Dark Matter from Axion Strings with Adaptive Mesh Refinement

Dark Matter from Axion Strings with Adaptive Mesh Reﬁnement

Malte Buschmann,1, ∗ Joshua W. Foster,2, 3, 4, † Anson Hook,5 Adam Peterson,6 Don E. Willcox,6 Weiqun Zhang,6 and Benjamin R. Safdi3, 4, ‡ 1Department of Physics, Princeton University, Princeton, NJ 08544, USA 2Leinweber Center for Theoretical Physics, Department of Physics, University of Michigan, Ann Arbor, MI 48109 3Berkeley Center for Theoretical Physics, University of California, Berkeley, CA 94720 4Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, CA 94720 5Maryland Center for Fundamental Physics, University of Maryland, College Park, MD 20742, U.S.A. 6Center for Computational Sciences and Engineering Lawrence Berkeley National Laboratory Berkeley, CA 94720 (Dated: August 13, 2021) Axions are hypothetical particles that may explain the observed dark matter (DM) density and the non-observation of a neutron electric dipole moment. An increasing number of axion laboratory searches are underway worldwide, but these efforts are made difficult by the fact that the axion mass is largely unconstrained. If the axion is generated after inflation there is a unique mass that gives rise to the observed DM abundance; due to nonlinearities and topological defects known as strings, computing this mass accurately has been a challenge for four decades. Recent works, making use of large static lattice simulations, have led to largely disparate predictions for the axion mass, spanning the range from 25 microelectronvolts to over 500 microelectronvolts. In this work we show that adaptive mesh refinement (AMR) simulations are better suited for axion cosmology than the previously-used static lattice simulations because only the string cores require high spatial resolution. Using dedicated AMR simulations we obtain an over three order of magnitude leap in dynamic range and provide evidence that axion strings radiate their energy with a scale-invariant spectrum, to within ∼5% precision, leading to a mass prediction in the range (40,180) microelectronvolts.

An outstanding mystery of the Standard Model of par- metry is restored. As the Universe expands the strings ticle physics is that the neutron electric dipole moment, shrink, straighten, and combine by emitting radiation which would cause the neutron to precess in the presence into axions. The contribution to the DM abundance from of an electric field, appears to be over ten billion times the string-induced axions has been heavily debated, with smaller than expected [1]. Axions were originally invoked some works claiming that string-induced axions play no as a dynamical solution to this problem; they would inter- important role [13, 14], with the DM abundance domi- act with quantum chromodynamics (QCD) inside of the nated by axions produced during the QCD phase transi- neutron so as to remove the electric dipole moment [2– tion, and others claiming these axions dominate the DM 5]. However, free-streaming ultra-cold axions may also be abundance [15–17]. produced cosmologically in the early Universe, and these The evolution of the axion string network in the early axions may explain the observed dark matter (DM) [6–8], Universe has been studied numerically and analytically which is known to govern the dynamics of galaxies and since the 1980’s [13–22] with increasingly complex and galaxy clusters. capable frameworks in recent years [23–28]. The earli- Multiple efforts are underway at present to search for est simulations were restricted computationally to lat- the existence of axion DM in the laboratory [9, 10], but tices of order 1503 sites [15], while modern-day static- these efforts are hindered by the fact that the mass of lattice simulations∼ have achieved 8, 0003 sites [25]. The the axion particle is currently unknown. The axion is approach we present in this work,∼ using adaptive mesh naturally realized as the pseudo-Goldstone boson of a refinement (AMR) simulations, provides an even larger global symmetry called the Peccei-Quinn (PQ) symme- jump in sensitivity by maintaining high resolution around arXiv:2108.05368v1 [hep-ph] 11 Aug 2021 try, which is broken at a high energy scale fa [2–5, 11]. If the string cores and lower resolution elsewhere [29]; to the PQ symmetry is broken after the cosmological epoch achieve the same resolution as our simulations using a 3 of inflation, then there is a unique axion mass ma that static grid would require a 65, 536 site lattice. Our un- leads to the observed DM abundance. (If the PQ sym- precedented dynamical range allows us to determine that metry is broken before or during inflation, then the DM radiation from axion strings prior to the QCD phase tran- abundance depends on the initial value of the axion field sition likely dominates the DM density. that is inflated [12].) However, computing this mass is difficult principally because after PQ symmetry breaking axion strings develop; at the string cores the full PQ sym- AMR SIMULATION FRAMEWORK

The axion a as the phase of the complex PQ scalar ∗ ia/fa [email protected] ﬁeld Φ = (r + fa)/√2e , with a = a(x) and r = † [email protected] r(x) real functions of spacetime x. The radial mode r ‡ [email protected] is heavy and is not dynamical at temperatures below its 2

Figure 1. (Top row) 3-D rendering of various simulation states from the initial state (left) to the ﬁnal state (right). Shown is the full simulation volume with the respective relative size of a Hubble volume indicated. The axion energy density is illustrated by the density of a 3-D media and string cores are overlaid in yellow. (Bottom row) Zoom in on a string segment. From left to right: Relationship between the string width and the number of reﬁnement levels as a function of time; 2-D slices of the radial mode and string radiation centered around a string element; string element enshrouded by axion energy density; and an illustration of the layout of the three coarsest grid levels around a string core (not to scale). Animations available here.

