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\small The dimensionless power spectrum of the axion dark matter overdensity field as it is expected to be produced by axion strings at $\log(m_r/H)\gg1$, as a function of the comoving momentum $k$ normalized to $k_\star = m_a a_\star$. The blue lines show the nontrivial (yet small) evolution of the spectrum around $H = H_\ell \simeq H_\star/200$, when the axions emitted from strings experience a nonlinear transient as the axion potential becomes relevant. This increases mildly the typical momentum with respect to a purely linear evolution (orange line). At the end of the nonlinear transient (black line), the spectrum is of order one at $k_{\rm wn}=Ck_\star$ with $C=\mathcal{O}(10)$ or slightly larger. At $k\ll k_{\rm wn}$, the spectrum acquires the white-noise form $(k/k_{\rm wn})^3$.

\small A sketch of the dimensionless power spectrum $\mathcal{P}=\mathcal{P}_{\rm ad}+\mathcal{P}_{\rm iso}$ around matter-radiation equality, shown as a function of comoving momentum $k$ in units of $k_\star=m_a a_\star\simeq 54 (m_a/10^{-20}\,\mathrm{eV})^{1/2}\,\mathrm{Mpc}^{-1}$. The isocurvature contribution $\mathcal{P}_{\rm iso}$ peaks at $k_{\rm wn}=Ck_\star$ with $C=\mathcal{O}(10)$. For $f_{\rm DM}=1$, axion free streaming suppresses the adiabatic component $\mathcal{P}_{\rm ad}$ at $k \gtrsim k_{\rm fs}$, visible as the dip in the dark blue curve, while leaving the white-noise tail of $\mathcal{P}_{\rm iso}$ unaffected. This suppression becomes increasingly smaller for subdominant axion fractions $f_{\rm DM}=0,0.1,0.5$ (orange, light blue, and purple curves, respectively). In addition, the axion velocity dispersion (classical pressure) and quantum pressure inhibit the growth of the axion fluctuations at $k \gtrsim k_J \propto (a/a_{\rm eq})^{1/2}\equiv y^{1/2}$ and $k \gtrsim k_j \propto y^{1/4}$ (light and dark gray regions).

\small {\bf{Left}}: The free-streaming transfer function $T^2_{\rm fs}$ for the adiabatic power spectrum $\mathcal{P}_{\rm ad}$ for different values of the fraction $f_{\rm DM}$ of dark matter in axions. For $f_{\rm DM}\ll1$, only the adiabatic perturbations in the (warm) axion dark matter component are affected by free streaming and $T^2_{\rm fs}\simeq 1-2f_{\rm DM}$ for $k\gtrsim k_{\rm fs}(a_{\rm eq})$; free-streaming effect is increasingly negligible. {\bf{Right}}: Sketch of the isocurvature transfer function for $f_{\rm DM}\ll1$ and at $a\gg a_{\rm eq}$, normalized that expected in pure CDM. For $k\gtrsim k_J^{\rm eq}$ the axion velocity dispersion mildly suppresses the growth of the axion+CDM fluctuations, leading to $T_{\rm iso}^2\propto 1/k$.

\small {\bf{Left}}: The free-streaming transfer function $T^2_{\rm fs}$ for the adiabatic power spectrum $\mathcal{P}_{\rm ad}$ for different values of the fraction $f_{\rm DM}$ of dark matter in axions. For $f_{\rm DM}\ll1$, only the adiabatic perturbations in the (warm) axion dark matter component are affected by free streaming and $T^2_{\rm fs}\simeq 1-2f_{\rm DM}$ for $k\gtrsim k_{\rm fs}(a_{\rm eq})$; free-streaming effect is increasingly negligible. {\bf{Right}}: Sketch of the isocurvature transfer function for $f_{\rm DM}\ll1$ and at $a\gg a_{\rm eq}$, normalized that expected in pure CDM. For $k\gtrsim k_J^{\rm eq}$ the axion velocity dispersion mildly suppresses the growth of the axion+CDM fluctuations, leading to $T_{\rm iso}^2\propto 1/k$.

95\% C.L. exclusion limits on the primordial amplitude $A_{\rm iso}$ of the isocurvature component of the power spectrum $\mathcal{P}_{\rm iso}(k)=A_{\rm iso}(k/k_{\rm cut})^3$ as a function of its UV cutoff $k_{\rm cut}$. The bounds originate from CMB+BAO measurements~\cite{Buckley:2025zgh} (orange), Lyman-$\alpha$ MIKE/HIRES data with EFT-based~\cite{Ivanov:2025pbu} (blue) and hydrodynamic simulations~\cite{Murgia:2019duy} (red), as well as the UVLF of HST galaxies (green) used in this work. The bounds assume all dark matter fluctuations evolve as CDM, i.e. with negligible velocity dispersion and quantum pressure.

