text:
\section{Introduction}
An era of cosmic inflation (see \cite{Baumann:2009ds} for a review) in the very early universe is an attractive paradigm to explain the origin and the properties of the primordial density perturbations that eventually seed the temperature fluctuations in cosmic microwave background (CMB) and the inhomogeneities in large scale structure (LSS). The simplest models of cosmic inflation predict that the primordial fluctuations are adiabatic, scale invariant and approximately Gaussian---all three of these properties are verified to a good extent by the observations \cite{Akrami:2018odb}. Importantly, such properties can be tested with even better precision with upcoming surveys and this would allow us to get a more detailed picture of the inflationary era. In particular, any detection of primordial non-Gaussianity (NG) can give us crucial insight into physics at the inflationary Hubble scale $H$, which could lie well beyond the reach of terrestrial experiments. The time dependent inflationary spacetime can produce new particles with masses $\sim H$. If such particles can decay into inflatons, they can give rise to a distinctive non-analytic momentum dependence and angular dependence of the three (and higher)-point correlation function of the primordial fluctuations. Through a careful measurement of such dependencies, one can infer the mass and spin of such heavy particles---a rare opportunity to probe \textit{on-shell} physics at very high energy scales. This is the focus of the ``cosmological collider physics'' program \cite{Chen:2009zp,Baumann:2011nk,Assassi:2012zq,Chen:2012ge,Noumi:2012vr,Pi:2012gf,Gong:2013sma,Arkani-Hamed:2015bza,Lee:2016vti,Chen:2016nrs,Chen:2016uwp,Chen:2016hrz,Chen:2017ryl,Kehagias:2017cym,An:2017hlx,Kumar:2017ecc,Baumann:2017jvh}. For various recent ideas in this direction see, \cite{Chen:2018xck,An:2018tcq,Chen:2018cgg,Kumar:2018jxz,Arkani-Hamed:2018kmz,Li:2019ves,Wu:2019ohx,Alexander:2019vtb,Lu:2019tjj,Hook:2019zxa,Hook:2019vcn}.

To understand the prospects of detection of such heavy particle-induced NG, we can
characterize the strength of NG by a dimensionless parameter, conventionally called $f_{\text{NL}}$ in the literature, whose definition will be given in sec. \ref{setup}. Current CMB constraints on $f_{\text{NL}}$ varies depending on the particular shape of NG under consideration and broadly, is given by $|f_{\text{NL}}|<\mathcal{O}(5-50)$ \cite{Akrami:2019izv}. Upcoming LSS experiments will improve this by having a precision $\sigma_{f_{\text{NL}}}\sim \mathcal{O}(1)$ \cite{Alvarez:2014vva} which will be useful to probe heavy particle-induced NG \cite{MoradinezhadDizgah:2017szk,MoradinezhadDizgah:2018ssw}.
The ultimate sensitivity in this regard will be provided by an only cosmic-variance-limited 21-cm experiment which can roughly achieve $\sigma_{f_{\text{NL}}}\sim 10^{-4}-10^{-3}$ \cite{Meerburg:2016zdz}. Thus we will consider $f_{\text{NL}}\sim 10^{-4}$ as the ultimate limiting strength of NG for observability.  Given this lower bound, with some rare exceptions \cite{Chen:2009zp}, many of the prime targets of the cosmological collider program such as, massive gauge bosons \cite{Chen:2016hrz,Chen:2016nrs,Chen:2016uwp,Kumar:2017ecc,Kumar:2018jxz}, charged scalars and fermions \cite{Chen:2016hrz,Chen:2016nrs,Chen:2016uwp,Kumar:2017ecc,Chen:2018xck,Hook:2019vcn,Hook:2019zxa}, Kaluza-Klein modes of the graviton \cite{Kumar:2018jxz} give rise to small, and sometimes even unobservable, strength of NG in the standard inflationary paradigm where the dynamics of inflation is \textit{explicitly} described in terms of scalar fields. The primary goal of the present work is to describe a simple, alternative paradigm, in which the above mentioned NGs are naturally orders of magnitude larger, and the associated targets can naturally be brought into the scope of the cosmological collider program. Let us first understand what suppresses such NG contributions in the standard paradigm.

The inflaton ($\phi$) is a light field (with mass $\ll H$) during inflation. To ensure that it remains light in the presence of potentially large radiative corrections due to heavy states, one normally imposes a shift symmetry $\phi\rightarrow \phi+c$, with $c$ being a constant, which is broken only weakly by its potential. Under the restriction of such a shift symmetry, the dominant coupling of $\phi$ to a generic operator $\mathcal{O}$ is schematically described within an effective field theory (EFT) framework as,
\begin{align}\label{hd}
    \frac{1}{\Lambda_{\phi}^{2n+\text{dim}(\mathcal{O})-4}}(\partial\phi)^n\mathcal{O},
\end{align}
where $\text{dim}(\mathcal{O})$ is the scaling dimension of the operator $\mathcal{O}$, and $\Lambda_{\phi}$ is a scale by which the EFT description must break down. Strengths of such non-renormalizable couplings are thus characterized by inverse powers of $\Lambda_{\phi}$, and a smaller value of $\Lambda_{\phi}$ implies a larger coupling, leading to a larger strength of NG. So from an observational perspective, it is important to ask how small $\Lambda_{\phi}$ can be while our description still remains in theoretical control. Several possible choices of $\Lambda_{\phi}$ exist, with varying levels of conservatism and assumptions about the ultra-violet (UV) physics. A schematic representation of the relevant scales is shown in fig. \ref{fig:energy-scales}.
\begin{figure}[h]
	\centering
	\includegraphics[width=0.25\linewidth]{energy-scales}
	\caption{Various energy scales discussed in this work. $H$ and $M_{\text{pl}}$ are respectively, the inflationary Hubble scale and the Planck scale. $V_{\text{inf}}^{1/4}$ and $\sqrt{\dot{\phi}_0}$ are respectively the potential and kinetic energy scales of the inflaton field. Similarly, $V_{\sigma}^{1/4}$ and $\sqrt{\dot{\sigma}_0}$ are respectively the potential and kinetic energy scales of the curvaton field. A sample set of values of the above scales can be obtained using the benchmark parameter point given in eq. \eqref{benchmark}.}
	\label{fig:energy-scales}
\end{figure}

Due to quantum gravity effects any EFT description is expected to break down at $M_{\text{pl}}$ and thus $\Lambda_{\phi} \sim M_{\text{pl}}$ can be a reasonable choice, assuming no new physics comes in between $H$ and $M_{\text{pl}}$. With such a high value of $\Lambda_{\phi}$, the strength of NG is small, but can be observable in a few cases \cite{Chen:2009zp,Arkani-Hamed:2015bza,Kumar:2018jxz}. Stronger NG can be obtained by taking $\Lambda_{\phi}\gtrsim
V_{\text{inf}}^{\frac{1}{4}}$ where $V_{\text{inf}}\sim H^2M_{\text{pl}}^2$ is the potential energy density that drives the inflationary expansion. Such a choice of $\Lambda_{\phi}$ is still conservative because $V_{\text{inf}}^{\frac{1}{4}}$ is the highest energy scale available during and after inflation. For example the reheat temperature, $T_R$, during the reheating stage at the end of inflation can be maximally $\sim V_{\text{inf}}^{\frac{1}{4}}$. Thus an EFT with $\Lambda_\phi\gtrsim V_{\text{inf}}^{\frac{1}{4}}$ is capable of describing the universe both during and after inflation.  Keeping this in mind, in this work we will take
\begin{align}\label{lambdaPE}
\Lambda_\phi\gtrsim V_{\text{inf}}^{\frac{1}{4}}>250 H ,
\end{align}
where we have used the Planck constraint \cite{Akrami:2018odb} $H/M_{\text{pl}} < 2.7\times10^{-5}$ on tensor-to-scalar ratio. We will show later that the above restriction \eqref{lambdaPE} implies that the NG mediated by heavy particles in several scenarios of interest are quite small or even unobservable.

We note that if one only demands theoretical control of the series of higher dimensional terms in the EFT which involve an expansion in $\frac{(\partial\phi)^2}{\Lambda_\phi^4}$, a weaker restriction of $\Lambda_{\phi}\gtrsim \sqrt{\dot{\phi}_0}\sim 60 H$
can be obtained \cite{Creminelli:2003iq}, where $\phi_0(t)$ denotes the homogeneous part of the inflaton field and we have used the fact that the scalar power spectrum implies $\frac{H^4}{\dot{\phi}_0^2}\approx 10^{-7}$ \cite{Akrami:2018odb}. However, such low $\Lambda_\phi$ does not explicitly capture the scalar field source of inflation, $V_{\text{inf}}$. We will not pursue this less conservative $\Lambda_\phi$ here.

Even more agnostically, the Goldstone effective theory of inflation \cite{Cheung:2007st}, which describes only the cosmological \textit{fluctuations} during inflation, but not the dynamics driving inflation, can avoid even the restriction $\Lambda_\phi\gtrsim \sqrt{\dot{\phi}_0}$. To ensure control over the description of just the fluctuations which probe energy scales $\sim H$, one need only have the EFT cut off $\Lambda_{\text{fluctuations}}\gtrsim H$. One can then have stronger couplings between the inflationary fluctuations and $H$-mass particles, leading to significantly larger NG than what is obtained by demanding \eqref{lambdaPE}.

This Goldstone description is completely agnostic about whether such low values of $\Lambda_{\text{fluctuations}}$ are consistent with higher scales such as $\sqrt{\dot{\phi}_0}$ and $V_{\text{inf}}^{1/4}$ since the latter scales, which control the homogeneous background, do not even appear in the Goldstone dynamics of inflationary fluctuations. The Goldstone description simply assumes that a suitable homogeneous inflationary background is given.
There do exist subtle mechanisms beyond the Goldstone description (e.g. \cite{Alishahiha:2004eh,Baumann:2011su}) which achieve compatibility between the two sets of scales.
In this work we explore a very simple alternative where such a compatibility is readily obtained. This is achieved by having an explicit separation of the field degrees of freedom responsible for the inflationary background and for seeding the density fluctuations.
We assume that along with the inflaton field, there exists a second light field $\sigma$, the ``curvaton'' \cite{Enqvist:2001zp,Lyth:2001nq,Moroi:2001ct}, whose role is to predominantly give rise to primordial fluctuations (today), whereas the role of the inflaton is reduced to just sourcing the background expansion (with subdominant fluctuations). Then it is completely consistent to have separate EFT cutoff scales for $\sigma$ and $\phi$, which automatically implies separate cutoffs for inflationary fluctuations and the homogeneous background. This can then naturally, but in a controlled way, lead to large NG if the EFT scale characterizing nonrenormalizable $\sigma$ interactions is parametrically smaller than the EFT cutoff of $\phi$ dynamics.

Concretely, one can have $\phi$ and $\sigma$ belong to two different sectors which are sequestered from each other (say, via having different locations in an extra dimension) with each having their own EFT cutoffs, $\Lambda_{\phi}$ and $\Lambda_{\sigma}$ respectively. Since $\sigma$ does not have to drive the background expansion, its energy density $V_\sigma$ can be $\ll V_{\text{inf}}$. Then an argument similar to the one leading to \eqref{lambdaPE}, will only imply
\begin{align}\label{lambdacurv}
\Lambda_{\sigma}\gtrsim V_{\sigma}^{\frac{1}{4}},
\end{align}
while still allowing $\Lambda_{\sigma}\ll V_{\text{inf}}^{\frac{1}{4}} <\Lambda_{\phi}$. Such a scenario with $\Lambda_{\sigma}\ll \Lambda_{\phi}$ can arise in several ways. One possibility is that the scale $\Lambda_{\sigma}$ could represent the masses of new mediator fields which couple to $\sigma$ and $H$-mass particles, but not to $\phi$. An example along this line will be studied in sec. \ref{curvatonparadigm}. Another possibility is that $\Lambda_{\sigma}$ could represent a compositeness/confinement scale for the $\sigma-$sector and/or heavy fields interacting with it. Via the AdS/CFT duality, this latter scenario is dual to an extra-dimensional set-up where $\phi$ and $\sigma$ are localized on two distinct ``branes'' and the extra-dimensional warping (redshifts) between the two branes explains why $\Lambda_{\sigma}\ll \Lambda_{\phi}$.

