# Taylor’s Frozen-in Hypothesis for Magnetohydrodynamic turbulence and Solar
Wind

Mahendra K. Verma<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology Kanpur, Kanpur 208016, India

(1 March 2022)

###### Abstract

In hydrodynamics, Taylor’s frozen-in hypothesis connects the wavenumber
spectrum to the frequency spectrum of a time series measured in real space. In
this paper, we generalize Taylor’s hypothesis to magnetohydrodynamic
turbulence. We analytically derive one-point two-time correlation functions
for Elsässer variables whose Fourier transform yields the corresponding
frequency spectra, $E^{\pm}(f)$. We show that $E^{\pm}(f)\propto|{\bf
U}_{0}\mp{\bf B}_{0}|^{2/3}$ in Kolmogorov-like model, and
$E^{\pm}(f)\propto(B_{0}|{\bf U}_{0}\mp{\bf B}_{0}|)^{1/2}$ in Iroshnikov-
Kraichnan model, where ${\bf U}_{0},{\bf B}_{0}$ are the mean velocity and
mean magnetic fields respectively.

##  I Introduction

Theoretical, experimental, and computational tools are employed for studying
turbulence. Kolmogorov (1941a, b) constructed one of the most popular model of
turbulence and showed that for homogeneous and isotropic turbulence, the
kinetic energy spectrum in wavenumber space is

$E_{u}(k)=K_{\mathrm{Ko}}\epsilon_{u}^{2/3}k^{-5/3},$ (1)

where $k$ is the wavenumber, $\epsilon_{u}$ is the dissipation rate of kinetic
energy, and $K_{\mathrm{Ko}}$ is Kolmogorov’s constant. Computation of
$E_{u}(k)$ requires three-dimensional velocity field, which is difficult to
measure in experiment. Instead, the velocity field is measured at select real-
space points. Taylor (1938) proposed an important conjecture that helps
connect $E_{u}(k)$ to the frequency spectrum, $E_{u}(f)$, of the measured time
series. Taylor (1938) hypothesized that turbulent fluctuations are advected by
the mean flow (velocity ${\bf U}_{0}$) as if fluctuations are frozen in the
flow. Under this assumption, using Kolmogorov’s spectrum of Eq. (1), we can
easily derive the frequency spectrum as

$E_{u}(f)=A(\epsilon_{u}U_{0})^{2/3}f^{-5/3},$ (2)

where $A$ is a constant. Thus, Taylor’s hypothesis plays an important role in
turbulence experiments and data analysis.

Kraichnan (1964) argued that the large-scale eddies too sweep the small-scale
fluctuations, a phenomenon called sweeping effect. Wilczek and Narita (2012)
and He et al. (2017) generalized Eq. (2) by taking into account the sweeping
effect. Recently, Verma et al. (2020) performed a detailed calculation of
temporal correlation function that includes sweeping effect and turbulent
diffusion; they showed that $E_{u}(f)\propto f^{-5/3}$ in the presence of a
mean flow, but $E_{u}(f)\propto f^{-2}$ when $U_{0}$ is small compared to the
fluctuations.

In magnetohydrodynamic (MHD) turbulence, the velocity and magnetic fields have
their mean values (${\bf U}_{0}$ and ${\bf B}_{0}$, respectively, in velocity
units) and fluctuations. Hence, Taylor’s hypothesis and Eq. (2) need
generalization, especially because Solar wind and solar corona are important
MHD turbulence laboratories where spacecrafts make in situ measurements of the
velocity and magnetic fields. Taylor’s hypothesis would help us derive the
wavenumber spectrum using the frequency spectrum of the time series. Note that
the typical speed of spacecrafts are much smaller than the solar wind speed
(250 km/s to 750 km/s). Hence, the spacecraft can be assumed to be stationary,
consistent with the assumptions of Taylor’s hypothesis. Another important
factor is that $B_{0}\ll U_{0}$ at distances 0.3 AU or higher, where most
spacecrafts are located. As a result, we can ignore the effects of the mean
magnetic field and employ Taylor’s hypothesis. In one of the first
applications of Taylor’s hypothesis to the solar wind, Matthaeus and Goldstein
(1982) deduced $k^{-5/3}$ spectrum for the solar wind by employing Eq. (2).
There are many more works on solar wind observations, e.g., Podesta et al.
(2007).

Recent spacecraft Parker Solar Probe (PSP) has come quite close the Sun (as
close as 8.86 solar radii). At this distance, $B_{0}$ is comparable to
$U_{0}$, and hence, we need to examine whether Taylor’s hypothesis can be
employed to PSP data. In addition, the solar wind is anisotropic that makes an
application of Taylor’s hypothesis problematic. Bourouaine and Perez (2018)
modelled the Fourier-transformed temporal correlation function of MHD
turbulence using a Gaussian model rather than a pure exponential model, and
argued that the decorrelation frequency is linearly related to the
perpendicular wavenumber. In a related work, Perez et al. (2021) investigated
the validity of Taylor’s hypothesis for the PSP data, and showed that the
frequency spectrum accurately represents the spectral indices associated with
the underlying spatial spectrum of turbulent fluctuations in the plasma frame.
In a another recent work, Kasper et al. (2021) computed the frequency spectrum
of the PSP data recorded at 13 million km above the photosphere, and observed
the spectral index to be closer to $-3/2$ than $-5/3$. Note that Kasper et al.
(2021) assume Taylor’s hypothesis for the solar wind even though $B_{0}$ and
$U_{0}$ are comparable in this regime.