\small{Linear matter power spectrum $P(k)=(2\pi^2/k^3)\mathcal{P}(k)$ at $z=0$ as a function of the comoving momentum $k$. The $\Lambda$CDM prediction (black) is compared to two benchmark models with additional isocurvature white-noise components (gray, dashed and dotted). Black points with error bars show the 95\% C.L. constraints on excess power inferred in this work from Hubble Space Telescope UVLF data. Blue points denote Lyman-$\alpha$ measurements from eBOSS DR14~\cite{eBOSS:2018qyj,Chabanier:2019eai}, and gray points the Lyman-$\alpha$ EFT analysis~\cite{Ivanov:2025pbu} (which constrains an isocurvature enhancement as well). The remaining error bars correspond to SDSS DR7 luminous red galaxies (purple)~\cite{2010MNRAS.404...60R} and the \textit{Planck} 2018 CMB power spectra—temperature (red), polarization (orange), and lensing (green)~\cite{Planck:2018vyg}.}

\small Halo mass function for standard CDM (black dashed) and for isocurvature enhancements corresponding to varying values of characteristic cutoff wavenumber, $k_{\rm cut}$, at $z=8$. The values $k_{\rm cut}=\{0.5,1.7,5.4,17.2,54.3\}$ $\rm Mpc^{-1}$ correspond to $\tilde{A}_{\rm iso}=\{1.2\cdot10^{-2},3.9\cdot10^{-4},1.2\cdot10^{-5},3.9\cdot10^{-7},1.2\cdot10^{-8}\}$ $\rm Mpc^{3}$, respectively. The lower panels show the fractional difference with respect to the standard CDM prediction.

\small UV luminosity function for standard CDM (black dashed) and for an additonal white-noise isocurvature component corresponding to varying values of its characteristic cutoff wavenumber, $k_{\rm cut}$, at redshift $z=8$. As in Fig.~\ref{fig:halmasscomparison}, the values $k_{\rm cut}=\{0.5,1.7,5.4,17.2,54.3\}$ $\rm Mpc^{-1}$ correspond to isocurvature components $\tilde{A}_{\rm iso}=\{1.2\times10^{-2},3.9\times10^{-4},1.2\times10^{-5},3.9\times10^{-7},1.2\times10^{-8}\}$ $\rm Mpc^{3}$, respectively. The black points correspond to the UVLF HST measurements at $z=8$, while the lower panels show the fractional difference with regards to the standard CDM prediction.

\small The UVLF function data points from the HST are plotted in the redshift range $z=4-10$, together with the corresponding best-fit predictions from our likelihood analysis (solid lines).

\small{Marginalized 68\% C.L. and 95\% C.L. contours obtained on the cosmological parameters and $\tilde{A}_\mathrm{iso}$ from our likelihood analysis performed for various cutoff wavenumbers, $k_{\rm cut}$. For visual clarity, only a subset of $k_{\rm cut}$ values is shown here; the full set of $\tilde{A}_\mathrm{iso}$ constraints is reported in Table~\ref{tab:Piso_bounds}.}

\small{Marginalized 68\% C.L. and 95\% C.L. contours obtained on the UVLF galaxy parameters and $\tilde{A}_\mathrm{iso}$ from our likelihood analysis performed for various cutoff wavenumbers, $k_{\rm cut}$.}

\small Constraints on the axion mass $m_a$ and decay constant $f_a$ in the post-inflationary scenario, including large-scale structure probes. Axions radiated by strings can overproduce dark matter (gray) or contribute as dark radiation $\Delta N_{\rm eff}$ (purple). CMB observations exclude very light $m_a$ at large $f_a$ via string-induced anisotropies (``CMB strings'', gray). Dashed red curves indicate $(m_a,f_a)$ for which axions make up a fraction $f_{\rm DM}$ of dark matter. For $f_{\rm DM}\simeq1$, free-streaming excludes sufficiently small $m_a$ (yellow). For smaller $f_{\rm DM}$, the most stringent bounds arise from the enhancement of CDM growth sourced by the axion white-noise power (CMB+BAO, Lyman-$\alpha$, and galaxy UVLF; see Fig.~\ref{fig:Piso}) and are presented for the benchmark $C=10$. The top axis shows the white-noise peak scale $k_{\rm wn}=C\,k_\star$.