Since the two sectors are decoupled (up to gravitational effects) and can undergo separate reheating, $\Lambda_{\sigma}\ll V_{\text{inf}}^{\frac{1}{4}}$ does not lead to a break down of the EFT description for $\sigma$ at the end of inflation.
If the $H-$mass particles now couple to $\sigma$, such couplings need only be suppressed by $\Lambda_\sigma$, instead of $\Lambda_\phi$, and thus one can then have orders of magnitude bigger NG of primordial density fluctuations. Compared to the Goldstone EFT of inflation framework, this scenario can successfully describe inflationary and post-inflationary dynamics as will be discussed in sec. \ref{curvatonreview}, giving us an example of a controlled field theoretic scenario having potentially larger NG than the standard paradigm.

This paper is organized as follows. In sec. \ref{setup}, we will set up the notation and discuss some essential aspects of the cosmological collider physics program. We will review the curvaton scenario and note the current set of observational constraints on it in sec. \ref{curvatonreview}. The detailed analysis of the EFT couplings of the heavy particles will then be carried out, both in the standard inflationary scenario, sec. \ref{inflatonparadigm}, and in the curvaton scenario, sec. \ref{curvatonparadigm}. Three types of NG contribution of heavy particles---tree-level effects of spontaneously broken charged scalars (Higgs bosons), and loop-level effects of unbroken charged scalars and charged Dirac fermions will be considered. We will discuss an issue of \textit{classical} tuning that arises in the inflationary paradigm and see that if we forego such tuning, the loop-mediated NG are unobservably small. While enhanced NG signals can be obtained with tuning, we will show that such tunings and enhancements are limited. The curvaton scenario will, on the other hand, have no such tunings but still gives rise to orders of magnitude larger NG compared to the standard inflationary scenario.
In appendices \ref{scalarloop} and \ref{fermionloop} we will calculate the NG contributions explicitly to confirm those statements. We conclude in sec. \ref{conclude}. Throughout this paper we work in the $(-,+,+,+)$ sign convention for the spacetime metric.

As this paper was being completed, ref. \cite{Lu:2019tjj} appeared which discusses how Higgs fluctuations, different from inflaton fluctuations, can source primordial density perturbations through reheating via Higgs-modulated inflaton decay. The curvaton scenario we present here allows a larger enhancement of NG signals compared to ref. \cite{Lu:2019tjj} and also makes the robust prediction of the strength of a ``local'' type of NG, $f_{\text{NL}}^{\text{loc}}=-5/4$, absent in ref. \cite{Lu:2019tjj}. Nevertheless, there are some structural similarities between the present work and ref. \cite{Lu:2019tjj}.

\section{Observables and cosmological collider physics}\label{setup}
We denote the gauge invariant curvature perturbation, which will be defined below, by $\zeta$, and use the primed notation to denote its momentum space correlation functions,
\begin{equation}
\langle \zeta(\vec{k}_1) \cdots \zeta(\vec{k}_n)\rangle  = (2\pi)^3\delta^3(\vec{k}_1+\cdots+\vec{k}_n)\langle \zeta(\vec{k}_1)\cdots\zeta(\vec{k}_n) \rangle^\prime.
\end{equation}
The power spectrum is denoted by,
\begin{align}
    P_k=\langle \zeta(\vec{k}) \zeta(-\vec{k})\rangle^\prime.
\end{align}
The dimensionless three point function parametrizing the strength of NG is defined as,
\begin{align}\label{F}
F\left(k_1,k_2,k_3\right)=\frac{\langle \zeta(\vec{k}_1) \zeta(\vec{k}_2)\zeta(\vec{k}_3)\rangle^\prime}{P_{k_1} P_{k_3}}.
\end{align}
The function $F$ defined above is in general momentum dependent and thus it is conventional in the literature to define an ``amplitude'' of NG in the equilateral limit $k_1=k_2=k_3$,
\begin{equation}
    f_{\text{NL}}=\frac{5}{18}F(k,k,k).
\end{equation}
Using the above definition of $f_{\text{NL}}$ one has a very rough estimate of the precision by which $f_{\text{NL}}$ can be measured in an only-cosmic-variance limited 21-cm experiment. Such a precision is controlled only by the number of modes $N_{\text{21-cm}}$ and is given by,
\begin{align}
    \sigma_{f_{\text{NL}}}\sim \frac{\langle\zeta^3\rangle}{\langle\zeta^2\rangle^{2}}\sim \frac{1}{\sqrt{N_{\text{21-cm}}}}\frac{1}{\zeta}.
\end{align}
Thus using the estimate $N_{\text{21-cm}}\sim 10^{16}$ \cite{Loeb:2003ya}, one has very roughly $\sigma_{f_{\text{NL}}}\sim 10^{-4}-10^{-3}$.

We will be interested in the so-called ``squeezed limit'' of $F$ in eq. \eqref{F} for which $k_3\ll k_1\approx k_2$. In that case, $F$ becomes a function of only $\frac{k_3}{k_1}$. In particular, heavy fields with $M\sim H$, can mediate non-analytic momentum dependence of $F$ of the type,
\begin{equation}\label{smallf}
F_{\text{non-analytic}}\overset{k_3\ll k_1}{=}f_s(M)\left(\frac{k_3}{k_1}\right)^{\Delta_s(M)} P_s(\cos\theta)+ c.c.
\end{equation}
where $\theta=\hat{k}_1\cdot\hat{k}_3$. The functions $\Delta_s(M),f_s(M),P_s(\cos\theta)$ depend on the mass $M$ and spin $s$ of the heavy particle and can be calculated given its coupling to the inflaton. The prospect of extracting the mass and spin of the such heavy fields via measuring $\Delta_s(M)$ and $P_s$, forms the basis of cosmological collider physics. In the following, we will quantify the strength of NG by the absolute value $|f_s(M)|$.

While the time dependent inflationary spacetime readily produces particles with masses $\sim\mathcal{O}(H)$, production of heavier particles are ``Boltzmann suppressed'' with $f_s(M)\sim e^{-\pi M/H}$ for $M\gg H$. Furthermore, for $M\ll H$, $F$ becomes analytically dependent on $k_3$ so that the distinctive non-analytic, on-shell information characterizing heavy-particle mediation is no longer apparent. For example, for a scalar particle, $\Delta_0(M)=3/2\pm i\sqrt{M^2/H^2-9/4}\rightarrow 0 \text{ or }3$ for $M\ll H$. Thus the cosmological collider program operates most efficiently in a window of heavy masses around $\sim H$ to give us on-shell mass and spin information.

\section{Curvaton paradigm}\label{curvatonreview}
\subsection{Cosmological history}
We will now briefly review the cosmological history in the curvaton paradigm and emphasize some of the important differences between it and the standard inflationary paradigm. For more details on the curvaton paradigm, the reader is referred to the original papers \cite{Enqvist:2001zp,Lyth:2001nq,Moroi:2001ct}.

We will model the curvaton field, $\sigma$, as a pseudo Nambu-Goldstone boson (pNGB) whose shift symmetry is broken (softly) by a mass term $m\ll H$,
\begin{equation}\label{curv-pot}
    V_\sigma = \frac{1}{2}m^2\sigma^2.

\end{equation}
Any significant interaction term involving the curvaton and another field will need to respect a shift symmetry $\sigma\rightarrow \sigma+c$ with $c$ being some constant. Furthermore, for simplicity, we will assume that $\phi$ and $\sigma$ belong to two separate sectors sequestered from each other (say, by different locations in an extra-dimensional geometry) and ignore any interaction between them. Thus our model is specified by the lagrangian,

\begin{equation}
\mathcal{L} = -\frac{1}{2}(\partial\phi)^2 - V_{\text{inf}}(\phi) +\mathcal{L}^{\text{int}}_\phi(\partial_\mu\phi,\{\chi\})- \frac{1}{2}(\partial\sigma)^2 - \frac{1}{2}m^2\sigma^2 + \mathcal{L}^{\text{int}}_\sigma(\partial_\mu\sigma,\{\chi\}),
\end{equation}
where $V_{\text{inf}}(\phi)$ is the inflaton potential and $\mathcal{L}^{\text{int}}_{\phi(\sigma)}$ captures the shift-symmetric interactions of the inflaton (curvaton) with a collection of the other heavy fields $\{\chi\}$ that we will specify in sec. \ref{curvatonparadigm}. During inflation, the potential energy is dominated by $V_{\text{inf}}(\phi) \gg \frac{1}{2}m^2\sigma^2$ so that $\phi$ drives the inflationary expansion and $\sigma$ acts as a spectator field.

To describe the fluctuations, we will split both the inflaton and the curvaton fields into homogeneous and fluctuating components: $\phi(t,\vec{x}) = \phi_0(t) + \delta\phi(t,\vec{x})$ and $\sigma(t,\vec{x}) = \sigma_0(t)+\delta\sigma(t,\vec{x})$. The equations of motion (EOM) for $\phi$ and $\sigma$ are decoupled (neglecting gravitational backreaction) and in particular, the homogeneous EOMs are given by \footnote{We will treat $\mathcal{L}^{\text{int}}_{\phi(\sigma)}$ in a perturbative manner so that they do not affect the free EOMs at the leading order.},
\begin{align}
    \ddot{\phi}_0+3H\dot{\phi}_0+V_{\text{inf}}'(\phi_0)=& 0,\\
    \ddot{\sigma}_0+3H\dot{\sigma}_0+m^2\sigma_0=& 0.
\end{align}
Assuming that the kinetic energy of the inflaton is much bigger than that of the curvaton, we get the standard relation,
\begin{equation}
  \epsilon\equiv -\frac{\dot{H}}{H^2}\approx\frac{\dot{\phi}_0^2}{2H^2M_{\text{pl}}^2}.
\end{equation}
Since $m^2\ll H^2$ and $\epsilon \ll 1$, the curvaton rolls very slowly along its potential, satisfying
\begin{equation}\label{sigmadot}
\dot{\sigma}_0 \approx -\frac{m^2}{3H}\sigma_0.
\end{equation}
Curvature fluctuations can be characterized by the gauge invariant quantity $\zeta$ defined by,
\begin{equation}\label{zetadef}
    \zeta = -\psi -H \frac{\delta\rho}{\dot{\rho}_0}.
\end{equation}
In the above, $\psi$ is a spatial metric fluctuation appearing as,
\begin{equation}
    ds^2 = ((1-2\psi)\delta_{ij}+\cdots)a^2(t)dx^i dx^j+\cdots,
\end{equation}
and we have split the density $\rho(t,\vec{x})=\rho_0(t)+\delta\rho(t,\vec{x})$ into a homogeneous and a fluctuation part. For brevity, we have not explicitly written the other scalar, vector and tensor fluctuations.
Since the inflaton dominates the energy density during inflation, the curvature perturbation when the relevant momentum modes exit the horizon, $\zeta_{\text{exit}}$, is sourced only by $\delta\phi$ to a good approximation and thus, in a gauge in which $\psi=0$,
\begin{equation}
\zeta_{\text{exit}} \approx  -H \frac{\delta\phi}{\dot{\phi}_0}.
\end{equation}
One of the important features of the curvaton paradigm is that the fluctuations of the inflaton are subdominant to those of the curvaton, and in particular $\ll 10^{-5}$, the characteristic size of the observed primordial fluctuations. For example, for the benchmark set of parameters given in eq. \eqref{benchmark}, $\zeta_{\text{exit}}\sim \frac{H^2}{2\pi\dot{\phi}_0}\sim 10^{-6}$. However, significant curvature perturbations can get generated after the end of inflation since there is a second light field $\sigma$ during inflation and thus $\zeta$ need not necessarily be conserved on superhorizon scales \footnote{This is to be contrasted with single-field inflation where quite generally $\zeta$ remains conserved on superhorizon scales \cite{Wands:2000dp,Weinberg:2003sw}.}. Let us now see how this happens.