In this paper, we extend the derivation of Verma et al. (2020) for
hydrodynamic turbulence to MHD turbulence. We express the “dressed” or
“renormalized” temporal correlation functions for the Elsässer variables in
the Fourier space. Then, an inverse Fourier transform of the above correlation
function yields two-point two-time correlation function. By setting the
distance between the two points to zero, we deduce one-point two-time
correlation functions, whose Fourier transform yields the frequency spectra
for MHD turbulence. For Kolmogorov’s and Iroshnikov-Kraichnan models
Iroshnikov (1964); Kraichnan (1965), we obtain respectively $f^{-5/3}$ and
$f^{-3/2}$ frequency spectra with prefactors that are functions of $|{\bf
U}_{0}\mp{\bf B}_{0}|$, which are the wave speeds of Alfvén waves. Thus, we
generalize Taylor’s hypothesis to MHD turbulence; our derivation is
mathematically more rigorous than earlier ones.

In the next two sections, we derive the correlation functions for linear and
turbulent MHD.

##  II Correlation functions for linear MHD

The equations of a magnetofluid moving with a mean velocity of
${\mathbf{U}}_{0}$ in a background of a mean magnetic field ${\bf B}_{0}$ are

$\displaystyle\frac{\partial{\bf u}}{\partial
t}+({\mathbf{U}}_{0}\cdot\nabla){\bf u}-({\mathbf{B}}_{0}\cdot\nabla){\bf
b}+({\bf u}\cdot\nabla){\bf u}-({\bf b}\cdot\nabla){\bf b}$ $\displaystyle=$
$\displaystyle-\nabla p+\nu\nabla^{2}{\bf u}+{\bf f},$ (3)
$\displaystyle\frac{\partial{\bf b}}{\partial
t}+({\mathbf{U}}_{0}\cdot\nabla){\bf b}-({\mathbf{B}}_{0}\cdot\nabla){\bf
u}+({\bf u}\cdot\nabla){\bf b}-({\bf b}\cdot\nabla){\bf u}$ $\displaystyle=$
$\displaystyle\eta\nabla^{2}{\bf b},$ (4) $\displaystyle\nabla\cdot{\bf
u}=0,~{}~{}~{}\nabla\cdot{\bf b}=0,$ (5)

where ${\bf u,b}$ are the velocity and magnetic field fluctuations, ${\bf f}$
is the external force, $p$ is the pressure, $\nu$ is the kinematic viscosity,
and $\eta$ is the magnetic diffusivity. In the above equations, the magnetic
field is velocity units, which is obtained by a transformation, ${\bf
B}_{\mathrm{CGS}}\rightarrow{\bf B}_{\mathrm{CGS}}/\sqrt{4\pi\rho}$, where
$\rho$ is the material density of the flow. In this paper, we assume that the
flow is incompressible.

Alfvén waves are the basic modes of linearized MHD equations, and they are
conveniently expressed in terms of Elsässer variables, ${\bf z^{\pm}=u\pm b}$.
Using Eqs. (3, 4, 5), we can derive the following equations for ${\bf
z^{\pm}}$:

$\displaystyle\frac{\partial{\bf z}^{\pm}}{\partial
t}+{\mathbf{Z}}_{0}^{\mp}\cdot\nabla{\bf z}^{\pm}+({\bf
z}^{\mp}\cdot\nabla){\bf z}^{\pm}$ $\displaystyle=$ $\displaystyle-\nabla
p+\nu_{\pm}\nabla^{2}{\bf z}^{\pm}+\nu_{\mp}\nabla^{2}{\bf z}^{\mp}+{\bf f},$
(6) $\displaystyle\nabla\cdot{\bf z}^{\pm}=0,$ (7)

where ${\mathbf{Z}}_{0}^{\mp}=({\mathbf{U}}_{0}\mp{\mathbf{B}}_{0})$, and
$\nu_{\pm}=(\nu\pm\eta)/2$. For simplification, in this paper, we take
$\nu=\eta$ that leads to $\nu_{+}=\nu=\eta$ and $\nu_{-}=0$. In Fourier space,
the equations for ${\bf z}^{\pm}$ are

$\displaystyle\left[\frac{\partial}{\partial
t}+i{\mathbf{Z}}_{0}^{\mp}\cdot{\bf k}+\nu k^{2}\right]{\bf z}^{\pm}({\bf k})$
$\displaystyle=$ $\displaystyle-i{\bf k}p({\bf k})-i\sum_{\bf p}[{\bf k\cdot
z^{\mp}({\bf q})]z^{\pm}({\bf p})}+{\bf f}({\bf k}),$ (8) $\displaystyle{\bf
k}\cdot{\bf z}^{\pm}({\bf k})=0,$ (9)

with ${\bf q=k-p}$. A linearized version of the above equations indicates that
the waves ${\bf z}^{+}({\bf k})$ and ${\bf z}^{-}({\bf k})$ move with speeds
$({\mathbf{U}}_{0}-{\mathbf{B}}_{0})\cdot{\bf k}$ and
$({\mathbf{U}}_{0}+{\mathbf{B}}_{0})\cdot{\bf k}$ respectively.

Using the linearised versions of Eq. (8), we derive the equations for the
Green’s functions

$\displaystyle\left[\frac{\partial}{\partial
t}+i{\mathbf{Z}}_{0}^{\mp}\cdot{\bf k}+\nu k^{2}\right]G^{\pm}({\bf
k},t,t^{\prime})$ $\displaystyle=$ $\displaystyle\delta(t-t^{\prime}),$ (10)

whose solutions are

$G^{\pm}({\bf k},\tau)=\theta(\tau)\exp{[-i{\mathbf{Z}}_{0}^{\mp}\cdot{\bf
k}\tau]}\exp{(-\nu k^{2}\tau)},$ (11)

where $\tau=t-t^{\prime}$, and $\theta(\tau)$ is the step function.

