Robust velocity-induced acoustic oscillations at cosmic dawn

The redshifted 21-cm line of hydrogen holds great potential for the study of cosmology, as it can probe otherwise unobservable cosmic epochs. In particular, measurements of the 21-cm power spectrum during cosmic dawn—the era when the first stars were formed—will provide us with a wealth of information about the astrophysics of stellar formation, as well as the origin of fluctuations intn our Universe. In addition to their usually considered density fluctuations, dark matter and baryons possess large relative velocities with respect to each other, due to the baryon acoustic oscillations suffered by the latter, which suppress the formation of stars during cosmic dawn, leaving an imprint on 21-cm observables during this era. Here we present 21cmvfast, a version of the publicly available code 21cmvfast modified to account for this effect. Previous work has shown that the inclusion of relative velocities produces an acoustic modulation on the large-scale 21-cm power spectrum during cosmic dawn. By comparing analytic calculations with simulations from 21cmvfast, here we demonstrate that this modulation takes the form of robust velocity-induced acoustic oscillations (VAOs), during both the Lyman-$\alpha$ coupling era and the subsequent epoch of heating. The unique shape of these VAOs, which is determined by the acoustic physics that generated the relative velocities, can be analytically computed and is easily distinguishable from the usual density-sourced fluctuations. We find that, under a broad range of astrophysical assumptions, the VAOs are detectable at high significance by the upcoming HERA interferometer, which could therefore confirm the presence of acoustic oscillations at cosmic dawn.

Figure 1

Standard deviation ${\sigma }_{\mathrm{cool}}$ of matter fluctuations in regions that will form haloes massive enough to cool and form stars, according to Eq. (11), at $z=20$ (top panel) and $z=30$ (bottom panel), normalized. Each line represents a different feedback assumption, as defined in Sec. 2. The kinks in the regular and low-feedback curves appear due to the transition from molecular- to atomic-cooling haloes, where one of the $v_{\mathrm{cb}}$ effects vanishes.

Figure 2

In the upper panel we show the HMF, and in the bottom panels the collapsed fraction of gas (in solid line) and all matter (in dashed lines) as a function of the minimum halo mass $M$ considered, for the cases of no feedback in the middle panel and regular feedback in the bottom, all at $z=20$. We show the results for $v_{\mathrm{cb}}=0$ in red, for $2v_{\mathrm{rms}}$ in green, and the average over velocities in black, where it is clear that larger relative velocities correspond to smaller collapsed fractions. We also overlay the $v_{\mathrm{cb}}=v_{\mathrm{rms}}$ case in the thin blue line, which is almost coincident with the average case. Finally, the vertical colored lines correspond to the minimum mass necessary to cool and form stars at this redshift for different assumptions, where each colored dotted line corresponds to a $v_{\mathrm{cb}}$ case mentioned above (assuming no feedback), whereas the dot-dashed lines represent the minimum cooling mass in the regular feedback case (black, for $v_{\mathrm{cb}}=0$) and for atomic cooling (gray).

Figure 3

We show the collapsed fraction of baryons to star-forming haloes as a function of redshift, for our regular feedback case, for patches with $v_{\mathrm{cb}}=0$ in the solid red line, $v_{\mathrm{cb}}=2v_{\mathrm{rms}}$ in the green line, and for the average over the sky in the black line. Additionally, we show the $v_{\mathrm{cb}}=v_{\mathrm{rms}}$ case in the light-blue long-dashed line, almost coincident with the average case. This plot includes the three effects of $v_{\mathrm{cb}}$ in the formation of stars, through the HMF, the gas fraction in haloes, and the minimum halo mass to form stars. For comparison, the red dotted and blue dash-dotted lines show the $v_{\mathrm{cb}}=0$ and $v_{\mathrm{cb}}=v_{\mathrm{rms}}$ results for the no-feedback case. In all feedback cases patches with large $v_{\mathrm{cb}}$ have a smaller collapsed fraction, and thus fewer stars.

Figure 4

Collapsed fraction ($F_{\mathrm{coll}}$) without (left) and with (right) the effect of DM-baryon relative velocities ($v_{\mathrm{cb}}$) at $z=20$, smoothed over different radii $R$, which shows how the fluctuations on $v_{\mathrm{cb}}$ are imprinted onto $F_{\mathrm{coll}}$. The top panels show a slice 1 Mpc thick with a 300 Mpc side of a simulation box, generated with a modified version of 21cmvfast [29, 30], where in the left we show the over/underdensities in matter, and on the right the fluctuations in the DM-baryon relative velocity $v_{\mathrm{cb}}$, both quantities evaluated at $z=20$. The second and third rows show the fraction $F_{\mathrm{coll}}$ of baryons collapsed to star-forming galaxies, as defined by Eq. (19), smoothed over $R=3$ and 20 Mpc, respectively, as typical of the mean-free path of x-ray photons with 0.2 and 0.5 keV energies [42]. The final row is also $F_{\mathrm{coll}}$, smoothed over distances of 100 Mpc, corresponding to the typical travel distance of a UV photon until it enters the Lyman-$\alpha$ resonance [13]. Note that in the left panels we have fixed $v_{\mathrm{cb}}=\overline{v_{\mathrm{cb}}}$, as opposed to $v_{\mathrm{cb}}=0$, in order to keep roughly the same average collapse fraction, the color scales are the same in both left and right $F_{\mathrm{coll}}$ plots, and in all cases we assume regular-strength Lyman-Werner feedback. For reference the horizontal black lines denote the smoothing distance $R$ in each panel.