We assume that at the end of inflation, the inflaton reheats into a radiation bath largely decoupled from $\sigma$. In the meantime, $\sigma$ keeps rolling very slowly along its potential until the Hubble scale $\lesssim m$, following which $\sigma$ starts oscillating around its minimum and dilutes like matter. At such a point the content of the universe comprises of radiation coming from the inflaton decay, having energy density $\rho_{\text{rad}}$, and matter due to the curvaton energy density $\rho_\sigma$.
Thus using eq. \eqref{zetadef} and using the gauge $\psi=0$ the curvature perturbation after inflaton reheating can be written as,
\begin{equation}
\zeta = \frac{1}{3}\frac{\delta\rho_\sigma}{\rho_\sigma} f_\sigma + \frac{1}{4}\frac{\delta\rho_{\text{rad}}}{\rho_{\text{rad}}}(1-f_\sigma),
\end{equation}
where $f_\sigma = \frac{3\rho_\sigma}{3\rho_\sigma+4\rho_{\text{rad}}}$ is related to the energy density in the curvaton field compared to the radiation energy density $\rho_{\text{rad}}$. $\frac{\delta\rho_\sigma}{\rho_\sigma}$ and $\frac{\delta\rho_{\text{rad}}}{\rho_{\text{rad}}}$ are respectively fluctuations corresponding to the curvaton and the radiation (in the $\psi=0$ gauge),
and they are conserved on super-horizon scales since the two fluids do not interact with each other, other than via gravity which is weak on these scales. Since the radiation bath originates from the inflaton decay, we will have $\frac{1}{4}\frac{\delta\rho_{\text{rad}}}{\rho_{\text{rad}}}=\zeta_{\text{exit}}$ which however is far subdominant in the curvaton scenario. Now, importantly since radiation dilutes faster than matter, assuming there is sufficient time between the start of curvaton oscillation and its decay, we will reach a stage at which $f_\sigma\approx 1$ when we can write,
\begin{equation}
\zeta \approx \frac{1}{3}\frac{\delta\rho_\sigma}{\rho_\sigma},
\end{equation}
which remains conserved on superhorizon scales subsequently. We assume all the relevant fluids during the later stage of evolution i.e. the SM photon, neutrinos, baryons and dark matter all originate from the decay of the curvaton. This way we do not generate any isocurvature fluctuations at a later stage. The differential evolution between $\sigma-$matter and radiation has converted the initial isocurvature fluctuations in the curvaton field into adiabatic ones.

We will now relate $\frac{\delta\rho_\sigma}{\rho_\sigma}$ to the quantum fluctuations of the curvaton field which will later help us to write expressions for NG of primordial density perturbations.
Since the curvature perturbation is negligible at the end of inflation and we are assuming a mass-only potential for the curvaton, both $\sigma_0$ and $\delta\sigma(t,\vec{x})$ dilutes in an identical way so as to give \cite{Lyth:2001nq,Lyth:2002my},
\begin{equation}
\frac{\delta\rho_\sigma}{\rho_\sigma}=2\frac{\delta\sigma}{\sigma_0} =2 \frac{\delta\sigma}{\sigma_0}\vert_*
\end{equation}
where $*$ denotes the fact that fluctuations are evaluated at the epoch of horizon exit. This then finally gives,
\begin{equation}\label{zetafin}
\zeta_{\text{final}}\approx\zeta_\sigma = \frac{2}{3}\frac{\delta\sigma}{\sigma_0}\vert_*,
\end{equation}
which relates the final curvature perturbation in terms of the quantum fluctuations of the curvaton field. It is in this limit that the adiabatic curvaton fluctuations can be identified as the Goldstone mode for spontaneous time translation breaking in the Goldstone effective theory of inflation \cite{Cheung:2007st}, the inflaton fluctuations having become completely subdominant.
Unless otherwise mentioned, in the following, we will omit the subscript in $\zeta_{\text{final}}$ and simply use $\zeta$ to denote the primordial density perturbations which act as ``initial'' conditions for the modes that subsequently re-enter the horizon after inflation. We are now in a position to note the present observational constraints on this paradigm.

\subsection{Observational constraints}
\paragraph{Scalar power spectrum.}

Due to the fact that $\sigma$ is a light spectator field during inflation,  its fluctuations $\delta\sigma$ acquire an approximately scale invariant spectrum. Thus the scalar power spectrum is given by,
\begin{equation}\label{power}
\langle \zeta(\vec{k}) \zeta(-\vec{k})\rangle'= \frac{4}{9\sigma_0^2}\langle \delta\sigma(\vec{k}) \delta\sigma(-\vec{k})\rangle'
 = \frac{4}{9}\frac{H^2}{2\sigma_0^2k^3},
\end{equation}
where the r.h.s. is evaluated at the time of horizon exit $k=aH$ for a given $k-$mode.
The amplitude of scalar power spectrum from Planck data \cite{Akrami:2019izv} then implies,
\begin{equation}\label{sigma0}
\frac{H}{\sigma_0} \approx 4.4\times 10^{-4}.
\end{equation}
\paragraph{Tilt of the scalar power spectrum.}
Defining $\Delta_\mathcal{\zeta} = \frac{k^3}{2\pi^2}\langle \zeta(\vec{k}) \zeta(-\vec{k})\rangle' = \frac{1}{9}\frac{H^2}{\pi^2\sigma_0^2},$
the tilt can be derived as,
\begin{equation}
\frac{d\ln\Delta_\mathcal{\zeta}}{d
\ln k} = -2\epsilon  +\frac{2}{3}\eta_\sigma.
\end{equation}
where $\eta_\sigma = \frac{m^2}{H^2}$ is fixed by the mass of the curvaton and $\epsilon\equiv-\frac{\dot{H}}{H^2}\approx\frac{\dot{\phi}_0^2}{2H^2 M_{\text{pl}}^2}$ is still determined by the homogeneous inflaton field. Planck data \cite{Akrami:2018odb} requires,
\begin{equation}
-2\epsilon  +\frac{2}{3}\eta_\sigma \approx -0.04.
\end{equation}
\paragraph{Tensor-to-Scalar ratio.}
The ratio of the power spectrum of tensor fluctuations to that of the scalar fluctuations, denoted by $r$, is given by,
\begin{equation}
r = \frac{\frac{8H^2}{M_{\text{pl}}^2}}{\frac{4H^2}{9\sigma_0^2}} =  \frac{18\sigma_0^2}{M_{\text{pl}}^2}.
\end{equation}
The upper bound $r<0.06$ from Planck data \cite{Akrami:2018odb} requires
\begin{align}
\sigma_0 < 0.06 M_{\text{pl}}.
\end{align}
\paragraph{Non-Gaussianity.}
A very stringent constraint on the curvaton paradigm comes from the upper bound on the ``local'' type of NG, defined as $\mathcal{\zeta} = \mathcal{\zeta}_g + \frac{3}{5}f_{\text{NL}}^{\text{loc}}\mathcal{\zeta}_g^2$, where $\zeta_g$ is a purely Gaussian field. Here the NG arises due to the fact that square of a Gaussian fluctuation is non-Gaussian. In the scenario when the curvaton dominates the energy density of the universe during before its decay, one can derive \cite{Lyth:2005fi,Bartolo:2003jx,Sasaki:2006kq},
\begin{equation}
    \mathcal{\zeta} = \frac{2}{3}\frac{\delta\sigma}{\sigma} - \frac{1}{3}\left(\frac{\delta\sigma}{\sigma}\right)^2.
\end{equation}
This has precisely the same form as the local type of NG defined above since $\delta\sigma$ is a Gaussian field, and in particular we have \footnote{Note that compared to \cite{Lyth:2005fi} our definition of $f_{\text{NL}}$ differs by an overall sign.},
\begin{equation}\label{fnlcurvaton}
f_{\text{NL}}^{\text{loc}} = -\frac{5}{4}.
\end{equation}
It should be noted that the above value of $f_{\text{NL}}^{\text{loc}}$ is parametrically larger than the slow-roll parameter suppressed $f_{\text{NL}}^{\text{loc}}$ in single-field inflationary models, as dictated by single-field consistency relations \cite{Maldacena:2002vr,Creminelli:2004yq} in the squeezed limit. Thereby in this curvaton scenario, even in the absence of heavy fields that will be considered below, a large $f_{\text{NL}}^{\text{loc}}$ is a tell-tale sign of beyond single-field inflationary dynamics. The above value of $f_{\text{NL}}^{\text{loc}}$ in eq. \eqref{fnlcurvaton} also serves as a crucial difference between the curvaton paradigm and Goldstone description of single-field inflation since $f_{\text{NL}}^{\text{loc}}$ is parametrically suppressed in the latter.

The above constraints can easily be satisfied. For example, one can choose a benchmark set of values:
\begin{equation}\label{benchmark}
    \sigma_0=5\times 10^{-3}M_{\text{pl}};\hspace{1em} H = 2.2\times 10^{-6}M_{\text{pl}}; \hspace{1em}\epsilon=0.02;\hspace{1em}\eta_\sigma=10^{-3},
\end{equation}
to get $r=4.5\times 10^{-4}$.
Although the robust prediction of  $f_{\text{NL}}^{\text{loc}}$ in eq. \eqref{fnlcurvaton} lies below the Planck upper bound on NG, quite excitingly, such a strength of NG will soon be tested by upcoming LSS observations \cite{Alvarez:2014vva}.

While it is true that the signal in the squeezed limit is dominated by the curvaton itself, the distinctive signaures of the heavy fields are imprinted in characteristic non-analytic ``oscillations'' in the squeezed limit as explained in eq. \eqref{smallf}. Multifield inflationary models having additional particles with masses $\ll H$ can not give such non-analytic momentum dependence. The observability of such oscillatory signals have been investigated in the literature in the context of single-field inflation, for example, in Refs. \cite{MoradinezhadDizgah:2017szk,MoradinezhadDizgah:2018ssw,Meerburg:2016zdz}. We expect that with some adaptations the above studies continue to be applicable in our case, but a detailed investigation lies beyond the scope of the present paper.

The fact that in the curvaton paradigm the background inflationary expansion and the (eventual) primordial density perturbations are sourced by two different fields, opens up an interesting possibility. In particular, during inflation both the kinetic and the potential energy stored in $\sigma$ can be much smaller than the kinetic and the potential energy stored in $\phi$, as can be checked by using the benchmark point in eq. \eqref{benchmark}. As will be explained below, this feature can allow significantly stronger coupling of some new degrees of freedom to $\sigma$ than to $\phi$ in light of  non-renormalizabality and shift symmetry of the couplings. We will illustrate this by considering the coupling of a charged scalar, with and without Higgsing, to both $\phi$ and $\sigma$, and the case of a Dirac fermion to both $\phi$ and $\sigma$.

\section{Charged heavy particles in the standard inflationary paradigm}\label{inflatonparadigm}
Since $\phi$ is a light field, we can model it as a pNGB in the low energy EFT, just like the curvaton, and impose a shift symmetry $\phi\rightarrow \phi+c$ which is broken only by its potential. This implies that the interaction of the inflaton will be characterized predominantly by a derivative expansion in $\frac{(\partial\phi)^2}{\Lambda_\phi^4}$ where $\Lambda_\phi$ is the EFT cutoff in the inflationary sector. As discussed in the introduction, theoretical control of such an expansion implies,
\begin{equation}\label{inflatonEFT}
\Lambda_\phi > \sqrt{\dot{\phi}_0} \sim  60H.
\end{equation}
A stronger restriction on $\Lambda_\phi$ can be placed if we demand that the EFT explicitly describes the scalar/gravity dynamics of inflation and reheating. All known descriptions of this refer to an inflaton potential as the source of inflationary expansion. Thus the control of such an EFT requires \eqref{lambdaPE},
\begin{align}\label{Vinf}
    \Lambda_\phi > V_{\text{inf}}^{\frac{1}{4}}>250 H.
\end{align}
In the following we will keep only the restriction in eq. \eqref{lambdaPE} in mind while considering the strengths of NG.

