# General Analytical Conditions for Inflaton Fragmentation: Quick and Easy
Tests for its Occurrence

Jinsu Kim<EMAIL_ADDRESS>Theoretical Physics Department, CERN, 1211 Geneva
23, Switzerland  John McDonald<EMAIL_ADDRESS>Department of
Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom

###### Abstract

Understanding the physics of inflaton condensate fragmentation in the early
Universe is crucial as the existence of fragments in the form of non-
topological solitons (oscillons or Q-balls) may potentially modify the
evolution of the post-inflation Universe. Furthermore, such fragments may
evolve into primordial black holes and form dark matter, or emit gravitational
waves. Due to the non-perturbative and non-linear nature of the dynamics, most
of the studies rely on numerical lattice simulations. Numerical simulations of
condensate fragmentation are, however, challenging and, without knowing where
to look in the parameter space, they are likely to be time-consuming as well.
In this paper, we provide generic analytical conditions for the perturbations
of an inflaton condensate to undergo growth to non-linearity in the cases of
both symmetric and asymmetric inflaton potentials. We apply the conditions to
various inflation models and demonstrate that our results are in good
agreement with explicit numerical simulations. Our analytical conditions are
easy to use and may be utilised in order to quickly identify models that may
undergo fragmentation and determine the conditions under which they do so,
which can guide subsequent in-depth numerical analyses.

††preprint: CERN-TH-2021-202

##  I Introduction

After inflation ends, a real scalar inflaton field starts to oscillate around
the minimum of its potential. An attractive inflaton self-interaction may then
result in growth of perturbations of the inflaton to non-linearity and, if
non-topological soliton (NTS) solutions exist, which will be the case in the
models we consider, subsequent fragmentation of the inflaton condensate111 It
is important to distinguish between the growth of perturbations of a scalar
field to non-linearity and condensate fragmentation. The latter requires that
discrete NTS solutions exist, so that the condensate may break up into
discrete NTS with vacuum between them. The condition for NTS to be possible in
models that grow to non-linearity is that the scalars are sufficiently massive
in the vacuum so that the energy per scalar in the NTS can be less than the
mass of the scalar in the vacuum. This will be generally true of the
potentials that we will consider. . The inflaton condensate fragments
correspond to oscillons, which are spherically symmetric quasi-stable NTS [1].
The longevity of oscillons can be understood by the conservation of adiabatic
charge [2].

The study of inflaton condensate fragmentation has gained much attention
partly due to the interesting consequences of the fragments. Examples include
primordial black hole formation due to the statistics of fragments [3, 4, 5],
gravitational wave signals associated with fragments [6, 7, 8, 9, 10], and
changes in reheating dynamics which lead to a different cosmic evolution [12,
11].

The evolution dynamics of the fragments often requires numerical lattice
simulations due to the non-perturbative and non-linear nature. Most of the
existing literature adopts numerical approaches and performs extensive lattice
simulations. Numerical simulations are, however, often technically challenging
and computationally demanding, and can obscure the underlying physics.
Furthermore, without knowing where to look in the model parameter space, or if
fragmentation is even possible in a given model, numerical studies can be
time-consuming. Having an analytical expression for the condition under which
the condensate is likely to undergo non-linear growth and hence, if discrete
NTS solutions exist, fragmentation, would be greatly beneficial for such in-
depth numerical analyses. Such a condition could serve as a starting point of
extensive numerical analyses, and it could also be used to quickly estimate
the likelihood of fragmentation in a given model. We expect that NTS will
exist in the models that we consider, therefore, in the following, we will
consider the growth of perturbations to non-linearity and fragmentation to be
equivalent.

The aim of this paper is to provide general analytical conditions for
perturbations of an inflaton condensate to grow to become non-linear and hence
for inflaton condensate fragmentation to be possible. In Ref. [13] we studied
the stability of an inflaton condensate, based on Ref. [1], for the case of a
general symmetric potential. We obtained analytical conditions on the self-
interaction couplings under which the condensate undergoes fragmentation.
However, the method of Ref. [13] cannot be used when the potential is
asymmetric. In this work, we generalise the analysis of Ref. [13] to encompass
asymmetric potentials, thus allowing the condition for fragmentation to be
determined for a wide range of inflaton potentials.

In the next section, we develop an analytical framework for studying the
growth of inflaton field perturbations and inflaton condensate fragmentation.
We then provide the analytical conditions under which the condensate
perturbations grow to non-linearity and fragment for both symmetric and
asymmetric potentials. The main results of the paper are summarised in Sec.
II.4. In Sec. III, we apply our analytical results to four examples for which
numerical results are available: the $\alpha$-attractor $T$-model, the
$\alpha$-attractor $E$-model, Starobinsky’s $R^{2}$ model, and the Palatini
$R^{2}$ model with a quadratic potential. In addition, we apply our results to
a model which has not yet been studied numerically, Higgs Inflation with a
general symmetry-breaking potential in both the metric and Palatini
formulations, and we perform lattice simulations to test our analytical
results. We show that our results are in good agreement with numerical
simulations in all cases, thus demonstrating their general effectiveness. We
conclude in Sec. IV.

##  II General Treatment

###  II.1 Inflaton condensate

Consider a real scalar field $\Phi$ whose equation of motion222 Throughout the
paper, we work in the Einstein frame with a canonically normalised kinetic
term, unless otherwise stated.  is given by

$\displaystyle\ddot{\Phi}+3H\dot{\Phi}-\frac{\nabla^{2}\Phi}{a^{2}}+V_{\Phi}=0\,,$
(1)

where the dot denotes the cosmic time derivative and $V_{\Phi}\equiv
dV(\Phi)/d\Phi$.

In our analysis, we will be considering a potential that is close to a
quadratic potential, so to a first approximation the field will undergo
coherent oscillations in a quadratic potential. We expect that any potential
that is capable of supporting NTS solutions will be approximately quadratic at
the minimum of its potential, as it is necessary for the scalars to have a
non-zero mass in the vacuum in order for the NTS solutions to have a lower
energy per scalar and so to be (meta)stable.

Defining $\Phi=(a_{0}/a)^{3/2}\phi$, where we have assumed that the coherent
oscillations start at $a=a_{0}$, gives

$\displaystyle\ddot{\phi}-\frac{\nabla^{2}\phi}{a^{2}}=-\frac{\partial
U(\phi)}{\partial\phi}\,,$ (2)

where

$\displaystyle\frac{\partial
U}{\partial\phi}\equiv\left(\frac{a}{a_{0}}\right)^{3/2}V_{\Phi}+\Delta_{H}\phi\,,$
(3)

with

$\displaystyle\Delta_{H}\equiv-\frac{3}{2}\left(\dot{H}+\frac{3}{2}H^{2}\right)\,.$
(4)

In the case of a symmetric potential, $V(\Phi)=V(-\Phi)$, the oscillatory
behaviour of the scalar field can well be described by
$\phi(t,\mathbf{x})=R\cos\Omega$, where $R$ and $\Omega$ are both functions of
$t$ and $\mathbf{x}$. The inflaton condensate fragmentation in symmetric
potentials is studied in Ref. [13]. However, for asymmetric potentials,
$V(\Phi)\neq V(-\Phi)$, the analysis of Ref. [13] cannot be applied as the
choice of $\phi=R\cos\Omega$ does not capture the asymmetric feature. In the
case of an asymmetric potential, we expect the amplitude of the oscillating
field to be different at $\Omega=0$ and $\Omega=\pi$. Assuming that the
potential is dominated by a quadratic term, one may model this by considering

$\displaystyle\phi=R(1+\epsilon\cos\Omega)\cos\Omega\,,$ (5)

where $\epsilon\ll 1$ parametrises the asymmetry of the potential. The form of
$\epsilon$ can be determined once the potential is specified. We explore the
case of an asymmetric potential in more detail in Sec. II.3. Note that the
symmetric potential case corresponds to the $\epsilon\rightarrow 0$ limit.
Thus, Eq. (5) may be used to describe the inflaton condensate motion for a
generic form of potential.

The equation of motion (2) can be expressed in terms of $R=R(t,\mathbf{x})$
and $\Omega=\Omega(t,\mathbf{x})$ as

$\displaystyle 0$
$\displaystyle=\ddot{R}(1+\epsilon\cos\Omega)\cos\Omega-2\dot{R}\dot{\Omega}(1+2\epsilon\cos\Omega)\sin\Omega$
$\displaystyle\quad-R\ddot{\Omega}(1+2\epsilon\cos\Omega)\sin\Omega-R\dot{\Omega}^{2}(\cos\Omega+2\epsilon\cos(2\Omega))$
$\displaystyle\quad-\frac{\nabla^{2}R}{a^{2}}(1+\epsilon\cos\Omega)\cos\Omega$
$\displaystyle\quad+\frac{2(\nabla
R\cdot\nabla\Omega)}{a^{2}}(1+2\epsilon\cos\Omega)\sin\Omega$
$\displaystyle\quad+\frac{R(\nabla^{2}\Omega)}{a^{2}}(1+2\epsilon\cos\Omega)\sin\Omega$
$\displaystyle\quad+\frac{R(\nabla\Omega)^{2}}{a^{2}}(\cos\Omega+2\epsilon\cos(2\Omega))+\frac{\partial
U}{\partial\phi}\,.$ (6)

Multiplying the equation of motion (II.1) by $\sin\Omega$ and averaging over
coherent oscillations, we obtain

$\displaystyle\ddot{\Omega}+2\dot{R}\dot{\Omega}/R-2\nabla
R\cdot\nabla\Omega/(a^{2}R)-(\nabla^{2}\Omega)/a^{2}=0\,.$ (7)

Similarly, multiplying the equation of motion by $\cos\Omega$ and averaging
over oscillations, we obtain

$\displaystyle\ddot{R}-R\dot{\Omega}^{2}-\nabla^{2}R/a^{2}+R(\nabla\Omega)^{2}/a^{2}+U_{\rm
eff}^{\prime}=0\,,$ (8)

where $U_{\rm eff}^{\prime}\equiv 2\langle\cos\Omega\frac{\partial
U}{\partial\phi}\rangle$. Here, we have assumed that $R$ and $\dot{\Omega}$ do
not change much over the period of oscillations, which is a good approximation
when the potential is dominated by the quadratic term. Furthermore, we can
assume that $\Delta_{H}$ term is small enough to be ignored, which is a good
approximation if $\omega$ is large compared to $H$, where $\omega$ is the
oscillation frequency, because in Eq. (2) $\Delta_{H}\sim H^{2}$ is
effectively a contribution to the frequency squared of the oscillations and so
is negligible if $\omega^{2}\gg H^{2}$. We note that Eqs. (7) and (8) have the
same form as those studied in Ref. [13]. We can thus follow the same procedure
as Ref. [13]. For completeness, we repeat the procedure.

