansion (for a more comprehensive treatment, see <cit.>)

    δ_h( x)=b_1δ_R( x)+b_2(δ_R( x)^2-⟨δ_R( x)^2⟩)+…

The bias coefficients can be approximated using the threshold model introduced in <cit.>.  We assume that halos form instantaneously at some redshift z_ coll, and that halos only form at points where the overdensity exceeds some critical threshold δ_c(z_coll).  We also neglect the evolution of halos after collapse.  In this model, the bias coefficients b_1 and b_2 are then

    b_1 = 2e^-δ_c^2/(2σ_R^2)/√(2π)σ_R erfc(δ_c/(√(2)σ_R))       b_2 =  e^-δ_c^2/(2σ_R^2)δ_c/√(2π)σ_R^3 erfc(δ_c/(√(2)σ_R))

where  σ_R^2 = ⟨δ_R( x)^2⟩.   In deriving our numerical results, we use δ_c = 4.215, which corresponds to δ_c(z_ coll)=1.686 at z_ coll = 1.5 <cit.>, and R = 3  Mpc/h.
 While a more sophisticated treatment of halo dynamics will change our precise numerical results, we do not expect them to impact our conclusions qualitatively.

We will be interested in computing the halo-halo power spectrum P_hh, 

    ⟨δ_h( k_1)δ_h( k_2)⟩ = P_hh(k_1)(2π)^3δ^3( k_1+ k_2)

as well as the halo-matter power spectrum P_hm,

    ⟨δ_h( k_1)δ_R( k_2)⟩ = P_hm(k_1)(2π)^3δ^3( k_1+ k_2).

The Gaussian contributions to P_hh and P_hm are found to be

    P_hh(k_s)|_G=b_1^2(H^2/ϕ̇_0)^2(2/5Ω_m H_0^2 R^2)^2k_sR^4|a^(i)_0|^2   P_hm(k_s)|_G=b_1(H^2/ϕ̇_0)^2(2/5Ω_m H_0^2 R^2)^2k_sR^4|a^(i)_0|^2.

At wavevectors of order (10^2  Mpc/h)^-1≪ R^-1, the most significant non-Gaussian contributions to P_hh are due to the squeezed and collapsed limits of the bispectrum (<ref>) and trispectrum (<ref>).  Then, plugging (<ref>) and  (<ref>) into (<ref>) and using (<ref>) and (<ref>) gives

    P_hh(k_s)/P_hh(k_s)|_G=1   -8(b_2/b_1)(H^2/ϕ̇_0)(2/5Ω_m H_0^2 R^2)(V”'/H)1/(k_sR)^2-α1/|a^(i)_0|^2        (cos(γlog k_sR)Re[J^*(α,γ)y^*(α,γ)a^(i)_0b^(i)*_1,s_-]         +sin(γlog k_sR)Im[J^*(α,γ)y^*(α,γ)a^(i)_0b^(i)*_1,s_-])   +8(b_2/b_1)^2(H^2/ϕ̇_0)^2(2/5Ω_m H_0^2 R^2)^2(V”'/H)^21/(k_sR)^4-2α1/|a^(i)_0|^2        (|J(α,γ)|^2|y(α,γ)|^2|b^(i)_1,s_-|^2         +cos(2γlog k_sR)Re[J(α,γ)^2y(α,γ)^2b^(i)_1,s_-b^(i)*_1,s_-^*]         -sin(2γlog k_sR)Im[J(α,γ)^2y(α,γ)^2b^(i)_1,s_-b^(i)*_1,s_-^*]).

where, to compactify notation, we have defined

    J(α,γ)   =1/2π^2∫_0^∞ dx   x^3-α-i γT(x/R)^2W(x/R)^2

The most significant non-Gaussian contribution to P_hm comes from the squeezed bispectrum:

    P_hm(k_s)/P_hm(k_s)|_G=1   -4(b_2/b_1)(H^2/ϕ̇_0)(2/5Ω_m H_0^2 R^2)(V”'/H)1/(k_sR)^2-α1/|a^(i)_0|^2        (cos(γlog k_sR)Re[J^*(α,γ)y^*(α,γ)a^(i)_0b^(i)*_1,s_-]         +sin(γlog k_sR)Im[J^*(α,γ)y^*(α,γ)a^(i)_0b^(i)*_1,s_-]).



In Figure <ref> we plot (<ref>) for the model parameters (<ref>) and (<ref>) and |f_ NL^ OQSFI|=10.  Note that at around (10^2 Mpc/h)^-1, P_hh begins to deviate from P_hh|_G.  The oscillations evident in Fig. <ref> are a consequence of the oscillatory squeezed bispectrum and collapsed trispectrum.

However, for |f_ NL^ OQSFI|∼ 10, the amplitude of the oscillations is quite small. Moreover, the non-Gaussian contributions to P_hh only begin to dominate over the Gaussian contribution at a scale of order (10^3  Mpc/h)^-1, which is unlikely to be detected experimentally in the near future. 