Figure 5

Sky-averaged 21-cm brightness temperature, obtained from our simulations, as a function of redshift for different assumptions. In the red long-dashed line we show the usual molecular-cooling case, without including the effect of velocities or Lyman-Werner feedback. In the purple dot-dashed line we show the result when including $v_{\mathrm{cb}}$, but not feedback, and the black heavy line shows the results of the full calculation, including both $v_{\mathrm{cb}}$ and LW feedback. The blue dashed line represents the result when fixing $v_{\mathrm{cb}}=v_{\mathrm{avg}}$, without fluctuations, whereas the green dotted line is the result if only atomic-cooling haloes form stars, as commonly assumed in the literature. The inclusion of both relative velocities and Lyman-Werner feedback delays the global-signal landmarks. Finally, the colored circles represent the redshifts at which we will show maps of $T_{21}$ in Figs. 7 and 9.

Figure 6

The 21-cm brightness temperature across cosmic dawn in the case without LW feedback or velocities (top), with $v_{\mathrm{cb}}$ but no feedback (middle), and the full case with (regular-strength) feedback and velocities (bottom). Each result is obtained from a simulation slice 300 Mpc in length and 1 Mpc in thickness, consistently evolved in redshift $z$. Blue and red represent 21-cm absorption and emission, respectively, and the anisotropies are primarily sourced by the formation of the first stars.

Figure 7

Fluctuations in the 21-cm brightness temperature (around their average) during the Lyman-$\alpha$ coupling era for the case of regular feedback. On the left panels we have fixed the DM-baryon relative velocity to its average value $v_{\mathrm{avg}}$, whereas on the right panels it fluctuates, with the initial conditions shown in the first row. The three redshifts are chosen to be on the onset of Lyman-$\alpha$ coupling ($z=28.3$), halfway through the Lyman-coupling era ($z=25.0$), and at the transition between Lyman-coupling and x-ray heating (where $⟨T_{21}⟩$ is minimum; at $z=20.3$). The black arrow represents the acoustic scale of 150 Mpc, for reference.

Figure 8

The 21-cm power spectrum as a function of wave number $k$ for the same three redshifts during the LCE as Fig. 7. The data points are the results of our simulations, including Poissonian error bars, whereas the solid lines are obtained with the fit in Eq. (36) for each redshift. The power here is driven by the anisotropic Lyman-$\alpha$ pumping and grows as more UV photons are emitted and couple the spin and gas temperatures. The arrow represents the periodicity of the VAOs, $\mathrm{\Delta }k=2\pi /r_{\mathrm{drag}}$, given by the acoustic scale $r_{\mathrm{drag}}=147\mathrm{Mpc}$.

Figure 9

The 21-cm fluctuations, as in Fig. 7, but during the EoH. In this case we choose the redshifts to be halfway through the EoH ($z=17.2$), at the transition to emission (where $⟨T_{21}⟩=0$; at $z_0=14.5$), and at the end of our simulation ($z=12.0$), where x-ray heating has saturated. The impact of $v_{\mathrm{cb}}$ during the EoH is apparent, for instance in the third row, where larger velocities produce deeper 21-cm absorption.

Figure 10

Same as Fig. 8 but during the EoH, for the same three redshifts as Fig. 9. Again our simulations are shown by data points with error bars, and the fits are solid lines. The source of power here is the inhomogeneous x-ray flux of the first stars, which heats up the hydrogen gas, reducing the amount of 21-cm absorption at lower redshifts.

Figure 11

The 21-cm power spectrum halfway through the EoH (at $z=17.2$) and the LCE (at $z=25.0$) for our regular-strength feedback model. The lighter colors and solid lines represent the case with velocity fluctuations, whereas the dark colors and dashed lines show the case with DM-baryon velocities fixed at $v_{\mathrm{avg}}$, without fluctuations. As before, data points with Poissonian errors are the results of our simulations, and the lines fit them using Eq. (36). The dotted lines show the VAO-only contribution at both redshifts, which can be added to the $v_{\mathrm{avg}}$ case to recover the total signal.