The equal-time correction functions, $C^{\pm}({\bf k},0)$, and two-time
correction functions, $C^{\pm}({\bf k},\tau)$, for ${\bf z^{\pm}(k)}$ are
defined as

$\displaystyle C^{\pm}({\bf k},0)$ $\displaystyle=$
$\displaystyle\langle|\mathbf{z}^{\pm}(\mathbf{k},t)|^{2}\rangle,$ (12)
$\displaystyle C^{\pm}({\bf k},\tau)$ $\displaystyle=$
$\displaystyle\langle\mathbf{z}^{\pm}(\mathbf{k},t)\cdot\mathbf{z}^{\pm*}(\mathbf{k},t+\tau)\rangle.$
(13)

Note that $C^{\pm}({\bf k},\tau)$ is a complex function. We define the
normalised correlation function as

$R^{\pm}(\mathbf{k},\tau)=\frac{C^{\pm}(\mathbf{k},\tau)}{C^{\pm}(\mathbf{k},0)}.$
(14)

A generalisation of fluctuation-dissipation theorem to MHD yields Kiyani and
McComb (2004)

$R^{\pm}(\mathbf{k},\tau)=G^{\pm}({\bf
k},\tau)=\theta(\tau)\exp{(-i{\mathbf{Z}}_{0}^{\mp}\cdot{\bf
k}\tau)}\exp{(-\nu k^{2}\tau)}.$ (15)

The above equation indicates that the normalised correlation functions exhibit
damped oscillations.

##  III Correlation functions for turbulent MHD

A magnetofluid becomes turbulent when $UL/\nu\gg 1$ and $UL/\eta\gg 1$, where
$U,L$ are the large-scale velocity and length respectively. There is no
definitive theory of MHD turbulence, rather it has many models Verma (2004,
2019); Beresnyak (2019). In this paper we will focus on two leading models.
Here, we focus on the shell spectra: $E^{\pm}(k)$, $E_{u}(k)$, $E_{b}(k)$,
which are defined as

$\displaystyle E^{\pm}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\langle|{\bf
z^{\pm}|^{2}}\rangle=\int E^{\pm}(k)dk,$ (16) $\displaystyle E_{u}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\langle|{\bf u}^{2}\rangle=\int
E_{u}(k)dk,$ (17) $\displaystyle E_{b}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\langle|{\bf b}^{2}\rangle=\int E_{u}(k)dk,$ (18)

where $E^{\pm},E_{u},E_{b}$ are the total energies per unit volume of ${\bf
z}^{\pm}$, ${\bf u}$, and ${\bf b}$ respectively.

  1. 1.

Komogorov-like MHD turbulence phenomenology: In this framework, the energy
spectra $E^{\pm}(k)$ are modelled as Marsch (1991); Verma et al. (1996); Verma
(2004)

$E^{\pm}(k)=K^{\pm}(\epsilon^{\pm})^{4/3}(\epsilon^{\mp})^{-2/3}k^{-5/3},$
(19)

where $\epsilon^{\pm}$ are the inertial-range energy fluxes or dissipation
rates of ${\bf z}^{\pm}$, and $K^{\pm}$ are constants, similar to Kolmogorov’s
constant for hydrodynamic turbulence. This phenomenology is also referred to
as imbalanced MHD.

In addition, Goldreich and Sridhar (1995) constructed a phenomenology for the
anisotropic MHD turbulence. Using critical balance between the time scales for
the nonlinear interactions and Alfvén wave propagation, they showed that the
modal energy

$\tilde{E}(k_{\perp},k_{\parallel})=K\epsilon^{2/3}k_{\perp}^{-10/3}g(k_{\parallel}/k_{\perp}^{2/3}),$
(20)

where $K$ is a constant, $\epsilon$ is the total dissipation rate, and
$k_{\parallel}$ and $k_{\perp}$ are respectively the wavenumber components
parallel and perpendicular to the mean magnetic field. Note that

$\int k_{\perp}dk_{\perp}dk_{\parallel}\tilde{E}(k_{\perp},k_{\parallel})=E,$
(21)

where $E$ is the total energy.

  2. 2.

Iroshnikov-Kraichnan phenomenologyIroshnikov (1964); Kraichnan (1965): In this
framework, the Alfvén time scale, $(kB_{0})^{-1}$, is the relevant time scale,
leading to the energy spectrum as

$E_{u}(k)\approx E_{b}(k)\approx K_{\mathrm{IK}}(\epsilon
B_{0})^{1/2}k^{-3/2},$ (22)

where $K_{\mathrm{IK}}$ is constant, and $B_{0}$ is the amplitude of the mean
magnetic field or that of large-scale magnetic field. In this phenomenology,
the kinetic and magnetic energies are equipartitioned.

Dobrowolny et al. (1980) showed that

$\epsilon^{+}=\epsilon^{-}=\frac{1}{B_{0}}E^{+}(k)E^{-}(k)k^{3}.$ (23)

For a special case when $E^{+}(k)=E^{-}(k)=E(k)$ ($E$ is the total energy), we
obtain

$E^{+}(k)=E^{-}(k)=E(k)=K^{\prime}(\epsilon B_{0})^{1/2}k^{-3/2}.$ (24)

Now, we model the correlation function for MHD turbulence following the
strategies adopted for the hydrodynamic turbulence. The most critical part is
the convective component. Using Eq. (8), we deduce the convective component to
be $\exp{(-i{\mathbf{Z}}_{0}^{\mp}\cdot{\bf k}\tau)}$ for ${\bf z}^{\pm}$. In
the following discussion we show that the convective part contributes most
significantly to the frequency spectra.

