\section{Introduction} \label{sec:intro}

The concordance model, $\Lambda$CDM, has been
very successful in describing the entire observed
history of the Universe. But we are now entering the era of precision cosmology where
the basic picture can be placed under closer scrutiny. Notably, as the
precision of measurements has improved, hints of cracks in the
concordance model have appeared.  The discrepancy of the present day
Hubble parameter $H_0$ derived from local measurements
\cite{riess11,Riess:2016jrr,Riess:2018byc,Bonvin:2016crt,2019MNRAS.484.4726B}
and from cosmic microwave background (CMB)
measurements \cite{Hinshaw:2012aka,planck15,Aghanim:2018eyx}
is perhaps the most notable indication of a problem. This  long-standing deviation (see e.g.~ref.~\cite{Freedman:2000cf}) has become stronger with
time and is currently
at $4.4\,\sigma$ \cite{Riess:2019cxk}.
This is worth taking seriously,
both for itself and as a template for how we will view current and
future cosmological measurements and their potential to probe new
physics.
While
the disagreement may ultimately be explained by experimental
systematics (see
e.g.~refs.~\cite{Rigault:2013gux,Rigault:2018ffm,Jones:2018vbn,Shanks:2018rka,Riess:2018kzi,Kenworthy:2019qwq})
or a statistical fluctuation, it is possible that we are
seeing the first signs of a theory beyond the standard cosmological
model.

The CMB is sensitive to $H_0$ through the
well-measured ratio $\theta_{\rm s} = r_{\rm s} / D_{\rm M}(z_*)$, where
$r_{\rm s}$ is the
comoving sound horizon at decoupling, and $D_{\rm M}(z_*)$ is the
comoving angular
diameter distance to the surface of last scattering. The sound horizon
is sensitive to early physics, whereas $D_{\rm M}$ is sensitive to physics
after decoupling,
\begin{align}
  r_{\rm s}
  &=
  \int_{z_*}^\infty
  \frac{c_{\rm s} dz'}{ H(z')},
  \\
  D_{\rm M}(z)
  &=
  \int_{0}^{z}
  \frac{dz'}{ H(z')}.
  \label{eq:rsDA}
\end{align}

Baryon acoustic oscillation (BAO) data at lower redshifts provide another measurement of
$r_{\rm s} / D_{\rm M}$, which asymptotes to $r_{\rm s} H_0/z$ for small $z$.
This implies that in order to accommodate a value of $H_0$ that is
compatible with the
local measurements (and larger
than the Planck $\Lambda$CDM\xspace best fit \cite{Aghanim:2018eyx}), $r_{\rm s}$ has to be
modified \cite{Evslin:2017qdn,2019ApJ...874....4A}.

One simple way to reduce $r_{\rm s}$ is to increase $H(z)$ before
decoupling. Since the integral for $r_{\rm s}$ is dominated by contributions
just before the epoch of CMB last-scattering, energy injection around that time is the most
efficient in reducing $r_{\rm s}$.
Indeed, energy injection that is peaked in a narrow window of redshift
around recombination minimizes
minimize the impact on other successful $\Lambda$CDM\xspace
predictions~\cite{Hojjati:2013oya,Poulin:2018dzj,Poulin:2018cxd,
2019ApJ...874....4A}. Along those lines, there have been a number of studies that try to resolve the Hubble
tension by extending $\Lambda$CDM\xspace by a new energy contribution with
distinctive time-dependence and properties \cite{DiValentino:2017rcr,DEramo:2018vss,Poulin:2018cxd,Pandey:2019plg,Vattis:2019efj,Renk:2017rzu,Khosravi:2017hfi,Kreisch:2019yzn}.

Evolving scalar fields feature prominently in models of inflation and quintessence (see e.g.~refs.~\cite{Ferreira:1997hj,Tsujikawa:2013fta}).
In this paper we present a class of scalar
field potentials and solutions which injects energy
in the requisite redshift window close to recombination to shrink the sound horizon.
We will present simple solutions that are
sufficient to provide the energy injection we need and are under
enough control that we can trace their evolution robustly. These
solutions arise from potentials of the form $V\propto \phi^{2n}$, with
$n$ determining whether the solutions are asymptotically oscillatory
or not. We will
show that these models fit the CMB and the SH0ES measurement
better than $\Lambda$CDM\xspace, with $n=2$ mildly preferred.

A set of potentials similar to the ones considered in this paper was
investigated
in~refs.~\cite{Karwal:2016vyq,Poulin:2018dzj,Poulin:2018cxd}.
In contrast the the analysis there, we use the full equations of
motion for the background as well as for the fluctuations and find
qualitatively different results.
We show that the coarse-grained approximation used
in~\cite{Poulin:2018dzj,Poulin:2018cxd}
is not valid for the initial oscillations.
Furthermore we identify solutions with asymptotically constant equation
of state for large $n$, where there are no oscillatory
solutions.

The outline of the paper is as follows.  In
section~\ref{sec:scaling_solns} we introduce the class of scalar
potentials which produce brief energy injections in the early
universe.  We classify the solutions to the scalar field equations of
motion and discuss their stability.  In
section~\ref{sec:cosmo_results} we describe the procedure we use to
fit the models to cosmological data and present the results, comparing
to alternate solutions.  We highlight unresolved issues in
section~\ref{sec:disc}, commenting on their potential resolution,
before concluding in section~\ref{sec:conc}.

\section{Classification of scalar field solutions in cosmology}\label{sec:scaling_solns}

CMB data require that
the ratio of energy density of the scalar
field to the background energy density should peak in a
relatively small window in
redshift~\cite{Hojjati:2013oya,Poulin:2018dzj,Poulin:2018cxd}.
The goal of this section is to demonstrate
potentials that give rise to this behavior.

The energy-momentum tensors for uncoupled fluids are separately
conserved.
A consequence of this conservation law is that the ratio of energy
densities is determined by the relative equation of
state parameters of the scalar field ($w_\phi$) and the background
($w_b$):
\begin{align}
  \frac{\rho_{\phi}}{\rho_{b}}
  &\propto
  \exp\left({-\int 3[w_{\phi}(a) - w_b(a)] d \ln a}\right)
  \,.
\end{align}
In order that the energy density in the scalar field does not come to
dominate that of the background either at early or late
times, $(w_{\phi} - w_b)$ must transition from
negative to positive.
This energy injection, which is required to be $\mathcal{O}(10\%)$ in order to resolve the Hubble tension, is
localized at the redshift when $w_\phi = w_b$.

One particularly simple class of such solutions has the scalar field
initially frozen on its potential, with $w_{\phi} \approx -1$, before
thawing onto a trajectory with a nearly constant equation of
state, $w_{\phi} > w_b$, which we will refer to as rolling solutions.  This behavior is in contrast with the
familiar case of, {\it e.g.}, axion dark matter, in which the field
and its equation of state oscillate rapidly after thawing.
The axion
oscillation frequency grows relative to the decreasing Hubble
parameter, so that it can become intractable to trace the oscillating
field evolution
numerically.
Cycle-averaged coarse-graining \cite{Turner:1983he} can provide an accurate description of
the fluid at late times, both in the case of axion dark matter as well
as more general potentials with oscillating equation of state.
However, the coarse-graining approximation does not apply
near the initial thawing phase when the field starts oscillating, and
this is precisely the regime in which the details of the
field evolution and fluctuations have the greatest impact on the
gravitational potential, and thence the CMB. Therefore, a
cycle-averaged description for such an oscillating solution might be
insufficient.

This is why we look for solutions that quickly asymptote to a constant
$w_{\phi}$.
The rolling trajectory is easiest to describe in a simplified limit,
in which the background cosmology has a constant equation
of state $w_b$. Further,
we work in the approximation where we neglect the back-reaction of the
scalar field.
This should be justified since the maximal energy injection
that we are interested in will be $\mathcal{O}(10\%)$.
We emphasize that we use these simplifications to elucidate the
physics of the scalar field and do not use them for any of our
numerical results.

\subsection{Emden-Fowler solutions}\label{subsec:E-F}
We look for a potential such that the late time solution
is
a trajectory where
the scalar field rolls with a constant equation of state, $w_{\phi}>w_b$,
\begin{align}
  \rho_\phi(a)
  &=
  \rho_0
  \left(\frac{a_0}{a}\right)^{3(1+w_{\phi})}.
  \label{eq:rhophi}
\end{align}
Recalling that $1+w_\phi = a^2 H^2 (\partial_a \phi)^2 / \rho_\phi$,
we can extract
formulae for $V(\phi)$ and $\partial_a{\phi}$
\begin{align}
  V(\phi) &= \frac{1-w_{\phi}}{2} \rho_\phi,
  \\
  \partial_a \phi &= \frac{\sqrt{(1+w_{\phi})\rho_\phi}}{a H}.
\end{align}
Using $3 H^2 M_{\rm Pl}^2 \approx \rho_b = \rho_{b0}
\left(\frac{a_0}{a}\right)^{3(1+w_b)}$, we can solve for $\phi(a)$,
\begin{align}
  \phi(a)
  &=
  c \left(\frac{a_0}{a}\right)^{\frac32 (w_\phi-w_b)},
  \qquad
  c = \frac{M_{\rm Pl}}{(w_\phi-w_b)}\sqrt{\frac{4(1+w_\phi)\rho_0}{3\rho_{b0}}},
\end{align}
where we have chosen the additive constant for the solution to correspond to
$\phi=0$ at $a=\infty$.
Since both the potential as well as the field have power-law
dependence on $a$, we see that the potential is a power-law in $\phi$ as
well. Explicitly,
\begin{align}
  V(\phi)
  &=
  \frac12(1-w_\phi) \rho_0 \left(\frac{\phi}{c}\right)^{2n},
  \quad n =
  \frac{1+w_\phi}{w_\phi-w_b}.
\end{align}
These $\phi^{2n}$ potentials have been previously studied in the
context of axiverse-motivated models~\cite{Karwal:2016vyq,Poulin:2018dzj}, but we emphasize that we
have arrived at them independently from theoretical motivation by
demanding the rolling solutions.
\begin{figure}[t]
  \centering
  \includegraphics[width=0.69\textwidth]{energy_injection_comp_n}\\
  \includegraphics[width=0.69\textwidth]{w_compared_n}
  \caption{
    \emph{Top panel:} Evolution of the energy density of the scalar field $\phi$
    relative to total energy density of the universe in a matter-dominated toy
    cosmology ($w_b=0$). Here, $\rho_{\rm tot}$ includes the contribution from the scalar field, and $a_c$ denotes the scalar factor at which the field starts rolling. We show the
    evolution for different choices of $n$ in the scalar potential
    $V\propto \phi^{2n}$, as well as an example of exponential potential,  $V\propto e^{-\lambda\phi}$.
    We choose values of $n$ to show an example each of an oscillating ($n=2$)
    and a rolling solution ($n=5$).  The asymptotics of the numerical solutions is
    shown to match well with the analytical results derived in the
    text, neglecting the back-reaction.  Note that the presence of slow oscillations around the rolling solution for $n=5$ is due to the transition from $w_{\phi} = -1$ to the asymptotic solution; the stability of the rolling solution guarantees that these oscillations are quickly damped. We note that the width of the energy injection grows larger as $n$ is increased. \emph{Bottom panel:} Equation of state of the scalar field as a function of the scale factor for the same potentials as in the left panel. For $n=2$, the cycle-average of $w_\phi$ (dark thick blue line) rapidly approaches $w_{\rm osc}=1/3$, while for $n=5$, it asymptotes to the rolling solution with $w_{\phi}=1/4$ instead of the cycle-averaged approximation ($w_{\rm osc}=2/3$). For the exponential potential, the scalar field approaches the well-known tracking behavior with $w_\phi=w_b$.
  }
  \label{fig:emden-fowler}
\end{figure}