The leading coupling of the inflaton to a scalar field $\chi$, charged under some gauge/global symmetry group, is given by a dimension-6 operator
\begin{align}\label{inf-higgs}
\mathcal{L}\supset\frac{1}{\Lambda_\phi^2}(\partial\phi)^2 \chi^\dagger \chi.
\end{align}
This term will also contribute to the mass of $\chi$ since,
$\frac{1}{\Lambda_\phi^2}(\partial\phi)^2 \chi^\dagger \chi\supset -\alpha\chi^\dagger \chi$ where $\alpha = \frac{\dot{\phi}_0^2}{\Lambda_\phi^2}$ is approximately constant in slow-roll inflation. In the presence of a ``bare'' mass $m_\chi$ and a quartic coupling $\lambda_\chi$, the lagrangian for $\chi$ then reads as,
\begin{align}
\mathcal{L}\supset -|\partial\chi|^2-(m_\chi^2+\alpha)\chi^\dagger \chi -\lambda_\chi (\chi^\dagger \chi)^2,
\end{align}
where the effective mass for $\chi$ is given by,
\begin{equation}\label{masstuning1}
m_{\chi,\text{eff}}^2=m_{\chi}^2+\alpha.
\end{equation}
Two scenarios arise depending on the sign of $m_{\chi,\text{eff}}^2$.

\subsection{Higgs exchange in the broken phase}
We first discuss the case when $m_{\chi,\text{eff}}^2 < 0$, leading to a Higgsing of the symmetry. Due to spontaneous symmetry breaking, one can now have inflationary couplings which are linear in the heavy field. Consequently, one can have tree level processes that can mediate NG and since such processes do not have the usual $\sim\frac{1}{16\pi^2}$ loop suppression, the associated NG are more readily observable. We consider a $U(1)$ symmetry group for simplicity. Given the coupling in eq. \eqref{inf-higgs}, one can expand $\chi = (0, \frac{1}{\sqrt{2}}(v+\tilde{\chi}))$ in the unitary gauge, to read off the vertices necessary for tree level NG. In the above, $v$ and $\tilde{\chi}$ are respectively the VEV and fluctuations of the Higgs field.

The details have been discussed in \cite{Kumar:2017ecc} and the summary is that one can have three types of diagrams giving rise to NG as shown in fig. \ref{fig:diagrams}.
\begin{figure}[h]
	\centering
	\includegraphics[width=0.8\linewidth]{diagrams.png}
	\caption{Massive Higgs mediated (in red) tree level ``in-in'' contributions to the inflaton (in black) three point function. Depending on the number of massive scalar propagators, these diagrams are labelled from left to right: (a) single exchange diagram, (b) double exchange diagram, (c) triple exchange diagram. $\eta$ denotes conformal time which ends at the end of inflation.}
	\label{fig:diagrams}
\end{figure}

The rough strengths of NG, in terms of $f$ defined eq. \eqref{smallf}, corresponding to each of the diagrams are \cite{Kumar:2017ecc},
 \begin{align}\label{treehiggsng}
|f_{\chi,\text{ single, tree}}|\sim\frac{\rho_1^2}{H^2}; \hspace{2em} |f_{\chi,\text{ double, tree}}|\sim\frac{\rho_1^2\alpha}{H^4}; \hspace{2em} |f_{\chi,\text{ triple, tree}}|\sim\frac{\rho_1^2\alpha}{H^4},
 \end{align}
where $\rho_1 = \frac{2\alpha v}{\dot{\phi}_0}$ denotes the quadratic mixing between $\phi$ and $\tilde{\chi}$.

For observable strengths of NG, we need the masses of the heavy particles $\sim H$, otherwise the cosmological production of the massive particle will be severely Boltzmann suppressed. Eq. \eqref{masstuning1} then implies that in the absence of any \textit{classical} tuning between $m_\chi^2$ and $\alpha$ we need to have both $\alpha\sim m_\chi^2\sim H^2$, in which case all three diagrams will give similar strength of NG, $|f_{\chi,\text{ natural, tree}}|\sim \rho_1^2/H^2$. Since a quadratic mixing between $\phi$ and $\tilde{\chi}$ also gives rise to a correction to the scalar power spectrum $k^3 P_k\sim \frac{H^4}{\dot{\phi}_0^2}(1+\mathcal{O}(\rho_1^2/H^2))$, we will require $\rho_1^2/H^2\lesssim 0.1$ for perturbativity of such corrections. Then we see that NG contributions are given by,
\begin{align}
    |f_{\chi,\text{ natural, tree}}|\lesssim 0.1.
\end{align}
The natural choice of $\alpha\sim H^2$ implies $\Lambda > V_{\text{inf}}^{\frac{1}{4}} > \sqrt{\dot{\phi}_0}$, also ensuring a controlled EFT description.

From eq. \eqref{treehiggsng} it is clear that by choosing a larger value of $\alpha$ and consequently fine tuning it against $m_\chi^2$ to obtain $m_{\chi,\text{eff}}^2\sim H^2$, a larger strength of NG can be obtained. However, one can not do this tuning to more than a percent level since the (slow) time evolution of $\alpha = \frac{\dot{\phi}_0^2}{\Lambda^2}$ will generically push $m_{\chi,\text{eff}}^2$ away from its tuned value $\sim H^2$ in a few Hubble times.

\subsection{Charged scalar exchange in the symmetric phase}
Here we assume $m_{\chi,\text{eff}}^2>0$ so that there is no spontaneous symmetry breaking and $\chi$ mediated NG appear only via loop diagrams.
The coupling to the inflaton is described by the same operator as above, namely $\frac{1}{\Lambda_\phi^2}(\partial\phi)^2 \chi^\dagger \chi$. With the symmetry being unbroken the relevant couplings between the inflaton and $\chi$ are given by,
\begin{equation}\label{inf-Higgs-coupling}
  \mathcal{L}_{\phi-\chi}\supset\frac{1}{\Lambda_\phi^2}(\partial\phi)^2 \chi^\dagger \chi = \left(-\alpha - \frac{2\alpha}{\dot{\phi}_0} \dot{\delta\phi}+\frac{\alpha}{\dot{\phi}_0^2}(\partial(\delta\phi))^2\right)\chi^\dagger \chi,
\end{equation}
with an effective $\chi$ mass given by eq. \eqref{masstuning1} and $\alpha=\frac{\dot{\phi}_0^2}{\Lambda_{\phi}^2}$.

Based on the couplings given in eq. \eqref{inf-Higgs-coupling}, there are two loop diagrams that can contribute to a three point function which we list in fig. \ref{fig:scalar-loop}.
\begin{figure}[h]
	\centering
	\includegraphics[width=0.7\linewidth]{loop-diagrams.png}
	\caption{Massive charged particle mediated (in red) loop level ``in-in'' contributions to the inflaton (in black) three point function. Depending on the number of massive charged particle propagators, these diagrams are labelled from left to right: (a) double exchange diagram, (b) triple exchange diagram. $\eta$ denotes conformal time which ends at the end of inflation.}
	\label{fig:scalar-loop}
\end{figure}
The associated NG can be estimated, in terms of $f$ defined eq. \eqref{smallf}, as
 \begin{align}\label{loopscalar}
|f_{\chi,\text{ double, loop}}|\sim\frac{1}{16\pi^2}\frac{\alpha^2}{\dot{\phi}_0^2}; \hspace{2em}
|f_{\chi,\text{ triple, loop}}|\sim\frac{1}{16\pi^2}\frac{\alpha^3}{H^2\dot{\phi}_0^2}.
 \end{align}
The above discussion of the ``classical'' tuning also applies here and in the natural case i.e. when $\alpha\sim H^2$, eq. \eqref{loopscalar} gives,
\begin{equation}\label{brokenscalrinf}
	|f_{\chi,\text{ natural, loop}}|\sim \frac{1}{16\pi^2}\frac{H^4}{\dot{\phi}_0^2}\sim 10^{-9},
\end{equation}
where we have used the fact that $\frac{H^4}{\dot{\phi}_0^2}\sim 10^{-7}$. Such a strength of NG is unobservably small.
Note already with $\alpha\sim H^2$, $m_{\chi,\text{eff}}^2$ receives $\mathcal{O}(1)$ ``contamination'' from the inflationary background and a measurement of $m_{\chi,\text{eff}}^2$ via NG would not have given us the underlying value of the ``pure'' mass $m_\chi^2$. For $\alpha \ll H^2$, such a contamination is small but the strength of NG becomes even smaller.

\subsection{Charged Dirac fermion}
We will consider a charged Dirac fermion coupled to the inflaton via a dimension-7 operator
$\frac{1}{\Lambda_\phi^3}(\partial\phi)^2 \bar{\Psi}\Psi$. A dimension-5 operator of type $\frac{1}{\Lambda_\phi}\partial_\mu\phi\bar{\Psi}\gamma^\mu\Psi$ can be eliminated by integration by parts and current conservation if the $\Psi$ couplings respect a $U(1)$ symmetry.

There could exist another dimension-5 operator involving the axial current, $\frac{\partial_\mu\phi\bar{\Psi}\gamma^\mu\gamma^5\Psi}{\Lambda_\phi}$ if it is not forbidden by parity. Such a coupling is special since it gives rise to an effective ``chemical potential'' $\lambda = \dot{\phi}_0/\Lambda_\phi$ for the fermion $\Psi$ once the inflaton is set to its background value \cite{Chen:2018xck,Hook:2019zxa,Hook:2019vcn}. It can help production of $\Psi$ even when the fermion mass, $m_\Psi$ is somewhat heavier than $H$ without paying significant Boltzmann suppression.
However, if we impose the restriction in \eqref{lambdaPE} i.e. $\lambda\lesssim 15 H$, and demand theoretical control of the calculation of NG in the squeezed limit, which forces $\frac{k_3}{k_1}<\frac{H}{\lambda}$ \cite{Hook:2019zxa}, we find $F\lesssim \text{few}\times 10^{-4}$ for the function defined in eq. \eqref{F}. We will see in the next section that the curvaton scenario will allow a larger strength of NG with just the analog of the dimension-7 operator defined above. Furthermore, in the regime where there is a substantial NG signal due to $\lambda$, the inflationary background also significantly contaminates the ``pure'' fermion mass $m_\Psi$, so that the non-analytic signatures are predominantly sensitive to $\lambda$, not $m_\Psi$.  Because of these reasons we will not consider the dimension-5 operator $\frac{\partial_\mu\phi\bar{\Psi}\gamma^\mu\gamma^5\Psi}{\Lambda_\phi}$ further.

In the coupling $\frac{1}{\Lambda_\phi^3}(\partial\phi)^2 \bar{\Psi}\Psi$, the VEV of the inflaton will give a contribution to the fermion mass as for charged scalars, and we can write the relevant couplings as,
\begin{equation}\label{inf-fermion-coupling}
    \mathcal{L}\supset\frac{1}{\Lambda_\phi^3}(\partial\phi)^2 \bar{\Psi}\Psi = \left(-\beta - \frac{2\beta}{\dot{\phi}_0} \dot{\delta\phi}+\frac{\beta}{\dot{\phi}_0^2}(\partial(\delta\phi))^2\right)\bar{\Psi}\Psi,
  \end{equation}
where $\beta = \frac{\dot{\phi}_0^2}{\Lambda_\phi^3}$. Thus the effective fermion mass becomes,
\begin{equation}\label{masstuning2}
m_{\Psi,\text{eff}}=m_{\Psi}+\beta.
\end{equation}
Naturalness requires $\beta\sim H$. Furthermore, even if we choose to tune, the requirement of $\Lambda_\phi > \sqrt{\dot{\phi}_0}$ already implies,
\begin{equation}
    \beta < \sqrt{\dot{\phi}_0}.
\end{equation}
Focusing on the natural case we can estimate the strength of NG for a fermion-mediated loop process given in fig. \ref{fig:scalar-loop}, where the internal lines now represent $\Psi$, in terms of $f$ defined eq. \eqref{smallf},
\begin{align}\label{loopfermion}
|f_{\Psi,\text{ double, loop}}|\sim\frac{1}{16\pi^2}\frac{\beta^2H^2}{\dot{\phi}_0^2}; \hspace{2em}
|f_{\Psi,\text{ triple, loop}}|\sim\frac{1}{16\pi^2}\frac{\beta^3H}{\dot{\phi}_0^2}.
 \end{align}
We see in the natural case i.e. $\beta \sim H$ such NG are again unobservably small with
\begin{equation}\label{fermioninf}
	|f_{\Psi,\text{ natural, loop}}|\sim 10^{-9}.
\end{equation}
As in the case of the charged scalar, if we demand the ``contamination'' to $m_{\Psi,\text{eff}}$ from the inflationary background to be small, the NG becomes even smaller.