−1 mass mr. The axion field, on the other hand, is massless The string width scale Γ is set by mr , while the maxi- until the QCD phase transition and thus is dynamical on mum physical length scale that may be resolved with the scales smaller than the cosmological horizon between the comoving lattice grows linearly with η. Thus, finer grids PQ and QCD epochs. The axion field acquires a small are needed to resolve Γ at later times. We start with 2 3 mass ma Λ /fa at temperatures T of order the a uniform grid of 2048 grid sites, with an initial state ∼ QCD QCD confinement scale ΛQCD from QCD instantons [30], based on a thermal distribution before the PQ phase though in our simulations we focus on temperatures T transition (see Methods). Extra refined grids are then ΛQCD where the mass may be neglected. added over time whenever the comoving string width Our simulation is based on the block-structured AMR drops below a certain threshold. We add the first four software framework AMReX [31]. The equations of mo- extra refinement levels when Γ is resolved by four grid tion (EOM) for Φ can be derived from the Lagrangian [32] sites at the respective finest level, with the fifth extra level added when Γ is resolved by three grid sites (see 2 2 2 Fig.1 and Supp. Fig. S1). In comparison, note that [27] 2 2 fa λT 2 PQ = ∂Φ λ Φ Φ , (1) resolves Γ by one grid site at the end of their simula- L | | − | | − 2 − 3 | | tion. Each extra level introduces eight times as many grid cells per volume as the previous level. Refined levels where λ is the PQ quartic coupling. (We fix λ = 1 with- are localized primarily around strings. This is achieved out loss of generality so that mr = √2fa.) The EOM by identifying grid cells that are pierced by a string core are solved using the strong-stability preserving Runge- using the algorithm described in [33]. The exact grid Kutta (SSPRK3) algorithm with a time step size that layout is periodically adjusted to track strings over time. satisfies the Courant–Friedrichs–Lewy condition on a lat- See Fig.1 for an illustration of the grid layout. tice defined in fixed comoving coordinates. Evolution takes place in rescaled conformal time η = R/R1 = 1/2 (t/t1) , where R is the scale factor of the Fried- STRING NETWORK EVOLUTION mann–Lemaˆıtre–Robertson–Walker metric, R1 R(t1), ≡ and t1 is a reference time such that H1 H(t1) = fa with Hubble parameter H. In these units≡ the PQ phase The axion string network is thought to evolve and transition takes place around η 1, and we chose a start- shrink with time by radiating axions so as to obey the ≈ ing time of ηi = 0.1 and a final time of ηf = 75.7. Our scaling solution, where the number of strings per Hubble simulation volume is a box with periodic boundary con- patch remains order unity as a function of time [20]. The ditions and comoving side length L = 120/(R1H1). This network evolution is illustrated in the top panels of Fig.1, 3 volume corresponds to 1200 Hubble volumes at ηi and with time slices labeled by log(mr/H) = log(2mrt). The 4 Hubble volumes at ηf . (Ref. [27] found that finite- energy density in axion radiation is overlaid on top of the volume∼ effects are not important for simulations ending string network and is strongest in the vicinity of areas of with 4 Hubble volumes.) large string curvature. ∼ 3

1.8 scale model (see Refs. [34–36]), which predicts that ξ 2 ξ = c 2/ log +c 1/ log +c0 + c1 log 1.6 − − should approach a constant at large log. On the other ξ = c + c log 1.4 0 1 hand, the observation that ξ grows linearly with log may 1.2 be naturally explained by the well-established logarith- 1.0 mic increase of the string tension with time, µ(t) ξ 2 ≈ 0.8 µ0 log mr/H with µ0 = πfa to leading order in large log 0.6 (see Supp. Fig. S2). A given string segment loses energy 0.4 at a constant rate that does not evolve with time [20], 0.2 and as a result energy builds up in the strings relative to 0.0 the situation where