Even when these rolling solutions exist, they may not be the unique
solutions. Fortunately,
the equation of motion for the $\phi^{2n}$ potential can be cast into
Emden-Fowler form, the solutions of which have been studied in detail
(see~\cite{10.2307/2029180} for a review).
The equation of motion of $\phi$ on a constant $w_b$ background, with
back-reaction neglected, is
\begin{align}
  \frac{\partial^2 \phi}{\partial(\log a)^2}
  + \frac32 (1-w_b) \frac{\partial \phi}{\partial (\log a)}
  + \frac{\partial_\phi V}{H^2(a)}
  =0.
\end{align}
Identifying $s \propto a^{\frac32(1-w_b)} $ and $y= s \phi$, the
above equation reduces to the Emden-Fowler form,
\begin{align}
  y''(s) + s^\sigma y^\gamma(s) &= 0,
\end{align}
with $\sigma =
\frac{4}{1-w_b}-2(n+1)$
and $\gamma = 2n-1$. The solutions of Emden-Fowler are characterized
by the values of $\sigma$ and $\gamma$. The different regimes of the solution are listed in table \ref{tab:Enden_Fowler_solns}.

\begin{table}
\begin{align}
  \begin{array}{|c|c|c|c|c|}
    \hline
    \text{Asymptotic solutions}& \text{Emdem-Fowler}
    &\text{Translation to} &
    \multicolumn{2}{c|}{\text{Background}}
    \\
    \cline{4-5}
    &\text{conditions} &\text{scalar field models}&
    \text{Radiation}
    &
    \text{Matter}
    \\
    \hline
    \text{Osc. only}
&
\sigma + 2 \geq 0
&
n < \frac{2}{1-w_b}
& n < 3 & n < 2\nonumber
\\
\vspace{-3mm}
&&&&\\
\text{Osc.
+ non-osc.}
&
\sigma +2 < 0 \leq \sigma + \frac{\gamma+3}{2}
&
\frac{3+w_b}{1-w_b} \geq n > \frac{2}{1-w_b}
& 5\geq n>3& 3\geq n>2
\\
\vspace{-3mm}
&&&&\\
    \text{Non-osc. only}
    &\sigma + \frac{\gamma+3}{2} < 0
& n > \frac{3+w_b}{1-w_b}
& n > 5
& n > 3
\\
\hline
\end{array}
\end{align}
\caption{Asymptotics of solutions to the Emden-Fowler equation, and
their translation to the parameterization used in this paper. The last two columns give the condition for a given type of solution to exist in either radiation- and matter-dominated cosmologies, respectively. \label{tab:Enden_Fowler_solns}}
\end{table}

The existence of these rolling trajectories is associated with the fact that $\partial^2_\phi V(\phi) /H(a)^2$ is an
$\mathcal{O}(1)$ constant on these solutions,
\begin{align}
  \frac{\partial_\phi^2 V(\phi)}{H(a)^2}
  &=
  \frac94 (1-w_\phi)(2+w_\phi+w_b)
  \,,
\end{align}
so that the curvature of the potential never grows large relative to the expansion rate of the universe.  This should be contrasted with the oscillatory case, \emph{e.g.}, $n=1$.  In this case $\partial_{\phi}^2 V(\phi)$ is initially much smaller than the expansion rate, and the field is correspondingly stuck on its potential.  As the expansion rate decreases, the field begins to roll when $\partial_{\phi}^2 V(\phi)$ and $H(a)^2$ become comparable, eventually oscillating rapidly at a frequency controlled by $\partial_{\phi}^2 V(\phi) \gg H(a)^2$.  The fact that $\partial_{\phi}^2 V(\phi)$ never becomes large compared to $H(a)^2$ on the rolling solutions thus prevents the onset of oscillations.

The lower bound on $n$ for non-oscillating solutions to exist agrees
with the restriction that $w_\phi \leq 1$, given that
$n = (1+w_{\phi})/(w_{\phi}-w_b)$ on the rolling solution.  As $n$ is
increased above this boundary, both oscillatory and rolling solutions
exist simultaneously.  However, they are not equally relevant:
as we will show in section~\ref{ssec:stability}, the rolling solutions
are unstable whenever both oscillatory and rolling solutions exist.
Rapidly oscillating solutions can be
modeled by a cycle-averaged description with the effective equation of state given by~\cite{Poulin:2018dzj}
\begin{align}
  w_{\rm osc}
  \simeq \frac{n-1}{n+1}.
\end{align}

While tracking these oscillatory solutions
could in principle prove problematic for numerical
evaluation, in practice these solutions never undergo more than
$\mathcal{O}(10)$s of cycles before the oscillations are cut off.
This is because of the transition from matter to dark energy
domination in the late Universe.  As demonstrated by the boundaries
given in table~\ref{tab:Enden_Fowler_solns}, when the background
equation of state approaches $w_b \rightarrow -1$, all solutions are
non-oscillatory for $n > 1$.  As a result, we are able to exactly
track these solutions in our numerical results in the following
sections, without relying on cycle-averaged approximations.

\subsection{Exponential potentials}

Exponential potentials are a special limiting case of the rolling solutions which occur when $w_\phi = w_b$, corresponding to the limit $n\rightarrow\infty$. In this
case, the field $\phi$ depends
logarithmically on $a$,
\begin{align}
  \phi(a)
  &= \phi_0 + \sqrt{\frac{3(1+w_b)\rho_0}{\rho_{b0}+\rho_0}}\log\frac{a}{a_0},
  \qquad\qquad(w_\phi=w_b).
\end{align}
In this case we have kept the back-reaction of the field since
it is possible to obtain a simple analytical solution even with the
back-reaction included.
This trajectory corresponds to an exponential potential,
\begin{align}
  V(\phi)
  &=
  V_0 \exp \left(-\lambda \frac{\phi}{M_{\rm Pl}}\right),
  \quad
  \lambda =
  \sqrt{
    3(1+w_b)
    \left(
      1+\frac{\rho_{b0}}{\rho_{0}}
    \right)
  },
  \,
  \qquad (w_\phi=w_b).
  \label{eq:exp-pot}
\end{align}
On this solution,
\begin{align}
  \frac{\partial_\phi^2 V(\phi)}{H(a)^2}
  &=
  \frac92(1-w_b^2),
  \qquad\qquad
  (w_\phi=w_b)
\end{align}
is an $\mathcal{O}(1)$ constant, as for the non-oscillatory solutions discussed in the previous subsection. These exponential potentials have previously been studied in the context of quintessence models~(see~\cite{Tsujikawa:2013fta} for a review).  However, since $w_\phi=w_b$, the energy injection for this potential does
not redshift relative to the background~(see figure~\ref{fig:emden-fowler}), which prevents these solutions from being ideal candidates to resolve the Hubble tension.  Therefore we will focus on the case of monomial potentials at finite $n$ for the remainder of the paper.

\subsection{Stability of fluctuations}
\label{ssec:stability}

As demonstrated in the previous subsection, for $\phi^{2n}$ potentials in the range $(3 + w_b)~\geq~n~(1 - w_b)~>~2$, rolling solutions are irrelevant at late times, since the field redshifts more quickly than the oscillation amplitude.  In fact, the rolling solutions are unstable for this range of $n$.  The homogeneous equation of motion for linear fluctuations about the rolling solution,
\begin{align}
  \frac{\partial^2 \delta \phi_k}{\partial(\log a)^2}
  + \frac32 (1-w_b) \frac{\partial \delta\phi_k}{\partial (\log a)}
  + \left[\frac{k^2}{a^2 H(a)^2} + \frac{9}{4} (1 - w_{\phi}) (2 + w_{\phi} + w_b) \right] \delta \phi_k
  =0,
\end{align}
has solutions which scale as
\begin{align}
\delta \phi_0 \sim a^{-\frac{3}{4}\left[(1 - w_b) \pm \sqrt{4 w_{\phi}^2 + (1 + w_b) (4 w_{\phi} + w_b - 7)}\right]}.
\end{align}
For $w_{\phi} < \sqrt{2} \sqrt{1+w_b} - (1+w_b)/2$, these solutions are oscillatory, and their envelope redshifts as $a^{-3(1-w_b)/4}$, while the rolling solution redshifts as $a^{-3(w_{\phi} - w_b)/2}$.  Thus the fluctuations grow relative to the rolling solution for $1 > w_{\phi} > (1 + w_b)/2$, corresponding to $2 < n~(1 - w_b) < 3 + w_b$ and coinciding with the entire range of $n$ for which both oscillatory and rolling solutions exist.  For larger $n$, these fluctuations redshift more quickly, and the rolling solutions are stable.

The oscillatory solutions, on the other hand, exhibit a somewhat more subtle instability.  For $n > 1$, $\partial^2_{\phi} V \propto \phi^{2 (n-1)}$ vanishes as $\phi \rightarrow 0$.  During each cycle of the oscillation, the fluctuations of the $\phi$ field become massless, leading to a short burst of particle production.  If the growth rate of the instability is sufficiently large, $\partial^2_{\phi} V \gg H^2$, the field will eventually fragment completely into modes with $k > 0$~\cite{Lozanov:2017hjm,Johnson:2008se}, invalidating the linear perturbation theory used in our analysis.

On the oscillating solutions,
\begin{align}
  \left.  \frac{\partial_{\phi}^2 V}{H^2}\right|_{osc}
  &\propto
  a^{-\frac{3}{n+1}\left(n(1 - w_b) - 3 - w_b\right)},
\end{align}
so that the growth rate increases with scale factor for $n < (3 + w_b)/(1 - w_b)$.  The case of $w_b = 0$ is most relevant for our numerical results, since the shape of the energy injection constrains most of the oscillations to occur during matter domination.  For $n=3$, the growth rate of the fluctuations remains smaller than the expansion rate at all times, so that they never become important.  For $n=2$, however, the growth rate accelerates, and it is a numerical question whether these fluctuations become important before the onset of dark ernergy domination, at which point the background solutions cease to be oscillatory.  Of course, a realistic potential may include a mass term for the scalar field which could prevent this instability entirely.  For our results in the following sections, we have not included such a mass term; rather, we have simply checked numerically that the scalar field fluctuations remain under control.

\subsection{Comparison with coarse-grained description}

The oscillatory solutions have a frequency which monotonically increases relative to the Hubble expansion rate, which makes it numerically challenging to track their evolution to late times. The usual method for addressing this issue is to coarse-grain these solutions, along with their fluctuations, replacing the numerically computed equation of state and rest-frame sound speed by their cycle-averaged values~\cite{Poulin:2018dzj},
\begin{align}
w_{\rm osc} \simeq \frac{n-1}{n+1}, &\hspace{1 cm} \langle c_s^2 \rangle \simeq \frac{2 a^2 (n - 1) \omega^2 + k^2}{2 a^2 (n+1) \omega^2 + k^2},
\end{align}
in which $\omega$ is the instantaneous oscillation frequency of the field fluctuations.  As demonstrated in figure~\ref{fig:emden-fowler}, however, this description is inaccurate near the onset of thawing, when the ratio of energy density of the scalar field to that of the background is near its maximum.  In fact, we will see (figure \ref{fig:energy_injection} below) that the oscillating solutions go through a short period of kination ($w \approx 1$) during the first oscillation cycle, rapidly depleting their energy density relative to the cycle-averaged solution.  For $n=2$ and $w_b = 0$, it takes several oscillation cycles before the oscillation frequency becomes large relative to the Hubble rate, at which point $\rho_{\phi} / \rho_{\text{tot}}$ has dropped by more than two orders of magnitude relative to its $\mathcal{O}(10\%)$ maximum.