Having discussed the case with inflationary couplings, let us see how things change when the heavy charged scalar and fermion fields are coupled to a curvaton instead of the inflaton as the dominant source of fluctuations.
\section{Charged heavy particles in the curvaton paradigm}\label{curvatonparadigm}
In the inflationary scenario, it is the same field $\phi$ that drives the inflationary expansion and also sources the observed primordial fluctuations. That is why the very high scales such as $\dot{\phi}_0^{1/2}$ and $V_{\text{inf}}^{1/4}$ inversely bound the strength of EFT couplings, making the associated NG small. However, in the curvaton scenario, $\sigma$ sources the observed fluctuations and $\phi$ drives the inflationary expansion. In particular, assuming $\phi$ and $\sigma$ belong to two different sectors, sequestered from each other, the scales $\dot{\phi}_0^{1/2}$ and $V_{\text{inf}}^{1/4}$ need not even be relevant for the couplings between $\sigma$ and heavy fields. The only relation that is relevant for control of the curvaton EFT is,
eq. \eqref{lambdacurv},
\begin{align}
    \Lambda_\sigma > V_\sigma^{\frac{1}{4}},
\end{align}
where $V_\sigma$ is the energy density in the curvaton.
To see how big $V_\sigma$ can be we write,
\begin{equation}\label{Vsigma}
V_\sigma= \frac{1}{2}m^2\sigma_0^2 \approx  2.5\times 10^6\times \eta_\sigma H^4,
\end{equation}
where we have used eq. \eqref{sigma0} and $\eta_\sigma = \frac{m^2}{H^2}$.
We then see that for the benchmark choice of $\eta_\sigma\sim 10^{-3}$ discussed in sec. \ref{curvatonreview} allows us to have using eq. \eqref{lambdaPE},
\begin{align}
\Lambda_\phi\gtrsim 250 H \gg \Lambda_\sigma \gtrsim V_\sigma^{1/4}\sim 10 H
\end{align}
This relation $\Lambda_\phi \gg \Lambda_\sigma \gtrsim V_\sigma^{1/4} $ plays a central role in giving significantly larger NG in the curvaton paradigm while also ensuring theoretical control of the set-up.
Compared to the standard inflationary paradigm, in the curvaton scenario there are no \textit{classical} tunings so long as we make sure $\dot{\sigma}_0\lesssim H^2 < \Lambda_\sigma^2$. As an example, the contribution of the curvaton to the mass term of the charged scalar field $\chi$,
$\frac{(\partial\sigma)^2}{\Lambda_\sigma^2}\chi^\dagger\chi\supset -\frac{\dot{\sigma}_0^2}{\Lambda_\sigma^2}\chi^\dagger\chi$ remains small for the above choices of $\dot{\sigma}_0$ and $\Lambda_\sigma$ and correspondingly there is no ``contamination'' to the ``pure'' scalar mass $m_\chi$.

\paragraph{Constraints from curvaton self-interaction mediated NG.}
Given the derivative expansion, we can also write a term $\frac{1}{\Lambda_\sigma^4}(\partial\sigma)^4$. This will contribute to both bispectrum and trispectrum. For bispectrum, the relevant couplings are,
\begin{equation}
\frac{1}{\Lambda_\sigma^4}(\partial\sigma)^4 = 4\frac{\dot{\sigma}_0}{\Lambda_\sigma^4}\dot{\delta\sigma}^3 -4\frac{\dot{\sigma}_0}{\Lambda_\sigma^4}\frac{1}{a(t)^2}\dot{\delta\sigma}(\partial_i\delta\sigma)^2+\cdots.
\end{equation}
From such a coupling we can do a naive estimate of curvaton self-interaction mediated NG, in terms of $f$ defined eq. \eqref{smallf},
\begin{equation}
    |f_{\text{curvaton, self-int.}}|\sim \frac{\dot{\sigma}_0 \sigma_0 H}{\Lambda_\sigma^4}.
\end{equation}
Thus with $\dot{\sigma}_0\sim H^2, \Lambda_\sigma\sim 4 H$, a choice which will be justified below, and using eq. \eqref{sigma0} we get, $f_{\text{curvaton, self- int.}}\sim 10$. This is below the current upper bound on orthogonal and equilateral type of NG which is the kind of NG induced by the above self-interaction. Doing a more careful calculation shows that $f_{\text{curvaton self int.}}$ is actually smaller than the above crude estimate, so that a choice of $\Lambda_\sigma\gtrsim 4 H$ is more than sufficient to avoid current constraints. With such a choice of $\Lambda_\sigma$, the trispectrum is also smaller than the current bound \cite{Akrami:2019izv}.
\paragraph{Cut-off of the effective theory and field range of the curvaton.}
With the above choice of $\eta_\sigma\sim 10^{-3}$, the requirement $\Lambda_\sigma > V_\sigma^{\frac{1}{4}}$ reduces to $\Lambda_\sigma \gtrsim  10 H$. However, even with $\Lambda_\sigma\sim 10 H$, we do still need $\sigma_0\gg \Lambda_\sigma$ to satisfy eq. \eqref{sigma0}. Thus one can ask whether it is problematic to have curvaton field range much bigger than the EFT cut-off, although both the derivative expansion
remains under full control and $\Lambda_{\sigma} > V_\sigma^{\frac{1}{4}}$. This is analogous to the problem of super-Planckian inflaton field range in high scale inflation models, given by the Lyth bound. Thus, one can borrow the mechanisms that are used to create large effective  field ranges, such as axion monodromy or multi-axion alignment. These mechanisms illustrate that having field ranges larger than the EFT cutoff can naturally emerge from controlled UV completions. In particular, following the bi-Axion mechanism \cite{Kim:2004rp,Choi:2014rja,Tye:2014tja,Ben-Dayan:2014zsa,Bai:2014coa,delaFuente:2014aca}, one can imagine having two axionic curvatons with potential,
\begin{equation}
V_{\text{curv}} = V_1\left(1-\cos\left(\frac{N \sigma_1}{f_1}+\frac{\sigma_2}{f_2}\right)\right) + V_2\left(1-\cos\left(\frac{\sigma_1}{f_1}\right)\right).
\end{equation}
With $V_1\sim V_2$, $f_1\sim f_2$ and $N\gg 1$, there is a heavier curvaton which gets stabilized at $\frac{N \sigma_1}{f_1}+\frac{\sigma_2}{f_2}\approx 0$ so that the effective light curvaton potential is given by,
\begin{equation}\label{curvpotfull}
    V_{\text{eff}} = V_2\left(1-\cos\left(\frac{\sigma_2}{Nf_2}\right)\right).
\end{equation}
We see that the light curvaton $\sigma_2$ has an effective field space $\sim Nf_2$ that is parametrically bigger than the fundamental field space $f_2$ and we can consistently have $Nf_2\gg\Lambda_\sigma\gtrsim f_2$ for a sufficiently large $N$. This UV completion then serves as a proof-of-principle that it is consistent to have field ranges that are significantly bigger than the EFT cutoff scales. The light curvaton $\sigma_2$ can still interact with heavy fields of interest with suppressions given by $\Lambda_\sigma\sim f_2$ rather than $Nf_2$, so that we can have stronger couplings leading to significant NG. Expanding eq. \eqref{curvpotfull} around its minima gives rise to the approximately mass-only potential for the curvaton considered in eq. \eqref{curv-pot} with $\sigma_2=\sigma$ and $m^2 = \frac{V_2}{N^2f_2^2}$.

\paragraph{Mediators and stronger EFT couplings.}
The curvaton will couple to heavy particles via higher dimensional operators and hence the NG will be proportional to multiple powers of $\frac{1}{\Lambda_\sigma}$, instead of multiple powers of $\frac{1}{\Lambda_\phi}$ as in the inflationary scenario. As we have discussed above, the strength of NG in the curvaton scenario can therefore be much stronger since $\Lambda_\sigma\ll\Lambda_\phi$ is consistent with EFT control. However, the restriction of $\Lambda_\sigma> 10 H$ still corresponds to somewhat suppressed NG. We will now show that in the presence of heavier ``mediator'' particles, the effective scale of $\Lambda_\sigma$ can be brought down from $\sim 10 H$ and we can obtain even stronger NG.

For example, one can have the following coupling between the curvaton, the mediator $\Sigma$ and the scalar $\chi$:
\begin{equation}
   \frac{1}{\Lambda_\sigma}(\partial\sigma)^2\Sigma+\mu_\Sigma\Sigma \chi^\dagger \chi.
\end{equation}
Upon integrating out the mediator $\Sigma$, we can get an effective dimension-6 operator,
\begin{equation}
  \frac{\mu_\Sigma}{M_\Sigma^2\Lambda_\sigma}(\partial\sigma)^2 \chi^\dagger \chi,
\end{equation}
which implies an effective cutoff,
\begin{equation}
    \Lambda_{\sigma,\text{eff}}^2 = \frac{M_\Sigma^2 \Lambda_\sigma}{\mu_\Sigma}.
\end{equation}
As an example, for $M_\Sigma = 3H, \mu_\Sigma=6H$ and $\Lambda_\sigma = 10 H$, one gets $\Lambda_{\sigma,\text{eff}} \approx 4 H < \Lambda_\sigma$.
A similar procedure can be repeated for fermionic couplings by starting with,
\begin{equation}
 \frac{1}{\Lambda_\sigma}(\partial\sigma)^2\Sigma+y\Sigma \bar{\Psi}\Psi
\end{equation}
to get (for $y=1$),
\begin{equation}
    \Lambda_{\sigma,\text{eff}}^3 = M_\Sigma^2 \Lambda_\sigma.
\end{equation}
This effective cutoff is again smaller than $\Lambda_\sigma$ for the same choice of $M_\Sigma$ and $\Lambda_\sigma$.

To summarize, demanding theoretical control of the curvaton derivative expansion and $\Lambda_\sigma > V_\sigma^{\frac{1}{4}}$ in the presence of somewhat heavy mediators, gives us $\Lambda_{\sigma,\text{eff}}\gtrsim 4H$. In the following, we will give parametric estimates of NG in terms of $\Lambda_{\sigma,\text{eff}}$ but for numerical results we will take
$\Lambda_{\sigma,\text{eff}} = 4H$. For brevity, in the rest of the paper we will denote $\Lambda_{\sigma,\text{eff}}$ by just $\Lambda_{\sigma}$.

\subsection{Higgs exchange in the broken phase}
The coupling of the curvaton with the charged scalar is given by the dimension-6 operator,
\begin{equation}\label{curv-chi-tree}
    \mathcal{L}_{\phi-\chi}\supset\frac{1}{\Lambda_\sigma^2}(\partial\sigma)^2 \chi^\dagger \chi .
\end{equation}
Expanding around the correct vacuum, $\chi=(0,\frac{1}{\sqrt{2}}(\tilde{\chi}+v))$ as in sec. \ref{inflatonparadigm}, we can get the relevant terms coupling $\tilde{\chi}$ to $\sigma$,
\begin{equation}\label{treescalar2}
\mathcal{L}_{\sigma-\chi}\supset\frac{1}{\Lambda_\sigma^2}(\partial\sigma)^2 \chi^\dagger\chi-\lambda_\chi(\chi^\dagger \chi)^2 \supset \left(-\frac{2\dot{\sigma}_0v}{\Lambda_\sigma^2}\dot{\delta\sigma}\tilde{\chi}-\frac{\dot{\sigma}_0}{\Lambda_\sigma^2}\dot{\delta\sigma}\tilde{\chi}^2+\frac{v}{\Lambda_\sigma^2}(\partial\delta\sigma)^2\tilde{\chi}-\lambda_\chi v \tilde{\chi}^3\right)+\cdots.
\end{equation}
The above couplings give NG mediated by diagrams given in fig. \ref{fig:diagrams} where the external legs represent fluctuations of the curvaton instead of the inflaton. The leading contribution comes from the single exchange diagram for which one can roughly estimate the NG, in terms of $f$ defined eq. \eqref{smallf} as
\begin{align}
|f_{\chi,\text{ tree}}|\sim 3\times 10^3 \times \frac{\rho_2^2}{\dot{\sigma}_0},
\end{align}
where we have denoted the quadratic mixing as $\rho_2 = \frac{2\dot{\sigma}_0v}{\Lambda_\sigma^2}$. Since $\dot{\sigma}_0\sim H^2$, compared to the result in the inflationary paradigm given in eq. \eqref{treehiggsng}, we can have orders of magnitude larger NG in the curvaton scenario, and importantly, \textit{without} any classical tuning.