µ does not increase logarithmically 4 5 6 7 8 9 with time. This increase in energy is manifest by a loga- log m /H r rithmically increasing ξ. (See Methods for details of this argument.) Figure 2. The string length per Hubble volume ξ increases with time in our simulation, indicating a logarithmic violation to the scaling solution [24], which would predict constant ξ. AXION RADIATION SPECTRUM At late times in the simulation (large log mr/H) the growth in ξ appears linear in log mr/H with coefficient c1 ≈ 0.25 as measured for the fit over the full log mr/H range shown, As the string network evolves in the scaling regime ax- 2 but including terms all the way down to c−2/ log . The fit 2 ions are produced at a rate Γa 2Hρs, where ρs = ξµ/t illustrated by the solid curve only includes terms down to q0 is the energy density in strings.≈ As we show later in but is limited to late times (log ∈ (7 5 9)); this fit leads to . , this Article, the DM density from string-induced ax- c1 ≈ 0.25 also. These fits indicate that at the beginning of ion radiation is proportional to the number density of the QCD phase transition, at log∗ ≈ 65, we expect ξ∗ ≈ 15. axions at log mr/H = log∗. To compute the number density we need to know the axion radiation spectrum from strings. We quantify the spectrum through The string length per Hubble volume is quantified the normalized distribution F (k/H) = d log Γa/d(k/H) through the parameter ξ, which is defined by ξ `t2/ ≡ V for physical momentum k. (See, e.g.,[24] for a review with ` the total string length in the simulation volume of the analytic aspects of the network evolution.) We . We determine ` by counting string-pierced plaquettes compute F numerically from our simulation ouput by inV our simulation using the algorithm described in [33]. 3 d 3 F (k/H) (1/R ) dt R ∂ρa/∂k , with ∂ρa/∂k the time- We illustrate ξ as a function of log mr/H in Fig.2. We dependent∝ differential axion energy density spectrum. compute ξ at points in time separated by a Hubble time The axion radiation is distributed in frequency be- (∆ log mr/H = log 2), since the network is strongly cor- tween the effective infrared (IR) cutoff, which is provided related on time scales smaller than a Hubble time. by H, and the effective ultraviolet (UV) cutoff set by the We verify that ξ increases linearly with log mr/H, string width mr. For momenta k well between these ∼ which was first suggested in [24, 27]. Ref. [27] con- two scales (H k mr) the radiation spectrum is ex- structed a suite of simulations on static grids of up to pected to follow a power-law. Below, we describe how we 3 4500 sites and out to at most log mr/H 7.9; they fit measure the index of this power law. 2 ∼ a model of the form ξ = c−2/ log +c−1/ log +c0 + c1 log, We calculate F via finite differences in nonuniform with log log mr/H, to their ξ data for log (4.5, 7.9) ∆t corresponding to uniform intervals in log mr/H. In ≡ ∈ 10 and found c1 = 0.24 0.02. Given mr 10 GeV our fiducial analysis, we calculate instantaneous emission ± ∼ and the QCD phase transition beginning at temperatures spectra using intervals of ∆ log mr/H = 0.25, which is of T 1 GeV, the string network is expected to evolve until order Hubble-time separations. At each log mr/H value, ∼ q log∗ 65, which is far beyond the dynamical range that we fit a power-law model F (k/H) 1/(k/H) to the in- ∼ ∝ may be simulated. stantaneous spectra between an IR cut-off kIR = xIRH In Fig.2 we illustrate our fit of the same functional and a UV cut-off kUV = mr/xUV, with the cut-offs cho- form as in [27] to our ξ data over the range log (4, 9); sen to be sufficiently far from the physical IR and UV ∈ we find c0 = 1.82 0.01 and c1 = 0.254 0.002 (see cut-offs. (See the methods for details of how this fit is − ± ± Methods for details). As a systematic test we fit the performed.) We chose xIR = 50 and xUV = 16 in order functional form ξ = c0 +c1 log to the ξ data over the lim- to be