The fluctuations are similarly poorly described by coarse-graining.  At the onset of thawing, $\partial_{\phi}^2 V \simeq H^2$, so that all subhorizon modes have $k^2 \gtrsim \partial_{\phi}^2 V$.  Thus the fluctuations are kinetic energy dominated, and their rest-frame sound speed is close to 1 for the entire relevant range of scale factors and $k$-modes.  By contrast, taking the above cycle-averaged formula gives $\langle c_s^2 \rangle \rightarrow w_{\rm osc} < 1$ over a large range of $k$, since $\omega$ becomes large relative to $H$ at late times.  The detailed evolution of these fluctuations qualitatively affects the fit of these scalar field models to the CMB data; therefore, in our numerical study, we have chosen not to make any coarse-graining approximations.  For the following results, we have tracked the individual oscillations to late times, checking that numerical errors remain under control until the energy density of the scalar field is sufficiently subdominant.

\section{Cosmological Analysis}\label{sec:cosmo_results}
\subsection{Parameterization}
In this section we present the fits for the models above to the
cosmological
data. We use the following parametrization for the model,
\begin{align}\label{eq:potential}
  V(\phi)
  &=
  V_0 \left(\frac{\phi}{M_{\rm Pl}}\right)^{2n} + V_{\Lambda},
\end{align}
where $V_{\Lambda}$ is a constant. In addition to $V_0$, $V_{\Lambda}$, and $n$, the scalar field initial conditions $\phi_i$ and $\partial_a\phi_i$ are
the two other
free parameters. Since we are primarily interested in field solutions that are initially frozen, we will always set the velocity of the scalar field
to zero at very early times.  Given $\phi_i \sim M_{\rm pl}$ initial
conditions, the field will then begin to thaw when $V_0 \sim \rho_b$.
Thus in order that the field begins to roll near matter-radiation
equality, we must have $V_0 \sim {\rm eV}^4$.  Note that the smallness
of the dimensionless coupling, $({\rm eV}/M_{\rm pl})^4$, is
consistent with a softly broken shift symmetry and therefore
technically natural.  For a discussion of the tuning required to
explain the absence of additional couplings, see
section~\ref{sec:disc}.  In our model, the value of $V_{\Lambda}$
determines the late-time dark energy density, and its value is set by
demanding that the Universe is spatially flat. The parameter $n$
determines the shape of
the energy injection.  We note that eq.~\eqref{eq:potential}
completely determines the evolution of the scalar field and its
perturbations. In particular, there is no extra freedom to
independently change the sound speed of the scalar field perturbations
in this model.

In principle, we could  perform our Markov Chain Monte Carlo (MCMC) analysis in terms  of $V_0$ and $\phi_i$. However, since our preliminary analysis shows that cosmological data are primarily sensitive to both the timing and the amount of energy injected by the rolling scalar field, we choose to parametrize the model using the approximate redshift at which the field starts rolling, $z_{\rm c, MC}$, and the approximate fraction of energy injected by the scalar field $f_{\phi,{\rm MC}}\approx (\rho_\phi / \rho_{\rm tot})|_{\rm max}$.
The initial field value $\phi_i$ and the $V_0$ parameter are then
determined through the relations $\partial^2_\phi V(\phi_i) \equiv
9 H^2(z_{\rm c, MC})$ and $V(\phi_i)\equiv 3 f_{\phi,{\rm MC}}
H^2(z_{\rm c, MC}) M_{\rm Pl}^2$. These latter relations should be interpreted as the definitions of $z_{\rm c, MC}$ and $f_{\phi,{\rm MC}}$.

We implement this model into a modified version of the \texttt{Class} code \cite{2011arXiv1104.2932L,2011JCAP...07..034B}. We perform parameter scans over $f_{\phi,{\rm MC}}$ and $\ln{(1+z_{\rm c, MC})}$ as well as the standard six flat $\Lambda$CDM parameters using the \texttt{MontePython} \cite{Brinckmann:2018cvx,Audren:2012wb} software package version 3.1.  We use flat prior distributions on our scalar field parameters, $f_{\phi,{\rm MC}}\in [10^{-4},0.3]$ and $\ln{(1+z_{\rm c, MC})}\in [7.5,9.5]$, and the same prior distributions for the other cosmological parameters as in the Planck analysis \cite{planck15}. We solve both the background and perturbed Klein-Gordon equations at each step to obtain the exact cosmological evolution of the scalar field and its perturbations, and use this solution to determine a posteriori the exact maximum fraction of energy injected by the rolling field, $f_\phi$, as well as the critical redshift $z_{\rm c}$ at which the energy injection peaks. In the following, we shall present our results in terms of these latter exact quantities.

\subsection{Datasets}
In our cosmological analysis, we use the following datasets:
\begin{itemize}
  \item {\bf Planck}:
    We use the Planck 2015\footnote{The 2018 likelihood is not yet publicly available.} CMB temperature and polarization likelihoods \cite{Aghanim:2015xee} for both low-$\ell$ and  high-$\ell$, including all nuisance parameters and using the same prior distributiona as in the Planck analysis \cite{planck15} for those. We also include the Planck lensing likelihood \cite{plancklens}.

\item {\bf BAO}: We include baryon acoustic oscillation (BAO) measurements from the CMASS and LOWZ galaxy samples of the Baryon Oscillation Spectroscopic Survey (BOSS) DR12 \cite{Alam:2016hwk}. These include both line-of-sight and transverse BAO measurements, as well as constraints on the growth of structure through the parameter $f\sigma_8$. We shall refer to these datasets as high-$z$ BAO. We also include low-$z$ BAO measurements from the 6dF Galaxy Survey \cite{BAO1} and from the Main Galaxy Sample (MGS) of SDSS \cite{BAO3}.

\item {\bf SH0ES}:
We include the recent\footnote{After our analysis was complete, a more recent measurement of $H_0$  \cite{Riess:2019cxk} was released. The latter further increases the tension with CMB-based estimate of the Hubble constant within the $\Lambda$CDM\xspace model, and may thus provide stronger evidence for new physics such as that discussed in this paper.} measurement \cite{Riess:2018byc} of the local Hubble parameter $H_0 = 73.52\pm1.62$ km/s/Mpc.

\item {\bf Pantheon}: We include the Pantheon compilation of type Ia supernovae (SN Ia) \cite{Scolnic:2017caz}, which includes 1048 luminosity distances in the redshift range $0.01<z<2.3$. We include in our fit the nuisance parameter $M$ describing the fiducial absolute magnitude of a SN Ia.

\end{itemize}

\subsection{Results: Fixed integer values of \texorpdfstring{$n$}{n}}\label{sec:res_fixed_n}

\begin{figure}[t!]
  \centering
  \includegraphics[width=\textwidth]{Planck_BAOhlz_HST_Lens_Panth.pdf}
  \caption{Marginalized posterior distributions for $V\propto\phi^{2n}$ models for three different values of $n$. Results are shown here for the data combination ``Planck + BAO + SH0ES + Pantheon''. }
  \label{fig:triangle_for_diff_n}
\end{figure}

We show posterior distributions for the most relevant parameters in figure \ref{fig:triangle_for_diff_n} for $n=2$, $3$, and $4$. The most striking feature is the similarity of the different distributions as $n$ is changed, with the notable exception of the injected energy fraction $f_\phi$. Of the models shown, the $n=2$ (corresponding to $V(\phi)\propto \phi^4$) posterior stands out for being more highly weighted towards larger values of the energy injection, which results in slightly larger values of the Hubble constant being preferred. Indeed, the inclusion of the SH0ES data in the analysis leads to a nearly 2$\,\sigma$ preference for a nonzero energy injection around $z_{\rm c}\simeq 3100$. As per our discussion in section \ref{sec:intro} (see also ref.~\cite{2019ApJ...874....4A}), the size of the baryon-photon sound horizon at the drag epoch $r_{\rm drag}$ is strongly anti-correlated with the injected energy fraction $f_\phi$. This smaller sound horizon is the main driver behind the reduction of the Hubble constant tension in these models.

The physical dark matter density ($\Omega_{\rm c}h^2$) is strongly correlated with $f_\phi$ in order to obtain the correct early integrated Sachs-Wolf effect, which fixes, in part, the height of the first CMB temperature acoustic peak. This larger dark matter abundance, combined with the extra energy injected by the rolling scalar field, leads to an increase of the CMB Silk damping scales as compared to the baryon-photon sound horizon since $r_{\rm damp}/r_{\rm s}\propto H^{1/2}$ \cite{Hou:2011ec}. This is compensated in the CMB fit by increasing the values of both the scalar spectral index $n_{\rm s}$ and fluctuation amplitude $A_{\rm s}$ as compared to their $\Lambda$CDM\xspace values. As a consequence of this change to the primordial spectrum of scalar perturbations, the late-time amplitude of matter fluctuations as probed by the $\sigma_8$ parameter are increased. Thus, obtaining a larger Hubble parameter within these scalar field models is correlated with an increase in the $\sigma_8$ value. This is a generic feature of models that inject energy near recombination, which might lead them to be in tension with late-time measurements of the matter power spectrum (see e.g.~ref.~\cite{Hikage:2018qbn}). We shall come back to this point in the discussion.

\begin{figure}[t!]
  \centering
  \includegraphics[width=0.484\textwidth]{Energy_injection_bestfit.pdf}
  \includegraphics[width=0.5\textwidth]{EoS_bestfit.pdf}
  \caption{\emph{Left panel:} Energy injection profile for the best-fit models for each value of $n$ as a function of redshift. Results are shown here for the data combination ``Planck + BAO + SH0ES + Pantheon''. For reference, we also show the amount of energy injected as compared to standard $\Lambda$CDM for the best fit $N_{\rm eff}$ model using the same data combination (corresponding to $\Delta N_{\rm eff}=0.27$). To guide the eye, we have indicated by vertical dashed lines the matter-radiation equality and baryon drag epochs in the standard $\Lambda$CDM model. \emph{Right panel:} The scalar field equation of state as a function of redshift for each value of $n$.}
  \label{fig:energy_injection}
\end{figure}

We show in figure~\ref{fig:energy_injection} the best-fit energy-injection profile and the EoS of the scalar field for each value of $n$ we consider. We observe that, for a fixed width of the primary energy injection peak, models with lower values of $n$ allow for more energy to be injected, resulting in a smaller value of the baryon-photon sound horizon and larger Hubble constant. However, as explained in section~\ref{sec:scaling_solns}, these low-$n$ models also have strong oscillatory solutions with secondary energy injection peaks post recombination, which adversely affect the fit to CMB data.
On the other hand, as was shown in figure~\ref{fig:emden-fowler}, models with large values of $n$ have a broader primary energy injection peak and slowly decaying tail, which means that it is difficult for these models to inject sufficient energy prior to recombination while not having significant residual energy towards low redshifts. The fit to cosmological data is thus driven by the competing effects of injecting the largest possible amount of energy in a relatively narrow window prior to recombination, while at the same time being able to get rid of this energy as quickly as possible in the post-recombination era.  In contrast with the results of section~\ref{subsec:E-F}, which were derived in a toy cosmology with $w_b = 0$, the best-fit solutions for $n=3$ and $n=4$ never achieve their asymptotic equations of state due to the effect of the cosmological constant at late times.  The salient feature, however, is the relatively large instantaneous value of $w_{\phi} \approx 1$ obtained shortly after the peak of the energy injection for all values of $n$.  This causes most of the energy density to quickly redshift away, so that the fit is relatively insensitive to the subsequent dynamics.  The speed of the transition from $w_{\phi} \simeq -1$ to $w_{\phi} \simeq 1$ determines the width of the energy injection; as shown in figure~\ref{fig:energy_injection}, this width is minimized at small $n$.