While the above is a rough estimate, the necessary ingredients for a precise calculation of the single exchange diagram in fig. \ref{fig:diagrams}, can be found in \cite{Arkani-Hamed:2015bza} using which we get,
\begin{align}\label{fscalartree}
&F_{\chi,\text{tree}}\left(m_\chi,\frac{k_3}{k_1}\right)=\frac{\langle \zeta(\vec{k}_1) \zeta(\vec{k}_2)\zeta(\vec{k}_3)\rangle^\prime}{\langle \zeta(\vec{k}_1) \zeta(-\vec{k}_1)\rangle^\prime \langle \zeta(\vec{k}_3) \zeta(-\vec{k}_3)\rangle^\prime}\nonumber\\
&=-\frac{3\sigma_0 \rho_2^2}{8\dot{\sigma}_0 H}
\left(\Gamma(\frac{1}{2} + i \mu)^2 \Gamma(-2 i \mu) (\frac{3}{2} + i \mu) (\frac{5}{2} +
i \mu) (1 + i \sinh(\pi \mu)) \left(\frac{k_3}{k_1}\right)^{\frac{3}{2} + i \mu} + \mu\rightarrow -\mu\right)\nonumber \\
&\equiv |f_{\chi,\text{tree}}(\mu)| \left(e^{i\delta_1(\mu)}\left(\frac{k_3}{k_1}\right)^{\frac{3}{2} + i \mu}+\mu\rightarrow-\mu\right),
\end{align}
where $\mu=\sqrt{m_\chi^2/H^2-9/4}$. In fig. \ref{fig:tree-higgs} we plot the function $|f_{\chi,\text{tree}}|$ which gives the strength of NG as a function of the scalar mass, $m_\chi$.
\begin{figure}[h]
	\centering
	\includegraphics[width=0.6\linewidth]{tree-higgs}
	\caption{The strength of NG for tree level Higgs exchange as a function of Higgs mass $m_\chi$ for $\rho_2=0.3H$ and $\dot{\sigma}_0=-H^2$. The function $|f_{\chi,\text{tree}}(\mu)|$ is defined in eq. \eqref{fscalartree}.}
	\label{fig:tree-higgs}
\end{figure}

\subsection{Charged scalar exchange in the symmetric phase}
The leading coupling is again given by a similar dimension-6 operator,
\begin{equation}\label{curv-higgs}
    \mathcal{L}\supset\frac{1}{\Lambda_\sigma^2}(\partial\sigma)^2 \chi^\dagger\chi.
  \end{equation}
In the absence of symmetry breaking, the relevant curvaton-$\chi$ interaction terms are given by,
\begin{equation}\label{curv-Higgs-coupling}
    \left( - \frac{2\dot{\sigma}_0}{\Lambda_\sigma^2} \dot{\delta\sigma}+\frac{1}{\Lambda_\sigma^2}(\partial(\delta\sigma))^2\right)\chi^\dagger\chi.
\end{equation}
Using eq. \eqref{curv-Higgs-coupling} we can estimate the strengths of NG. The relevant diagrams are still given by fig. \ref{fig:scalar-loop} except the external legs now represent curvaton fluctuations instead of inflaton fluctuations. The leading NG is given by the double exchange diagram in fig. \ref{fig:scalar-loop} whose
parametric strength, in terms of $f$ defined eq. \eqref{smallf}, is given by,
\begin{equation}
|f_{\chi,\text{loop}}|\sim \frac{1}{16\pi^2}\frac{\dot{\sigma}_0 H^2}{\Lambda_\sigma^4}\times\frac{\sigma_0}{H}.
\end{equation}
Since $\Lambda_\sigma \ll \sqrt{\dot{\phi}_0}$, compared to the result in the inflationary paradigm given in eq. \eqref{brokenscalrinf}, we can have orders of magnitude bigger NG in the curvaton scenario, and again, \textit{without} any classical tuning.

The dimensionless three point function, defined in eq. \eqref{F}, due to the double exchange diagram in fig. \ref{fig:scalar-loop} will be calculated in appendix \ref{scalarloop}. The result is given by eq. \eqref{scalarloopF},
\begin{align}\label{fscalarloop}
F_{\chi,\text{loop}}\left(m_\chi,\frac{k_3}{k_1}\right)&=\frac{\langle \zeta(\vec{k}_1) \zeta(\vec{k}_2)\zeta(\vec{k}_3)\rangle^\prime}{\langle \zeta(\vec{k}_1) \zeta(-\vec{k}_1)\rangle^\prime \langle \zeta(\vec{k}_3) \zeta(-\vec{k}_3)\rangle^\prime}\nonumber\\
&=-\frac{3\sigma_0}{2}\frac{2\dot{\sigma}_0 H}{\Lambda_\sigma^4} \left( \frac{\frac{1}{16\pi^5}\Gamma(-i\mu)^2\Gamma(3/2+i\mu)^2}{\frac{1}{4\pi^{5/2}}\Gamma(-3/2-2i\mu)\Gamma(3+2i\mu)}\mathcal{F}\left(3+2i\mu,\frac{k_3}{k_1}\right)+\mu\rightarrow -\mu\right)\nonumber \\
&\equiv |f_{\chi,\text{loop}}(\mu)| \left(e^{i\delta_2(\mu)}\left(\frac{k_3}{k_1}\right)^{3+2i\mu}+\mu\rightarrow-\mu\right),
\end{align}
where $\mathcal{F}$ is defined in eq. \eqref{curlyF} and $\mu=\sqrt{m_\chi^2/H^2-9/4}$.
In fig. \ref{fig:loop-higgs} we plot the function $|f_{\chi,\text{loop}}(\mu)|$ as a function of the scalar mass, $m_\chi$.

\begin{figure}[h]
	\centering
	\includegraphics[width=0.6\linewidth]{Loop-higgs}
	\caption{The strength of NG for loop level scalar exchange as a function of scalar mass $m_\chi$ for $\Lambda_{\sigma}=4H$ and $\dot{\sigma}_0=-H^2$. The function  $|f_{\chi,\text{loop}}(\mu)|$ is defined in eq.  \eqref{fscalarloop}.}
	\label{fig:loop-higgs}
\end{figure}

\subsection{Charged Dirac fermion}
We again consider a charged Dirac fermion, and its coupling to the curvaton is given by a similar dimension-7 operator,
\begin{equation}
    \frac{1}{\Lambda_\sigma^3}(\partial\sigma)^2\bar{\Psi}\Psi.
\end{equation}
As before, there can be a dimension-5 axial coupling $\frac{\partial_\mu\sigma\bar{\Psi}\gamma^\mu\gamma^5\Psi}{\Lambda_\sigma}$ between $\sigma$ and $\Psi$ if it is not forbidden by parity. However, such a coupling does not give rise to a significant chemical potential as in the inflationary scenario, and correspondingly there is no substantial enhancement of NG signals. We leave a detailed study of this coupling for future work, focusing only on the dimension-7 operator in the present work.

Through a reasoning identical to the case of scalars, one can check that the mass correction to the fermion due to the curvaton coupling is negligible with similar choice of parameters as above. The relevant curvaton-fermion interaction terms are given by,
\begin{equation}\label{curv-fermion-coupling}
    \left( - \frac{2\dot{\sigma}_0}{\Lambda_\sigma^3} \dot{\delta\sigma}+\frac{1}{\Lambda_\sigma^3}(\partial(\delta\sigma))^2\right)\bar{\Psi}\Psi.
\end{equation}
Using eq. \eqref{curv-fermion-coupling} we can estimate the strengths of NG for which the relevant diagrams are still given by fig. \ref{fig:scalar-loop}, except the external legs now represent curvaton fluctuations instead of inflaton fluctuations. The leading NG is given by the double exchange diagram in fig. \ref{fig:scalar-loop} whose
parametric strength, in terms of $f$ defined eq. \eqref{smallf}, is given by,
\begin{equation}
    f_{\Psi,\text{loop}}\sim \frac{1}{16\pi^2}\frac{\dot{\sigma}_0 H^4}{\Lambda_\sigma^6}\times \frac{\sigma_0}{H}\vert_*.
\end{equation}
Since $\Lambda_\sigma \ll \sqrt{\dot{\phi}_0}$, compared to the result in the inflationary paradigm given in eq. \eqref{loopfermion}, once again we can have orders of magnitude bigger NG in the curvaton scenario \textit{without} any classical tuning.

The dimensionless three point function, defined in eq. \eqref{F}, due to the double exchange diagram in fig. \ref{fig:scalar-loop} will be calculated in appendix \ref{fermionloop}. The result is given by eq. \eqref{fermionloopF},
\begin{align}\label{ffermionloop}
F_{\Psi,\text{loop}}\left(m_\Psi,\frac{k_3}{k_1}\right)&=\frac{\langle \zeta(\vec{k}_1) \zeta(\vec{k}_2)\zeta(\vec{k}_3)\rangle^\prime}{\langle \zeta(\vec{k}_1) \zeta(-\vec{k}_1)\rangle^\prime \langle \zeta(\vec{k}_3) \zeta(-\vec{k}_3)\rangle^\prime}\nonumber\\
&=-\frac{3\sigma_0}{2}\frac{2\dot{\sigma}_0H^3}{\Lambda_\sigma^6} \left( \frac{-\frac{3}{\pi^5}\frac{\Gamma(1/2-i\tilde{\mu})^2\Gamma(2+i\tilde{\mu})^2}{(1+2i\tilde{\mu})}}{\frac{1}{4\pi^{5/2}}\Gamma(-5/2-2i\tilde{\mu})\Gamma(4+2i\tilde{\mu})}\mathcal{F}\left(4+2i\tilde{\mu},\frac{k_3}{k_1}\right)+\tilde{\mu}\rightarrow -\tilde{\mu}\right)\nonumber\\
&\equiv |f_{\Psi,\text{loop}}(\tilde{\mu})| \left(e^{i\delta_3(\tilde{\mu})}\left(\frac{k_3}{k_1}\right)^{4+2i\tilde{\mu}}+\tilde{\mu}\rightarrow-\tilde{\mu}\right),
\end{align}
where $\mathcal{F}$ is defined in eq. \eqref{curlyF} and $\tilde{\mu} = m_\Psi/H$.
In fig. \ref{fig:loop-fermion} we plot the function $|f_{\Psi,\text{loop}}|$ as a function of fermion mass, $m_\Psi$.

\begin{figure}[h]
	\centering
	\includegraphics[width=0.6\linewidth]{Loop-fermion}
	\caption{The strength of NG for loop level charged fermion exchange as a function of fermion mass $m_\Psi$ for $\Lambda_{\sigma}=4H$ and $\dot{\sigma}_0=-H^2$. The function  $|f_{\Psi,\text{loop}}(\tilde{\mu})|$ is defined in eq.  \eqref{ffermionloop}.}
	\label{fig:loop-fermion}
\end{figure}

\section{Conclusions and future directions}\label{conclude}
In the standard inflationary scenario, the inflaton-heavy particle couplings are suppressed by (at least) inverse powers of an EFT cutoff scale $\Lambda_{\phi}$, which has to satisfy $\Lambda_\phi\gtrsim V_{\text{inf}}^{\frac{1}{4}}>250 H$ if we demand that the EFT explicitly describes the scalar field source of inflation $V_{\text{inf}}$. Such a high value of $\Lambda_\phi$ leads to small or even unobservable strengths of NG in various otherwise well-motivated scenarios of cosmological collider physics.