We expand $R$ and $\Omega$ into their background parts and perturbations parts
as

$\displaystyle R(t,\mathbf{x})=R_{0}(t)+\delta
R(t,\mathbf{x})\,,\;\Omega(t,\mathbf{x})=\Omega_{0}(t)+\delta\Omega(t,\mathbf{x})\,.$
(9)

with $R_{0}$ and $\Omega_{0}$ being the solutions of Eqs. (7) and (8). It is
straightforward to show that the perturbations satisfy

$\displaystyle\ddot{\delta
R}-2R_{0}\dot{\Omega}_{0}\dot{\delta\Omega}-\dot{\Omega}_{0}^{2}\delta
R-\frac{\nabla^{2}\delta R}{a^{2}}+U_{\rm eff}^{\prime\prime}\delta R$
$\displaystyle=0\,,$ (10)
$\displaystyle\ddot{\delta\Omega}-\frac{\nabla^{2}\delta\Omega}{a^{2}}-\frac{2\dot{R}_{0}\dot{\Omega}_{0}}{R_{0}^{2}}\delta
R+\frac{2\dot{R}_{0}}{R_{0}}\dot{\delta\Omega}+\frac{2\dot{\Omega}_{0}}{R_{0}}\dot{\delta
R}$ $\displaystyle=0\,,$ (11)

where we assumed that the background solutions $R_{0}$ and $\Omega_{0}$ are
homogeneous and we considered terms only up to the first order in $\delta R$
and $\delta\Omega$. The solution for the growth modes takes the form [1]

$\displaystyle\delta R,\delta\Omega\propto
e^{S(t)-i\mathbf{k}\cdot\mathbf{x}}\,,$ (12)

where we may assume that $s\equiv\dot{S}=\text{constant}$ on timescales short
compared to the expansion time since $|\dot{s}/s|\sim H$ [1]. Thus, assuming
the oscillation time scale is short compared to the expansion time $H^{-1}$,
we obtain

$\displaystyle 0=\left[s^{2}+\frac{k^{2}}{a^{2}}+U_{{\rm
eff}}^{\prime\prime}-\dot{\Omega}_{0}^{2}\right]\left[s^{2}+\frac{k^{2}}{a^{2}}\right]+4s^{2}\dot{\Omega}_{0}^{2}\,.$
(13)

One can solve the equation for $s^{2}$,

$\displaystyle s^{2}$
$\displaystyle=-\frac{k^{2}}{a^{2}}-\frac{3\dot{\Omega}_{0}^{2}+U_{{\rm
eff}}^{\prime\prime}}{2}$
$\displaystyle\quad\pm\left(\frac{|3\dot{\Omega}_{0}^{2}+U_{{\rm
eff}}^{\prime\prime}|}{2}\right)\left(1+\frac{16(k^{2}/a^{2})\dot{\Omega}_{0}^{2}}{(3\dot{\Omega}_{0}^{2}+U_{{\rm
eff}}^{\prime\prime})^{2}}\right)^{1/2}\,.$ (14)

For small $k/a$ values,

$\displaystyle s^{2}$
$\displaystyle\approx-\frac{k^{2}}{a^{2}}-\frac{3\dot{\Omega}_{0}^{2}+U_{{\rm
eff}}^{\prime\prime}}{2}$ (15)
$\displaystyle\pm\left(\frac{|3\dot{\Omega}_{0}^{2}+U_{{\rm
eff}}^{\prime\prime}|}{2}\right)\left(1+\frac{8(k^{2}/a^{2})\dot{\Omega}_{0}^{2}}{(3\dot{\Omega}_{0}^{2}+U_{{\rm
eff}}^{\prime\prime})^{2}}-\frac{36(k^{4}/a^{4})\dot{\Omega}_{0}^{4}}{(3\dot{\Omega}_{0}^{2}+U_{{\rm
eff}}^{\prime\prime})^{4}}\right)\,.$

For a growing mode, $s^{2}>0$, and thus we take the $+$ sign, provided that
$3\dot{\Omega}_{0}^{2}+U_{{\rm eff}}^{\prime\prime}>0$, which gives

$\displaystyle s^{2}$
$\displaystyle\approx\frac{k^{2}}{a^{2}}\frac{1}{3\dot{\Omega}_{0}^{2}+U_{{\rm
eff}}^{\prime\prime}}\left[\dot{\Omega}_{0}^{2}-U_{{\rm
eff}}^{\prime\prime}-16\frac{k^{2}}{a^{2}}\frac{\dot{\Omega}_{0}^{4}}{(3\dot{\Omega}_{0}^{2}+U_{{\rm
eff}}^{\prime\prime})^{2}}\right]\,.$ (16)

The largest possible value for $k^{2}/a^{2}$ is then given by

$\displaystyle\frac{k_{{\rm
max}}^{2}}{a^{2}}\approx\dot{\Omega}_{0}^{2}-U_{{\rm eff}}^{\prime\prime}\,.$
(17)

Note that $\dot{\Omega}_{0}^{2}=U_{{\rm eff}}^{\prime}/R$ from the background
equation of motion. We therefore obtain

$\displaystyle s^{2}\approx\frac{k^{2}}{a^{2}}\left(1-\frac{k^{2}}{k_{{\rm
max}}^{2}}\right)\frac{U_{{\rm eff}}^{\prime}/R-U_{{\rm
eff}}^{\prime\prime}}{3U_{{\rm eff}}^{\prime}/R+U_{{\rm
eff}}^{\prime\prime}}\,.$ (18)

We now can integrate $s$ to obtain the total growth factor,

$\displaystyle
S(k,a(t))=\int_{t_{0}}^{t}dt\,s(k,t)=\int_{a_{0}}^{a(t)}da\,\frac{s(k,a)}{aH}\,.$
(19)

As a condition for fragmentation, we require $S\sim\ln(R_{0}/\delta R_{0})$,
which results in non-linear perturbations, i.e., $\delta R/R\sim 1$. The
primordial inflaton perturbation, $\delta R_{0}/R_{0}\sim 10^{-4}$, can be
used as the initial perturbation. Thus, the condition for growth to non-
linearity and so fragmentation is given by

$\displaystyle S(k,a(t))>10\,.$ (20)

The discussion above is applicable for a general potential. The polynomial
expansion of a general potential is expected to have a leading-order quartic
or cubic interactions for a symmetric potential or an asymmetric potential,
respectively. The fragmentation condition (20) can then be converted into a
bound on the model parameters. In the following two subsections, we discuss
the case of a symmetric and an asymmetric potential respectively and obtain
the condition on the model parameters for fragmentation to occur.

###  II.2 Symmetric potentials

Let us consider

$\displaystyle V(\Phi)=\frac{1}{2}m^{2}\Phi^{2}-A\Phi^{4}\,,$ (21)

where $A>0$. The quadratic term is assumed to be dominant. The leading-order
quartic self-interaction is expected to naturally arise in the polynomial
expansion of a general symmetric potential333 For the potential of the form
$V(\Phi)=\frac{1}{2}m^{2}\Phi^{2}-A|\Phi|^{3}$, one may refer to Ref. [13]. .
The condition for fragmentation to occur with the quartic self-interaction was
studied in detail in Ref. [13]. For completeness, we repeat the calculation
here.

As the potential is symmetric under $\Phi\rightarrow-\Phi$, we may set
$\epsilon=0$ in Eq. (5) and obtain the following effective potential,

$\displaystyle U_{\rm
eff}(R)=\frac{1}{2}m^{2}R^{2}-\frac{3}{4}A\left(\frac{a_{0}}{a}\right)^{3}R^{4}\,.$
(22)

The upper limit of the comoving wavenumber, $k_{\rm max}$, is then given by
Eq. (17),

$\displaystyle k_{\rm max}\approx\sqrt{6A}Ra_{0}^{3/2}a^{-1/2}\,.$ (23)

We note that $k_{\rm max}\propto a^{-1/2}$. For a mode $k<k_{\rm max}$, the
mode is initially growing. Eventually, the mode $k$ will become larger than
$k_{\rm max}$ since $k_{\rm max}$ decreases as $a$ increases. Therefore, the
mode will stop growing.

The maximum possible growth of a mode can be found by first considering
perturbations with $k<k_{\rm max,0}$ at an initial scale factor $a_{0}$. We
then calculate the growth of the perturbations from $a_{0}$ to a later scale
factor $a_{\rm f}$, $S(k,a_{\rm f})$, noting that the perturbation growth for
a given $k$ will stop once $k>k_{\rm max}(a_{\rm f})$. We maximise $S(k,a_{\rm
f})$ with respect to $k$ for each $a_{\rm f}$, corresponding to $k(a_{\rm
f})$, and finally we maximise $S(k(a_{\rm f}),a_{\rm f})$ with respect to
$a_{\rm f}$. In this way we determine the maximum possible growth of a mode
starting at $a_{0}$.

Using Eq. (19), we obtain the growth factor from $a_{0}$ to $a_{\rm f}$ as
follows:

$\displaystyle S(k,a_{\rm
f})=\sqrt{\frac{3AR^{2}}{2m^{2}}}\frac{k}{a_{0}H_{0}}\int_{1}^{x_{\rm
f}}x^{-2}\left(1-\frac{k^{2}}{k_{\rm max,0}^{2}}x\right)^{1/2}dx\,,$ (24)

where $x_{\rm f}\equiv a_{\rm f}/a_{0}$. A conservative analytical result for
the growth factor can be obtained by setting $x=x_{\rm f}$ in the square root
factor, as it overestimates the suppression of the growth factor. Then we
find, with $k_{\rm max,f}\equiv k_{\rm max}(a_{\rm f})$,

$\displaystyle S(k,a_{\rm
f})=\sqrt{\frac{3AR^{2}}{2m^{2}}}\frac{k}{a_{0}H_{0}}\left(1-\frac{k^{2}}{k_{\rm
max,f}^{2}}\right)^{1/2}\left(1-\frac{a_{0}}{a_{\rm f}}\right)\,.$ (25)

Maximising the growth factor with respect to $k$ gives a maximum at $k(a_{\rm
f})$, where

$\displaystyle k(a_{\rm f})=\frac{k_{\rm
max,f}}{\sqrt{2}}=\left(\frac{a_{0}}{a_{\rm f}}\right)^{1/2}\frac{k_{\rm
max,0}}{\sqrt{2}}\,.$ (26)

Thus,

$\displaystyle S(k(a_{\rm f}),a_{\rm
f})=\sqrt{\frac{3AR^{2}}{2m^{2}}}\frac{k_{\rm
max,0}}{2a_{0}H_{0}}\left(\frac{a_{0}}{a_{\rm
f}}\right)^{1/2}\left(1-\frac{a_{0}}{a_{\rm f}}\right)\,.$ (27)

Maximising this with respect to $a_{\rm f}$ gives

$\displaystyle a_{\rm f}=3a_{0}\,.$ (28)

Therefore,

$\displaystyle S_{\rm
max}=\frac{1}{3\sqrt{3}}\sqrt{\frac{3AR^{2}}{2m^{2}}}\frac{k_{\rm
max,0}}{a_{0}H_{0}}=\frac{\sqrt{2}ARM_{\rm P}}{m^{2}}\,,$ (29)

where we used the fact that the potential is dominated by the quadratic term
so that $H_{0}=mR/(\sqrt{6}M_{\rm P})$. We should choose $R\equiv\Phi_{0}$,
where $\Phi_{0}$ is the amplitude of the field oscillations at $a_{0}$, to be
as large as possible while being consistent with the assumption that the
potential is dominated by the quadratic term. We define the ratio between the
quartic part and the quadratic part of the potential by $r_{S}$,

$\displaystyle
r_{S}=\frac{A\Phi_{0}^{4}}{m^{2}\Phi_{0}^{2}/2}=\frac{2AR^{2}}{m^{2}}\,.$ (30)

In terms of $r_{S}$, we then obtain

$\displaystyle S_{\rm max}=(r_{S}A)^{1/2}\frac{M_{\rm P}}{m}\,.$ (31)

Therefore, the fragmentation condition (20), $S_{\rm max}>10$, will be
satisfied if

$\displaystyle A>1000\left(\frac{0.1}{r_{S}}\right)\left(\frac{m}{M_{\rm
P}}\right)^{2}\,.$ (32)

We consider $r_{S}\approx 0.1$ in order to be consistent with our assumption
that the quadratic potential term is dominant while still making $r_{S}$
sufficiently large to give the smallest possible lower bound on the
coefficient of the quartic potential term for fragmentation to occur 444This
also assumes that $r_{S}>0.1$ at the end of inflation, which is typically
true..