The finding that lower $n$ values provide a better fit to cosmological data is supported by examining the individual best-fit $\chi^2$ values. We present these in table \ref{tab:best_fit_chi2} for $n=2$, $3$, and $4$, as well as for $\Lambda$CDM. We find that the $n=2$ model provides that best overall improvement of the fit as compared to $\Lambda$CDM. We observe, however, that the improved fit to the SH0ES measurement of $H_0$ is partially offset by a degradation of the goodness-of-fit to the high-$\ell$ CMB tail and to the CMB lensing spectrum. We note that the results shown in table \ref{tab:best_fit_chi2} are significantly different than those presented in ref.~\cite{Poulin:2018cxd} due to their use of a coarse-grained effective fluid approach. As discussed in section \ref{sec:scaling_solns} above, the coarse-grained approach is a reasonable approximation for $n=2$ deep in the matter-dominated era (see right panel of figure \ref{fig:energy_injection} for $z<500$), but it fails to accurately capture the most important part of scalar field evolution near the peak of the energy injection. The coarse-grained approach is inaccurate for $n\geq3$ since the models undergoes at most two oscillations before the onset of dark energy domination, resulting in very different fits once the exact evolution of the scalar field is taken into account.

\begin{table}
\centering
\begin{tabular}{|c | c | c | c | c | c |}
\hline
\hline
Datasets & $\Lambda$CDM & $n=2$ & $n=3$ & $n=4$ & $N_{\text{eff}}$ \\
\hline
\hline
{\it Planck} high-$\ell$ & 2448.6 & 2449.3 & 2447.3 & 2446.2 & 2449.2 \\
{\it Planck} low-$\ell$ & 10495.6 & 10494.4 & 10494.9 & 10495.6 & 10495.0 \\
{\it Planck} lensing & 9.3 & 9.9 & 10.2 & 9.2 & 10.1 \\
BAO - low $z$ & 1.9 & 1.8 & 1.4 & 1.8 & 2.7 \\
BAO - high $z$ & 1.8 & 1.9 & 1.9 & 1.8 & 2.0 \\
Pantheon & 1027.1 & 1027.0 & 1027.1 & 1027.0 & 1027.2 \\
SH0ES & 10.3 & 3.5 & 6.5 & 7.4 & 4.2 \\
\hline
Total $\chi^2_{\text{min}}$ & 13994.7 & 13987.8 & 13989.2 &13989.0 & 13990.3 \\
$\Delta \chi^2_{\text{min}}$ & 0 & -6.9 & -5.5 & -5.7 & -4.4 \\
\hline
\end{tabular}
\caption{Best-fit $\chi^2$ values for each individual dataset used in our cosmological analysis. \label{tab:best_fit_chi2}}
\end{table}

\begin{table}
\centering
\begin{tabular}{|c | c | c | c |}
\hline
\hline
Parameter & $\Lambda$CDM & $n=2$ & $N_{\text{eff}}$ \\
\hline
\hline
$100\ \Omega_{\rm b} h^2$ & 2.238 (2.236) \small${}^{+0.014}_{-0.015}$ & 2.261 (2.264) \small${}^{+0.021}_{-0.020}$ & 2.254 (2.269) $\pm 0.018$ \\
\hline
$\Omega_{\rm c} h^2$ & 0.1180 (0.1177) $\pm 0.0012$ & 0.1264 (0.1267) \small${}^{+0.0044}_{-0.0043}$ & 0.1220 (0.1213) \small${}^{+0.0027}_{-0.0028}$ \\
\hline
100 $\theta_{\rm s}$ & 1.0420 (1.0422) $\pm 0.0003$ & 1.0415 (1.0417) $\pm 0.0004$ & 1.0414 (1.0413) \small$^{+0.0004}_{-0.0005}$ \\
\hline
$\tau_{\rm reio}$ & 0.074 (0.077) \small${}^{+0.013}_{-0.012}$ & 0.072 (0.081) \small${}^{+0.013}_{-0.012}$ & 0.075 (0.080) \small${}^{+0.013}_{-0.012}$ \\
\hline
$\ln (10^{10} A_{\rm s})$ & 3.079 (3.080) \small${}^{+0.024}_{-0.021}$ & 3.091 (3.105) \small${}^{+0.026}_{-0.023}$ & 3.089 (3.100) \small${}^{+0.025}_{-0.022}$ \\
\hline
$n_{\rm s}$ & $0.968\ (0.969) \pm 0.004$ & 0.978 (0.981) $\pm 0.007$ & $0.977\ (0.977)$ \small${}^{+0.006}_{-0.007}$ \\
\hline
$f_\phi$ / $\Delta N_{\text{eff}}$ & - & 0.064 (0.073) \small${}^{+0.031}_{-0.028}$ & $0.26\ (0.27) \pm 0.16$\\
\hline
$z_{\rm c}$ & - & 3040 (3160) \small${}^{+330}_{-630}$ & - \\
\hline
$\sigma_8$ & 0.819 (0.819) \small${}^{+0.009}_{-0.008}$ & 0.835 (0.841) $\pm 0.012$ & 0.831 (0.832) $\pm 0.011$ \\
\hline
$\Omega_{\rm m}$ & 0.304 (0.301) $\pm 0.007$ & 0.304 (0.302) $\pm 0.007$ & 0.299 (0.293) \small${}^{+0.007}_{-0.008}$ \\
\hline
$r_{\text{drag}}$ [Mpc] & 147.6 (147.7) $\pm 0.3$ & 143.2 (142.9) \small${}^{+2.0}_{-2.3}$ & 145.1 (145.1) $\pm 1.5$ \\
\hline
$H_0$ [km/s/Mpc] & $68.2\ (68.3) \pm 0.5$ & 70.1 (70.5) \small${}^{+1.0}_{-1.2}$ & $69.7\ (70.2) \pm 1.1$ \\
\hline
\end{tabular}
\caption{Mean values and $68\%$ confidence intervals for key cosmological parameters using the data combination  ``Planck + BAO + SH0ES + Pantheon''. The numbers in parentheses are the best-fit values for each model. \label{tab:confidence_interval}}
\end{table}

We show in table \ref{tab:confidence_interval} the mean, 1$\sigma$
uncertainty, and best fit values for the most relevant cosmological
parameters and quantities.  The $n=2$ best-fit model has the largest
Hubble constant value of all models studied in this paper, which is a
direct consequence of it having the largest energy injection fraction
(see figure \ref{fig:energy_injection}), and therefore the smallest
photon-baryon drag horizon. The physical dark matter density is also
significantly larger for the $n=2$ best-fit model, as is the amplitude
of the matter power spectrum as captured through the $\sigma_8$
parameter, for reasons explained above. We will perform a thorough
comparison between the $n=2$ model and $N_{\rm eff}$ in section
\ref{sec:N_eff} below.

In figure \ref{fig:CMB_residuals}, we show the CMB residuals between our best-fit $\phi^{2n}$, $\Lambda$CDM\xspace, and $N_{\rm eff}$ models using the data combination ``Planck + BAO + SH0ES + Pantheon'', and a reference $\Lambda$CDM\xspace model fitted to ``Planck + BAO + Pantheon''  data. This allows us to isolate the impact of the SH0ES data on the residuals.  All models shown display, on average, less power in $C_\ell^{TT}$ at $\ell > 500$ compared to our reference $\Lambda$CDM\xspace model. This is caused by the well-known \cite{Hou:2011ec} competition between having a large enough Hubble constant to fit the SH0ES measurement while ensuring that there is enough power in the CMB temperature Silk damping tail. We note, however, that the variations between the different models shown are typically smaller than the size of the error bars.

\begin{figure}[t!]
  \centering
  \includegraphics[width=0.495\textwidth]{ClsTT_residuals_highell.pdf}
  \includegraphics[width=0.495\textwidth]{ClsEE_residuals_highell}
  \caption{High-$\ell$ CMB residuals between the ``Planck + BAO +
  SH0ES + Pantheon'' best fit models for $\Lambda$CDM\xspace, $N_{\rm eff}$, and $\phi^{2n}$, and a best-fit $\Lambda$CDM\xspace reference model obtained using the data combination ``Planck + BAO + Pantheon''. The left panel shows the temperature residuals, while the right panel shows the E-mode polarization residuals. }
  \label{fig:CMB_residuals}
\end{figure}

\begin{figure}[b!]
  \centering
  \includegraphics[width=0.495\textwidth]{f_vs_n_Planck_BAO_SH0ES_Panth.pdf}
  \includegraphics[width=0.49\textwidth]{H0_vs_n_Planck_BAO_SHOES_Panth.pdf}
  \caption{ Marginalized posterior in the $f_\phi$-$n$ plane (left panel), and the $H_0$-$n$ plane (right panel). The gray band shows the SH0ES measurement \cite{Riess:2018byc}. Results are shown here for the data combination ``Planck + BAO + SH0ES + Pantheon.'' Clearly, larger values of the Hubble constant require lower values of $n$.}
  \label{fig:free_n}
\end{figure}

\subsection{Results: Promoting \texorpdfstring{$n$}{n} to a free parameter }
So far, we have considered only fixed integer values of the power-law
index $n$ parameterizing the scalar potential. This was effectively
restricting the behavior of the energy injection to have a specific
shape specified by our choice of $n$. In this section, we relax the
assumption of fixed integer values of $n$ and allow the power-law
index to float freely in the fit to cosmological data (writing our
  potential as $V\propto |\phi|^{2n}$), hence exploring a broader range
  of energy injection shapes and scalar field behaviors. While we
  adopt this simple phenomenological point-of-view, we note that
  models with noninteger $n$ values may be problematic from a
  model-building perspective. We consider a flat prior on $n\in[2,6]$.
  The main results of this analysis are shown in figure
  \ref{fig:free_n}.  As discussed above, the data tends to favor lower
  values of the index $n$ since they allow a larger peak energy
  injection fraction. The right panel clearly illustrates that larger
  Hubble constant values are more likely for the lowest possible index
  $n$, which, given our choice of priors, corresponds to $n\sim2$
  (that is, $V\propto\phi^4$).

Figure \ref{fig:free_n} suggests that one should consider even lower values of $n$ in order to fit the large Hubble constant from the SH0ES measurement, despite these solutions being increasingly oscillatory. To explore this possibility, we extend our prior range on $n$ to include values all the way down to $n=1.5$, a compromise between exploring low values of $n$ while at the same time ensuring that the oscillatory solutions are tractable numerically. We find that it is indeed possible to obtain larger values of the Hubble constant (our best for with $n< 2$ has $H_0=71.74$ km/s/Mpc for $n=1.89$), but at the price of significantly degrading the fit to the CMB damping tail. Overall, we find that none of the models with $n< 2$ provides a better global fit to cosmological data than the $n=2$ model presented in tables \ref{tab:best_fit_chi2} and \ref{tab:confidence_interval}.