We have studied a non-standard inflationary paradigm in which the space-time expansion and the predominant production of primordial fluctuations are due to two \textit{different} fields. In particular, we have focused on the curvaton scenario, where the above two roles are played by the inflaton and the curvaton respectively. Heavy particles can then couple to the curvaton suppressed only by inverse powers of $\Lambda_\sigma\gtrsim V_{\sigma}^{\frac{1}{4}}$, where $V_\sigma$ is the energy density contained in the curvaton field which can be as low as $\sim (10 H)^4$ for the benchmark parameter choice considered in this work. With the choice $V_{\sigma}^{\frac{1}{4}}\lesssim\Lambda_\sigma \ll \Lambda_\phi$, the heavy particles can couple to the primordial fluctuations much more strongly and lead to orders of magnitude larger NG while still ensuring controlled EFT description of the inflationary and reheating dynamics. To illustrate this fact, we have considered NG mediated by charged scalars, both in the Higgs phase and in the unbroken phase, as well as charged Dirac fermions. In particular, we have shown that even loop-level NG effects are observable!

Several future directions remain open. We have considered a scenario in which the curvaton and the inflaton belong to two decoupled sectors (up to gravitational effects), which can naturally be achieved, for example, by having an extra dimension in which the two fields are localized on two different branes. Since the presence of Kaluza-Klein (KK) gravitons is a robust feature of such an extra-dimensional set-up, it would be interesting to see whether appreciable KK-graviton mediated NG can be generated using our scenario. In particular, this may be applicable to the strongly motivated case of orbifold unification \cite{Kawamura:1999nj,Kawamura:2000ev,Hall:2002ea} along the lines of \cite{Kumar:2018jxz}.

We have restricted our attention to bispectra of primordial scalar fluctuations which required us to use a non-zero value of $\dot{\sigma}_0$ in the ``in-in'' diagrams. Demanding $\dot{\sigma}_0\sim H^2$ then implied $V_\sigma\sim (10H)^4$ via eqs. \eqref{Vsigma}, \eqref{sigmadot} and \eqref{sigma0}. Considering trispectra (four-point correlation functions) on the other hand, such insertions of $\dot{\sigma}_0$ are not essential, and consequently one can have even smaller values of $V_\sigma$. This will then allow smaller $\Lambda_\sigma$ and larger NG. It would also be very interesting to analyze all the ``heavy-lifted'' SM signals \cite{Chen:2016hrz,Chen:2016nrs,Chen:2016uwp,Kumar:2017ecc}, especially the signal due to the loops of massive $W$ bosons, in the curvaton scenario that was presented in this work.

\section*{Acknowledgements}
The authors would like to thank Anson Hook for helpful discussions. This research was supported in part by the NSF grants PHY-1620074 and PHY-1914731, and by the Maryland Center for Fundamental Physics (MCFP). RS acknowledges the hospitality of the Kavli Institute for Theoretical Physics, UC Santa Barbara, during the ``Origin of the Vacuum Energy and Electroweak Scales'' workshop, and the support by the NSF grant PHY-174958.


Foreach by reference does not honour scope.
And they were closed with bogus, saying to look at the manual.
Well, the manual doesn't talk about this issue at all, infact it seems to say that it should work as expected.
Also, that var should not exists beyond the scope of the foreach loop, and the developer that closed the bug seems to agree (he said for whatever weird reason, and honestly, php should cater to the normal reasons, not the weird ones, and anyone can get the same responce out of setting it manually in the foreach themselves, rather then allowing the engine to do something weird).
In anycase, this should at the very least get fixed up in the documentation, but I still consider this a engine bug.
in bug 29992 read the "gardan at gmx dot de" comment for an explanation about why this is not bug.
I also consider this a bug, here's a simpler test case. I consider an unset() should be executed on the "value" variable before starting the foreach.


text:
Orchard Bros: Serving the West Wight through five generations
More from Features
Tuesday, October 6th, 2020 10:35am
By Amy Shephard @words_by_amy
Originally built as a private residence in 1845, Freshwater Bay's Alexandra House has been in the hands of the Orchard family for more than 150 years.
Becoming a family grocers in 1865, the business has been operated by five generations of Orchards and to this day, continues to be a thriving local business, loved and visited by many.
The fascinating history of this iconic Isle of Wight business spans a century and a half. Originally a branch of Misslebrook and Weston, grocers, based in Newport, the premises were rented to William Orchard from 1865 for the annual rent of £50.
In 1867, William Orchard had the opportunity to purchase the business outright when it was put up for auction at Murrow's Hotel, Freshwater Bay – now known as the Albion Hotel. With the help of his son, 23 year old William Thomas Orchard, the Orchards started to expand the business and so it was named 'Orchard and Son'.
By 1870 the population of Freshwater Bay had grown to such an extent that demand existed for a local post office. Pioneer photographer, Julia Margaret Cameron, wrote to the Post Office authorities and recommended William Thomas Orchard as a suitable candidate for local Postmaster.
"Mr Tennyson's correspondence is very large as you know & my correspondence averages 300 letters every month, my fortnightly correspondence to Ceylon being about 10s to 12s every month. This I tell you to convince you that it is important to us to have a central and convenient Post Office," she wrote.
"Mr Orchard the other man who applies is at the very foot of Mr Tennyson's Park close also to me and close also to the two large hotels close also to the lodging houses at the Bay & Mr Tennyson himself & I & the landlords of the two hotels all signed a paper to request that Mr Orchard of Freshwater might be appointed Postmaster as he is a very respectable & businesslike man. He has a separate room & hall at Alexandra House fitted for the reception of letters & his locality is the most desirable."
The 1871 Census return for Alexandra House gives an idea of the state of the business during the late Victorian era. William and Mary Orchard were living on the premises, along with their son, William Thomas, a grocer's assistant, a telegraph clerk, a baker, a general servant and a domestic servant.
In 1882, the shop obtained a licence to sell wine, spirits and beer, with various types of drink bottled on the premises. Around this time, the family were also acting as coal merchants and house letting agents. By 1883, Orchard & Son became agents of Capital and Counties Bank, later part of Lloyds Bank.
The third generation of Orchards entered the business in 1887, when the founder's grandson, another William, left school in London at the age of 15 and headed back to the Isle of Wight to help run the family business. His grandfather was seriously ill and died soon after, aged 72. Will started work in the post office, learning Morse code to send and receive telegrams and delivering them all over the West Wight. In his memoirs he said: "Before I was 16 years of age I was sending and receiving telegrams on the Morse Printer...When a lad I had sometimes to deliver telegrams – have walked over to Brook House lots of times , and several times to Pitt Place, Brighstone – which was our longest delivery – a boy and cart were used sometimes but this was not always available and you often had to walk, no cycles except the penny farthings and a small lad couldn't ride them – Can you see a boy nowadays walking to deliver telegrams six miles - and return?"
Founder, William Orchard
At the age of 26, Will became postmaster, succeeding his father, William Thomas – and incredibly remained in the post until his retirement in 1954 at the age of 82!
In his memoirs he recalled handling telegrams between Queen Victoria at Osborne House near East Cowes and the poet Tennyson at nearby Farringford House:
"I remember receiving quite a number of telegrams over the wire from Queen Victoria to the poet – they were always signed by Minnie Cochrane, or one of the Gentlemen in waiting and usually they were to ask Tennyson to lunch or dine. Only one, I remember, was from the Queen herself direct – it was just after the rather sudden death of the Duke of Clarence [1892] and was as follows – 'Thank you for your beautiful lines on the death of my dear Grandson' – Victoria R.I."
William Thomas, the founder's son, died age 64 in 1908 and his sons Will and Fred inherited the business. Will ran the post office and the house letting agency whilst Fred ran the grocery business. The shop was renamed 'Orchard Bros', a name that has continued to this day. Fred died aged just 53 in 1927, leaving the business in the sole hands of Will.
Around 1933, the fourth generation entered the business. Fred's son, Dick Orchard, was to remain actively involved in the business up to his death in 2005, apart from his years of service in the Royal Navy during the Second World War.
Dick and Barbara Orchard
During the 1930s the old stables were converted into a new bake-house as motor vehicles took over the deliveries from horse- drawn vehicles. Two bakers were employed and Orchards supplied a wide range of breads, cakes and pastries including wedding and birthday cakes to order. The steam ovens were the most modern on the Island.
In 1953, after the Second World War, Will was awarded the Coronation Medal in recognition of his long service in the post office - some sixty four years. He retired the following year in 1954, aged 82, and the post office was transferred from Orchard Bros to Woolf and Geer's stationery shop across the road.
Dick Orchard took over the running of the business and when his five children were old enough his wife Barbara joined him at his side.
Current proprietor and Dick Orchard's son, Mark joined the family business straight from school in 1968; the fifth generation to be involved in the running of the shop.
Dick and Mark Orchard
Today, Orchard Bros offers a wide range of everyday products including groceries, fresh fruit and vegetables, beers, wines and spirits, confectionery, newspapers and National Lottery tickets. Orchards also offer local delivery and customer and business accounts and will make up Christmas hampers to order.
Their renowned home- cooked ham is regularly commended across the Isle of Wight, along with other local produce which is available to purchase.
Orchards Bros to this day provides a friendly, personal service to the local community – something which has been their hallmark for the past 155 years.
Orchard Bros is situated at the junction of Victoria Road and Bedbury Lane, near the thatched church of St Agnes, about three miles south of Yarmouth. www.orchardbros150.com
Orchard Bros in around 1930

redpajama_set_name: RedPajamaCommonCrawl


Follow this list to discover and track the stocks that were sold the most by activist hedge funds in the last quarter.
Virtual reality can be a rush to the senses, but not if you have vision problems. Microsoft researchers are trying to solve that problem via a tool kit for Unity VR developers to help players with low vision. Since vision problems vary by individual, the research team at Microsoft decided on a tool kit approach so players can customize their experience.
This post was done in partnership with Wirecutter. When readers choose to buy Wirecutter's independently chosen editorial picks, it may earn affiliate commissions that support its work. Read Wirecutter's continuously updated list of deals here.
Is Disney Spending Enough on Disney+ Original Content?
One analyst thinks the entertainment giant's plan will fall short of its subscriber goal.
Sometimes, the market gives up on businesses that still have some life left in them.
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text:
No profiting from hate: Online bookstores pull manifesto by mass shooter Breivik after outcry
22 Mar, 2019 13:54
The cover of Breivik's manifesto / Andres Breivik © Global Look Press/ZUMAPRESS.com/Alexander Widding
At least two online bookstores stopped selling the manifesto compiled by Norwegian mass murderer Anders Breivik, who reportedly inspired the mosque massacres in Christchurch, New Zealand last week.
Breivik, an extremist white supremacist, is currently serving a maximum 21-year sentence for slaughtering 77 people and injuring over 300 in an act of political violence. He timed his 2011 bomb and gun attack with the release of an anti-Muslim manifesto explaining his motivations. The 1,518-page text is mostly a compilation of what other people wrote and Breivik liked, and is available freely online.
Until recently, one could also order a hard copy of the manifesto from Amazon. This was the case until the British news outlet the Times highlighted this fact on Sunday in an accusatory report. A text by Dylann Roof, the white supremacist who murdered nine people at a black church in Charleston in 2015, was also available for purchase, the newspaper said, accusing the international giant of profiting from hate literature.
Also on rt.com Norway appeals Breivik 'inhuman treatment' ruling: How does his prison regime compare?
"Online retailers – just like social media companies – need to stop playing into the hands of terrorists by giving them the notoriety they crave and even selling their so-called manifestos," the Times was told by Brendan Cox, the husband of Labour MP Jo Cox, who was murdered by a white supremacist in June 2016. "Too many corporates are actively making future attacks more likely."
Amazon did not respond to a request for comment, but pulled down the controversial products soon after the story was published.
Scandinavian book retailer Adlibris was also found selling Breivik's manifesto by Norwegian broadcaster NRK. Adlibris quickly withdrew the three-volume book and explained that the text showed up among its products due to an automated publication process. The product was created through a self-publishing website and was not pre-moderated.
"As a rule, we read the incoming titles. Of course, we cannot control all 15 million titles, but we actively remove them," Adlibris Nordic sales manager Sakari Luovio told NRK. He added the records showed not a single copy of the book had been sold and that if the opposite were true, the firm would have donated all proceeds to charity.
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Breivik's manifesto returned to the public spotlight after a terrorist attack on two mosques in New Zealand last week, which left 50 people dead. The suspect got inspiration from the Norwegian murderer's crimes, or at least that's what he claimed in his own manifesto, which he published before going on a gun rampage in Christchurch.
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text:
List: Great Graphic Novels for Teens
Author: Gengoroh Tagame
Rainbow Book List - Graphic Novels (1)
By Gengoroh Tagame
After the passing of Ryoji, his twin brother, Yaichi, and his now widowed husband, Mike, come together to learn more about the loved one they have both lost.
Yaichi is a work-at-home suburban dad in contemporary Tokyo; formerly married to Natsuki, father to their young daughter, Kana. Their lives suddenly change with the arrival at their doorstep of a hulking, affable Canadian named Mike Flanagan, who declares himself the widower of Yaichi's estranged gay twin, Ryoji. Mike is on a quest to explore Ryoji's past, and the family reluctantly but dutifully takes him in. What follows is an unprecedented and heartbreaking look at the state of a largely still-closeted Japanese gay culture: how it's been affected by the West, and how the next generation can change the preconceptions about it and prejudices against it.
Great Graphic Novels for Teens 2019, Rainbow Book List - Graphic Novels 2019