This scale is smaller than the scale at which the non-Gaussian contributions to P_hh begin to dominate in single-isocurvaton QSFI theories with f_NL∼ 10 and α small <cit.>.  The reason is the integrals involving the transfer functions, J(α,γ), are oscillatory in multi-isocurvaton QSFI when γ≠ 0, which washes them out.  This makes the coefficients of the non-Gaussian contributions smaller in multi-isocurvaton QSFI than in single-isocurvaton QSFI.



§ CONCLUSION

In this paper, we studied quasi-single field inflation with multiple isocurvatons.  
Multi-isocurvaton QSFI includes the interaction ρ_IJσ̇_I σ_J which can give rise to novel inflationary dynamics.  In particular, the mode functions of π and σ_I can exhibit late time log oscillations that decay slowly as η→ 0.  Due to these late time oscillations, the primordial non-Gaussianities of ζ exhibit log-oscillatory behavior in ratios of wavevector magnitudes.

For example, the bispectrum is proportional to (k_s/k_l)^-3+αcos(γlog k_s/k_l) in the squeezed limit, which means for small α it experiences nearly cubic growth while oscillating.  This behavior cannot be achieved in single-isocurvaton QSFI.  
Furthermore, the collapsed trispectrum goes as (k_s/k_l)^-3+2α(a + bcos(2γlog k_s/k_l)), i.e. there is a term that does not oscillate in k_s/k_l as well as one that does with frequency 2γ.

In models where α≲ 0.5, the contributions of the squeezed bispectrum and collapsed trispectrum to the halo-halo power spectrum P_hh(k_s) and the halo-matter power spectrum P_hm(k_s) can dominate over the Gaussian contributions at k_s ∼ (10^3  Mpc/h)^-1. When γ≠0, P_hh and P_hm oscillate logarithmically in k_s R.



§ ACKNOWLEDGEMENTS


We would like to thank Mark Wise for useful discussions. This work was supported by the DOE Grant DE-SC0011632. We are
also grateful for the support provided by the Walter Burke Institute for Theoretical Physics.





§ COMMUTATOR CONSTRAINTS


The η integrals of (<ref>) are potentially IR divergent because of the factors of 1/η^4 in the integrands.  It can be shown that all potentially IR divergent terms are zero.  Eq. (<ref>) implies that the σ mode functions can be written in series form as

    σ_I^(i)(η)    = b^(i)_I,s_-(-η)^s_- + b^(i)_I,s_-^*(-η)^s_-^* + b^(i)_I,2(-η)^2 + b^(i)_I,2+s_-(-η)^2+s_- + b^(i)_I,2+s_-^*(-η)^2+s_-^*                  + b^(i)_I,3-s_-(-η)^3-s_- + b^(i)_I,3-s_-^*(-η)^3-s_-^* + b^(i)_I,3 (-η)^3 + …

The equal time commutation relation [π,σ_I] = 0 holds order by order in powers of η and implies the following relations among the power series coefficients

    π^(i)(0)b^(i)*_I,s_- = (π^(i)(0)b^(i)*_I,s_-^*)^*,  π^(i)(0)b_I,s_-^(i)* =  (π^(i)(0)b_I,s^*_-^(i)*)^*
         π^(i)(0)b_I,3-s_-^(i)* =  (π^(i)(0)b_I,3-s^*_-^(i)*)^*,    Im(π^(i)(0)b^(i) *_I,2) = 0

where the sum over mode label i is implicit.  Furthermore, the mode equations (<ref>) imply

    b^(i)_I,2+s_- = c b^(i)_I ,s_-    b^(i)_I,2+s^*_- = c^
    *  b^(i)_I,s^*_-.

Combining the first relation in (<ref>) with (<ref>) yields

    π^(i)(0)b^(i)*_I,2+s_- = (π^(i)(0)b^(i)*_I,2+s_-^*)^*

The leading infrared behavior of (<ref>) is then

    y(α,γ)    = 2  Im[π^(i)(0)b^(i)*_1,3]∫_-∞^0 dη/(-η)^1-α-iγ Re[π^(j)(0)σ^(j)*_1(η)].

It is straightforward to fit the σ^(i)_1 mode functions to (<ref>) and determine the numerical coefficient in (<ref>).  The method to evaluate y(α,γ) is to choose an -η_IR < 1 and numerically integrate from -∞ < η < η_IR using (<ref>) and then integrate from η_IR≤η < 0 using (<ref>).