The other two parts of the correlation function are the effective diffusion
parameters and the sweeping effect. In hydrodynamic turbulence, field-
theoretic treatment shows that the renormalized viscosity or effective
viscosity ($\nu(k)$) is

$\nu(k)=\nu_{*}\epsilon^{1/3}k^{-4/3},$ (25)

where $\nu_{*}$ is a nondimensional constantYakhot and Orszag (1986); McComb
(1990); Verma (2004). However, MHD turbulence has two diffusion parameters,
viscosity and magnetic diffusivity, that depend on the cross helicity, Alfvén
ratio, and mean magnetic field. We do not yet have general formulas for these
renormalized parameters, even though they have been solved for special cases
(see Verma (2001, 2003a, 2003b)). In this paper, we simplify the calculation
by assuming that both the renormalized parameters are equal (i.e.,
$\nu(k)=\eta(k)$), and that for Kolmogorov-like phenomenology,

$\nu^{\pm}(k)=\nu^{\pm}_{*}(\epsilon^{\pm})^{1/3}k^{-4/3},$ (26)

and for Iroshnikov-Kraichnan phenomenology,

$\nu^{\pm}(k)=\nu^{\prime}_{*}(\epsilon B_{0})^{1/4}k^{-5/4}.$ (27)

Here, $\nu^{\pm}_{*}$ and $\nu^{\prime}_{*}$ are constants. As we show in the
next section, the terms with $\nu^{\pm}(k)$ get integrated in $E^{\pm}(f)$.
Hence, a precise form of $\nu^{\pm}(k)$ may not be critical for the derivation
of $E^{\pm}(f)$.

In addition, according to the sweeping effect, large-scale flow structures
sweep the inertial-range fluctuations. For hydrodynamic turbulence, Kraichnan
(1964), Wilczek and Narita (2012), Verma et al. (2020) and others have
constructed models for the sweeping effect. For MHD turbulence, we follow the
prescription of Verma et al. (2020) who added a random large-scale velocity
field, $\tilde{\bf U}_{0}$, to the mean velocity field ${\bf U}_{0}$. These
corrections are added in the correlation function for the linear equation [Eq.
(15)].

Under the above assumptions, we arrive at the following expressions for the
correlation functions of MHD turbulence Verma et al. (2020):

$\displaystyle
R^{\pm}(\mathbf{k},\tau)=\frac{C^{\pm}(\mathbf{k},\tau)}{C^{\pm}(\mathbf{k})}$
$\displaystyle=$ $\displaystyle\exp{(-i{\mathbf{Z}}_{0}^{\mp}\cdot{\bf
k}\tau)}\exp(-i{\bf\tilde{U}_{0}\cdot k}\tau)\exp[-\nu^{\pm}(k)k^{2}\tau].$
(28)

Note that the correlations functions depend on both ${\bf U}_{0}$ and ${\bf
B}_{0}$. In the next section, we will relate the above functions to Taylor’s
hypothesis for MHD turbulence.

##  IV Taylor’s hypothesis for MHD turbulence

Using Eq. (28), we first derive the two-point two-time correlation functions,
after that we derive one-point two-time correlation functions, whose Fourier
transform yields the frequency spectra of real-space time series of ${\bf
z}^{\pm}$.

Using Eq. (28), we derive the following two-point two-time correlation
functions for ${\bf z}^{\pm}$:

$\displaystyle C^{\pm}({\bf r},\tau)$ $\displaystyle=$ $\displaystyle\int
d{\bf k}C^{\pm}({\bf k})\exp[-\nu(k)k^{2}\tau-i{\bf Z^{\mp}_{0}\cdot
k}\tau]\exp[-i{\bf k}\cdot\tilde{\bf U}_{0}({\bf k})\tau]\exp(i{\bf
k}\cdot{\bf r}),$ (29)

where ${\bf Z}^{\mp}_{0}={\bf U}_{0}\mp{\bf B}_{0}$. We ensemble average
$C^{\pm}({\bf r},\tau)$ for random $\tilde{\bf U}_{0}$ (assuming isotropic, as
in Kraichnan (1964)) that yields Kraichnan (1964); Wilczek and Narita (2012);
Verma et al. (2020)

$\displaystyle C^{\pm}({\bf r},\tau)$ $\displaystyle=$ $\displaystyle\int
d{\bf k}C^{\pm}({\bf k})\exp[-\nu^{\pm}(k)k^{2}\tau-i{\bf Z^{\pm}_{0}\cdot
k}\tau]\langle\exp[-ick\tilde{U}_{0}(k)\tau]\rangle\exp(i{\bf k}\cdot{\bf r})$
(30) $\displaystyle=$ $\displaystyle\int d{\bf k}C^{\pm}({\bf
k})\exp[-\nu^{\pm}(k)k^{2}\tau-i{\bf Z^{\pm}_{0}\cdot
k}\tau]\exp[-c^{2}k^{2}\\{\tilde{U}_{0}(k)\\}^{2}\tau^{2}]\exp(i{\bf
k}\cdot{\bf r}).$

For simplicity, we assume that the constant $c\approx 1$, and set ${\bf r}=0$
to compute one-point two-time correlation functions $C^{\pm}({\bf
r}=0,\tau)=C^{\pm}(\tau)$. The Gaussian model for the sweeping effect has been
reported earlier by Kraichnan (1964); Wilczek and Narita (2012), and
Bourouaine and Perez (2018).

Now we derive $C^{\pm}(\tau)$ for the Kolmogorov-like phenomenology. Following
Pope Pope (2000), we take $C^{\pm}({\bf k})$:

$C^{\pm}({\bf k})=\frac{E^{\pm}(k)}{2\pi k^{2}}=\frac{1}{2\pi
k^{2}}f_{L}(kL)f_{\eta}(k\eta)K^{\pm}k^{-5/3}\frac{(\epsilon^{\pm})^{4/3}}{(\epsilon^{\mp})^{2/3}}$
(31)

where

$\displaystyle f_{L}(kL)$ $\displaystyle=$
$\displaystyle\left(\frac{kL}{[(kL)^{2}+c_{L}]^{1/2}}\right)^{5/3+p_{0}},$
(32) $\displaystyle f_{\eta}(k\eta)$ $\displaystyle=$
$\displaystyle\exp\left[-\beta\left\\{[(k\eta)^{4}+c_{\eta}^{4}]^{1/4}-c_{\eta}\right\\}\right]$
(33)

are respectively the forcing and dissipative components of the energy spectra,
and $c_{L},c_{\eta},p_{0},\beta$ are constants. We employ Eq. (26) for
$\nu(k)$, and $\tilde{U}_{0}(k)=\epsilon^{1/3}k^{-1/3}$ with $\epsilon$ as the
total dissipation rate (see Verma et al. (2020)). In addition, we ignore the
constants for brevity. After the above substitutions in Eq. (30) with ${\bf
r}=0$, we obtain