###  II.3 Asymmetric potentials

Next, we consider a typical asymmetric potential which, in the small field
region where inflaton starts to oscillate around the minimum of the potential
after inflation ends, can generically be described by the leading-order cubic
term,

$\displaystyle V(\Phi)=\frac{1}{2}m^{2}\Phi^{2}\pm A\Phi^{3}\,.$ (33)

The quadratic term is assumed to be the dominant term in the potential as
before. The sign of the cubic term is chosen in such a way that $A$ takes a
positive value.

In the case of asymmetric potentials, $\epsilon$ in Eq. (5) takes a non-zero
value. To find $\epsilon$, let us assume that the field reaches maximum at
$\Phi=\Phi_{\pm}$ with $\dot{\Phi}=0$, where
$\Phi_{\pm}=\pm(\Phi_{0}\pm\delta\Phi)$ for the $-$ sign in the potential (33)
and $\Phi_{\pm}=\pm(\Phi_{0}\mp\delta\Phi)$ for the $+$ sign in the potential
(33). Here, $\Phi_{0},\delta\Phi>0$. Note that $\delta\Phi\rightarrow 0$ as
$A\rightarrow 0$. From $V(\Phi_{+})=V(\Phi_{-})$, we find that
$\delta\Phi=A\Phi_{0}^{2}/m^{2}=AR^{2}/m^{2}$, where $\Phi_{0}$ is identified
with $R$. Thus, the parameter $\epsilon$ can be determined as

$\displaystyle\epsilon=\pm\frac{AR}{m^{2}}\,,$ (34)

with the $+$ ($-$) sign for the $-$ ($+$) sign in the potential (33),
representing the deviation from the quadratic term.

The effective potential now takes

$\displaystyle U_{{\rm
eff}}=\frac{1}{2}m^{2}R^{2}-\frac{9A^{2}}{8m^{2}}\left(\frac{a_{0}}{a}\right)^{3/2}R^{4}\,.$
(35)

The upper limit of the comoving wavenumber, $k_{\rm max}$, in this case is
given by

$\displaystyle k_{\rm max}\approx\frac{3AR}{m}a_{0}^{3/4}a^{1/4}\,.$ (36)

We see that $k_{\rm max}\propto a^{1/4}$. Thus, a wavenumber that is initially
larger than the maximum wavenumber, $k>k_{\rm max}$, will eventually become
smaller than $k_{\rm max}$, and the growth of the mode starts. Therefore, the
maximum growth corresponds to a mode $k=k_{\rm max,0}$, for a given $a_{0}$.
The growth factor is given by

$\displaystyle S(k_{\rm max,0},a)=\frac{3AR}{2m^{2}}\frac{k_{{\rm
max,0}}}{a_{0}H_{0}}\int_{1}^{x}x^{-5/4}\sqrt{1-x^{-1/2}}\,dx\,,$ (37)

where $x\equiv a/a_{0}$. We take the limit $x\rightarrow\infty$. Then, the
maximum growth factor is given by

$\displaystyle S_{{\rm max}}=\frac{9\sqrt{6}\pi A^{2}RM_{{\rm P}}}{2m^{4}}\,,$
(38)

where we used the fact that the potential is dominated by the quadratic term
so that $H_{0}=mR/(\sqrt{6}M_{{\rm P}})$.

Therefore, the fragmentation condition (20), $S_{\rm max}>10$, will be
satisfied if

$\displaystyle\frac{A^{2}RM_{{\rm P}}}{m^{4}}>\frac{20}{9\sqrt{6}\pi}\approx
0.29\,.$ (39)

We may take the value of $R$ as large as possible while keeping the cubic term
in the potential smaller than the quadratic term. We parametrise it as
$r_{A}$, similarly to the $r_{S}$ in the symmetric case (30),

$\displaystyle R=r_{A}\frac{m^{2}}{2A}\,.$ (40)

Thus, the condition for growth to non-linearity and so fragmentation (39) can
be written as

$\displaystyle A>\frac{400}{9\sqrt{6}\pi}\left(\frac{m^{2}}{M_{{\rm
P}}}\right)\left(\frac{0.1}{r_{A}}\right)\approx
5.78\left(\frac{m^{2}}{M_{{\rm P}}}\right)\left(\frac{0.1}{r_{A}}\right)\,.$
(41)

Similarly to the symmetric potential case, we take $r_{A}\approx 0.1$ in order
to be consistent with our assumption that the quadratic potential term is
dominant while giving the smallest possible lower bound on the coefficient of
the cubic potential term for fragmentation to occur.

###  II.4 Summary of analytical conditions

We summarise the general analytical fragmentation conditions:

  * •

Symmetric potentials $V=\frac{1}{2}m^{2}\Phi^{2}-A\Phi^{4}$:

$\displaystyle A>1000\left(\frac{m}{M_{\rm
P}}\right)^{2}\left(\frac{0.1}{r_{S}}\right)\,.$ (42)

  * •

Asymmetric potentials $V=\frac{1}{2}m^{2}\Phi^{2}\pm A\Phi^{3}$:

$\displaystyle A>\frac{400}{9\sqrt{6}\pi}\left(\frac{m^{2}}{M_{{\rm
P}}}\right)\left(\frac{0.1}{r_{A}}\right)\,.$ (43)

These are the main results of the paper.

Strictly speaking, these conditions are sufficient conditions for
fragmentation, as they show that if the condensate forms it will fragments,
hence fragmentation is guaranteed if our conditions are satisfied. In our
analysis, we are restricting the initial value of the field to be close enough
to the minimum of the potential that the potential is dominated by the
quadratic term. However, it is possible that fragmentation could occur more
rapidly and at larger field values due to tachyonic preheating, in which case
it is possible that fragmentation could still occur even if our conditions are
not satisfied. In practice, as we will demonstrate via a range of examples
below, we find that quite generally our conditions accurately predict the
conditions under which fragmentation occurs in numerical simulations.

In the next section we will apply our analytical conditions to various
examples of inflation models and compare them with the known numerical results
in order to demonstrate the effectiveness of our analytic fragmentation
conditions.

##  III Applications

###  III.1 $\alpha$-attractor $T$-model

The $\alpha$-attractor $T$-model [14, 15, 16] with $n=1$ has the following
potential in the Einstein frame:

$\displaystyle V=\lambda\tanh^{2}\left(\frac{\Phi}{\sqrt{6\alpha}M_{\rm
P}}\right)\,.$ (44)

Around the minimum, the potential can be expanded as

$\displaystyle V\approx\frac{\lambda}{6\alpha}\left(\frac{\Phi}{M_{\rm
P}}\right)^{2}-\frac{\lambda}{54\alpha^{2}}\left(\frac{\Phi}{M_{\rm
P}}\right)^{4}\,.$ (45)

We thus find, by comparing with Eq. (21),

$\displaystyle m=\sqrt{\frac{\lambda}{3\alpha M_{\rm P}^{2}}}\,,\quad
A=\frac{\lambda}{54\alpha^{2}M_{\rm P}^{4}}\,.$ (46)

Therefore, by substituting $m$ and $A$ into Eq. (42), we see that the
fragmentation condition is satisfied if

$\displaystyle\alpha\lesssim 5.6\times
10^{-5}\left(\frac{r_{S}}{0.1}\right)\,.$ (47)

Thus, in the $\alpha$-attractor $T$-model, fragmentation of the inflaton
condensate will occur for $\alpha\lesssim 5\times 10^{-5}$. Our analytical
result is in agreement with the numerical analysis of Ref. [12], which
demonstrates fragmentation for an explicit example with $\alpha\approx
10^{-5}$, but finds no fragmentation for larger values of $\alpha$. A recent
analytical and numerical study of tachyonic preheating in plateau inflation
models has also obtained a similar bound [17, 18].

###  III.2 $\alpha$-attractor $E$-model

The Einstein-frame potential of the $\alpha$-attractor $E$-model [14, 15, 16]
with $n=1$ is given by

$\displaystyle
V=\lambda\left[1-\exp\left(-\sqrt{\frac{2}{3\alpha}}\frac{\Phi}{M_{\rm
P}}\right)\right]^{2}\,.$ (48)

The potential can be expanded around the minimum as

$\displaystyle V\approx\frac{2\lambda\Phi^{2}}{3\alpha M_{\rm
P}^{2}}-\frac{2}{3}\sqrt{\frac{2}{3}}\frac{\lambda}{\alpha^{3/2}}\frac{\Phi^{3}}{M_{\rm
P}^{3}}\,.$ (49)

Comparing with Eq. (33), we find

$\displaystyle m=\sqrt{\frac{4\lambda}{3\alpha M_{\rm P}^{2}}}\,,\quad
A=\frac{2}{3}\sqrt{\frac{2}{3}}\frac{\lambda}{\alpha^{3/2}M_{\rm P}^{3}}\,.$
(50)

Therefore, by substituting $m$ and $A$ into Eq. (43), we see that the
fragmentation condition is satisfied if

$\displaystyle\alpha\lesssim 5\times
10^{-3}\left(\frac{r_{A}}{0.1}\right)^{2}\,.$ (51)

Thus, in the $\alpha$-attractor $E$-model, fragmentation of the inflaton
condensate will occur for $\alpha\lesssim 5\times 10^{-3}$. This analytical
result agrees with a numerical analysis performed in Ref. [19], where it is
concluded that fragmentation will occur for $\alpha\lesssim 10^{-3}$.

###  III.3 Starobinsky $R^{2}$ model

Starobinsky’s $R^{2}$ inflation model [20] has the following potential in the
Einstein frame,

$\displaystyle
V=\lambda\left[1-\exp\left(-\sqrt{\frac{2}{3}}\frac{\Phi}{M_{\rm
P}}\right)\right]^{2}\,.$ (52)

We note that this potential can be obtained from the $\alpha$-attractor
$E$-model potential (48) by setting $\alpha=1$. From Eq. (51), we see that
$\alpha=1$ does not satisfy the fragmentation condition. Therefore, we
conclude that the fragmentation does not occur in the $R^{2}$ model. This is
consistent with the results of Ref. [21].