§ LOOP CONTRIBUTION TO THE COLLAPSED TRISPECTRUM


Recall that the tree-level collapsed trispectrum, which has one internal line, has terms that oscillate with frequency 2γ.  We now show that a loop contribution to the collapsed trispectrum, which has two internal lines, can contain terms that oscillate with frequency 4γ.  The loop contribution to the collapsed trispectrum then induces terms in the halo-halo power spectrum that oscillate as cos (4 γ logk_s R).[Contributions to the halo-halo power spectrum due to primordial non-Gaussianities sourced by quantum loops in the context of single-isocurvaton QSFI were considered in <cit.>.]




Consider a theory in which the interaction Hamiltonian is composed of a single σ_1^4 interaction:

    H_int(τ) = 1/(H τ)^4∫ d^3 xV””/4!σ_1(x)^4.

Inserting two factors of (<ref>) into (<ref>) yields the 1-loop contribution to the trispectrum:

    T_ζ^coll(k_l1,   k_l2,k_s) = 4 (H^2/ϕ̇_0)^4V””^21/k_l1^3k_l2^3∫_-∞^0dη/η^4∫_-∞^k_l2/k_l1ηdη'/η'^4∫d^3p/(2π)^31/p^3| p+ k_s|^3
       ×Im[(π^(i)(0)σ_1^(i)*(η))^2] Im[(π^(j)(0)σ_1^(j)*(η'))^2(σ_1^(l)(p/k_l1η)σ_1^(l)*(p/k_l2η')).
       .                                       ×(σ_1^(m)(| p+ k_s|/k_l1η)σ_1^(m)*(| p+ k_s|/k_l2η'))]

As described in <cit.>, for k_s≪ k_l, the loop diagram can give a large contribution to the collapsed trispectrum.  In this limit, the loop integral is dominated by the region in which p ∼ k_s, which means the mode functions on the bottom line of (<ref>) can be expanded in p/k_l i.  Defining the integral

    Z(ϵ_1,ϵ_2)≡∫d^3u/(2π)^3u^-3+2ϵ_1| u+x̂|^-3+2ϵ_2,

where x̂ is an arbitrary unit vector, the loop contribution can be written:

    T_ζ^coll(   k_l1,k_l2,k_s)=  2(H^2/ϕ̇_0)^4V””^21/k_l1^9/2k_l2^9/2(k_s^2/k_l1k_l2)^-3/2+2α                                          [Re[c_1]+Re[c_2]cos(2γlogk_l2/k_l1)+Re[c_3] cos(2γlogk_s/k_l1)+Re[c_3] cos(2γlogk_s/k_l2)   -Im[c_3]sin(2γlogk_s/k_l1)-Im[c_3] sin(2γlogk_s/k_l2)+Re[c_4] cos(2γlogk_s^2/k_l1k_l2)   -Im[c_4]sin(2γlogk_s^2/k_l1k_l2)]

where

    c_1    = 2y(2α,0)^2(|b^(i)_1,s_-|^2|b^(j)_1,s_-^*|^2Z(α,α)+|b^(i)_1,s_-b^(i)*_1,s_-^*|^2Z(α+iγ,α-i γ))
        c_2    =|y(2α,2γ)|^2((|b^(i)_1,s_-|^2)^2+(|b^(i)_1,s_-^*|^2)^2)Z(α,α)
        c_3     = 4y(2α,0)y(2α,2γ)b^(i)_1,s_-b^(i)*_1,s_-^*(|b^(j)_1,s_-|^2+|b^(j)_1,s_-^*|^2)Z(α,α+i γ)
        c_4     = 2|y(2α,2γ)|^2(b^(i)_1,s_-b^(i)*_1,s_-^*)^2Z(α+iγ,α+iγ).

Note that the loop contribution to the trispectrum has terms that do not oscillate with log k_s/k_li, terms that oscillate with frequency 2γ, and terms that oscillate with frequency 4γ.

The loop contribution to P_hh due to the quartic σ_1^4 interaction can also be computed:

    P_hh(k_s)|_loop/P_hh(k_s)|_G =      2(b_2/b_1)^2(H^2/ϕ̇_0)^2(2/5Ω_mH_0^2 R^2)^2V””^21/(k_s R)^4-2α1/|a^(i)_0|^2                         (Re[c_1 J(2α,0)^2+c_2|J(2α,2γ)|^2]   +2cos(2γlog k_s R)Re[ c_3J(2α,0)J(2α,2γ)]   -2sin(2γlog k_s R)Im[ c_3J(2α,0)J(2α,2γ)]   +cos(4γlog k_s R)Re[ c_4J(2α,2γ)^2]   -sin(4γlog k_s R)Im[ c_4J(2α,2γ)^2])



While it appears that the terms oscillating with frequency 4γ in (<ref>) could induce unique observable features in P_hh(k_s), it turns out their coefficients are suppressed relative to those of the non-oscillating terms since |J(2α,2γ)|≪ J(2α, 0).  This can be understood by noting that while the magnitudes of the integrands of J(2α,2γ) and J(2α, 0) are the same, the integrand of J(2α,2γ) is oscillatory and washes J(2α,2γ) out.  


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