$\displaystyle C^{\pm}(\tau)$ $\displaystyle=$ $\displaystyle
K^{\pm}\frac{(\epsilon^{\pm})^{4/3}}{(\epsilon^{\mp})^{2/3}}\int
dkk^{-5/3}f_{L}(kL)f_{\eta}(k\eta)\exp(-i{\bf Z^{\mp}_{0}\cdot k}\tau)\times$
(34)
$\displaystyle\exp[-(\epsilon^{\pm})^{1/3}k^{2/3}\tau]\exp[-\epsilon^{2/3}k^{4/3}\tau^{2}].$

The above integral is quite complex, but it can be simplified in the
asymptotic case.

For Taylor’s frozen-in hypothesis to work, we assume that
$\mathbf{Z}^{\mp}_{0}\cdot\mathbf{k}\gg\nu^{\pm}(k)k^{2}$ and
$\mathbf{Z}^{\mp}_{0}\cdot\mathbf{k}\gg k\tilde{U}_{0}(k)$. In addition, for
$C^{+}(\tau)$ and $C^{-}(\tau)$, we choose the $z$ axis to be along the
direction of ${\bf Z}^{-}_{0}$ and ${\bf Z}^{+}_{0}$ respectively. For
simplification, we make a change of variable,
$\tilde{k}_{\pm}=Z^{\mp}_{0}k\tau$, and use $\epsilon=U^{2}/T$ ($U$ is the rms
speed). As a result, we obtain

$\displaystyle C^{\pm}(\tau)$ $\displaystyle\approx$ $\displaystyle
K^{\pm}(Z^{\mp}_{0}\tau)^{2/3}\frac{(\epsilon^{\pm})^{4/3}}{(\epsilon^{\mp})^{2/3}}\int
d{\tilde{k}_{\pm}}\tilde{k}_{\pm}^{-5/3}f_{L}[\tilde{k}_{\pm}(L/Z^{\mp}_{0}\tau)]f_{\eta}[\tilde{k}_{\pm}(\eta/U_{0}\tau)]\frac{\sin(Z^{\mp}_{0}k\tau)}{Z^{\mp}_{0}k\tau}\times$
(35)
$\displaystyle\exp[-\tilde{k}_{\pm}^{2/3}(U/Z^{\mp}_{0})^{2/3}(\alpha^{\pm}\tau/T)^{1/3}-\tilde{k}_{\pm}^{4/3}(U/Z^{\mp}_{0})^{4/3}(\tau/T)^{2/3}],$

where $\epsilon^{\pm}=\epsilon\alpha^{\pm}$. For $\tau$ in the inertial range,
$L/Z^{\mp}_{0}\tau\gg 1$ and $\eta/Z^{\mp}_{0}\tau\ll 1$. Consequently,
$f_{L}(\tilde{k}_{\pm}(L/Z^{\mp}_{0}\tau))\approx 1$, and
$f_{\eta}(\tilde{k}_{\pm}(\eta/Z^{\mp}_{0}\tau)\approx 1$. Therefore,

$\displaystyle C^{\pm}(\tau)$ $\displaystyle\approx$ $\displaystyle
K^{\pm}(Z^{\mp}_{0}\tau)^{2/3}\frac{(\epsilon^{\pm})^{4/3}}{(\epsilon^{\mp})^{2/3}}\int
d{\tilde{k}_{\pm}}\tilde{k}_{\pm}^{-5/3}\frac{\sin\tilde{k}_{\pm}}{\tilde{k}_{\pm}}\times$
(36)
$\displaystyle\exp[-\tilde{k}_{\pm}^{2/3}(U/Z^{\mp}_{0})^{2/3}(\alpha^{\pm}\tau/T)^{1/3}-\tilde{k}_{\pm}^{4/3}(U/Z^{\mp}_{0})^{4/3}(\tau/T)^{2/3}]$
$\displaystyle=$ $\displaystyle
A^{\pm}K^{\pm}(Z^{\mp}_{0}\tau)^{2/3}\frac{(\epsilon^{\pm})^{4/3}}{(\epsilon^{\mp})^{2/3}},$

where $A^{\pm}$ are the values of the nondimensional integrals. The Fourier
transform of the above $C^{\pm}(\tau)$ yields the following frequency spectra:

$\displaystyle E^{\pm}(f)$ $\displaystyle\approx$ $\displaystyle\int
C^{\pm}(\tau)\exp(-i2\pi f\tau)d\tau=\int
A^{\pm}K^{\pm}(Z^{\mp}_{0}\tau)^{2/3}\frac{(\epsilon^{\pm})^{4/3}}{(\epsilon^{\mp})^{2/3}}\exp(-i2\pi
f\tau)d\tau$ (37) $\displaystyle=$ $\displaystyle A^{\prime\pm}(|{\bf
U}_{0}\mp{\bf
B}_{0}|)^{2/3}\frac{(\epsilon^{\pm})^{4/3}}{(\epsilon^{\mp})^{2/3}}f^{-5/3},$

where $A^{\prime\pm}$ are constants. Thus, we obtain $-5/3$ frequency spectra
for Kolmogorov-like MHD turbulence phenomenology. Note that $E^{+}(f)$ and
$E^{-}(f)$ are functions of $|{\bf U}_{0}-{\bf B}_{0}|$ and $|{\bf U}_{0}+{\bf
B}_{0}|$, respectively, which are the respective speeds of ${\bf z^{+}(k)}$
and ${\bf z^{-}(k)}$ in the linear approximation. In comparison to Eq. (2),
$E^{\pm}(f)$ has $U_{0}\rightarrow|{\bf U}_{0}\mp{\bf B}_{0}|$ and
$\epsilon_{u}\rightarrow(\epsilon^{\pm})^{4/3}(\epsilon^{\mp})^{-2/3}$.