\subsection{Results: Comparison with \texorpdfstring{$N_{\rm eff}$}{Neff}}\label{sec:N_eff}

It is informative to compare the scalar field models with the more standard $N_{\rm eff}$ extension to the $\Lambda$CDM model. The amount of energy injected by the best-fit $N_{\rm eff}$ model is shown in figure \ref{fig:energy_injection} (corresponding to $\Delta N_{\rm eff}=0.27$). The addition of relativistic species leads to a broadband energy injection, in contrast to the relatively narrow energy injection peaks of the $n=2$ model. Despite these differences, we see in table \ref{tab:best_fit_chi2} that the fit to the high-$\ell$ CMB data is similar to the $n=2$ scalar field model. We note however that these two fits are significantly worse than that achieved in the reference $\Lambda$CDM\xspace model fitted only to  ``Planck + BAO  + Pantheon.''  The main differences between the two models is the physical amount of dark matter required to obtain the correct magnitude of the early ISW effect. Since the $N_{\rm eff}$ model injects less relativistic energy near matter-radiation equality than the $n=2$ model, a smaller amount of dark matter is required to leave the early ISW effect invariant.  In turn, this leads to a smaller $\Omega_{\rm m}$ than in $n=2$, which worsens the fit to late-time probes such as BAO. Thus, while $N_{\rm eff}$ models can accommodate larger values of the Hubble constant without degrading the global fit, they do so by compromising the fit to the low-$z$ BAO data as shown in the last column of table \ref{tab:best_fit_chi2}.

\begin{figure}[t!]
  \centering
  \includegraphics[width=\textwidth]{Compare_phi4_Neff_LCDM_Planck_BAO_HST_Lens_Panth.pdf}
  \caption{Comparison between the $n=2$ model, the $N_{\rm eff}$ extension of the standard cosmological model, and plain $\Lambda$CDM. All posterior distributions shown here were obtained using the data combination ``Planck + BAO + SH0ES + Pantheon''. }
  \label{fig:triangle_comp_Neff_phi4}
\end{figure}

By comparison, the $n=2$ scalar field model has a similar fit to the CMB high-$\ell$ tail as the $N_{\rm eff}$ model, while at the same time typically allowing for larger Hubble constant and better fit to BAO data. We must however keep in mind that the $\phi^{2n}$ models introduce two free parameters ($f_\phi$, $z_{\rm c}$) to the fit, compared to the single parameter in $N_{\rm eff}$ models. We can use the Aikake information criterion (AIC)  \cite{AIC}, which can be defined as $\Delta {\rm AIC} = \Delta\chi^2_{\rm min} + 2\Delta k$, where $\Delta k$ is the difference in the number of free parameters between the two models considered. Using the $\Lambda$CDM fit to the ``Planck + BAO + SH0ES + Pantheon'' dataset as our reference model, we obtain $\Delta {\rm AIC}_{n=2} = -2.9$, while $\Delta{\rm AIC}_{N_{\rm eff}} = -2.4$, indicating that the $n=2$ scalar field model provides a slightly better fit to the data as compared to $N_{\rm eff}$, even after accounting for the extra free parameter. In comparison, the $n=3$ and the $n=4$ models provide a worse global fit to the data than $N_{\rm eff}$ after accounting for the different numbers of free parameters. This reinforces the fact that models with lower values of the power-law index $n$ are favored by the data.   On the other hand, $N_{\rm eff}$ models are theoretically somewhat simpler.  UV completions of $\phi^{2n}$ models must explain not only the absence of additional self-couplings, but also the surprising coincidence between the timing of the energy injection and the redshift of matter-radiation equality demanded by the fit to CMB data.

A quantitative comparison between the $n=2$ scalar field model and the $N_{\rm eff}$ extension of $\Lambda$CDM is shown in figure \ref{fig:triangle_comp_Neff_phi4}. We first observe that the scalar field model generally allows for smaller values of the baryon drag horizon $r_{\rm drag}$ than the $N_{\rm eff}$ model. Naively, following the argument presented in ref.~\cite{2019ApJ...874....4A}, this should allow the scalar field model to accommodate a larger Hubble constant as compared to $N_{\rm eff}$. Instead, we see that the two models have fairly similar posterior distribution for $H_0$, with the scalar field model allowing only slightly larger values of the expansion rate. This difference between our naive expectation and what is shown in figure \ref{fig:triangle_comp_Neff_phi4} is caused by the different amount of dark matter needed in each model, as described above. This larger dark matter abundance pulls down on the Hubble constant value in the cosmological fit, partially compensating for the increase coming from a smaller baryon sound horizon. The scalar field models with $\phi^{2n}$ potentials are thus less efficient at turning a given reduction of the sound horizon into an increase of $H_0$, as compared to $N_{\rm eff}$.

Both $n=2$ and $N_{\rm eff}$ models have larger values of $\sigma_8$ which are largely caused by the change to the primordial spectrum of fluctuations ($A_{\rm s}$ and $n_{\rm s}$) necessary to bring the CMB damping tail in agreement with the data. As discussed in section \ref{sec:res_fixed_n}, this is a generic feature of models that inject energy near the epoch of recombination. The effect is larger for the $n=2$ model since more energy is injected overall, leading to a slightly larger value of $\sigma_8$ as compared to $N_{\rm eff}$.

\section{Discussion}\label{sec:disc}
We have seen that a scalar field with $V\propto \phi^{2n}$ leads to an energy injection which is localized in time, potentially alleviating the tension between local measurements of the Hubble constant and that inferred from CMB observations. One important worry is the slight degradation of the fit to CMB data when Planck and SH0ES measurements are analyzed simultaneously.  Does this indicate that the CMB alone disfavors the energy injection brought by the rolling scalar field? To answer this question, we redo our MCMC analyses for the $\phi^{2n}$ models but this time without the SH0ES likelihood. In figure \ref{fig:H0_tension} we show the normalized posterior of $H_0$ resulting from these analyses for $\Lambda$CDM\xspace, $N_{\rm eff}$, and $n=2$. While both the $n=2$ and $N_{\rm eff}$ posteriors are
wider than $\Lambda$CDM\xspace, that of $n=2$ is peaked at marginally higher values of
$H_0$ than either alternative model, with $N_{\rm eff}$ actually peaking below $\Lambda$CDM\xspace. This suggests that the energy injection brought by the evolving scalar field can naturally accommodate larger Hubble constants in the CMB fit, even without prior information from late-time measurements. While the $n=2$ model does not entirely remove the $H_0$ tension, it can significantly reduce it to the $\sim2\sigma$ level.  Given the longer tail of the posterior distribution, future local measurements of $H_0$ which reduce the error bars could more strongly distinguish the scalar field model from $N_{\rm eff}$.

\begin{figure}[t!]
  \centering
  \includegraphics[width=0.6\textwidth]{1D_H0_posterior.pdf}
  \caption{Normalized $H_0$ posteriors obtained using the data combination ``Planck + BAO + Pantheon'', that is, without including the local Hubble constant measurement from ref.~\cite{Riess:2018byc}. }
  \label{fig:H0_tension}
\end{figure}

It is also interesting to examine the impact of the $\phi^{2n}$ models on late-time cosmology. We show in figure \ref{fig:BAO_Hubble} the changes to the cosmology at $z<1$ for the different models studied in this paper. The left panel shows the fit to the BAO measurements used in our analyses. For clarity, we illustrate the quantity $D_V(z)/r_{\rm drag}$ for the best-fit parameters of the different models considered, normalized by that of the ``Planck + BAO + Pantheon''  $\Lambda$CDM\xspace model. We see that all models have lower values of $D_V(z)/r_{\rm drag}$ over the redshift range shown.  Expanding around $z = 0$,
\begin{align}
  \frac{D_V (z)}{D_{V,{\rm fid}} (z)}
  &\approx
  \frac{H_{0,{\rm fid}}}{H_0} \left[1 + \left(\Omega_{\rm m, fid} - \Omega_{\rm m}\right) z + \mathcal{O}(z^2)\right].
\end{align}
In all of the models considered, $\Omega_{\rm m}$ is decreased relative to the best-fit ``Planck + BAO + Pantheon'' $\Lambda$CDM reference model, despite the relative increase in $\Omega_{\rm c} h^2$.  This produces a positive slope for the $D_V$ ratio at late times, guaranteeing that those models which fit the high-$z$ BAO data at redshift 0.5 will be in mild tension with the MGS measurement at lower redshift. Since $N_{\rm eff}$ has the smallest $\Omega_{\rm m}$ of all models shown, this effect is more severe for this model.

The right panel of figure \ref{fig:BAO_Hubble}  shows the Hubble expansion history at $z<1$ for our best-fit $n=2$ model, and compares it to that of the best-fit $\Lambda$CDM\xspace and $N_{\rm eff}$ models (for the data combination ``Planck + BAO + SH0ES + Pantheon''). To illustrate how the scalar field model helps reconciling the CMB, BAO, and SH0ES measurements of the expansion rate, we also show two sets of line-of-sight BOSS DR12 BAO measurements \cite{Alam:2016hwk}. The first set is calibrated using the sound horizon from $n=2$ scalar field fit to the CMB ($r_{\rm drag} =142.9$ Mpc), while the second uses the sound horizon from the best-fit $\Lambda$CDM\xspace model ($r_{\rm drag} = 147.7$ Mpc). We observe in the right panel that the $n=2$ model fits the properly calibrated BAO data points nearly as well as $\Lambda$CDM\xspace, while at the same time providing a much better fit to the SH0ES measurement.

\begin{figure}[t!]
  \centering
  \includegraphics[width=0.495\textwidth]{BAO_D_V_plot.pdf}
  \includegraphics[width=0.495\textwidth]{Hubble_evolution.pdf}
  \caption{\emph{Left panel:} The BAO distance ladder as expressed through the ratio $D_V(z)/r_{\rm drag}$ normalized to the best-fit ``Planck + BAO + Pantheon'' $\Lambda$CDM\xspace reference model. \emph{Right panel:} Hubble expansion history for $z<1$. We show the $H_0$ measurement from ref.~\cite{Riess:2018byc}, as well as two sets of line-of-sight BOSS DR12 BAO measurements \cite{Alam:2016hwk}: one calibrated to the sound horizon of the best-fit $\Lambda$CDM\xspace model (black triangles), and the other calibrated to that of the best-fit $n=2$ model (blue diamonds).  All curves shown in this figure are best-fit models to the data combination ``Planck + BAO + SH0ES + Pantheon''.}
  \label{fig:BAO_Hubble}
\end{figure}

As discussed in section \ref{sec:res_fixed_n}, rolling scalar field
models that inject energy in the period just prior to recombination
require larger values of the scalar spectral index and amplitude of primordial fluctuations. This causes the late-time amplitude of matter fluctuations to be larger than their $\Lambda$CDM\xspace counterparts. This is potentially problematic since several measurements of the quantity $S_8\equiv \sigma_8\sqrt{\Omega_{\rm m}/0.3}$ have returned slightly lower values than the $\Lambda$CDM\xspace expectation based on its CMB fit. We illustrate in figure \ref{fig:S8_H0} the joint marginalized posterior of $S_8$ and $H_0$ for both the $n=2$ and $N_{\rm eff}$ models. Since the $n=2$ scalar field model has slightly larger values of both $\sigma_8$ and $\Omega_{\rm m}$ as compared to $N_{\rm eff}$, we find that it worsens the tension with the value of $S_8$ measured from weak lensing observations (e.g.~ref.~\cite{Hikage:2018qbn}).

Another possible issue with these scalar field models is that the larger values of the Hubble constant is correlated with a larger reionization optical depth $\tau_{\rm reio}$, as can be seen in figure \ref{fig:triangle_comp_Neff_phi4}. Such large values of $\tau_{\rm reio}$ could require new high-redshift sources in order to reionize the Universe at a higher redshift \cite{Robertson:2015uda}. We note that this larger reionization optical depth is a generic feature of all models having an early energy injection.

\begin{figure}[t!]
  \centering
  \includegraphics[width=0.6\textwidth]{H0_Sigma8_Planck_BOSS_HST_Lens_Panth.pdf}
  \caption{Marginalized posterior distributions in the $S_8$-$H_0$ plane obtained using the data combination ``Planck + BAO + SH0ES + Pantheon''. The gray vertical band shows the local Hubble constant measurement from ref.~\cite{Riess:2018byc}, while the horizontal green band shows the $S_8$ measurement from ref.~\cite{Hikage:2018qbn}. }
  \label{fig:S8_H0}
\end{figure}

Up to now, we have considered pure monomial potentials, $V \propto
\phi^{2n}$, in which the more relevant $\phi^{2 k}$ terms with $k < n$
have been set to zero.  In realistic UV completions, however, one
expects that these terms would be present at some order, and their
absence would then reflect some degree of tuning in the model.  For a
sufficiently small mass term, the dominant effect is that the residual
energy density in the scalar field comes to behave like an additional
dark matter component once the scalar field amplitude has decayed such
that $\langle m^2 \phi^2 \rangle \sim \rho_{\phi}$.  At larger
amplitudes, the field behaves as described in
section~\ref{sec:scaling_solns}.  This additional dark matter slightly
modifies the distance-redshift relationship, in particular decreasing
the distance to the surface of last scattering.  As long as the dark
matter energy density is increased by less than one part in
$\mathcal{O} (10^{-3})$, the effects are within our stated error bars
and the above results will hold. For our best fit $n=2$ model, this
corresponds to a mass below $\sim 10^{-4} {\rm eV}^4 / M_{\rm pl}^2$,
representing at most a 1\% tuning given the size of the quartic
coupling.  We have neglected to include such a term for simplicity,
and including it could actually improve the fit.  While the best fit
model has more dark matter than is found in $\Lambda$CDM\xspace, the increase
in $H_0$ causes $\Omega_m$ to be slightly smaller, representing an
increase in the redshift of dark energy domination.  Including a small
amount of additional dark matter in the late universe relative to its
abundance near matter-radiation equality would then improve the fit to
low-z BAO and supernovae data, at the cost of an increase in the
required numerical precision to track the rapid field oscillations.
We leave a detailed study to future work.