###  III.4 Palatini $R^{2}$ model with a quadratic potential

Let us consider the quadratic inflation model in the Palatini formulation with
an $R^{2}$ term added [22, 23]. The action is given in the Jordan frame by

$\displaystyle S$ $\displaystyle=\int d^{4}x\,\sqrt{-g_{\rm
J}}\,\bigg{[}\frac{M_{\rm P}^{2}}{2}g_{\rm J}^{\mu\nu}R_{{\rm
J}\mu\nu}+\frac{\alpha^{2}}{4}(g_{\rm J}^{\mu\nu}R_{{\rm J}\mu\nu})^{2}$
$\displaystyle\qquad\qquad\qquad\qquad-\frac{1}{2}g^{\mu\nu}_{\rm
J}\partial_{\mu}\varphi\partial_{\nu}\varphi-\frac{1}{2}m^{2}\varphi^{2}\bigg{]}\,,$
(53)

where we put the subscript J to denote the Jordan frame. The action can be
brought into the Einstein frame via Weyl rescaling555 As our aim is to apply
our general analytical conditions to a specific model, we do not discuss the
detailed computational steps. Readers may refer to Refs. [22, 23]. ,

$\displaystyle S=\int d^{4}x\,\sqrt{-g_{\rm E}}\,\left[\frac{M_{\rm
P}^{2}}{2}g_{\rm E}^{\mu\nu}R_{{\rm E}\mu\nu}-\frac{1}{2}g^{\mu\nu}_{\rm
E}\partial_{\mu}\Phi\partial_{\nu}\Phi-V\right]\,,$ (54)

where the subscript E stands for the Einstein frame, $\Phi$ is the canonically
normalised field, and $V$ is the Einstein-frame potential. Near $\varphi=0$,
the canonically normalised field can be approximated as $\Phi\approx\varphi$,
and the Einstein-frame potential is given by [24]

$\displaystyle V\approx\frac{1}{2}m^{2}\Phi^{2}-\alpha\left(\frac{m}{M_{\rm
P}}\right)^{4}\Phi^{4}\,.$ (55)

Thus, it belongs to the symmetric case (21) with

$\displaystyle A=\alpha\left(\frac{m}{M_{\rm P}}\right)^{4}\,.$ (56)

Substituting the expression of $A$ into Eq. (42), we see that the
fragmentation condition is satisfied if

$\displaystyle\alpha\left(\frac{m}{M_{\rm
P}}\right)^{2}>1000\left(\frac{0.1}{r_{S}}\right)\,.$ (57)

To match the magnitude of the primordial curvature power spectrum, $2\times
10^{-9}$, the inflaton mass needs to be $m\simeq 1.4\times 10^{13}$ GeV. The
prediction for the fragmentation condition is then [24]

$\displaystyle\alpha\gtrsim 2.94\times
10^{13}\left(\frac{0.1}{r_{S}}\right)\,.$ (58)

The same lower bound of $10^{13}$ was later found in Ref. [23] by performing a
numerical analysis of tachyonic preheating in this model, and more recently by
an analytical and numerical study in Ref. [17].

###  III.5 Higgs Inflation

In the previous sections we considered models for which a numerical analysis
of fragmentation already exists, and we demonstrated that our simple
analytical conditions can reproduce the conditions for fragmentation that were
previously obtained numerically. In this section, we consider examples for
which no numerical analysis at present exists, namely Higgs Inflation with a
symmetry-breaking potential in both the metric and Palatini formalisms, where
the inflaton is oscillating around the symmetry-breaking minimum of its
potential. We derive the analytical conditions for fragmentation and then
confirm numerically that these conditions are correct.

Higgs Inflation is described the following action in the Jordan frame:

$\displaystyle S$ $\displaystyle=\int d^{4}x\,\sqrt{-g_{\rm
J}}\,\bigg{[}\frac{M_{\rm P}^{2}}{2}F(\varphi)g_{\rm J}^{\mu\nu}R_{{\rm
J}\mu\nu}$ $\displaystyle\qquad\qquad\qquad\qquad-\frac{1}{2}g_{\rm
J}^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi-V_{\rm
J}(\varphi)\bigg{]}\,,$ (59)

where the subscript J denotes the Jordan frame,
$F(\varphi)=(M^{2}+\xi\varphi^{2})/M_{\rm P}^{2}$ is the non-minimal coupling
term with $M$ being a mass parameter, which reproduces the Planck mass at
today, i.e., $M^{2}+\xi v^{2}\equiv M_{\rm P}^{2}$, $v$ is the vacuum
expectation value of the $\varphi$ field, and $V_{\rm
J}(\varphi)=\lambda(\varphi^{2}-v^{2})^{2}/4$ is the Higgs potential in the
Jordan frame. In the Einstein frame, the action is given by

$\displaystyle S=\int d^{4}x\,\sqrt{-g_{\rm E}}\left[\frac{M_{\rm
P}^{2}}{2}g_{\rm E}^{\mu\nu}R_{{\rm E}\mu\nu}-\frac{1}{2}g_{\rm
E}^{\mu\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi-V_{\rm E}(\varphi)\right]\,,$
(60)

where the subscript E denotes the Einstein frame, $\Phi$ is the canonically
normalised field, and $V_{\rm E}$ is the scalar potential in the Einstein
frame. The Einstein-frame field and potential are related to the Jordan-frame
field and potential as

$\displaystyle\frac{d\Phi}{d\varphi}$
$\displaystyle=\sqrt{\frac{1}{F}+\sigma\frac{3M_{\rm
P}^{2}}{2F^{2}}\left(\frac{dF}{d\varphi}\right)^{2}}\,,$ (61) $\displaystyle
V_{\rm E}$ $\displaystyle=\frac{V_{\rm J}}{F^{2}}\,,$ (62)

where $\sigma$ parametrises the metric ($\sigma=1$) and the Palatini
($\sigma=0$) formulations. We shall omit the subscript E in the following. For
inflationary physics of the Higgs Inflation in the metric and the Palatini
formulations, see, e.g., Refs. [25, 26, 27].

Assuming that $|\varphi-v|\ll v$, we can expand $V(\varphi)$ as

$\displaystyle V(\varphi)$
$\displaystyle=\frac{\lambda}{4F^{2}}(\varphi+v)^{2}(\varphi-v)^{2}$
$\displaystyle=\frac{\lambda}{4F^{2}}[(\varphi-v)+2v]^{2}(\varphi-v)^{2}$
$\displaystyle\approx\frac{\lambda
v^{2}}{F^{2}}(\varphi-v)^{2}\left(1+\frac{\varphi-v}{v}\right)\,.$ (63)

We can also expand $F$ as

$\displaystyle M_{\rm P}^{2}F$ $\displaystyle=M_{\rm
P}^{2}+\xi(\varphi+v)(\varphi-v)$ $\displaystyle=M_{\rm
P}^{2}+\xi[2v+(\varphi-v)](\varphi-v)\,,$ (64)

and thus we find

$\displaystyle\frac{1}{F^{2}}\approx 1-\frac{4\xi v}{M_{\rm
P}^{2}}(\varphi-v)-\frac{2\xi}{M_{\rm P}^{2}}\left(1-\frac{6\xi v^{2}}{M_{\rm
P}^{2}}\right)(\varphi-v)^{2}\,.$ (65)

Thus, to leading order in $(\varphi-v)$, the potential $V(\varphi)$ can be
expressed as

$\displaystyle V(\varphi)\approx\lambda
v^{2}(\varphi-v)^{2}\left[1-\left(\frac{4\xi v^{2}}{M_{\rm
P}^{2}}-1\right)\frac{\varphi-v}{v}\right]\,.$ (66)

For $|\varphi-v|\ll v$, the relation between $\varphi$ and $\Phi$ becomes

$\displaystyle\Phi\approx\sqrt{1+\frac{6\sigma\xi^{2}v^{2}}{M_{\rm
P}^{2}}}(\varphi-v)\,,$ (67)

where we define $\Phi=0$ at $\varphi=v$. This follows since, to leading order
in $(\varphi-v)$, we can set $\varphi=v$ on the right-hand side of Eq. (61).
Thus, in terms of $\Phi$, the Einstein-frame potential for $|\varphi-v|\ll v$
becomes

$\displaystyle V(\Phi)=\frac{\lambda v^{2}}{1+6\sigma\xi^{2}v^{2}/M_{\rm
P}^{2}}\Phi^{2}\mp\frac{\lambda v|4\xi v^{2}/M_{\rm
P}^{2}-1|}{(1+6\sigma\xi^{2}v^{2}/M_{\rm P}^{2})^{3/2}}\Phi^{3}\,.$ (68)

This is of the form of Eq. (33) with

$\displaystyle m^{2}=\frac{2\lambda v^{2}}{1+6\sigma\xi^{2}v^{2}/M_{\rm
P}^{2}}\,,\quad A=\frac{\lambda v|4\xi v^{2}/M_{\rm
P}^{2}-1|}{(1+6\sigma\xi^{2}v^{2}/M_{\rm P}^{2})^{3/2}}\,.$ (69)

Applying the fragmentation condition for an asymmetric potential, (43), we
obtain the general condition for fragmentation in Higgs Inflation with a
symmetric-breaking potential as follows:

$\displaystyle\frac{v}{M_{\rm
P}}\lesssim\frac{9\pi}{400}\sqrt{\frac{3}{2}}\left(\frac{r_{A}}{0.1}\right)\frac{|4\xi
v^{2}/M_{\rm P}^{2}-1|}{(1+6\sigma\xi^{2}v^{2}/M_{\rm P}^{2})^{1/2}}\,.$ (70)

(i) Case $M^{2}\ll\xi v^{2}$: In this case, $M_{\rm P}^{2}\approx\xi v^{2}$,
corresponding to non-minimally coupled inflation in the induced gravity limit.
In this limit, the condition for fragmentation becomes

$\displaystyle\frac{(1+6\sigma\xi)^{1/2}}{\sqrt{\xi}}\lesssim
0.26\left(\frac{r_{A}}{0.1}\right)\,.$ (71)

For the metric case, $\sigma=1$, and, assuming that $\xi\gg 1$, the condition
becomes

$\displaystyle\sqrt{6}\lesssim 0.26\left(\frac{r_{A}}{0.1}\right)\,,$ (72)

which is not satisfied. Therefore, fragmentation does not occur for metric
Higgs Inflation in the induced gravity limit. For the Palatini case,
$\sigma=0$, and the condition becomes

$\displaystyle\xi\gtrsim 14.83\left(\frac{0.1}{r_{A}}\right)^{2}\,.$ (73)

This is easily satisfied since we expect that $\xi\gg 1$. Therefore, we expect
that the inflaton condensate will fragment in the case of Palatini Higgs
Inflation in the induced gravity limit, but it will not fragment in the case
of metric Higgs Inflation in this limit.

(ii) Case $M^{2}\gg\xi v^{2}$: In this case, $M_{\rm P}^{2}\approx M^{2}$,
corresponding to conventional Higgs Inflation. In addition, since
$|\varphi-v|\ll v$, we also have $\xi\varphi^{2}\ll M_{\rm P}^{2}$. In this
limit, the condition for fragmentation becomes

$\displaystyle\frac{v}{M_{\rm P}}\lesssim
0.087\left(\frac{0.1}{r_{A}}\right)\left(1+\frac{6\sigma\xi^{2}v^{2}}{M_{\rm
P}^{2}}\right)^{-1/2}\,.$ (74)

For the metric case, $\sigma=1$, and the condition becomes

$\displaystyle\frac{v}{M_{\rm P}}\left(1+\frac{6\xi^{2}v^{2}}{M_{\rm
P}^{2}}\right)^{1/2}\lesssim 0.087\left(\frac{0.1}{r_{A}}\right)\,.$ (75)

There are two cases, corresponding to $6\xi^{2}v^{2}/M_{\rm P}^{2}\ll 1$ and
$6\xi^{2}v^{2}/M_{\rm P}^{2}\gg 1$. In the case where $6\xi^{2}v^{2}/M_{\rm
P}^{2}\ll 1$, the condition for the fragmentation becomes

$\displaystyle\frac{v}{M_{\rm P}}\lesssim
0.087\left(\frac{0.1}{r_{A}}\right)\,.$ (76)

In this case, the non-minimal coupling plays no role and we simply have the
condition for fragmentation for a minimally coupled scalar with a broken-
symmetry potential. This is satisfied as long as $v$ is not very close to the
Planck scale.