The calculation for the anisotropic MHD turbulence is more complex. However,
the complexity is likely to be in the integral computation, which will reflect
in the constants $A^{\pm}$. For anisotropic MHD turbulence, we expect that the
form of $E^{\pm}(f)$ will the same as in Eq. (37).

The above analysis can be extended to Iroshnikov-Kraichnan phenomenology,
where the correlation functions are

$C^{\pm}({\bf k})=\frac{E^{\pm}(k)}{2\pi k^{2}}\sim(\epsilon
B_{0})^{1/2}k^{-7/2}.$ (38)

We substitute Eq. (38) into Eq. (30), employ $\nu^{\pm}(k)$ of Eq. (27), and
set ${\bf r}=0$. Following the same steps as above, we obtain

$\displaystyle C^{\pm}(\tau)\approx(\epsilon B_{0}Z^{\mp}_{0}\tau)^{1/2}.$
(39)

Fourier transform of the above $C^{\pm}(\tau)$ yields the following frequency
spectra:

$\displaystyle E^{\pm}(f)=A_{\mathrm{IK}}(\epsilon B_{0}|{\bf U}_{0}\mp{\bf
B}_{0}|)^{1/2}f^{-3/2},$ (40)

where $A_{\mathrm{IK}}$ is a constant. Thus, we obtain $f^{-3/2}$ frequency
spectrum for the Iroshnikov-Kraichnan phenomenology.

The prefactors for $E^{\pm}(f)$ are functions of $Z^{\pm}_{0}$. However, the
prefactors for $E_{u}(k)$ and $E_{b}(k)$ would be more complex because

$E^{\pm}(f)=E_{u}(f)+E_{b}(f)\pm 2H_{c}(f),$ (41)

where $H_{c}=(1/2)\langle{\bf u\cdot b}\rangle$. Clearly, derivation of
$E_{u}(k)$, $E_{b}(k)$, and $H_{c}(k)$ requires further inputs, e.g.,
relationships among these functions. Based on these complexities,

##  V Discussion and Conclusions

In this paper, we extend Taylor’s frozen-in hypothesis to MHD turbulence. From
the first principle, we derive one-point two-time correlation functions for
MHD turbulence, whose Fourier transform yields the corresponding frequency
spectra. The main predictions of our quantitative calculations are as follows:

  1. 1.

The spectral indices for $E^{\pm}(k)$ and $E^{\pm}(f)$ are the same.

  2. 2.

The prefactors of $E^{\pm}(f)$ are proportional to $|{\bf U_{0}\mp
B_{0}}|^{2/3}$ in Kolmogorov-like phenomenology, but proportional to
$B_{0}^{1/2}|{\bf U_{0}\mp B_{0}}|^{1/2}$ in Iroshnikov-Kraichnan
phenomenology. In contrast to $E_{u}(f)$ for hydrodynamic turbulence,
$U_{0}\rightarrow|{\bf U_{0}\mp B_{0}}|$ in $E^{\pm}(f)$ of MHD turbulence.

  3. 3.

The kinetic and magnetic energy spectra, $E_{u}(f)$ and $E_{b}(f)$, are
expected to have more complex prefactors.

  4. 4.

When $B_{0}\ll U_{0}$, the frequency spectrum of Eq. (2) can be employed for
all the fields.

The above predictions are important for the time series analysis of the solar
wind and solar corona when $U_{0}$ and $B_{0}$ are comparable, e.g., for
Parker Solar Probe (PSP) when it is close to the Sun. Hence, PSP’s data
provides an unique opportunity for testing the above predictions. We hope to
validate these predictions in near future.

Solar wind data reveal another interesting property. Many authorsMatthaeus and
Goldstein (1982); Podesta et al. (2007); Kasper et al. (2021) observe that the
kinetic energy spectrum ($E_{u}(k)$) is steeper than the magnetic energy
spectrum ($E_{b}(k)$). This relative steepening of $E_{u}(k)$ with relative to
$E_{b}(k)$ is attributed to the energy transfers from the kinetic energy to
the magnetic energyVerma et al. (2021); Verma (2022). Note that these energy
transfers are critical for the magnetic field generation or dynamo.
Interestingly, $E^{\pm}(k)$ do not suffer from such steepening due to an
absence of cross transfer between ${\bf z}^{+}$ and ${\bf z}^{-}$ (see Verma
(2004, 2019)). This is another reason why $E^{\pm}(k)$ are more reliable
energy spectra compared to $E_{u}(k)$ and $E_{b}(k)$. It will be interesting
to quantitatively compare the solar wind observations with the theoretical
predictions made in this paper.

###### Acknowledgements.

The author thanks Rupak Mukherjee for useful suggestions and discussions. This
work is partially supported by the project CRG/2021/00l097 by Science and
Engineering Research Board (SERB), India.

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11institutetext:  Gran Sasso Science Institute, viale F. Crispi 7, 67100
L’Aquila, Italy 22institutetext: INFN - Laboratori Nazionali Gran Sasso, via
G. Acitelli 22, 67100 Assergi (AQ), Italy 33institutetext: Observatorio Pierre
Auger, Av. San Martín Norte 304, 5613 Malargüe, Argentina

# Pierre Auger Observatory and Super Heavy Dark Matter

Roberto Aloisio for the Pierre Auger Collaboration
roberto.aloisio<EMAIL_ADDRESS>
A complete list of authors can be found at
https://www.auger.org/archive/authors_2022_09.html112233

###### Abstract

We briefly discuss the connection of the Pierre Auger Observatory data with a
large class of dark matter models based on the early universe generation of
super heavy particles, their role in the solution of the dark matter problem,
highlighting the remarkable constraining capabilities of the Auger
observations.