\section{Conclusions}
\label{sec:conc}
We see that scalar fields are good candidates for localized energy
injection in the early universe that can in principle produce better
combined fits to the CMB, BAO, and SH0ES than, \emph{e.g.}, $N_{\rm eff}$. We have considered models in which the scalar field spends a long time, spends a long time, spends a long lonely, lonely, lonely, lonely, lonely time frozen before beginning to roll near matter-radiation equality.  In addition:

\begin{itemize}
\item We have classified the scalar field solutions for $\phi^{2n}$
potentials and highlighted a class of rolling solutions whose equation
of state asymptotes to a constant value.

\item Oscillating solutions exist only for $n\leq3$ in a
matter-dominated universe.
\item A coarse-grained description of the cosmology is insufficient (and
unnecessary), leading to our results being significantly different than
those of others in the literature.

\item These models predict larger values of $\sigma_8$. Improvements in this measurement as well as other, more
direct measurements of the matter power spectrum will help test
these solutions in the future. We expect these models to be
distinguishable at large $k$.

\item A weakness of the model as it stands is the requirement that
thawing occurs very close to matter-radiation equality at $T\sim$ eV.  A more complete model
would also explain this, presumably through a triggering of the onset of rolling
by the change in the background evolution.

\item Our analysis requires a modest tuning.  A small mass term which does not
affect the physics around the energy injection might improve the fit to low-$z$ BAO data.
It might also be interesting to search for models in which the absence of a mass term is natural.

\item For now, $N_{\rm eff}$ might still be a more compelling model since
there is neither fine tuning nor a coincidence problem. However, if
future measurements of the Hubble
parameter have smaller error bars with the same central value, scalar field models could become favored as they
allow larger values for $H_0$.

\end{itemize}

\noindent Clearly the Hubble tension is one of the most intriguing
discrepancy in the cosmological model today. Improved measurements
of the CMB, $H(z)$, BAO at various redshifts, and the matter power
spectrum (at both $\sigma_8$ and higher $k$), will ultimately give us
greater insight into the existence of physics beyond $\Lambda$CDM.

\acknowledgments
We thank Daniel Grin, Julian Mu\~noz, Mustafa Amin, Manuel Buen-Abad,
Martin Schmaltz, Daniel Eisenstein, and Vivian Poulin for useful
discussions. L.R.~wishes to thank Adam Riess for drawing our attention to
ref.~\cite{Poulin:2018cxd} and Matthias Steinmetz for an invitation to
the symposium  ``The Hubble constant controversy: Status,
implications, and solutions'', which inspired this work. D.P. wishes
to thank Boston University for hospitality during completion of this
work. We thank the Institut Henri Poincar\'e, where part of this work
was carried out, for their hospitality. F.-Y. C.-R.~acknowledges the
support of the National Aeronautical and Space Administration (NASA)
ATP grant NNX16AI12G at Harvard University.  D.P.~is supported by DOE
grants DE-SC-0013607 and DE-SC-0010010 as well as NASA ATP grant
NNX16AI12G.  L.R.~is supported by NSF grant PHY-1620806, the Kavli
Foundation grant ``Kavli Dream Team,'' the Simons Fellows Program, the
Guggenheim Foundation, and an IHES CARMIN fellowship. The work of PA is supported by the NSF grants
PHY-0855591 and PHY-1216270.
PA thanks the Galileo Galilei
Institute for Theoretical Physics for their hospitality and
the INFN for partial support during the completion of
this work.
The computations
in this paper were run on the Odyssey cluster supported by the FAS
Division of Science, Research Computing Group at Harvard University.


\section{Introduction}

In a recent publication~\cite{Chang:2011eb} henceforth referred to as
{\texttt{CPY}}\ we examine recent claims that the cosmic ray muon flux can be
responsible for generating the modulation signals seen by
the DAMA~\cite{Bernabei:2008yi,Bernabei:2010mq} and, more recently, by the
CoGeNT~\cite{Aalseth:2011wp} Dark Matter (DM) direct detection
experiments. By examining the time series of the reported DAMA and
CoGeNT signals we find the data sets differ in phase, likely in
amplitude, and potentially in their power spectrum from measurements
of the underground muon flux. A correlation analysis in {\texttt{CPY}}\ reveals
that the data sets are incompatible and that the muon flux as the sole
source of the reported signals is excluded at high confidence.

In this note we restrict the discussion of the muon hypothesis to DAMA
which is the only direct detection experiment which claims detection
of a firm DM signal. It is situated at Gran Sasso, Italy, in the
underground LNGS laboratory. The collected data encompasses two major
runs, DAMA/NaI (Dec 1995 - July 2002) and DAMA/LIBRA (Sept 2003 - Sept
2009). The residual count rate reported by the collaboration exhibits
annual modulation compatible with what is expected from generic DM
models.

TeV-scale cosmic ray muons produced at stratospheric altitude levels
can reach deep underground and induce spallation reactions in the
detector and nearby.  The seasonal variations of the underground muon
flux have been observed on the northern and southern hemisphere
alike. For DAMA, the relevant measurement at the LNGS site is the one
from LVD~\cite{Selvi} which overlaps in time with DAMA/LIBRA cycles
1--5. The integral muon intensity underground reads $\langle I_{\mu}
\rangle \simeq 3\times 10^{-4}\,\mathrm{m}^{-2}\,\sec^{-1}$ and
exhibits an annual variation of $\sim 2\%$ in amplitude. In
Fig.~\ref{fig:lngs} we show the percent residuals of the muon flux
when binned in concordance with DAMA with the annual mean count rate
subtracted. In addition, the (2--4)\,keVee\ bin of the DAMA/LIBRA
experiment are shown; a baseline rate of $\bar s = 1.15$~cpd/kg/keV\
has been assumed. The seeming similarity in time and amplitude is
tantalizing and has lead to the notion that both signals may in fact
be measurements of the very same cosmic ray
phenomenon~\cite{ralston,nygren,Blum:2011jf}.  In the following we
recap and further develop on the arguments which challenge this
hypothesis.

\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.7\textwidth]{TS_DAMA+MUONSresiduals.eps}
\end{center}
\caption{Percent annual residuals of the LVD measured muon flux when
  binned in accordance with the DAMA/LIBRA runs~1--5. The latter
  residuals are shown for the 2--4\,keV bins assuming a baseline count
  rate of $\bar s = 1.15$~cpd/kg/keV.}
\label{fig:lngs}
\end{figure}

\section{The Null Hypothesis: periods, binning, and subtraction of means}
\label{sec:null}

A spectral analysis will most readily reveal periodic behavior in a
time series $\{t_i, X_i\}_{i=1,\dots N}$. In case the times $t_i$ are
unevenly spaced, it is most convenient to make use of the Lomb-Scargle
(LS) periodogram~\cite{1982ApJ...263..835S},
\begin{eqnarray}
  \label{eq:LS}
  P_X(\omega) =
\frac{1}{2 \sigma_X^2} \left\{ \frac{\left[ \sum_i (X_i - \bar X) \cos\omega (t_i-\tau) \right]^2}{ \sum_i \cos^2 \omega (t_i-\tau)} +
 \frac{\left[ \sum_i (X_i - \bar X) \sin\omega (t_i-\tau) \right]^2}{ \sum_i \sin^2 \omega (t_i-\tau)} \right\} .
\end{eqnarray}
Here $\tau $ is obtained from
$  \tan(2\omega \tau) = {\sum_i \sin 2\omega t_i}/{\sum_i \cos 2\omega t_i }$,
and it ensures that $P_X(\omega)$ is invariant to shifts in the time
origin; $\bar X$ and $\sigma^2_X$ are the arithmetic mean and the
variance of the data $\{ X_i \}$, respectively.  A Bayesian derivation
and generalization of (\ref{eq:LS}) can be found in {\texttt{CPY}}.  Noise is
rejected at the $1-\alpha$ confidence level by demanding that the
power at a sampled frequency is in excess of
\begin{eqnarray}
  \label{eq:sig}
  z_\alpha =-\ln \left[ 1-(1-\alpha)^{1/M} \right] .
\end{eqnarray}
A statistical penalty has been included for examining $M$ independent
frequencies.  In order to obtain a meaningful statistical
interpretation we have to choose $M$ carefully. In {\texttt{CPY}}\ we chose to
sample the data at frequencies $\omega_n = n \omega_F$ with
$n=1,\dots, N$, $M=N$, and with the fundamental frequency $\omega_F =
2\pi/T$ where $T$ is roughly the range of years covered by the data
set. In this note we instead oversample the data using a finer grid of
frequencies.  Since $X_i$ are not grossly unevenly spaced in $t_i$,
$M\approx N $ holds~\cite{Horne:1986bi} as long as we only sample
$X_i$ above the average Nyquist frequency, $\nu_c = 1/2\Delta$ where
$\Delta$ denotes the average spacing between $t_i$. We checked that
both approaches agree on coinciding frequencies.

\begin{figure}[t]
\begin{center}
\includegraphics[width=0.47\textwidth]{TS_LSdamaPROC.eps}%
\hfill
\includegraphics[width=0.47\textwidth]{TS_LSmuonsPROC.eps}
\end{center}
\caption{ \textit{Left:} LS spectrum for DAMA/LIBRA residual count
  rates as a function of the period.  The horizontal gray line marks
  the power above which noise can be rejected with $99\%$ confidence
  for any frequency. \textit{Right:} Power spectra of the LVD data
  which has temporal overlap with DAMA/LIBRA (cycles 1--5). The solid
  line is for unbinned LVD data; the dotted line is obtained after
  binning the data in concordance with the time bins of DAMA/LIBRA
  \textit{and} after re-scaling the resulting power spectrum to match
  significance levels of the solid curve: the dotted line is no LS but
  can directly be compared with the other lines. The dashed line is
  obtained from unbinned LVD data but the yearly mean intensity has
  been subtracted. The subtraction results in damping of the long
  period modes; binning damps power on all time-scales. Both effects
  make the muon data appear more similar to DAMA.}
\label{fig:damalvd}
\end{figure}

The left panel of Fig.~\ref{fig:damalvd} shows the DAMA/LIBRA power
spectrum for the residual count rates as reported
in~\cite{Bernabei:2010mq} and which are obtained by subtracting the
mean count rate in each cycle~\cite{belli:2012} for the various energy
bins as labeled. Above the horizontal line, noise is rejected at
$99\%$~C.L.  With an approximate power $P_X(2\pi/1\,\mathrm{yr}) \sim 12$
at a period of $1$~yr, the null hypothesis is disfavored by at least
$4.5\sigma$ at a single frequency and at $3.6\sigma$ according to
(\ref{eq:sig}). Notably, there is significant power at a period at
$480\,$days in the $(2-4)\,~{\rm keV_{ee}}$ and $(2-6)\,~{\rm keV_{ee}}$ bins and a
consistent power in all three energy bins with period~1/3\,yr
corresponding to a triannual mode; for the latter only the
$(2-5)\,~{\rm keV_{ee}}$ bin exceeds the $99\%$~C.L.~level.