sufficiently far into the power law regime of k. ited range log (7.5, 9) and determine c0 1.05 and In the top panel of Fig.3 we illustrate F computed ∈ ≈ − c1 0.252. Importantly, the parameter c1, which governs at log mr/H = 8.75 for our fiducial choice of xIR and ≈ the large log behavior of ξ, agrees between the two meth- xUV as well as two systematic variations on the choice ods and agrees with the measurement in [27]. Assuming of fitting range, extending to xIR = 30 (“Extended IR that the QCD phase transition begins at log∗ (60, 70) Data”) and xUV = 12 (“Extended UV Data”). The best- we estimate that at the beginning of the phase transition∈ fit power-law models are also illustrated. In the bottom ξ = ξ∗ (13, 17). The linear growth of ξ with log mr/H panel, we show the evolution of the index q as a func- ∈ does not support the analytic velocity-dependent one- tion of log mr/H, for both our fiducial analysis and for 4

Fiducial Fit Fiducial Data contribute to the diﬀerence in q is that Ref. [27] used Extended IR Fit Extended IR Data xUV = 4; in Supp. Fig. S4 we show that using xUV = 4 Extended UV Fit Extended UV Data in our ﬁts also leads to positive q1 at non-trivial signif-

) [arb.] icance (see Supp. Tab. S2); however, as illustrated in 100 Supp. Fig. S8 at large log mr/H and xUV = 4 the ﬁts k/H (

F become visibly poor at large k/H because the spectrum ( ) begins to drop rapidly for . The fact log mr = 8.75 F k/H k mr H that [27] is only resolving the string cores∼ by around one 0 100 200 300 400 500 grid site at large log mr/H may also play a role. We test k/H the importance of the string-core resolution by perform- 1.50 Gorghetto et al. ∆ log = 0.25 (Fid.) ∆ log = log 2 ing an alternate simulation where we do not add extra refinement levels after log mr/H 5.3, such that Γ is 1.25 ≈ q resolved by one grid site at log mr/H 8.1 (see Supp. 1.00 Fig. S1). As illustrated in Supp. Fig. ≈S10, in this case the spectrum becomes distinctly biased towards larger q 0.75 7.0 7.5 8.0 8.5 9.0 at larger log, where the string-core resolution is low. mr log H Our result that q1 is consistent with zero is robust to changes to xUV (Supp. Fig. S4 and Tab. S2), for 32 xUV 8, to xIR (Supp. Fig. S5 and Tab. S1), for Figure 3. (Above) Example fits to the instantaneous emis- ≥ & sion spectrum calculated at log mr/H = 8.75. In our fiducial the range 30 xIR 100 that we consider, to the ∆ log analysis, the instantaneous emission spectra are calculated size used in computing≤ ≤ F (Supp. Fig. S6 and Tab. S3), for using a timestep corresponding to ∆ log mr/H = 0.25, and 0.125 ∆ log log 2, and to the method used for regu- a power-law model is fit to the data at k between the IR lating≤ the string≤ cores when computing F (Supp. Fig. S7 and UV cutoffs of kIR = 50H and kUV = mr/16. The data and Tab. S4). included in this fit range is shown in grey with the best-fit power law depicted in black. We also illustrate two systematic variations, one in which we extend our IR cutoff down to DARK MATTER DENSITY kIR = 30H (“Extended IR Data”), and another where we extend our UV cutoff upward to kUV = mr/12 (“Extended UV data”). For clarity, the data are down-binned by a factor of 2 The axion EOM during the QCD epoch generi- in k/H.(Below) The evolution of the fitted power-law index cally violates number density conservation. In partic- q as a function of log mr/H. The best fit indices obtained in ular, the non-linear axion potential is a function of our fiducial analysis are shown in black, with red showing the cos(a/fa), which implies that non-linear terms in the indices computed using ∆ log mr/H = log 2. In our fiducial EOM are important if a/fa π. Given the in- analysis we constrain q = 1.02 ± 0.03, which is shaded. For | | & comparison, the best fit linear growth of q obtained in [27] is stantaneous spectrum F (k/H) we may compute the shown in dotted grey. average field value squared at a given time t by 2 t 0 0 0 0 2 0 (a/fa) 4π (dt /t)ξ(t ) (H /k ) log mr/H , with h(H0/k0)i2 ≈being the expectedh valuei of H/k at time th0 computedi fromR the distribution F (k/H) (see Meth- a systematic variation where we use ∆ log mr/H = log 2 ods and note that this is accurate to leading order in when computing F . We compare our results to the best- 2 log mr/H). We expect (H/k) to be proportional to fit model obtained in [27], who claimed evidence that 2 2 h i H /kIR, with kIR/H √ξ being the effective IR cut- q evolves logarithmically in time, with q > 1 at late off for F (k/H) that arises∝ from the typical separation of times. In particular, Ref. [27] fit the evolution model −1 strings kIR ; note that this implies that as ξ(t) grows q(t) = q1 log(mr/H) + q0 to their q data and found evi- with time,∼ the effective IR cut-off moves towards the UV dence for non-zero q1, claiming q1 = 0.053 0.005. Fit- ± like √ξ because the strings become more closely packed ting this model to our q data (see Methods for details) together. Let us define a dimensionless coefficient β by yields q1 = 0.04 0.08 and q0 = 1.36 0.69, which is 2 −1 − ± ± the relation (H/k) = β ξ; a fit of this functional in tension with the results in [27]. (The best-fit model form to theh spectrali data leads to β = 840 70 for in that work is inconsistent at the level 1.8σ with our ± ∼ q = 1.06 (see Supp. Fig. S9). Note that smaller values of measured q values). Given that we do not find evidence q lead to larger values of β and that q = 1.06 is the maxi- for logarithmic growth of q, we impose q1 = 0 and find mum value of q allowed at 1σ from our analysis. In terms q = 1.02 0.04, which is interestingly consistent with 2 0 of this coefficient (a/fa) (4π/β) log mr/H . 1.1 the scale invariant± spectrum q = 1, suggested in [13], to h i ≈ 0 (for log mr/H . 70), which implies that non-linear num- within 5%. An additional argument in favor of q0 = 1 is ∼ ber changing processes are at most marginally relevant. that the string loops appear logarithmically distributed (Non-linear corrections to the linearized force are at most in size, as shown in Fig. S3 and as expected for a network 15%.) This justifies our use of number density conser- of intersecting strings (see Methods). vation∼ below in estimating the DM abundance. One difference between [27] and this work that may To compute the axion number density we need to com- 5

200 t & t∗ axion number density is conserved. Assuming ax- δ √ξ 175 × ion number density conservation allows us to relate the string 150 present-day DM abundance to the expression for na at (see Methods): 125 t∗ 1 − i 100 1.17 k H fa 107 ξ∗ log h str −2 ∗ (2) 75 Ωa 0.12 h 10 . ≈ 4.3 10 GeV δ r17 70 50 · Axions produced from domain wall and misalignment dy- 25 q = 1.06 δ = 113 7 namics during the QCD phase transition provide a sub- 0 ± QCD 6.0 6.5 7.0 7.5 8.0 8.5 dominant contribution to the DM density [26]: Ωa −2 10 1.17 ≈ log mr/H 0.017 h (fa/4.3 10 GeV) . (Note that we assume a domain wall number· of unity so that domain walls are Figure 4. The inverse expectation value hH/ki−1 is com- unstable, but see e.g. [37].) The DM abundance as mea- puted using the instantaneous axion spectrum F (k/H) by sured by the Planck Observatory using the cosmic mi- −2 numerically integrating the spectrum to k/H = xmax = 50 crowave background is ΩDM = (0.12 0.0012)h , with and then analytically integrating the power law distribution h the Hubble rate scaling factor [38].