In the case where $6\xi^{2}v^{2}/M_{\rm P}^{2}\gg 1$, the condition for
fragmentation becomes

$\displaystyle\frac{\xi v^{2}}{M_{\rm P}^{2}}\lesssim
0.035\left(\frac{0.1}{r_{A}}\right)\,.$ (77)

In this case, the non-minimal coupling plays a role, even though
$\xi\varphi^{2}\ll M_{\rm P}^{2}$. This is because once $\varphi>M_{\rm
P}/(\sqrt{6}\xi)$, the inflaton kinetic term in the Einstein frame in the
metric case is modified. The allowed range of $\xi v^{2}/M_{\rm P}^{2}$ is
$1\gg\xi v^{2}/M_{\rm P}^{2}\gg 1/(6\xi)$. Therefore, the fragmentation
condition will be satisfied for most of the allowed range of $\xi v^{2}/M_{\rm
P}^{2}$ except for when $\xi v^{2}$ approaches $M_{\rm P}^{2}$.

In the case of Palatini Higgs Inflation, $\sigma=0$, and the fragmentation
condition becomes

$\displaystyle\frac{v}{M_{\rm P}}\lesssim
0.087\left(\frac{0.1}{r_{A}}\right)\,.$ (78)

This is simply the fragmentation condition for a minimally coupled scalar with
a broken-symmetry potential. This is expected as in the limit $\xi v^{2}\ll
M_{\rm P}^{2}$ (and so $\xi\varphi^{2}\ll M_{\rm P}^{2}$), the non-minimal
coupling plays no role in Palatini inflation.

Figure 1: Lattice simulation for the Palatini Higgs Inflation model. We
present the quantity $\rho/\langle\rho\rangle=10$ at $t=500m^{-1}$. The choice
of parameters are as follows: $N_{{\rm grid}}=128^{3}$, $L=50m^{-1}$,
$\xi=1.0\times 10^{3}$, and $\sqrt{\xi}v/M=1.0\times 10^{3}$. We used the
field value at the end of inflation, $\Phi_{{\rm end}}=(M_{{\rm
P}}/2\sqrt{\xi})\sinh^{-1}(4\sqrt{2\xi})$, as the initial condition.

Figure 2: Lattice simulation for the Higgs Inflation model. We present the
quantity $\rho/\langle\rho\rangle=10$ at $t=40m^{-1}$. The choice of
parameters are as follows: $N_{{\rm grid}}=128^{3}$, $L=30m^{-1}$,
$\xi=1.0\times 10^{4}$ and $\sqrt{\xi}v/M=0.1$. We used the field value at the
end of inflation, $\Phi_{{\rm end}}=(M_{{\rm
P}}/2\sqrt{\xi})\sinh^{-1}(4\sqrt{2\xi})$, as the initial condition.

In summary, we find significantly new results for the case of metric and
Palatini Higgs Inflation with a broken-symmetry potential in the induced
gravity limit, and for metric Higgs Inflation with a broken-symmetry potential
in the limit where $v>M_{\rm P}/(\sqrt{6}\xi)$, with all other cases reducing
to a conventional minimally coupled scalar with a symmetry-breaking potential.
In particular, we find that fragmentation will occur for Palatini Higgs
Inflation in the induced gravity limit, but will not occur for metric Higgs
Inflation in this limit.

To our knowledge, there is no numerical study of this particular setup. In
order to verify our analytical study, we perform numerical simulations in 1+3
dimensions by using the public lattice simulation code LATTICEEASY [28]. We
present results of numerical simulations for the following two cases:

  * •

Palatini Higgs inflation in the $\xi v^{2}\gg M^{2}$ limit and

  * •

Metric and Palatini Higgs inflation with $\xi v^{2}\ll M^{2}$, together with
the condition $\xi^{2}v^{2}\ll M^{2}$.

In both cases, our analytical results predict that fragmentation will occur.
Note that, in the second case, the fragmentation conditions are same in both
formalisms as in this limit both models reduce to a minimally coupled scalar
with a symmetry-breaking potential, and there is no difference between the
formalisms.

For the numerical simulation, we set $N_{{\rm grid}}=128^{3}$, $L=50m^{-1}$
($30m^{-1}$), and $\delta t=0.1m^{-1}$ for the first (second) case, where
$m\equiv\sqrt{2\lambda}v$. For both cases, we used the field value at the end
of inflation, $\Phi_{{\rm end}}=(M_{{\rm
P}}/2\sqrt{\xi})\sinh^{-1}(4\sqrt{2\xi})$ [26], as the initial condition. The
result of the numerical simulation for the case of $\xi v^{2}/M^{2}\gg 1$ is
shown in Fig. 2. We present the energy density $\rho/\langle\rho\rangle=10$,
where $\langle\rho\rangle$ is the averaged energy density over the lattice. In
Fig. 2, we plotted the energy density $\rho/\langle\rho\rangle=10$ for the
case of $\xi v^{2}/M^{2}\ll 1$. We see that the inflaton condensate fragments
after inflation ends in both cases. Our analytical results are thus in
agreement with the numerical analysis.

##  IV Conclusion

In this paper, we have derived general analytical conditions under which the
inflaton condensate will fragment for the case of both symmetric and
asymmetric potentials. The robustness of our results was demonstrated by
applying our analytical fragmentation conditions to a range of models for
which the result is known numerically, including the $\alpha$-attractor $T$
and $E$ models, Starobinsky’s $R^{2}$ model, and the Palatini $R^{2}$ model
with a quadratic potential. In all cases, we find that the analytically
predicted condition on the model parameters for fragmentation to occur are in
complete agreement with the results of the numerical analyses.

In addition, we have applied our results to Higgs Inflation with a broken-
symmetry potential in both the metric and Palatini formulations and derived a
general condition for fragmentation to occur in these models. We then carried
out numerical simulations of these models using LATTICEEASY and found complete
agreement between our analytical predictions for fragmentation to occur and
the results of the numerical simulations.

The conditions we have derived provide a quick and simple method to check
whether any model which can support non-topological soliton solutions (which
requires that the scalars have a mass at the potential minimum and therefore
that the potential is approximately quadratic at the minimum) will undergo
fragmentation and to determine the range of model parameters for which this is
possible.

The physics of the inflaton condensate fragmentation has many interesting
phenomenological consequences, including the formation of primordial black
holes and gravitational wave signals. The evolution dynamics generally
requires extensive numerical lattice simulations. We believe that our findings
may serve as a starting point for such numerical analyses.

## Acknowledgements

J.K. would like to thank Kaloian D. Lozanov for useful discussion on the use
of LATTICEEASY.

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# Predicting and Visualizing Daily Mood of People Using Tracking Data of
Consumer Devices and Services

Christian Reiser  University of Stuttgart
Stuttgart, Germany
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

Users can easily export personal data from devices (e.g., weather station and
fitness tracker) and services (e.g., screentime tracker and commits on GitHub)
they use but struggle to gain valuable insights. To tackle this problem, we
present the self-tracking meta app called InsightMe, which aims to show users
how data relate to their wellbeing, health, and performance. This paper
focuses on mood, which is closely associated with wellbeing. With data
collected by one person, we show how a person’s sleep, exercise, nutrition,
weather, air quality, screentime, and work correlate to the average mood the
person experiences during the day. Furthermore, the app predicts the mood via
multiple linear regression and a neural network, achieving an explained
variance of 55% and 50%, respectively. We strive for explainability and
transparency by showing the users p-values of the correlations, drawing
prediction intervals. In addition, we conducted a small A/B test on
illustrating how the original data influence predictions. The source
code111Sourcecode app: https://github.com/christianreiser/insightme
Sourcecode backend: https://github.com/christianreiser/correlate and
app222Playstore: https://play.google.com/store/apps/details?id=com.insightme
Appstore: https://apps.apple.com/de/app/insightme/id1522480765 are available
online.

###### Index Terms:

machine learning, visualization, mood, mobile, quantified-self

##  I Introduction

We know that our environment and actions substantially affect our mood,
health, intellectual and athletic performance. However, there is less
certainty about how much our environment (e.g., weather, air quality, noise)
or behavior (e.g., nutrition, exercise, meditation, sleep) influence our
happiness, productivity, sports performance, or allergies. Furthermore,
sometimes, we are surprised that we are less motivated, our athletic
performance is poor, or disease symptoms are more severe.

This paper focuses on daily mood. Although negative moods have essential
regulating functions like signaling the need for help or avoiding harmful
behavior like going on buying sprees, taking risks, or making foolish
investments [1], other studies show that bad moods can also have unfavorable
consequences like less resistance to temptations, especially to unhealthy
food[2], impaired learning capabilities[3], and inhibited creative
thinking[4].

Our ultimate goal is to know which variables causally affect our mood to take
beneficial actions. However, causal inference is generally a complex topic and
not within the scope of this paper. Hence, we started with a system that
computes how past behavioral and environmental data (e.g., weather, exercise,
sleep, and screentime) correlate with mood and then use these features to
predict the daily mood via multiple linear regression and a neural network.
The system explains its predictions by visualizing its reasoning in two
different ways. Version A is based on a regression triangle drawn onto a
scatter plot, and version B is an abstraction of the former, where the slope,
height, and width of the regression triangle are represented in a bar chart.
We created a small A/B study to test which visualization method enables
participants to interpret data faster and more accurately.

The data used in this paper come from inexpensive consumer devices and
services which are passive and thus require minimal cost and effort to use.
The only manually tracked variable is the average mood at the end of each day,
which was tracked via the app.

##  II Related Work

This section provides an overview of relevant work, focusing on mood
prediction (II-A) and related mobile applications with tracking, correlation,
or prediction capabilities. (II-B).

###  II-A Prediction of Mood

In the last decade, affective computing explored predicting mood, wellbeing,
happiness, and emotion from sensor data gathered through various sources.

Harper and Southern[5] investigate how a unimodal heartbeat time series,
measured with a professional EGC device, can predict emotional valence when
the participant is seated.

Choudhury et al. detect major depressive disorder of Twitter users who posted
more than 4500 tweets on average with an average accuracy of ~70%[6].

Several studies estimated mood, stress, and health with data from multimodal
wearable sensors, a smartphone app, and daily manually reported behaviors such
as academic activities and exercise, claiming maximum accuracies of 68.48%[7],
74.3%[8], 82.52%[9], all with a baseline of 53.94. Another study scored 78.7%,
with a baseline of 50.4%[10].

All the studies mentioned above are less practical for non-professional users
committed to long-term everyday usage because expensive professional
equipment, time-consuming manual reporting of activity durations, or frequent
social media behavior is needed. Therefore, we focus on cheap and passive data
sources, requiring minimal attention in everyday life.

One study meeting these criteria shows that mood can be predicted from passive
data, specifically, keyboard and application data of mobile phones with a
maximum accuracy of 66.59% (62.65% if without text)[11]. However, this project
simplifies mood prediction to a classification problem with only three
classes. Furthermore, compared to a high baseline of more than 43% (due to
class imbalance), the prediction accuracy of about 66% is relatively low.

###  II-B Related Apps

Several apps allow users to track their moods but lack correlation and
prediction features [12][13] [14][15] [16][17][18][19]. Some health apps allow
correlating symptoms with food and behavior but still do not allow for
prediction [20][21].

Apps capable of prediction are, and [22] which estimates the activity of the
sympathetic nervous system from heart rate and heart rate variability and [23]
which calculates stress, energy, and productivity levels from heart data as
well[23]. Further, FitBit allows for logging how the user feels and computes a
‘Stress Management’ score taking the manually logged feeling, data about
sleep, electro-dermal activity, and exercise into account[24]. While these
apps are capable of prediction, they are specialized in a few data types,
which exclude mood, happiness, or wellbeing.