The right panel of Fig.~\ref{fig:damalvd} shows the LS periodogram
obtained from LVD data where only those data points were included
which have actual temporal overlap with the bins of the DAMA/LIBRA
cycles, namely, full cycles 1--5.
We only show a fraction of the full power spectrum with smaller
periods omitted. The horizontal line is calculated according to
(\ref{eq:sig}) from all sampled independent frequencies. Since
$N_{\mathrm{LVD}}\gg N_{\mathrm{DAMA}}$ the required power to reject
the null hypothesis is larger when compared to the right plot
---despite the similar period intervals shown.
The solid line is obtained from the unbinned LVD data set. Aside from
the significant yearly modulation, the spectrum exhibits temporal
variations on time-scales smaller \textit{and} greater than one
year. At first sight this is in stark contrast to the DAMA spectra
which barely exhibit power at periods other than at
$T=1\,$yr. However, in order to better compare these data sets one
should (1) bin the muon data over the same time spans DAMA/LIBRA bins
their signal and (2) subtract the yearly mean of muon flux. The latter
mimics the procedure employed by the DAMA collaboration when they
present their residual count rates. To isolate the effects of (1) and
(2) we carry out each procedure separately and show its effect on the
power spectrum.

We first observe that binning reduces the number of data points from
$N\sim 1.3$k to $N=38$ when considering DAMA/LIBRA cycles 1--5. For a
given significance $\alpha$ this changes the condition on the minimum
power in Eq.~(\ref{eq:sig}). Therefore, we first bin the LVD data and
subsequently scale the resulting power spectrum in such a way that for
a given $y$-value the significance agrees with the one from the
unbinned data. The result is shown by the dotted line in
Fig.~\ref{fig:damalvd}. Though the dotted line is not a LS anymore,
one can now directly compare the dotted with solid line.  We observe
that the generic effect of binning is to mask power on all time
scales.
The effect of subtracting the mean intensity from the data on a yearly
basis is shown by the dashed line instead.  This time the power
spectrum is not scaled and the dashed line is indeed a LS
periodogram. In contrast to binning, this procedure largely preserves
power on periods $T\leq 1\,$yr. However, for larger periods we again
observe that variations are masked due to the particular treatment of
the data.

In contrast to the LVD spectrum, the Fourier transformed DAMA
residuals in the left panel of Fig.~\ref{fig:damalvd} show much less
significant power at periods smaller and greater than a year. As shown
above, this may simply be an artifact of the way the DAMA/LIBRA
modulation signal is obtained. It is unfortunate since a Fourier
analysis could otherwise be used to discriminate between background
induced effects and real signal.  It is not clear how to assign a
quantitative significance to this difference without knowledge of the
full, unsubtracted time series. This calls for a release of the yearly
baseline count rates by the DAMA collaboration.

The DAMA collaboration has recently responded to the criticism that
baselines are subtracted per cycle~\cite{Bernabei:2012wp}. Instead of
providing the actual values, the LS power spectrum of the joint
DAMA/NaI and DAMA/LIBRA baselines is presented. No significant power
which would point towards long-term behavior is observed. However, the
baselines are a mere set of 13~numbers. It is not surprising that no
statistically significant power is observed from such a small data
set. Similarly, we find that the LVD baseline spectrum does not yield
any significant power when calculated according to~(\ref{eq:LS}), too.
This casts doubt that a presentation of the baseline-LS precludes the
existence of variations in DAMA with periods larger than one year,
similarly to what is observed in muons; see
also~\cite{UCLAjosef,UCLAkfir}.

\section{The phase of muons: a potentially stretched concept}
\label{sec:phase}

Given an isotropic DM velocity distribution $f(v)$ in the earth's
vicinity, the differential recoil rate $dR/dE_R$ of DM scattering in
the detector is predicted to peak on June 2$^{\rm nd}$, corresponding
to a phase of $t_0 =152.5$ days after January 1$^{\rm st}$,
\begin{eqnarray}
  \label{eq:drder}
  \frac{dR}{dE_R}& \propto &\int^\infty_{v_{\mathrm{min}}} \frac{f(v)}{v}dv
  \approx
  c_0 + c_1 \cos{\left[\omega(t-t_0)\right]} .
\end{eqnarray}
Here, $v_{\mathrm{min}}$ is the minimum required relative velocity
between DM and target which produces a nuclear recoil of energy
$E_R$. In the canonical case, the annual mode is at the percent level,
$c_1/c_0 = \mathcal{O}(10^{-2})$; higher harmonic corrections to
(\ref{eq:drder}) have first been pointed out in~{\texttt{CPY}}.
The clear-cut prediction on $t_0$ arises from the geometric setup of the
earth's rotation around the sun in conjunction with the movement of
the solar system relative to the Galactic coordinate system. Hence,
apart from the period, the second most important characteristic of the
oscillations observed by the DAMA experiment is the phase of the
signal.

On the other hand, when considering environmental factors as potential
background, the variation in time will in general not be modulated
with a strict period of a year. Additionally, we cannot expect a
sinusoidal behavior similar to (\ref{eq:drder}). Both assertions can
be tested on the LVD and DAMA data sets. For this we model the data by
a sinusoid, minimize the usual $\chi^2$ function, and assess the
goodness-of-fit. Confidence regions in $t_0$ and $T$ can be inferred
from a $\Delta \chi^2$-method which is equivalent to constructing a
profile likelihood.
Fixing the period to one year we find that the DAMA/LIBRA
$(2-4)\,~{\rm keV_{ee}}$ and the full LVD data do not agree with respective values
\begin{align}
  \label{eq:phase-posterior}
\mathrm{DAMA/LIBRA:}  & \quad t_0  = (130\pm 8)\,\mathrm{days}, \quad \chi^2/dof = 37.5/41 ,  \\
\mathrm{full\ LVD:} & \quad t_0  = (185\pm 1.5)~\mathrm{days}  , \quad \chi^2/dof =  3450/1971 .
\end{align}
The uncertainty quoted is statistical. An additional uncertainty of
$\pm 2$~days may be attributed to digitization. A few comments are in
order: 1) The sinusoid provides a very poor description of the LVD
data with prohibitively large $\chi^2$. This supports our expectation
that a complex phenomenon such as the underground muon flux cannot be
adequately described by this simple model. 2) Even if we scale up the
error bars for LVD to the point where $\chi^2/dof\simeq 1$, the
stringent error on the phase remains below $\pm 2$~days. 3) An error of $\pm
15$~days on $t_0$ as quoted by LVD~\cite{Selvi} is certainly
incorrect. 4) Once $T$ is allowed to float, the confidence regions in
$t_0$ and $T$ become sensitive to the time-origin; for further details
we refer the reader to {\texttt{CPY}}. These results have recently been
confirmed in~\cite{FernandezMartinez:2012wd}. One major shortcoming of
this approach is that the data has to be subjected to the simple model
of sinusoidal variation, before any conclusions can be drawn. Indeed,
the power-spectrum of the LVD data makes it clear that significant
power exists in modes with periods larger and smaller than one
year. The notion of a single phase must therefore be treated with
caution.

\section{A correlation analysis: the conclusive approach}
\label{sec:correlation}

A glance at Fig.~\ref{fig:lngs} suggests that the DAMA/LIBRA and LVD
data sets are correlated. Indeed, when evaluating Pearson's
coefficient of correlation, $r\in[-1,1]$, one finds for the DAMA and
LVD sets as presented in Fig.~\ref{fig:lngs}
\begin{eqnarray}
r_{\mathrm{LVD, DAMA}} = 0.44. \label{eq:corr}
\end{eqnarray}
This value excludes the no-correlation hypothesis with a confidence
level greater than 99\%. However, just because the data sets are
correlated does not imply that they are causally connected. This can
only be answered on a model-by-model basis.

Clearly, a muon background model for DAMA must encompass the
stochastic nature of the underlying process. Even if the timing
between muons and DAMA are incommensurate at first sight, could it be
that such a Poisson smearing alleviates the observed tension in the
annual phase? That this is indeed the case has been shown
in~\cite{Blum:2011jf} where a simple, linear model for how a
muon-sourced background may be realized is employed. Taking from this
that the muon hypothesis for DAMA would become viable again is however
not correct.

In {\texttt{CPY}}\ we generate $10^4$ realizations for DAMA based on the generic
model presented in~\cite{Blum:2011jf} which connects the muon flux
measured in LVD with the count rate observed in DAMA/LIBRA. The
question which has to be asked is now the following: How likely is it
that one realization out of the mock data indeed induces the signal
observed in DAMA. To answer this question we can again look at the
correlation coefficient. It has the merit that we do not impose any
functional form on the data. As expected, the generated mock data
exhibits a high degree of correlation with LVD. In fact, the
correlation with the actual muon data is substantially \textit{higher}
than the one which is actually observed in~(\ref{eq:corr}). This rules
out the model at $99\%$~C.L.. Given the generality of the model, it
also strongly disfavors the hypothesis that muons can be the origin
for the count rate variation seen in the DAMA detectors.

\section{Conclusions}
\label{sec:conclusions}

In {\texttt{CPY}}\ we lay out a detailed time-series analysis of DM direct
detection data as well as related datasets. In particular, we find no
evidence to support the claim that atmospheric muons are responsible
for the signal that the DAMA collaboration observes. We identify
difficulties in phase, amplitude, power spectrum and degree of implied
correlation between the data sets. Here we offer some additional
comments:
\begin{list}{\labelitemi}{\leftmargin=1em}
\item When binning data, fine-grained information on the time-series
  is lost. In addition, we find that when the LVD measured muon flux
  is binned in time in the same way as DAMA/LIBRA presents their data,
  significant power is lost on \textit{all} time scales. This has the
  consequence that the LVD and DAMA data sets become more alike in the
  frequency domain with a dominating peak at a period of one
  year. Thus binning seems to diminish discriminating potential
  between the data sets. This calls for an unbinned analysis of the
  DAMA data set.
\item The uncertainty on the phase of the underground muon flux
  reported by LVD, $t_0 = (185 \pm 15)$ days is erroneous. In {\texttt{CPY}}, an
  uncertainty of $ \pm 2$~days was obtained, but the discrepancy had
  not been emphasized. An independent analysis
  in~\cite{FernandezMartinez:2012wd} confirms our value.
\item A recent paper by the DAMA collaboration presents the LS spectra
  of the average count rates in the DAMA/NaI and DAMA/LIBRA cycles. No
  significant power indicating potential long-term variation
  ($T>1$~yr) is observed. Following this procedure for LVD data, we
  can reach analogous conclusions---in seeming contradiction with the
  power observed in the right panel of Fig.~\ref{fig:damalvd}. One
  cannot expect statistical significance from a LS-transform of
  $\mathcal{O}(10)$ numbers without further insight into the
  normalization of (\ref{eq:LS}).
\item The same paper confirms our previous findings in {\texttt{CPY}}\ that a
  putative cosmic ray activation can only lead to a maximum phase
  shift of a quarter year in the subsequently modulated decay. Hence
  it is not possible to overcome the $\sim 11$ month discrepancy
  between muons and DAMA.
\item A recent paper~\cite{FernandezMartinez:2012wd} incorporates
  additional data on the Gran Sasso underground muon flux. Frequentist
  fits to period and phase improve on the already significantly
  disjoint LVD and DAMA regions presented in {\texttt{CPY}}. This corroborates
  the statement that both data sets are seemingly
  incompatible. However, as argued in Sec.~\ref{sec:correlation} and
  at great length in {\texttt{CPY}}\ reliance on the inferred phase can be
  misleading. Indeed in~\cite{Blum:2011jf} and {\texttt{CPY}}\ it was shown that
  Poisson smearing can sufficiently alleviate the tension between the
  phases of the muon and DAMA data sets. It is the correlation
  analysis which reveals that the data sets remain nevertheless
  incompatible. However, the latter conclusion depends on the model
  which connects the muon flux with the DAMA count rate and it can
  only be achieved considering data sets which have actual temporal
  overlap. From this perspective, no further gain of information with
  respect to DAMA is obtained by enlarging the set of muon
  measurements as done in~\cite{FernandezMartinez:2012wd}.
\end{list}
As in the case of DAMA, in {\texttt{CPY}}---employing a similar line of
analyses---we find no significant correlation between the CoGeNT data
set and the yearly variation of the muon flux. In summary, cosmic ray
muons cannot be the sole source for the observed modulation signals in
these experiments.