"Housing subdivision" redirects here. It is not to be confused with Housing development.
In the 1920s, the Coolidge administration formed the Advisory Committee on City Planning and Zoning, which undertook as its first task the publication of The Standard State Zoning Enabling Act in 1926, model enabling legislation for use by state legislatures. This was followed by publication of the Standard City Planning Enabling Act (SCPEA) in 1928. The SCPEA covered six subjects: (1) the organization and power of planning commissions, which was directed to prepare and adopt a master plan; (2) the content of the master plan; (3) provisions for a master street plan; (4) provisions for approval of all public improvements by the planning commission; (5) control of private subdivision of land; and (6) provisions for the creation of regional planning commissions.
for the purpose of sale or of building development: Every division of a piece of land into two or more lots, parcels or parts is, of course, a subdivision. The intention is to cover all subdivision of land where the immediate or ultimate purpose is that of selling the lots or building on them. The object of inserting a definition in the text of the act is to avoid the inclusion, within the planning commission's control, of such cases as a testator's dividing his property amongst his children, partners' dividing firm property amongst themselves on dissolution, or cases of that nature.
A subdivision does not need to be sold, in whole or in part, for its resulting pieces to be considered separate parcels of land. A subdivision plat approved by a local planning commission, once recorded in a registry of deeds, is generally deemed to have created the parcels of land identified on the plat itself.
The problem of testamentary division of property was identified by the SCPEA in the footnote to the definition of subdivision, but not fully clarified by it. In some jurisdictions, a testamentary division of property does not constitute a legal subdivision for purposes of separate conveyancing of the "subdivided" parcels.
Furthermore, the SCPEA's definition leaves ambiguous the notion of 'building development' and whether the identification of multiple construction sites on a single parcel of land constitutes a subdivision subject to the review and approval authority of the planning commission. Interpretations of this vary among American jurisdictions. Subdivision developers may use an architect's services only once, with the rest of the tract houses using the same master template: the resulting houses all look similar as in the above photograph of Markham, Ontario.
… the design work of lot layout, the construction of drainage structures, the construction of buildings or public use areas, the planning and construction of public streets and public roads, and the placement of public utilities.
In the Philippines, subdivisions are areas of land that have been subdivided into individual residential plots. Whereas some subdivisions comprise exclusive gated communities, others are merely demarcations denoting a specific neighborhood. Some subdivisions may conduct autonomous security, or provide basic services such as water and refuse management. Most subdivisions are governed by associations made up of members who are residents of the subdivision. In the Philippines, subdivisions are also known as villages.
In Alberta, subdivision is the dividing of a single parcel of land into two or more parcels, each to be given a separate title. Subdivision is also used for existing lot line adjustments. Notwithstanding a few exceptional circumstances, subdivision approval and endorsement by the local municipality must always be received before the subdivision can be registered at the Land Titles Office and titles issued (including bare land condominiums). Exceptions may occur with parcels of land that contain more than one quarter section, a river lot, a lake lot, or some settlement lots created prior to July 1, 1950.
^ U.S. Department of Commerce (DOC), Washington, D.C.(1926). "A Standard State Zoning Enabling Act." Document no. BH5.
^ a b DOC (1928). "A Standard City Planning Enabling Act." Document no. BH11.
^ See, e.g., In re Estate of Sophia Sayewich, 120 N.H. 237 (1980)).
^ Derossi, Dean. "Definition of Subdivisions and How to Change Land Classification". Land Century. Retrieved 23 December 2015.
Kone, D. Linda (2006). Land Development (10th ed.). BuilderBooks.com. ISBN 9780867186093.
Standard State Zoning Enabling Act and Standard City Planning Enabling Act - American Planning Assoc.


text:
Ouch! Ruth Reichl Takes Down Alain Ducasse's Restaurant Jules Verne
The six saddest lines from her recent write-up.
by Daniela Galarza Oct 10, 2014, 4:44pm EDT
Via Paris: One Sad Night [Ruth Reichl]
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Former Gourmet magazine Editor in Chief/novelist Ruth Reichl has avoided critiques of restaurants since her days as the restaurant critic for the New York Times. Typically, her blog posts are glowing notes about chefs and restaurants that she loves. But a recent meal at Alain Ducasse's one Michelin-starred Restaurant Jules Verne, which is inside the Eiffel Tower, caused the influential food writer to take to her blog with a scathing review. She writes: "It was, from start to finish, a miserable experience." Here now, the six worst lines from Reichl's blog post.
1. If you're not seated at a window, "you are in a room, dark as a nightclub, facing a black wall, so close to the surrounding tables you can hear them wishing each other happy birthday... I felt as if I had been transported back in time to one of those depressing seventies discos."
2. The service is awful: "I have never had such bad service in an expensive restaurant... It took forever for the wine to arrive. The amuse bouche - if you could call it that — was a little dish of stone cold gougeres. When we told the surly waitress that we thought they would taste better hot, she said testily that 'they can be served hot or cold, but if you'd like them hot I can warm them up.'"
3. Ducasse's promise of a fine, memorable meal — he is quoted on the menu: "I want to... ensure that the experience at Jules Verne remains in the memory of everyone. More than a restaurant, it is a place of dreams and memories." — makes Reichl "mad," because the food reminds her "of nothing so much as first class airline food."
4. "The bread was sad." Sad bread in France?!
5. Reichl describes the fish as gelatinous. The artichokes served on the side? They "were still clad in their indigestible outer leaves."
6. The first wine Reichl and company ordered had been 86'd. When their second pick was brought to the table, Reichl writes that it "was going through a second fermentation, and when we mentioned that the sommelier frowned and said repressively, 'This wine was chosen by M. Ducasse himself.'"
The best part of the meal? A tiny chocolate pastry that came at the very end.
Parisian restaurateurs: Heads up, Ruth Reichl is in town.


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In this season, Kris starts prioritizing business over family. Kim pushes for everyone to be more supportive of her new relationship. Kim takes Kendall to New York for a modeling gig but ends up pushing her to her breaking point.
Stream Keeping Up with the Kardashians - Season 6 2011 Free Online Episodes - Yify TV.


text:
NFL player releases book about fathers bonding with daughters
Marcus Smith Jr. notes that he was inspired by Kobe Bryant
Christian Spencer
Free agent linebacker Marcus Smith Jr. has released a bedtime story for children and fathers.
The 28-year-old, who has a two-year-old daughter named Sarai, released the new book called "Bathtime with Rai," which is about the bond between father and daughter, NBC News reported.
Read More: NFL star DeAndre Baker gets charges dropped in armed robbery case
In an interview with Access, Smith Jr. said the book's goal was, "written for the modern Black dads of the world and for the little brown girls of the world."
A post shared by Marcus Smith II (@moneymarc91)
"I just want to put on for the Black fathers to let us know that we are here and for the little young girls out there, there are fathers out there that are trying to become the best fathers they can possibly be," the NFL star continued.
"From nostalgic moments to laughs and giggles, Bathtime with Rai delightfully chronicles all the exchanges of love amidst a favorite nighttime ritual," the synopsis of the book says.
Smith Jr. cited Kobe Bryant as the inspiration behind the book. Smith Jr. said the late Los Angelos Lakers player's public affection with his daughters had a big influence on him.
"I look at him and his daughters' relationship, and I feel like that's what a daughter/father relationship should look like," Smith Jr. said. "I try to go off some of the same morals and principles that Kobe did."
Former #NFL Player's New #Children's Book Depicts What It's Like Being a 'Modern #Black #Father'
'I wanted to #translate that experience into something that was relatable for Black #fathers and Black #daughters everywhere.'
Details: https://t.co/Lc5DTMt5Vz pic.twitter.com/ELcE53qlTF
— Tamara C. Daniels (@TamaraCDaniels) October 11, 2020
READ MORE: Ex-NFL star Dana Stubblefield sentenced to 15 years to life for rape
"I am a BLACK MAN! And black father! I build …. I don't tear down other BLACK MEN! ….I have felt the pain of being torn down and I have decided I will be deliberate about building others! All too often, we men find it easier to criticize each other, instead of building each other up," Smith Jr. wrote on IG. "With all the negativity going around let's do something positive!!"
Hs book will be available on December 2.
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text:
Nipping fraud in the bud saves victims millions
Wednesday 20th February 2019 12:58 pm
The Banking Protocol trains bank staff to spot when someone is about to fall victim to a scam ()
A RAPID-response scheme set up by banks and the police prevented potential fraud victims in Hampshire from being scammed out of £1.4million in 2018.
The Banking Protocol trains bank staff to spot when someone is about to fall victim to a scam and try to prevent them from withdrawing cash to give to a fraudster, after which they can request an immediate police response to the branch.
A total of seven arrests were made by Hampshire Constabulary last year through the initiative, while in total police received 247 emergency calls from those taking part.
The latest figures reveal that across the country the scheme has prevented £38 million of fraud and led to 231 arrests in 2018.
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The average age of a customer helped by the Banking Protocol nationally last year was 71, showing how fraudsters are often targeting more elderly victims with these types of scams.
The Banking Protocol was rolled out by Hampshire Constabulary in June 2017.
Since March 2018, the scheme has been implemented by all 45 police forces across the UK.
Fifty-two payment service providers, including all the main high street banks and the Post Office, are now fully signed up to the Banking Protocol and have trained their front-line branch staff in the steps that need to be taken when a customer is at risk.
Katy Worobec, managing director of economic crime at UK Finance, said: "Bank branch staff are on the front line in the fight against fraud, as increasingly sophisticated gangs target consumers directly and trick them into withdrawing large sums of cash.
"This rapid response scheme is giving bank staff the tools they need to protect vulnerable customers from scams, while helping local police catch fraudsters and bring them to justice.
"The banking industry will keep taking action on all fronts to combat fraud, working closely with our partners in law enforcement to crack down on the criminal gangs responsible."
Detective Sergeant Sarah Dring, of Hampshire Constabulary's Volume Fraud Investigation Team, also supports the initiative.
"It is very important bank staff can recognise the signs and feel confident enough to call 999, as often the victims do not know they are being scammed," she said.
"All bank employees have now received fraud loss prevention training and we will continue to work with them to make sure people aren't conned into handing over their hard-earned cash.
"The types of fraud we see through the Banking Protocol range from online romance fraud to scams where people have been called by scammers impersonating police officers or council officials.
"Anyone who suspects they have been a victim of fraud should call 101 for advice or report direct to Action Fraud (www.actionfraud.police.uk)."
The initiative, which was developed in partnership with National Trading Standards, trains bank branch staff to spot the signs of a scam and what steps need to be taken when a customer is deemed at risk.
If a staff member suspects a customer is being tricked by a fraudster, for example if someone is making an unusually large cash withdrawal, they will take them aside to ask additional questions.
Then if their suspicions are confirmed, the staff member can invoke the Banking Protocol and contact the local police, who will send a priority response to the branch and investigate the suspected fraud.
As well as stopping attempted fraud, the scheme ensures extra support is provided to those customers affected to help prevent them falling victim to similar scams in the future. This can include referrals to social services, expert fraud prevention advice and additional checks on future transactions.
The advice is to never give details out to someone unless you are sure of their credentials and remember if a financial arrangement seems too good to be true, it probably is.
HampshireHampshire ConstabularyPost Office