± Adding in the con- −q log F (x) ∝ x from xmax to the UV cut-off at k/H ∼ e ∗ for QCD tribution from the QCD phase transition Ωa , and as- log ≈ 65. For q > 1 the expectation value does not strongly ∗ suming q (0.98, 1.06), we find that the fa that gives rise depend on the UV cut off but is instead a function of the effec- ∈ −1 √ to the observed DM abundance should be in the range tive IR cut-off, which is set by ξ such that hH/Ki = δ ξ for 10 11 fa (3.1 10 , 1.4 10 ) GeV (ma (40, 180) µeV), some parameter δ, which we determine by fitting this model ∈ × × ∈ to the numerical data as illustrated here. Smaller values of where for the lower fa bound we have conservatively al- δ correspond to larger axion number densities and thus large lowed for the possibility that at t∗ the remaining energy axion DM densities. Here, we illustrate the result for the density in strings is instantaneously deposited into axions maximum allowed q of 1.06, which leads to the smallest δ with spectrum F , raising the string-induced DM density consistent with our simulation results. by a factor of 3/2, though in actuality this contribution is likely smaller since the spectrum shifts towards the UV as ma(t) increases. If the index is scale invariant (q = 1), pute the expectation value H/k over the distribution then we predict ma = 65 6 µeV. F (k/H). Following the justificationh i in the previous para- ± graph we may parameterize this expectation value in terms of the IR cut-off and thus ξ, H/k −1 = δ√ξ, DISCUSSION for a dimensionless parameter δ. In Fig.h 4 wei illustrate the H/k −1 data, assuming q = 1.06, as a function of h i In this work we provide the largest and highest- log mr/H along with the best fit model, which leads to resolution simulation of the axion string network to-date δ = 113 7; note that smaller values of q lead to larger by making use of a novel AMR framework that allows values of±δ. To compute H/k −1 (and also (H/k)2 −1) h i h i us to resolve the axion string cores while maintaining we numerically integrate the spectrum up to k/H = lower resolution over the majority of the simulation vol- xmax, with xmax = 50, and then analytically integrate ume. Our AMR approach may be used in the future to q the power-law functional form F (k/H) 1/k from xmax simulate the axion dynamics at the QCD epoch where log ∝ to k/H e ∗ , with log 60 70. The axion number ∼ ∗ ∼ − domain walls form and the string network collapses [26] density at the epoch of the QCD phase transition is then, and to study axion-like particle string networks that pro- string 2 to leading order in log , n (8πf H/δ)√ξ∗ log . ∗ a ≈ a ∗ duce gravitational wave radiation [36, 39, 40]. If the spectrum is exactly scale invariant at large k, Our results have important implications for axion di- such that q = 1, then δ log(mr/H). Defining δ = rect detection experiments, as our preferred mass range ∝ δ1 log(mr/H) in this case we compute δ1 = 6.2 0.4. of (40, 180) µeV is higher than that which may be probed string± The axion number density from strings is then na by two of the main dedicated experiments that are aim- 2 ≈ (8πfa H/δ1)√ξ∗. At 1σ we find that q could be as low as ing to test this cosmological scenario, ADMX [41] and q 0.98. For q < 1 the quantity δ increases for increasing HAYSTAC [42]. On the other hand, this mass range ≈ 1−q UV cut-offs like (mr/H) ; in particular, for q = 0.98 may be probed by ADMX with future searches [43], by and log mr/H = 70 we calculate δ = 820 50. Thus, the MADMAX experiment [44, 45], and by the proposed ± accounting for the uncertainty on q from our simulations plasma haloscope [46]. Our work motivates focusing ex- we find that δ is in the range δ (106, 870). perimental efforts on this mass range. The dominant ∈ Let us more precisely define the time t∗ as the time source of uncertainty on ma in our estimates arises from when the axion field becomes dynamical, which is when the index q, which we find does not evolve with log mr/H 3H(t∗) = ma(t∗), for a time-dependent mass ma(t) and is in the range (0.98, 1.06); this range is statistics- that is increasing rapidly during the QCD phase tran- limited and will shrink with future simulation efforts us- sition [30]. The axion string network is observed to col- ing AMR, leading to more precise predictions that can in lapse around t∗ (see, e.g.,[26]), meaning that at times turn better inform experimental efforts. 6