The product description of the smartwatch app ‘Happimeter’ states to “get your
body signals to predict your mood with machine learning”[25]. However, we
could not test the app as it requires the operating system wearOS and the app
has a user rating of only 1.5 of 5 stars on Google Play and was not updated
for more than a year[25].

##  III Data sources

This project aims to use non-intrusive, inexpensive sensors and services that
are robust and easy to use for a few years. Meeting these criteria, we tracked
one person with a FitBit Sense smartwatch, indoor and outdoor weather
stations, screentime logger, external variables like moon illumination,
season, day of the week, manual tracking of mood, and more. The reader can
find a list of all data sources and explanations in the appendix (Section
VIII).

##  IV Data Exploration and Processing

This section describes how the data processing pipeline aggregates raw data,
imputes missing data points, and exploits the past of the time series.
Finally, we explore conspicuous patterns of some features.

###  IV-A Pre-processing

The sampling rates of the raw data typically vary between five minutes (e.g.,
heart rate) to about weekly (e.g., Bodyweight and $VO_{2}Max$).

####  IV-A1 Data Aggregation

The goal is to have a sampling rate of one sample per day. In most cases, the
sampling rate is greater than $1/24h$, and we aggregate the data to daily
intervals by taking the sum, fifth percentile, 95th percentile, and median. We
use these percentiles instead of the minimum and maximum because they are less
noisy and found them more predictive.

####  IV-A2 Data Imputation

The sampling frequency of Bodyweight and $VO_{2}Max$ is usually $<1/24h$.
Because Bodyweight and $VO_{2}Max$ represent physical entities that change
relatively slowly, we assume a linear change, allowing linear interpolation of
consecutive measurements to obtain the 24h frequency. If there are days or
features where many values are missing, we drop these days or features,
respectively. Otherwise, data imputation fills missing values with the
feature’s average.

####  IV-A3 Time-series

As the dataset is a time series, and yesterday’s features could also affect
today’s mood, we added all of yesterday’s features to the set of today’s
predictors. We also include the mood of the last days until there is no new
significant information about autocorrelation, given the mood of the previous
days. As shown in Figure 1, computing the partial autocorrelation[26]
determines these days, when including all days from the left until the first
insignificant day. In our case, this means the values of one to four days ago.

####  IV-A4 Standardization

Standardization rescales the features to have a mean of 0 and unit variance.

Figure 1: Partial autocorrelation of mood. Values outside of the blue are
within the 95% confidence interval, thus statistically significant.

###  IV-B Data Exploration

The dataset has many outliers because the sensors and services are cheap
consumer devices. For example, the estimated metabolic energy output, shown in
Figure 2 has values at about 1000 kcal and above 4000 kcal.

Figure 2: The distribution of the estimated metabolic energy output. Values
below 1600 kcal and above 3500 kcal are outliers.

Moreover, Figure 3 shows a suspicious $CO_{2}$ spike at 5000 ppm. A closer
look into the raw sensor data depicted in Figure 4 indicates an improbable
plateau at 5000 ppm. The causal explanation is an ending sensor range at 5000
ppm, which falsely counts all values greater than 5000 to 5000 ppm.

Figure 3: Distribution of $CO_{2}$ data. Note the frequent occurrence of
5000ppm as it is the sensor’s maximum range.
$0$$50$$100$$150$$200$$0$$1{,}000$$2{,}000$$3{,}000$$4{,}000$$5{,}000$TimestampCO2
in ppmCO2 Sensor data Figure 4: The actual CO2 level is higher than 5000 ppm,
but the sensor’s maximum is 5000 ppm.

The distribution of the wakeup time looks gaussian except for one suspicious
spike at 320 minutes after midnight. However, an alarm clock at 5:20 am
indicates the plausibility of this spike.

Figure 5: Wakeup time in minutes after midnight. Note the spike at 320, which
is caused by the alarm clock

.

Improbable values in the dataset are not corrected manually because we do not
have access to data in the actual mobile application due to our strict privacy
policy. Instead, we exploit robust statistics by aggregating the data via the
fifth and 95th percentile instead of the maxima. Our experiments have shown
that these percentiles are more predictive than the maxima.

The app computes the Pearson correlation coefficient and p-values between all
attributes. Because comparing each attribute with every other, we correct the
p-values according to the Benjamini-Hochberg procedure[27] to control the
false discovery rate due to multiple testing. We declare a result as
significant for p $<$ 0.05. Users can visually explore the data via a plotted
time series with a seven-day moving average and manually inspect the
relationship between two variables through scatter plots.

###  IV-C Train-test-split

Because there is a temporal dependency between observations, standard cross-
validation, which assigns samples randomly to the train or test set, would
lead to using some data from the future to forecast the past, which is not
possible in the real-life application. We use time-series-splits to avoid this
fallacy. However, many splits are inefficient w.r.t. data use because only the
last split uses all training data. Ultimately, we create only one split, which
is a simple train test split, where the training set contains old data points,
and the test split the most recent ones.

###  IV-D Multiple Linear Regression

The estimation parameters for multiple linear regression are computed via the
training dataset while applying elastic net regularization and sample-
weighting.

####  IV-D1 Regularization

We use a combined $L_{1}$ and $L_{2}$ weight penalty called elastic net
regularization[28] with

$\min_{w}\frac{1}{2n_{\text{samples
}}}\|Xw-y\|_{2}^{2}+\alpha\rho\|w\|_{1}+\frac{\alpha(1-\rho)}{2}\|w\|_{2}^{2}$
(1)

being the objective function to minimize, where $\rho$ is the $L_{1}$ ratio,
$\alpha$ the strength of the penalty terms, $w$ the weights, $X$ the features,
and $y$ the predictions. Specific cases are for $\rho=0$, which simplifies to
ridge regularization and $\rho=1$ to lasso regularization. We search for the
optimal $\rho$ and $\alpha$ via cross-validation.

####  IV-D2 Sample-weighting

We assume that the factors influencing a person’s mood change over time. To
account for it, we use exponential sample-weight decay, as shown in Figure 6.
The formula is

$sample\\_weight_{i}=max(14.7498-13.2869(i+30)^{0.0101585},0),$ (2)

which is exponential decay fitted to a value of 1 for today’s datapoint and
reaches zero after about 82 years. $max(x,0)$ ensures that the sample weight
never becomes a negative value.

Figure 6: Exponential sample-weight decay discounting older data as they might
be less valuable to predict today’s mood.

###  IV-E Neural Network

The neural network has two fully connected hidden layers, each with a leaky
rectified linear unit[29] as an activation function. The first and second
hidden layers have 16 and 8 neurons, respectively. Because we want to regress
on a scalar, the output layer has one unit. AdamW[30] is the optimizer,
minimizing the mean squared error for 4125 epochs with a learning rate of
$10^{-4}$ and a weight decay of one. The number of epochs is determined by
early stopping using cross-validation[31]. We searched for the best neural
network architecture and hyper-parameters manually through cross-validation.

##  V Results

Of 198 variables, 77 correlate significantly with mood. All correlation
coefficients are in Table III in the appendix.

###  V-A Multiple Linear Regression

Elastic net regularization with a penalty strength $\alpha=0.12$ and
$L_{1}$-ratio $\rho=1$ leads to the best prediction performance on our
dataset.

TABLE I: Regression weights. All other features have a weight of zero. Feature | Regression Weight
---|---
HumidInMax() | -0.116
HeartPoints | 0.101
CO2Median() | -0.092
MoodYesterday | 0.065
NoiseMax()Yesterday | -0.053
PressOutMin()Yesterday | 0.050
BodyWeight | 0.046
VitaminDSup | 0.032
DistractingScreentime | -0.009

Table I shows all regression weights $w$ for $w\neq 0$. Note that lasso
regularization selected only nine features to predict the mood. The 95%
prediction interval is $\pm 2.3$ on the original scale from 1 to 9 and $\pm
1.2$ on the standardized scale with unit variance. The average mean squared
error on the standardized test set is 0.45, meaning 55% of the original
variance can be explained. The effect of sample weighting is negligible.

###  V-B Neural Network

The average mean squared error is 0.50 on the standardized test set, meaning
half of the original variance can be explained.

###  V-C Explainability, Visualization, and A/B Test

(a) Users can explore data via time-series with a seven-day moving average
trend-line.

(b) A scatter plot with linear regression and Pearson correlation coefficient
to visualize how variables relate to each other.

Figure 7: Screenshots showing how users can explore data.

Screenshot (a) of Figure 7 shows how the app visualizes each feature via time
series to allow the user to spot changes over time and trends through a seven-
day moving average. Screenshot (b) of Figure 7 shows an example of a scatter
plot enabling exploration of how variables visually relate to each other. In
addition, it draws a linear regression line and indicates the degree of the
linear relationship by visualizing the correlation coefficient in a bar.

Figure 8: Mood prediction: Each row represents the contribution of a feature
that was selected by $L_{1}$ regularization. The row contains a red or green
bar if the contribution is negative or positive, and the size of the bar
indicates the magnitude of the contribution. The black box with whiskers is
the mood estimate with its 95% prediction interval.

Figure 8 shows a black box with whiskers on a scale, which is the mood
estimate with the 95% prediction interval. This provides the users not only
the prediction but also how much they can rely on the accuracy. Above the
prediction in screenshot8, we explain to the user how multiple linear
regression calculates the predictions. Each row represents the contribution of
the selected features. The row contains a red or green bar if the contribution
is negative or positive, respectively, and the size indicates the magnitude of
the contribution. The final mood prediction is simply the sum of all
contributions.

Figure 9: Two methods explain how multiple linear regression computes the
contribution of a single feature on the final prediction. We call the method
(a) ‘bar chart’ because it contains a chart with three bars and (b) ‘triangle
chart’, because it contains a red or green triangle inside a scatter plot.

To understand more about the contribution of a feature, the user can tap on
one and see either a bar chart or a regression triangle drawn onto the scatter
plot.

The bar chart shows that the contribution (green or red bar) of a feature is
the product of the weight of the feature (deep teal) and the difference from
today’s value to its average value (light teal).

The triangle drawn onto the scatter plot shows the same information but with
more context. The triangle’s horizontal line represents the difference between
the average and today’s value of the feature. The triangle’s slope represents
the feature’s weight, which is a regularized regression line of the two
variables. Finally, the vertical length between both lines depicts the
contribution.

The bar and triangle chart contain redundant information. Therefore, we
conducted a small A/B test with 10 participants to determine which chart
conveys the information more accurately and faster.

Seven of the participants are male, and three are female. The age ranges from
19 to 32, and all of them have an engineering background. The participants
answered four single choice questions concerning the bar chart (A) and four
similar but not identical questions about the triangle chart (B). We assured
proper testing by passively observing them. Each participant worked on both
part A and part B; however, 50% of the participants first completed part A and
vice versa to control for the order. We measured the accuracy and time
required to answer the questions of each part and reported their average
w.r.t. the number of participants and number of questions. As shown in Table
II, the ‘bar chart’ results in a slightly higher accuracy of 90% and 43
seconds a marginally faster completion, compared to the ‘regression triangle
chart’ with an accuracy of 85% and 45 seconds. While these results are not
significant, the users also commented that they favor the ‘bar chart’ as the
length of the bar representing the weight is more accessible than the slope of
the triangle, especially if the slope or triangle is small.