##  1 Introduction

Ultra-high energy cosmic rays (UHECR) are the most energetic particles ever
observed with energies larger than $10^{20}$ eV. These extreme energies, as
high as $10^{11}$ GeV, eleven orders of magnitude above the proton mass and
"only" eight below the Planck mass, are a unique workbench to probe new ideas,
models and theories beyond the Standard Model (SM) of particle physics, which
show their effects at energies much larger than those ever obtained, or
obtainable in the future, in accelerator experiments. This is the case of
theories with Lorentz invariance violations PierreAuger:2021tog ;
Addazi:2021xuf ; Aloisio:2000cm ; Aloisio:2002ed  or models of Dark Matter
(DM) with super heavy particles PierreAuger:2022ibr ; PierreAuger:2022wzk ;
Aloisio:2006yi ; Guepin:2021ljb ; Aloisio:2015lva ; Aloisio:2007bh , that
connect UHECR observations with the Dark Sector (DS) and the cosmological
evolution of the early universe.

The leading paradigm to explain DM observations is based on the Weakly
Interactive Massive Particle (WIMP) hypotheses, a stable DM particle with mass
in the range of $0.1\div 10$ TeV Bergstrom:2000pn . Searches for WIMPs
provided no evidence of a clear candidate and alternative solutions to the DM
problem should be considered. An alternative to WIMP models is represented by
the scenarios based on long lived super-heavy relics, that can be produced by
several mechanisms taking place during the inflationary phase or just after,
in the re-heating phase PierreAuger:2022ibr ; PierreAuger:2022wzk ;
Aloisio:2006yi ; Aloisio:2015lva ; Aloisio:2007bh . Once created in the early
universe, the abundance of the long-lived super-heavy particles can evolve to
match the DM density observed today, the so-called Super Heavy Dark Matter
(SHDM). This conclusion can be drawn under three general hypotheses: (i) SHDM
in the early universe never reaches local thermal equilibrium; (ii) SHDM
particles have mass $M_{X}$ of the order of the inflaton mass or higher; and
(iii) SHDM particles are long-living particles with a lifetime exceeding the
age of the universe, $\tau_{X}\gg t_{0}$.

By far the largest experiment devoted to the observation of UHECR is the
Pierre Auger Observatory (Auger) in Argentina. UHECR can be observed only
indirectly through the detection of their interaction products with the
Earth’s atmosphere. An UHE particle interacting with a nucleus of the
atmosphere produces both hadronic and electromagnetic cascades of particles,
collectively called Extensive Air Shower (EAS). In Auger the EAS observation
is performed by detecting the EAS particles that reach the ground and by the
(coincident) detection of the fluorescence emission produced in the
atmosphere, the so-called hybrid events.

As always in the case of cosmic rays, the information we can gather
experimentally concerns: energy spectrum, mass composition and arrival
direction anisotropy. The Auger observations of UHECR clarified several
important facts Coleman:2022abf ; AlvesBatista:2019tlv ; Aloisio:2017ooo ;
Aloisio:2017qoo ; PierreAuger:2022uwd : (i) UHECR are charged particles, with
severe upper limits on UHE photon and neutrino fluxes (see later); (ii) the
spectrum observed at the Earth shows a slight flattening at energies around
$5\times 10^{18}$ eV (called the ankle) with (iii) an instep at $10^{19}$ eV
and (iv) a steep suppression at the highest energies; (v) mass composition is
dominated by light particles (proton and helium) at energies around $10^{18}$
eV becoming progressively heavier starting from energies around $3\times
10^{18}$ eV, with lacking of light particles toward the highest energies; (vi)
at energies larger than $8\times 10^{18}$ eV UHECR show a strong signal of
dipole anisotropy in the arrival directions, with $6\%$ amplitude and the
phase pointing toward the direction of the galactic anti-center.

In section 2, we will briefly introduce SHDM models discussing their
connection to the Auger observations and the related constraining
capabilities. We conclude in section 3.

##  2 Super-heavy dark matter models and constrains

Recently, using the LHC measurements of the masses of the Higgs boson and Top
quark, it has been pointed out that the Higgs potential is stable until very
high energy scales $\Lambda_{I}\simeq 10^{10}\div 10^{12}$ GeV and it might be
possible to extrapolate the SM to even higher energies up to the Planck mass
$M_{P}=10^{19}$ GeV Buttazzo:2013uya . This fact implies that the mass
spectrum of the DS is restricted to ultra-high energies and SHDM could play a
role. In the following, we will consider SHDM with masses in the range between
$10^{8}$ GeV up to the Planck mass.

Apart from the gravitational interaction, SHDM can be coupled to ordinary
matter also through some super-weak coupling, driven by a high energy scale
$\Lambda>\Lambda_{I}$ as the Grand Unification Scale ($\Lambda_{GUT}\simeq
10^{16}$ GeV).

Figure 1:  [Left Panel] Distribution function of the SM products in the SHDM
decay. [Right Panel] Auger experimental limits on the fluxes of gamma rays and
neutrinos. Figures taken from PierreAuger:2022ibr .