\section*{Acknowledgments}
It is a pleasure to thank S.~Chang and I.~Yavin for collaboration on
this subject. The author also acknowledges useful discussions with
K.~Blum and E.~Fernandez-Martinez.


\section*{Supplementary Material}

\subsection{The dielectric constant}

 Here we elaborate on the understanding of the dielectric constant $\epsilon_c$. In BKT theory, the vortex system is descibed by the Hamiltonian
\begin{eqnarray}
\frac{{\cal H}_v}{k_BT}=&-&\pi K\int d^2{\mathbf r}  \int d^2{\mathbf r'}n(\mathbf r)n(\mathbf r')\log\frac{|{\mathbf r}-{\mathbf r}'|}{R_0} \nonumber\\
&-&\log y \int d^2{\mathbf r} n^2(\mathbf r),
\end{eqnarray}
where the stiffness $K=n_s\hbar^2/4mk_BT$ and the vortex fugacity $y=e^{-E_c/k_BT}$ obey the renormalization group (RG) equations [\onlinecite{Kosterlitz74, Jose77}]
\begin{eqnarray}
\frac{d}{dl}K^{-1}(l)&=&4\pi^3y^2(l),\nonumber\\
\frac{d}{dl}y(l)&=&[2-\pi K(l)]y(l).
\label{KTRG}
\end{eqnarray}
Here $l=\ln(r/\xi)$ is the RG scale, $\xi$ is the coherence length, and $E_c$ is the vortex core energy.

\begin{figure}
\begin{centering}
\includegraphics[width=0.6\linewidth]{EcC.pdf}
\includegraphics[width=0.48\linewidth]{Ec.pdf}
\includegraphics[width=0.48\linewidth]{Ecec.pdf}
\end{centering}
\caption{(a): The dielectric constant $\epsilon_c$ as function of the dimensionless vortex core energy $C$. The dashed line is a fit to the power law behavior. (b): Renormalization of the dielectric constant $\epsilon(r)$, for different temperatures  $T=T_{\rm BKT}, 0.95T_{\rm BKT}, 0.9T_{\rm BKT}$ (from top to bottom). Here $\epsilon_c=90, C=0.0599$. (c): The ratio of vortex core energy and BKT transition temperature as function of the dielectric constant, $E_c/k_BT_{\rm BKT}= (A^{1/\theta}/2\pi)\epsilon_c^{-(1-\theta)/\theta}$, with $\theta=0.83$. }
\label{Ec}
\end{figure}

One can define a scale-dependent dielectric constant $\epsilon(r)=K(0)/K(l)$, which measures the renormalization of the stiffness $K$ due to the screening of vortex-antivortex pairs. Without screening, $K$ takes the bulk value $K(0)=\Phi_0^2d/16\pi^3\lambda^2_{\rm b}(T)k_BT$, with $\lambda_{\rm b}$ the bulk penetration depth. Including the effect of screening, $K$ changes with the scale $r$.  One of the most important experimental consequencies of the BKT theory is that, at the BKT transition temperature, the renormalized $K$, i.e. $K(l=\infty)$, approaches a universal value  [\onlinecite{Nelson77}], which can be read out directly from the above RG equations to be $K(\infty)=2/\pi$. At $T=T_{\rm BKT}, r=\infty$, the scale-dependent dielectric constant becomes of the form $\epsilon(r=\infty,T_{\rm BKT})=\Phi_0^2d/32\pi^2\lambda^2_{\rm b}(T_{\rm BKT})k_BT_{\rm BKT}\equiv\epsilon_c$. $\epsilon_c$ is a nonuniversal number. It takes different values for different systems. For conventional superconductors, e.g. InO$_x$, it is typically 1.1 to 1.9. For YBa$_2$Cu$_3$O$_7$ thin films, it is much larger, $\epsilon_c\simeq$ 4.6 [\onlinecite{Firoy88}] or 6 [\onlinecite{Matsuda93}] .

 The penetration depth is correspondingly renormalized with respect to the bulk value, with $\lambda^{-2}=\lambda^{-2}_{\rm b}/\epsilon(r=\infty)$. At the transition, the renormalized penetration depth satisfies the relation  [\onlinecite{Nelson77}] $k_BT_{\rm BKT} =\Phi_0^2d/32\pi^2\lambda^2$ (Eq.~(4) in the main text), which is universal in the sense that, different from $\epsilon_c$, this relation is identical for different systems. Thus to determine whether a superconducting transition is of the BKT type, it is crucial to measure the penetration depth $\lambda$, and to check whether such universal relation between $\lambda$ and $T_{\rm BKT}$ is satisfied. Such relation has been observed in superfuid helium thin films [\onlinecite{Bishop78}].

We can parameterize the vortex fugacity in term of a dimensionless quantity $C$, with $y(0)=\exp[-CK(0)/4]$ [\onlinecite{Davis90}]. $C$ is directly proportional to the vortex core energy, with $E_c=E_0C$ and $E_0=\Phi_0^2d/64\pi^3\lambda^2_{\rm b}=(\epsilon_c/2\pi)k_BT_{\rm BKT}$. The vortex core energy can be written as $E_c=(C\epsilon_c/2\pi)k_BT_{\rm BKT}$. From the above RG equations, one can see that the renormalized fugacity vanishes at the transition, i.e. $y(r=\infty, T_{\rm BKT})=0$.

Now we proceed to quantify the relation between the vortex core energy $E_c$ (or its dimensionless counterpart $C$) and the dielectric constant $\epsilon_c$. With the initial condition $K(0)=2\epsilon_c/\pi$, $y(0)=e^{-CK(0)/4}$ and the final condition $K(\infty)=2/\pi$, $y(\infty)=0$, we can numerically solve the RG equations. We find that  $\epsilon_c=2, 4.6, 6, 90$ corresponds to $C=7.27, 2.24, 1.583, 0.0599$ respectively (see Fig.~\ref{Ec}(a)). Following the RG flow (Fig.~\ref{Ec}(b)), one can see that, only very close to the transition temperature, the dielectric constant changes substantially with scale. When moving away from $T_{\rm BKT}$, $\epsilon(r)$ quickly settles down to its infared value $\epsilon_{\infty}$, and $\epsilon_{\infty}$ decreases significantly with decreasing temperature [\onlinecite{Davis90}].

 It is interesting to notice that for $\epsilon_c\gtrsim 5$, $\epsilon_c$ and $C$ has a power law scaling, $\epsilon_c\simeq AC^{-\theta}$, with the coefficient $A\simeq 8.62$ and the power $\theta\simeq 0.83$ (see Fig.~\ref{Ec}(a)). The dielectric constant and the vortex core energy thus has the relation $\epsilon_c\simeq A(E_c/E_0)^{-\theta}$. A large dielectric constant corresponds to a small vortex core energy. For $\epsilon_c=90, C=0.0599$, the vortex core energy $E_c=(C\epsilon_c/2\pi)k_BT_{\rm BKT}\simeq (2.7/\pi) k_BT_{\rm BKT}$ \footnote{In BCS theory, the vortex core energy can be estimated as the loss of condensation energy within the vortex core, $E_c\simeq \pi \xi^2d\epsilon_{\rm cond}$, with the condensation energy density $\epsilon_{\rm cond}=N(0)\Delta^2/2$, the density of states at the Fermi level $N(0)\simeq 3n/2v_F^2m$, the BCS gap $\Delta$, and the coherence length $\xi=\hbar v_F/\pi\Delta$. Assuming $n_s=n$ at $T=0$, we have $E_c\simeq (1.9/\pi)k_BT_{\rm BKT}$ (see e.g. [\onlinecite{Mondal11}]). In XY-model, one has instead $E_c\simeq \pi k_BT_{\rm BKT}$ [\onlinecite{Nagaosa99}].
}. Taking $T_{\rm BKT}\simeq 1.6K$, one obtains $E_c\simeq 0.13 {\rm meV}$. For YBCO thin films [\onlinecite{Matsuda93}], we have $E_c\simeq (1.583\times 6/2\pi)\times 7{\rm meV}\simeq 10.6{\rm meV}$, which is one order of magnitude larger than that of heavy fermion superlattice [\onlinecite{Mizukami11}].

For large $\epsilon_c$, we have $E_c/k_BT_{\rm BKT}\simeq (A^{1/\theta}/2\pi)\epsilon_c^{-(1-\theta)/\theta}$  (see Fig.~\ref{Ec}(c)). Due to the small power $(1-\theta)/\theta\simeq 1/5$, for a given $T_{\rm BKT}$, a small change in the vortex core energy leads to significant change in the dielectric constant. Increasing $\epsilon_c$ from 5 to 90, the vortex core energy only changes from $1.54 k_BT_{\rm BKT}$ to $0.85 k_BT_{\rm BKT}$.

In the presence of competing orders, the vortex core energy is reduced, $E_c=E_c^{(0)}-|\delta E_c|$. As shown in the main text, $|\delta E_c|$ increases as one approaches the QCP. The dielectric constant becomes a function of the distance to the QCP,
\begin{equation}
\epsilon_c=A\left[ \frac{E_c^{(0)}-V_0e^{-2\sqrt{a}}(3+6\sqrt{a}+4a)}{E_0} \right]^{-\theta},
\end{equation}
where $a=\alpha\lambda^4/g^2\mu_B^2\Phi_0^2$ and $\alpha$ is the distance to the QCP. $V_0$ and $a$ depends on the material specific parameters $g, \gamma$. In order to determine quantitatively the evolution of the dielectric constant near the QCP, more material specific microscopic calculations are needed.

\subsection{Effect of the interface}

 At the interface, the Yb ions disorder (due to cross diffusion and displacements) and act as nonmagnetic impurities to locally suppress superconductivity in CeCoIn$_5$ layers [\onlinecite{Bauer11}]. The superconducting order parameter is strongly suppressed near the impurity sites, and it recovers the bulk value over the distance on the order of the coherence length [\onlinecite{Franz97,Xiang95,Franz96}], $\xi(T)\simeq \nu \xi_0/\sqrt{1-T/T_{c0}}$,
with $T_{c0}$ the bulk superconducting transition temperature, $\xi_0$ the BCS coherence length, and $\nu$ a number of order unity. When the thickness of the CeCoIn$_5$ layers is large, $d>\xi(T)$, the areas of defect-depressed order parameter do not overlap, and the gap is not affected by the defects. When the thickness of CeCoIn$_5$ layers become smaller than $\xi(T)$, the depressed areas will start to overlap, and the superconducting gap in the CeCoIn$_5$ layers will be suppressed.

At low temperatures with $T\ll T_{c0}$, $\xi(T)$ is of order $\xi_0$, which is about the thickness of four layers of CeCoIn$_5$. So we expect that for $n\gg 4$, gap has the same value as the bulk material; while for $n\lesssim 4$, gap gets suppressed. This explains the experimental observation that the Pauli-limited upper critical field, which is a direct measure of the gap, retains the bulk value for $n=5,7$, and is suppressed for $n=3$.

\begin{figure}[h]
\begin{centering}
\includegraphics[width=0.48\linewidth]{defect1.pdf}
\includegraphics[width=0.48\linewidth]{defect2.pdf}
\end{centering}
\caption{Illustration of the effect of Yb ions as pair breaking nonmagnetic impurities for $d>\xi$ and $d<\xi$. In the shaded regions of size the coherence length $\xi$ around the Yb ions, superconductivity is suppressed. }
\end{figure}

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