TABLE II: Average required time and accuracy of single choice questions about the bar and triangle chart. The required time is relatively long because it includes the time to read the question. | bar chart | triangle chart
---|---|---
average accuracy | 90% | 85%
average required time | 43 sec | 45 sec

##  VI Discussion

####  VI-1 Multicollinearity

Although 77 variables correlate significantly, there is multicollinearity. The
principal component analysis shows in Figure 10 that only 25 components
explain more than 1% of the total variance. Examples of correlated predictors
are

  * •

all the fourfold aggregated variables (i.e., the mean, median, fifth-, and
95th percentile of a variable)

  * •

weather indoor and outdoor

  * •

time in bed and time asleep

  * •

walking minutes, heart points, exertion points

Figure 10: The principal component analysis shows that there are only 25
components that explain more than 1% of the variance, indicating highly
dependent features.

####  VI-2 Multiple linear regression versus neural network

Multiple linear regression performed better on the test set than the neural
network. Neural networks can have an advantage over the linear multiple linear
regression method because they can approximate nonlinear relationships. Still,
the downside is the need for more training data to optimize additional
parameters. In our case, the training set is probably too small, leading to
overfitting and difficulty generalizing to new data. We expect the neural
network will increase its performance with a growing dataset. Besides, an
enhanced architecture and improved hyperparameters could lead to better
predictions of the neural network. An advantage of multiple linear regression
is good explainability, as illustrated in Figure9, which is less intuitive for
neural networks [32].

###  VI-A Limitations

####  VI-A1 Large unexplained variance

Predictions of the multiple linear regression and neural network leave about
half the original variance unexplained. Factors limiting performance are:

  * •

unmeasured variables which influence a person’s mood

  * •

sensor data can be noisy, data imputation of missing values non-optimal

  * •

multiple linear regression assumes a linear relationship between features and
mood and neglects nonlinear mechanisms

  * •

the training set might be too small, especially for the neural network

####  VI-A2 Assumed linear mood scale

Our method of asking the users’ mood on an absolute scale from 1 to 9 assumes
a linear relationship of these values. However, the genuine relationship might
not be linear because there may be a higher degree of change on a comparative
scale than on an absolute scale for extreme values[33]. Comparative surveys
would reduce these biases. However, we decided against them because they
require more of the user’s time.

####  VI-A3 Recency and fading affect bias

Asking the user’s average mood at the end of the day suffers from a potential
recency bias[34], where recent events of the evening have a stronger effect
than more distant events in the morning. Furthermore, it is prone to the
fading affect bias, where negative memories fade faster than positive
ones[35]. Asking for a rating multiple times per day would reduce these
biases; however, we decided against it because it requires more effort and is
less sustainable over long periods.

####  VI-A4 Survey

The results of the survey are not conclusive because of the small number of
participants. Furthermore, the selection of participants might not represent
the actual distribution of the users, w.r.t. age, gender, and background.

####  VI-A5 Causal inference

Features correlating strongly with mood and predictors of multiple linear
regression could potentially causally affect mood. While this could be the
case and interesting to know, correlation does not imply causation, and good
predictors are not necessarily causes. For example, the positive correlation
between mood and exercise could be caused by exercise causing better mood, or
that good mood leads someone to exercise more. Furthermore, both could affect
each other, or there could be a confounder like good weather, which causes
someone to exercise more and improve mood independently of each other. In some
instances, we can remove directions from a causal graph. For example, when
there is a positive correlation between the variable weekend and good mood, it
is unlikely that the good mood causes the day to be a weekend. Unfortunately,
this study does not allow for causal inference. Today, randomized controlled
trials are still the gold standard for establishing causal conclusions[36].

###  VI-B Future Work

####  VI-B1 Stronger feature selection and weaker weight penalty

Figure 11: Sample predictions: Predictions are in red, ground truth values in
green. The blue bar represents the 95% prediction interval. It seems that the
predictions go in the right direction but are too close to the mean at 6.2.

Figure 11, shows sample predictions. It seems that predictions tend to go in
the right direction but are too close to the mean at 6.2. We hypothesize that
the weight penalty is too strong, which pushes predictions to the mean, but
feature selection is critical because the overall regularization strength
$\alpha\neq 0$. Further evidence is elastic’s-net optimal $L_{1}$-ratio of
$\rho=1$, meaning that performance is best with the lowest allowed weight
penalty while maximizing feature selection.

Fortunately, the equation of elastic net regularization, which operates
between the $L_{1}$ and $L_{2}$-norm, can be generalized to the $L_{p}$-norm

$\|x\|_{p}=\left(\sum_{i=1}^{n}\left|x_{i}\right|^{p}\right)^{1/p},$ (3)

with $0<p<\infty$, allowing an even more extreme feature-selection to weight
penalty ratio. Figure 12 shows how feature selection becomes stronger and the
penalty for large parameters weaker as p gets closer to zero. Future work
could explore if prediction performance improves for $p<1$ like applying a
regularization with the $L_{0.5}$-norm. However, this has the downside of
leaving convex optimization[37].

Figure 12: Visualization of regularization with the $L_{p}$-norm. For $p\to
0$, the penalty is proportional to the number of non-zero parameters. For
$p\to\infty$, the penalty has the size of the largest parameter. Elastic net
regularization works only for $1\leq p\leq 2$. Figure cropped and reprinted
following the publisher’s copyright[38].

####  VI-B2 Improved data imputation

A common problem in long-term studies is that sensor data is missing. While we
impute missing values with the average or linear interpolation, we plan to
impute with a deep multimodal autoencoder to enable better mood
prediction[39].

####  VI-B3 Forecasting tomorrow’s mood

This project explored predicting the mood of the same day, but we also plan to
forecast tomorrow’s mood. A study shows it is possible with a mean absolute
error of 10.8 for workers and 17.8 for students, while the mood’s standard
deviation is 17.14[40].

###  VI-C Conclusion

In this work, we presented a meta app that can import data from consumer
devices and services and allows for manual tracking. The app allows the user
to explore data via plotted time-lines and the relationship between variables
through scatter plots and correlation coefficients.

The app predicts the mood or any other chosen target variable by automatically
aggregating the data into daily features and selecting the best ones to
predict the user’s mood. This project shows that multiple linear regression
can explain more than half of the original variance.

We strive for transparency by conveying information about confidence through
p-values and prediction intervals and created the app as an open-source
project.

We hope the app helps users understand themselves better and improve their
wellbeing, health, and physical & cognitive performance.

##  VII Acknowledgments

I thank Benedikt V Ehinger and Tanja Blascheck for helpful discussions and
suggestions.

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##  VIII Appendix

Data sources are:

  * •

Moon Illumination

  * •

daytime (the time between sunrise and sunset, which is shorter in the
fall/winter and longer in the spring/summer)

  * •

OpenWeatherMap, which gives outside weather measurements GPS location

    * –

temperature

    * –

heat index (human-perceived equivalent temperature, which includes humidity
and windspeed)

    * –

air pressure

    * –

humidity

    * –

windspeed

    * –

cloud cover

    * –

precipitation (rain, snow)

  * •

Phone Location

    * –

longitude

    * –

latitude

  * •

Day of the week (Monday, Tuesday, etc.)

  * •

the severity of covid lockdown measures

  * •

commits on GitHub

  * •

Google Fit

    * –

Heart Points (duration and intensity exercise estimate)

    * –

walking minutes

    * –

running minutes

    * –

meditating minutes

    * –

Sleep efficiency

    * –

Sleep starting time

    * –

sleep end time

    * –

minutes in bed

    * –

minutes asleep

    * –

binary: was a nap taken?

    * –

minutes of light sleep

    * –

minutes of deep sleep

    * –

minutes of REM sleep

    * –

minutes awake in bed

  * •

Fitbit Charge 3 Smartwatch, which estimate:

    * –

steps taken

    * –

burned calories

    * –

heart rate

    * –

resting heart rate

    * –

VO2Max (maximum rate of oxygen consumption)

    * –

sleep revitalization

    * –

sleep duration score

    * –

sleep restlessness

    * –

vertical meters exercised (measured in floors)

  * •

Later Upgrade to Fitbit Sense Smartwatch, which extends the following
estimates:

    * –

Responsiveness points: a proprietary assessment of how well the sympathetic
and parasympathetic nervous system are in balance which takes Heart Rate
Variability (HRV), Elevated Resting Heart Rate (RHR), Sleeping Heart Rate
above RHR, and Electrodermal Activity (EDA) into account

    * –

Exertion points (similar to heart points)

    * –

the temperature of the wrist during sleep

    * –

sleep points (sleep rating)

    * –

stress points (average of sleep-, exertion-, and responsiveness points)

  * •

Netatmo indoor weather station

    * –

Temperature

    * –

Humidity

    * –

CO2 in ppm

    * –

Noise in dB

    * –

air pressure in Pa

  * •

Nutrition tracking with MyFitessPal

    * –

Carbohydrates intake

    * –

Fat intake

    * –

Protein intake

    * –

Sodium intake

    * –

fiber intake

    * –

sugar intake

    * –

cholesterol intake

    * –

total calories intake

  * •

Bodyweight (Renpho Scale)

