\rho_a &= \frac{1}{2a^2}\dot{\phi_0}^2 + \frac{m_a^2}{2}\phi_0^2, \\
    P_a &= \frac{1}{2a^2}\dot{\phi_0}^2 - \frac{m_a^2}{2}\phi_0^2,
\end{align}
where the dot denotes the derivative with respect to conformal time and $\phi_0$ is the homogeneous background field. The equation of motion of the field for the field reads
\begin{align}
    \ddot{\phi_0} + 2\mathcal{H} \dot{\phi_0} + \frac{1}{2}a^2 m_a^2 \phi_0 =0,
\end{align}
where $\mathcal{H}=aH$ is the conformal Hubble factor. At early times ($H\gg m_a$), the field is slowly rolling with $\dot{\phi}\approx 0$, the axion equation of state is $w_a \equiv P_a/\rho_a\approx -1$, and the field behaves as a dark energy component. At late times ($H\ll m_a$), the field's equation of state oscillates around zero and the energy density of the field obeys the scaling $\rho_a\propto a^{-3}$. It is therefore a dark matter component after the field started oscillating around the potential minimum. The value of the scale factor when this transition happens is denoted $a_\mathrm{osc}$ and defined such that
\begin{align}
    3H(a_\mathrm{osc}) \approx m_a.
\end{align}
This implies that the time of transition of the field is given as a function of its mass. For $m_a\lesssim 10^{-28}$ eV, and $a_\mathrm{osc}>a_\mathrm{eq}$ the field does not behave as dark matter at matter-radiation equality and cannot constitute the entirety of the dark matter. We refer to those axions as dark-energy-like axions. For any mass, the mean axion density depends on $a_\mathrm{osc}$ and is given by
\begin{align}
    \rho_a = a_\mathrm{osc}^3\bigg[\frac{1}{2a^2}\dot{\phi_0}^2 + \frac{m_a^2}{2}\phi_0^2 \bigg]_{m_a=3H}.
\end{align}
   
   
   
   
   

One of the main signatures of axions is their lack of clustering on small scales. Axions have a characteristic Jeans scale $k_{J}$ below which the growth of structure is suppressed\footnote{See Ref.~\cite{Khlopov1985GravitationalInstability} for a study of the Jeans instability in the presence of self-interactions.}. For the linear matter power spectrum with dark-matter-like axions, the suppression is frozen in at matter-radiation equality and we have~\cite{Hu2000FuzzyCold}
\begin{align}
    k_{J,\mathrm{eq}} \approx 9 \bigg(\frac{m_a}{10^{-22}\;\mathrm{eV}}\bigg)^{1/2} \mathrm{Mpc}^{-1}.
\end{align}

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{figures/linear_pk_changing_frac.pdf}
    \caption{Linear power spectrum for different axion masses and densities. We note that the level of suppression increases with the axion density while the scale at which the axion power spectrum deviates from the CDM ($\Omega_a=0$) case is dependent on the axion mass. The grey shaded area highlights the Fourier modes probed by our analysis between $k_\mathrm{min}=0.025\;h/$Mpc and $k_\mathrm{max}=0.23\;h/$Mpc. The black arrow indicates the Jeans scale for the only axion mass for which $a_\mathrm{osc} < a_\mathrm{eq}$ among the three showed. For reference, the massive neutrino characteristic scale of $k_\mathrm{nr}$ for a neutrino with a mass of $150$ eV would correspond to the Jeans scale of axions of mass $10^{-26}$ eV.}
    \label{fig:linear_pk}
\end{figure}
This holds when assuming that the axions behave as dark matter at matter-radiation equality or that $m_a\gtrsim 3H(a_\mathrm{osc})$. In the alternate case, the field behaves as a dark energy component until it starts oscillating. The Jeans scale at matter-radiation equality is not the optimal reference scale for these axions. We need instead to use the scale at horizon crossing when the field started oscillating which we can calculate as~\cite{Amendola2006DarkMatter,Arvanitaki2010StringAxiverse,Bauer2021IntensityMapping}
\begin{align}
    k_m = a_\mathrm{osc} H\left(a_\mathrm{osc}\right).
\end{align}
We note that in both cases, the loss in power on small scales is dependent on the axion fraction $\Omega_a/\Omega_d$ where $\Omega_d$ is the total dark matter density (combining CDM and axions). The level of suppression may then only be partial when fixing the axion mass and varying the axion concentration with the $\Omega_a\to 0$ limit recovering the pure CDM result. This deviation from CDM in the linear power spectrum illustrated in Fig.~\ref{fig:linear_pk} is the main signature we use in this study to constrain axions. The matter power spectra including axionic effects are computed using the adapted Boltzmann code \texttt{axionCAMB}~\cite{Hlozek2015AData}. It is worth noting that the code solves for the evolution of the axion perturbations using an effective fluid approach. It has been shown to match closely the exact solution, although biases could occur if the sensitivity of the experiments used approaches the cosmic variance limit~\cite{Cookmeyer2020HowSound}.

\section{Model}
\label{sec:model}
Our datasets consist of the multipole moments of the CMASS ($z_\mathrm{eff}=0.57$) and LOWZ ($z_\mathrm{eff}=0.32$) catalogs from the BOSS survey~\cite{Beutler2017TheClustering}. We use the North and South Galactic Caps denoted NGC and SGC for our analysis. Given galaxy positions, the local galaxy overdensity is defined as
\begin{align}
    \delta_g(\mathbf{x}) \equiv \frac{n_g(\mathbf{x})}{\bar{n}_g}-1,
\end{align}
where $n_g(\mathbf{x})$ is the number density of galaxies at the comoving position $\mathbf{x}$ and $\bar{n}_g$ is the mean number density for the survey volume $V$. After taking the Fourier transform $\delta_g(\mathbf{k})$ of the real-space overdensity, the multipole moments $P_\ell$ of the power spectrum are calculated by integrating over the angle between the line-of-sight and the wavevector $\mathbf{k}$. It is customary to decompose the latter into $k$ which is the scalar amplitude of the vector and $\mu$ which is the cosine of angle between the wavevector and the line-of-sight. The integral expression for the galaxy multipoles can then be written as (using the Yamamoto estimator~\cite{Yamamoto2006AMeasurement})
\begin{align}
    P_\ell (k) = \left\langle \frac{2\ell+1}{V}\int\frac{d\Omega_k}{4\pi} \delta_g(\mathbf{k})\delta_g(-\mathbf{k}) \mathcal{P}_\ell\left(\hat{\mathbf{k}}\cdot\hat{\mathbf{z}}\right)\right\rangle,
    \label{eq:P_ell}
\end{align}
where the hat denotes unit vectors and $\mathcal{P}_\ell$ are the Legendre polynomials of degree $\ell$. In the data, the shot noise $P_\mathrm{shot} = 1/\bar{n}_g$ is subtracted from the monopole. For the purpose of this study, we will focus on the monopole ($\ell=0$) and quadrupole ($\ell=2$) moments of the power spectrum. 


\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{figures/mon_quad_changing_frac.pdf}
    \caption{Monopoles and quadrupoles of the galaxy clustering obtained from the linear power spectra of Fig.~\ref{fig:linear_pk}. The model is evaluated with fixed cosmological and bias parameters at redshift $z=0.57$. We highlight a slight increase in the quadrupole on small scales indicating that the structure suppression from axions also has anisotropic effects.}
    \label{fig:mon_quad_model}
\end{figure}



Our model for the multipoles begins with the linear matter power spectrum generated from a set of standard flat $\Lambda$CDM cosmological parameters and a set of two axion parameters. We first obtain the