Figure 12

Window functions required to account for photon propagation in the case of x-ray and Lyman-$\alpha$ photons, computed as in Eqs. (31) and (35), at redshifts (from left to right) $z=\{22.3,25.0,28.3\}$ for the Lyman-$\alpha$ case and $z=\{14.5,17.2,20.3\}$ for the x-ray case. The x-ray case shows a less-marked suppression, as x-ray photons typically travel smaller distances than their UV counterparts.

Figure 13

Amplitude of fluctuations of ${\delta }_{v^2}\equiv \sqrt{3/2}[(v_{\mathrm{cb}}/v_{\mathrm{rms}}{)}^2-1]$ as a function of wave number $k$, under three assumptions. In the dashed black line we show the local power spectrum (i.e., without any window function), whereas we account for photon propagation by multiplying this quantity by the window function squared for the x-ray and Lyman-$\alpha$ cases in the purple and green lines, corresponding to 21-cm fluctuations at $z=17.2$ and 25.0, halfway through the EoR and LCE, respectively.

Figure 14

Amplitude of the 21-cm VAOs as a function of redshift. In the top panel we show the VAO-induced variance, computed from Eq. (39), obtained from our simulations in the black line and from analytic estimates in the red dashed line. As a consistency check we show the result from a simulation without $v_{\mathrm{cb}}$ fluctuations in the grey dotted line, which should be consistent with zero, providing an estimate of the error in our fitting procedure. In the bottom panel we show the VAO-only power spectrum for the same case, at a reference scale $k_{\mathrm{ref}}=0.14{\mathrm{Mpc}}^{-1}$, typically observable by interferometers.

Figure 15

Global 21-cm signal for the four feedback cases we study. From right to left we show the cases of no feedback, as well as low-, high-, and regular-feedback strengths, in all cases including the effect of $v_{\mathrm{cb}}$.

Figure 16

The 21-cm power spectrum halfway through the epoch of heating (where $⟨T_{21}⟩=0.5\times {⟨T_{21}⟩}_{\min }$), for each feedback case. From top to bottom we show the cases of no feedback, and low-, regular-, and high-feedback strengths, with corresponding redshifts $z=\{19.1,17.6,17.2,17.2\}$. The vertical gray lines are separated by $\mathrm{\Delta }k=2\pi /r_{\mathrm{drag}}$, where $r_{\mathrm{drag}}=147\mathrm{Mpc}$ is the acoustic scale, as the shape and periodicity of the VAOs do not depend on the feedback strength.

Figure 17

Same as Fig. 16 but for the half-point of the Lyman-$\alpha$ coupling era, with $⟨T_{21}⟩=0.5\times {⟨T_{21}⟩}_{\min }$, which corresponds to $z=\{28.3,25.5,25.0,25.5\}$, going from lowest to highest feedback strength.

Figure 18

Same as the bottom panel of Fig. 14, albeit for the four feedback cases, from top to bottom: no feedback, low, regular, and high feedback. Higher feedback produces a smaller VAO amplitude, as stellar production in molecular-cooling haloes is less important.

Figure 19

The 21-cm power spectrum as a function of wave number $k$, where we have increased the minimum energy of the x-ray spectrum to $E_0=0.5\mathrm{keV}$. The redshifts correspond to the landmarks in the EoH: postheating ($z=12.0$), at the transition to emission ($z_0=14.8$), and halfway through heating ($z=17.2$). Solid lines are the fits to the simulations, using the modified window function for the new x-ray spectrum. Light colors indicate the inclusion of $v_{\mathrm{cb}}$ fluctuations, whereas darker colors and dashed lines have a fixed $v_{\mathrm{cb}}=v_{\mathrm{avg}}$.

Figure 20

Relative difference in the 21-cm power spectrum between the case with and without velocities, considering only haloes that can form stars through atomic cooling ($T_{\mathrm{cool}}={10}^4\mathrm{K}$). We show results at $z=16.8$ and 23.5, which correspond to the halfway points of the EoH and the LCE, where $⟨T_{21}⟩=0.5\times {⟨T_{21}⟩}_{\min }$. The black dashed line follows the shape of ${\mathrm{\Delta }}_{v^2}^2(k)$ divided by the fiducial 21-cm power spectrum, shown for comparison.

Figure 21

The 21-cm power spectrum for standard 21cmmc (in dark dashed lines) and 21cmvfast with $v_{\mathrm{cb}}=0$ (in solid lines), assuming atomic cooling only. The minor differences arise from the slightly different background evolution in the two cases.

Figure 22

Cross spectra of ${\delta }_v^{(n)}$ and ${\delta }_v^{(m)}$, as defined in Eq. (c5), normalized at $k=0.1{\mathrm{Mpc}}^{-1}$, as a function of wave number $k$. Different colors correspond to different values of $n$ and $m$ from 1 to 4, where the black thick line is the baseline $n=m=2$ case. This shows that the power spectrum of any generic function $f(v_{\mathrm{cb}})$ of the DM-baryon relative velocity has the same VAO shape.