  * •

Screentime tracking, with RescueTime

    * –

total screen time

    * –

productive screentime

    * –

distracting screen time

    * –

neutral screen time

TABLE III: Correlation coefficients, Benjamini–Hochberg corrected p-values and regression weights of all features. Sorted by p-values, which are bold when below 0.5. feature | corr. coeff. | p-value | reg. weight
---|---|---|---
MoodYesterday | 0.348 | 3.3E-18 | 0.06
HumidInMax() | -0.406 | 4.6E-09 | -0.12
BodyWeight | 0.248 | 4.6E-09 | 0.05
MoodEreyesterday | 0.236 | 2.9E-08 | 0.00
BodyWeightYesterday | 0.235 | 3.0E-08 | 0.00
Covid_lifestyleYesterday | 0.211 | 1.1E-06 | 0.00
Covid_lifestyle | 0.210 | 1.1E-06 | 0.00
HumidInMedian() | -0.342 | 1.4E-06 | 0.00
HumidInMax()Yesterday | -0.329 | 4.2E-06 | 0.00
CO2Median() | -0.321 | 7.3E-06 | -0.09
HumidOutMin()Yesterday | -0.186 | 2.2E-05 | 0.00
NoiseMax()Yesterday | -0.303 | 2.9E-05 | -0.05
TempInMedian() | -0.301 | 3.0E-05 | 0.00
HumidOutMean()Yesterday | -0.181 | 3.3E-05 | 0.00
HumidOutMax()Yesterday | -0.179 | 3.8E-05 | 0.00
HumidOutMin() | -0.177 | 5.1E-05 | 0.00
NoiseMax() | -0.288 | 6.8E-05 | 0.00
Daytime | 0.172 | 8.1E-05 | 0.00
DaytimeYesterday | 0.171 | 8.1E-05 | 0.00
TempOutMax()Yesterday | 0.171 | 8.7E-05 | 0.00
HumidOutMean() | -0.168 | 1.2E-04 | 0.00
HumidInMedian()Yesterday | -0.276 | 1.4E-04 | 0.00
TempInMax() | -0.270 | 2.0E-04 | 0.00
HumidOutMax() | -0.162 | 2.0E-04 | 0.00
TempOutMean()Yesterday | 0.160 | 2.4E-04 | 0.00
TempOutMax() | 0.157 | 3.2E-04 | 0.00
HRMean()Yesterday | -0.155 | 3.8E-04 | 0.00
TempInMax()Yesterday | -0.258 | 3.8E-04 | 0.00
HumidInMin() | -0.257 | 3.8E-04 | 0.00
TempInMedian()Yesterday | -0.250 | 6.2E-04 | 0.00
TempOutMean() | 0.148 | 7.8E-04 | 0.00
TempFeelOutMax()Yesterday | 0.147 | 7.8E-04 | 0.00
TempFeelOutMax() | 0.145 | 9.6E-04 | 0.00
TempInMin() | -0.240 | 1.0E-03 | 0.00
VitaminDSupYesterday | 0.144 | 1.0E-03 | 0.00
VitaminDSup | 0.143 | 1.1E-03 | 0.03
TempOutFeelMean()Yesterday | 0.142 | 1.1E-03 | 0.00
PressureInMin()Yesterday | 0.234 | 1.3E-03 | 0.00
TempOutFeelMean() | 0.134 | 2.5E-03 | 0.00
TempOutDelta()Yesterday | 0.133 | 2.6E-03 | 0.00
CO2Max() | -0.220 | 2.8E-03 | 0.00
PressureInMedian()Yesterday | 0.221 | 2.8E-03 | 0.00
Fiber | 0.195 | 3.6E-03 |
TempFeelOutMin()Yesterday | 0.129 | 3.8E-03 | 0.00
TempOutMin()Yesterday | 0.127 | 4.5E-03 | 0.00
PressureInMax()Yesterday | 0.209 | 4.9E-03 | 0.00
FiberYesterday | 0.188 | 5.3E-03 |
TempOutDelta() | 0.124 | 5.3E-03 | 0.00
SleepRevitalizationYesterday | 0.123 | 6.3E-03 | 0.00
HeartPoints | 0.121 | 6.6E-03 | 0.10
HighLongitude | 0.120 | 6.6E-03 | 0.00
CO2Min() | -0.199 | 7.0E-03 | 0.00
CO2Median()Yesterday | -0.199 | 7.2E-03 | 0.00
TempFeelOutMin() | 0.117 | 8.6E-03 | 0.00
SleepLight | -0.117 | 8.6E-03 | 0.00
HumidInMin()Yesterday | -0.194 | 8.9E-03 | 0.00
TempOutMin() | 0.115 | 9.2E-03 | 0.00
CloudOutMean() | -0.114 | 1.0E-02 | 0.00
CloudOutMean()Yesterday | -0.114 | 1.0E-02 | 0.00
CloudOutMax()Yesterday | -0.113 | 1.1E-02 | 0.00
NoiseMedian()Yesterday | -0.188 | 1.1E-02 | 0.00
TempInMin()Yesterday | -0.185 | 1.3E-02 | 0.00
NoiseMedian() | -0.182 | 1.4E-02 | 0.00
StressManagement | 0.179 | 1.5E-02 | 0.00
CO2Max()Yesterday | -0.181 | 1.5E-02 | 0.00
DistractingScreentimeYesterday | 0.110 | 1.5E-02 | 0.00
PressureInMin() | 0.174 | 2.0E-02 | 0.00
LowLongitude | 0.103 | 2.2E-02 | 0.00
ExertionPointsYesterday | 0.169 | 2.2E-02 | 0.00
PressureInMedian() | 0.165 | 2.9E-02 | 0.00
CarbsYesterday | 0.149 | 2.9E-02 |
feature | corr. coeff. | p-value | reg. weight
---|---|---|---
CO2Min()Yesterday | -0.164 | 3.0E-02 | 0.00
SleepDeep | 0.094 | 3.8E-02 | 0.00
PressureInMax() | 0.153 | 4.6E-02 | 0.00
HighLongitudeYesterday | 0.089 | 5.4E-02 | 0.00
SleepStartYesterday | -0.089 | 5.4E-02 | 0.00
HRResting | -0.085 | 6.5E-02 | 0.00
CloudOutMax() | -0.084 | 7.2E-02 | 0.00
RunningMin | 0.082 | 7.8E-02 | 0.00
LowLongitudeYesterday | 0.082 | 7.8E-02 | 0.00
SleepPoints | 0.134 | 8.3E-02 | 0.00
ScreentimeYesterday | 0.083 | 8.3E-02 | 0.00
StressManagementYesterday | 0.131 | 9.0E-02 | 0.00
NeutralScreentimeYesterday | 0.080 | 9.5E-02 | 0.00
SodiumYesterday | 0.118 | 9.6E-02 |
SleepRevitalization | 0.077 | 1.1E-01 | 0.00
Sodium | 0.113 | 1.2E-01 |
HRMax() | 0.072 | 1.3E-01 | 0.00
SleepLightYesterday | -0.072 | 1.4E-01 | 0.00
LowLatitude | -0.071 | 1.4E-01 | 0.00
RunningMinYesterday | 0.071 | 1.4E-01 | 0.00
ExertionPoints | 0.108 | 1.8E-01 | 0.00
SugarYesterday | 0.099 | 1.8E-01 |
HRMax()Yesterday | 0.065 | 1.9E-01 | 0.00
CloudOutMin() | -0.065 | 1.9E-01 | 0.00
Steps | 0.060 | 2.4E-01 | 0.00
SleepWake | -0.060 | 2.4E-01 | 0.00
SleepEndYesterday | 0.059 | 2.5E-01 | 0.00
SleepREM | 0.056 | 2.8E-01 | 0.00
MeditatingMinYesterday | 0.056 | 2.9E-01 | 0.00
Floors | 0.056 | 2.9E-01 | 0.00
KCalIn | 0.083 | 2.9E-01 |
HRMin()Yesterday | -0.054 | 3.0E-01 | 0.00
SleepREMYesterday | 0.052 | 3.3E-01 | 0.00
RainSnowYesterday | -0.051 | 3.4E-01 | 0.00
PressOutDelta()Yesterday | -0.051 | 3.4E-01 | 0.00
MeditatingMin | 0.050 | 3.5E-01 | 0.00
InBedMin | -0.050 | 3.5E-01 | 0.00
HRRestingYesterday | -0.050 | 3.5E-01 | 0.00
HRMin() | -0.048 | 3.8E-01 | 0.00
Sunday | 0.047 | 3.9E-01 | 0.00
SaturdayYesterday | 0.047 | 3.9E-01 | 0.00
Carbs | 0.069 | 4.0E-01 |
Saturday | -0.044 | 4.2E-01 | 0.00
FridayYesterday | -0.044 | 4.2E-01 | 0.00
SleepStart | -0.043 | 4.3E-01 | 0.00
SleepEfficiencyYesterday | -0.043 | 4.3E-01 | 0.00
SleepWakeYesterday | -0.043 | 4.3E-01 | 0.00
NoiseMin()Yesterday | 0.071 | 4.4E-01 | 0.00
PressOutDelta() | -0.042 | 4.4E-01 | 0.00
CloudOutMin()Yesterday | -0.041 | 4.5E-01 | 0.00
HeartPointsYesterday | 0.041 | 4.5E-01 | 0.00
PressOutMin()Yesterday | 0.041 | 4.5E-01 | 0.05
HighLatitude | -0.041 | 4.5E-01 | 0.00
NoiseMin() | 0.068 | 4.5E-01 | 0.00
ProductiveScreentimeYesterday | 0.041 | 4.6E-01 | 0.00
ProteinYesterday | 0.060 | 4.6E-01 |
ResponsivenessPoints | 0.065 | 4.6E-01 | 0.00
LowLatitudeYesterday | -0.039 | 4.6E-01 | 0.00
WindOutMean()Yesterday | 0.039 | 4.6E-01 | 0.00
InBedMinYesterday | -0.039 | 4.6E-01 | 0.00
FatYesterday | -0.058 | 4.7E-01 |
KcalOut | 0.038 | 4.7E-01 | 0.00
Monday | -0.037 | 4.7E-01 | 0.00
SundayYesterday | -0.037 | 4.7E-01 | 0.00
WindOutMin()Yesterday | -0.037 | 4.7E-01 | 0.00
HRMean() | -0.037 | 4.7E-01 | 0.00
Cholesterol | 0.056 | 4.7E-01 |
AsleepMin | -0.035 | 5.0E-01 | 0.00
NeutralScreentime | 0.035 | 5.1E-01 | 0.00
KCalInYesterday | 0.052 | 5.2E-01 |
Sugar | 0.051 | 5.2E-01 |
DistractingScreentime | 0.034 | 5.2E-01 | -0.01
VO2MaxYesterday | 0.034 | 5.2E-01 | 0.00
RainSnow | -0.033 | 5.2E-01 | 0.00
Fat | 0.050 | 5.2E-01 |
KcalOutYesterday | -0.032 | 5.3E-01 | 0.00
feature | corr. coeff. | p-value | reg. weight
---|---|---|---
PressOutMean()Yesterday | 0.032 | 5.3E-01 | 0.00
VO2Max | 0.030 | 5.7E-01 | 0.00
CommitsYesterday | -0.030 | 5.7E-01 | 0.00
WalkingMinYesterday | -0.029 | 5.8E-01 | 0.00
SleepRestless | 0.029 | 5.9E-01 | 0.00
AsleepMinYesterday | -0.027 | 6.2E-01 | 0.00
SleepDeepYesterday | 0.026 | 6.4E-01 | 0.00
StepsYesterday | -0.025 | 6.4E-01 | 0.00
Protein | 0.037 | 6.5E-01 |
CholesterolYesterday | 0.036 | 6.6E-01 |
PressOutMax()Yesterday | 0.022 | 6.9E-01 | 0.00
SleepEfficiency | 0.022 | 6.9E-01 | 0.00
Commits | 0.022 | 6.9E-01 | 0.00
Screentime | 0.022 | 6.9E-01 | 0.00
SleepPointsYesterday | 0.035 | 6.9E-01 | 0.00
WindOutMax()Yesterday | 0.021 | 7.0E-01 | 0.00
SleepDurationScore | 0.021 | 7.0E-01 | 0.00
PressOutMin() | 0.020 | 7.1E-01 | 0.00
TempWristNight | 0.031 | 7.3E-01 | 0.00
HighLatitudeYesterday | -0.018 | 7.3E-01 | 0.00
WindOutMean() | 0.018 | 7.4E-01 | 0.00
SleepDurationScoreYesterday | 0.018 | 7.5E-01 | 0.00
Thursday | 0.017 | 7.5E-01 | 0.00
WednesdayYesterday | 0.017 | 7.5E-01 | 0.00
ThursdayYesterday | 0.015 | 7.8E-01 | 0.00
Friday | 0.015 | 7.8E-01 | 0.00
WindOutMin() | -0.012 | 8.4E-01 | 0.00
WalkingMin | 0.012 | 8.4E-01 | 0.00
ProductiveScreentime | 0.011 | 8.6E-01 | 0.00
SleepRestlessYesterday | 0.010 | 8.6E-01 | 0.00
FloorsYesterday | 0.010 | 8.7E-01 | 0.00
TempWristNightYesterday | 0.015 | 8.8E-01 | 0.00
PressOutMean() | 0.009 | 8.8E-01 | 0.00
MondayYesterday | 0.008 | 8.8E-01 | 0.00
Tuesday | 0.008 | 8.8E-01 | 0.00
MoonIlluminationYesterday | 0.007 | 9.0E-01 | 0.00
MoonIllumination | 0.006 | 9.1E-01 | 0.00
Wednesday | -0.006 | 9.1E-01 | 0.00
TuesdayYesterday | -0.006 | 9.1E-01 | 0.00
SleepEnd | -0.005 | 9.2E-01 | 0.00
WindOutMax() | 0.004 | 9.5E-01 | 0.00
PressOutMax() | 0.003 | 9.5E-01 | 0.00
NapYesterday | 0.002 | 9.8E-01 | 0.00
Nap | 0.000 | 9.9E-01 | 0.00
ResponsivenessPointsYesterday | 0.001 | 9.9E-01 | 0.00