To assure long-lived particles, the interaction SHDM-SM should be suppressed
by some power $n$ of the high energy scale $\Lambda$, with a decay SHMD
$\rightarrow$ SM characterised by a lifetime
$\tau_{X}\simeq(M_{X}\alpha_{X\Theta})^{-1}(\Lambda/M_{X})^{2n-8}$, being
$\alpha_{X\Theta}$ the reduced coupling constant between SHDM and SM particles
PierreAuger:2022ibr . In the case in which SHDM interacts with SM particles
only through the gravitational interaction111Note that, in this case, a SHDM
particle can be also called Planckian-Interacting Massive Particle (PIDM).,
assuming the very general case of a DS characterised by its own non-abelian
gauge symmetry, SHDM can decay only through non-perturbative effects as the
instanton-induced decay PierreAuger:2022ibr . In this case, the lifetime of
SHDM follows from the corresponding instanton transition amplitude that, being
exponentially suppressed, provides long-living particles. Considering the
zeroth order contribution, the instanton-induced lifetime can be written as
$\tau_{X}\simeq M_{X}^{-1}\exp{(4\pi/\alpha_{X})}$, being $\alpha_{X}$ the
reduced coupling constant of the hidden gauge interaction in the DS.

Under very general assumptions PierreAuger:2022ibr ; PierreAuger:2022wzk ;
Aloisio:2006yi ; Guepin:2021ljb ; Aloisio:2015lva ; Aloisio:2007bh , we can
determine the composition and spectra of the standard model particles produced
by the SHDM decay. Typical decay products are couples of quark and anti-
quark222For a discussion of alternative (leptonic) decay patterns see
Guepin:2021ljb . that, through a cascading process (jets), give rise to
neutrinos, gamma rays and nucleons. As follows using the DGLAP equations
Aloisio:2006yi , these particles exhibit a flat spectrum, that, as shown in
the left panel of figure 1, at the relevant energies, can be approximated as
$dN/dE\propto E^{-1.9}$, independently of the particle type, with a
photon/nucleon ratio of about $\gamma/N\simeq 2\div 3$ and a neutrino nucleon
ratio $\nu/N\simeq 3\div 4$, quite independent of the energy range
Aloisio:2006yi . Therefore, the most constraining limits on the SHDM mass and
lifetime are those coming from the (non) observation of UHE photons and, even
to a lesser extent, neutrinos.

Photons with energies larger than $10^{8}$ GeV have an absorption length
roughly of the size of our galaxy, increasing up to the scale of the local
group ($1\div 10$ Mpc) at energies larger than $10^{10}$ GeV. Moreover, as
follows from numerical simulations of structure formation, DM is clustered in
galactic halos with an over-density at the level of $10^{5}$ Aloisio:2006yi ,
it implies that only UHE photons from the decay of SHDM in our our own galaxy
play a role. The observation of photons with energy larger than $10^{8}$ GeV
is affected by the background of cosmogenic photons, i.e. UHE diffuse photons
produced by the propagation of UHECR. This background depends on the mass
composition of UHECR and on the details of their astrophysical sources, it can
be estimated to range (at most) around $10^{-2}$ km-2 sr-1 yr-1 above $10^{8}$
GeV and around $10^{-3.5}$ km-2 sr-1 yr-1 above $10^{9}$ GeV
PierreAuger:2022uwd . These cosmogenic diffuse fluxes are two orders of
magnitude below the sensitivity of Auger and can be safely neglected. The
search for UHE photon events in the Auger dataset exploits the well known
differences in the EAS initiated by photons or hadrons: EAS generated by
photons develop deeper in the atmosphere and are characterised by a reduced
number of muons, with a substantial detectable difference in the lateral
distribution of EAS particles PierreAuger:2022uwd . In the right panel of
figure 1 we plot the very stringent limits on the photon and neutrino fluxes
as extracted from the Auger data (2022 update) in the energy range $10^{8}\div
10^{11}$ GeV PierreAuger:2022ibr ; PierreAuger:2022wzk ; PierreAuger:2022uwd .

Figure 2: Limits fixed by the Auger observations on the SHDM mass $M_{X}$ and:
(left panel) the coupling SHDM-SM $\alpha_{X\Theta}$ for different values of
the suppression scale order $n$, (central panel) the coupling of the hidden DS
gauge interaction $\alpha_{X}$ in the case of the instanton-induced decay,
(right panel) the universe expansion rate at the end of inflation $H_{inf}$.
In the right panel, we plot the limits for different values of the reheating
efficiency $\epsilon$ (see text). Figures taken from PierreAuger:2022ibr .

Assuming a DM density in our galaxy as in the Navarro-Frenk-White profile
Navarro:1995iw , we can determine the expected flux of UHE photons produced by
the decay of SHDM. Requiring that this flux does not exceed the Auger
experimental limits, it is possible to constrain the mass and lifetime of
SHDM. In the left and central panels of figure 2, we plot the limits on
$M_{X}$ and the coupling SHDM-SM $\alpha_{X\Theta}$, in the case of direct
coupling (left panel), and the coupling of the hidden DS gauge interaction
$\alpha_{X}$, in the case of the instanton-induced decay (central panel).
Assuming the class of models based on SHDM creation during the reheating
phase, the limits on $M_{X}$ and $\tau_{X}$ can be rewritten in terms of the
cosmological parameters characterising the generation mechanism of SHDM. In
the right panel of figure 2, we plot the limits coming from Auger data on the
age of the universe at the end of inflation $H_{inf}^{-1}$ for different
values of the reheating efficiency $\epsilon=(\Gamma_{\phi}/H_{inf})^{1/2}$,
being $\Gamma_{\phi}=g_{\phi}^{2}M_{\phi}/8\pi$ the inflaton decaying
amplitude into SHDM ($M_{\phi}$ inflaton mass; $g_{\phi}$ inflaton coupling
with SHDM) PierreAuger:2022ibr ; PierreAuger:2022wzk .

##  3 Conclusions

We conclude by stating the importance of Auger observations on UHE photons and
neutrinos, as they enable to constrain models of new physics connected to the
grand unification scale and the reheating phase of the universe. This is the
case of SHDM that, as discussed above, being a viable alternative to the WIMP
paradigm for DM, can be tested only through UHECR observations, eventually
connected with cosmological observations.

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