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METHODS the coarse level, the time step is ∆η0 = 0.02, satis- fying the Courant–Friedrichs–Lewy (CFL) condition at A. Simulation Framework ∆η0/∆x0 1/3. The laplacian in the EOM is computed to sufficient≈ accuracy by a second-order finite difference method. We decompose the complex PQ scalar field as Φ = 3 A grid of N0 = 2048 cells will not be able to resolve (φ1 +iφ2)/√2 and assume a radiation-dominated cosmological background. In this notation the axion field is string cores at late times. To maintain resolution we periodically refine a volume around strings, which means given by a(x) = f arctan2(φ , φ ) and the radial mode a 2 1 decreasing the grid spacing by a factor of 2 in a local by ( ) = 2 + 2 . The EOM can be derived from r x φ1 φ2 fa volume (see Fig.1). We refer to the volumes with dif- the Lagragian in (1)− and expressed in the dimensionless ferent resolutions as levels ` with the coarse level be- fields ψ = pφ/f as a ing level ` = 0. Each level differs from each other not ` 2 only in spatial resolution, ∆x` = ∆x0/2 , but also in 00 2 0 ¯ 2 2 2 2 T1 ψ1 + ψ1 ψ1 + λψ1 η ψ1 + ψ2 1 + = 0, temporal resolution to locally satisfy the CFL condition, η − ∇ − 3f 2 a ∆η = ∆η /2`. The higher-resolution lattice on level 2 ` 0 00 2 0 ¯ 2 2 2 2 T1 ` is determined by fourth-order spatial interpolation of ψ2 + ψ2 ψ2 + λψ2 η ψ1 + ψ2 1 + = 0 , η − ∇ − 3f 2 the coarser level ` 1 if no data at that location and a (3) level exists. Since− different grid spacing and time step with T1 defined as the temperature when H(T1) = fa. sizes are used simultaneously, each level is evolved inde- Here, primes denote derivatives with respect to η while pendently and then synchronized appropriately. This is the spatial gradient ¯ is taken with respect tox ¯ = known as the subcycling-in-time approach and requires ∇ R1H1x. We chose λ = 1 without loss of generality and fourth-order spatial interpolation and second-order tem- 2 the ratio (T1/fa) is given by poral interpolation during synchronization. The simulation is insensitive to the exact order of the interpolation T 2 1012 GeV 1 8.4 105 . (4) used. See the AMReX documentation [31] for more infor- f ≈ × f mation about the technical details of the AMR approach. a a We add an additional level each time the string width Note that the PQ breaking scale fa is degenerate with Γ drops below four grid sites at the current finest level, the choice of physical box size and dynamical range L i.e. at η 3, 6, 12, and 24,1 leading to a total number in . This implies that one has to perform only a single η of 5 levels.≈ A 6th level is added at η 642 when the simulation, which can be reinterpreted through trivial ≈ string width drops to 3∆x`=4. See Supp. Fig. S1 for rescaling for different axion masses. an illustration of the respective string core resolution at Using an AMR technique means that some parts of different times in our simulation, compared to the res- our simulation volume are run at a higher spatial (and olution achieved in the static lattice simulation in [27]. temporal) resolution than other parts. Our implementa- Note that to match the resolution of the finest level on a tion is based on AMReX [31], a software framework for uniform grid would require a stunning 65, 5363 cells. block-structured AMR. We use a tagging algorithm to decide on which local Our simulation starts out with a uniform grid of = N0 volumes to refine, with cells tagged ensured to be within 20483 cells, which we will refer to as the coarse level. We a refined volume. In total we use three different tagging generate thermal initial conditions with wavenumber up criteria that target (i) string cores, (ii) large gradients to 25 in each spatial direction at an initial time η = i in ψ, and (iii) short wave-length radiation emitted by 0 1. See [26] for details of how the initial state for Φ is . strings: generated from the thermal correlation functions. The comoving box length of our simulation volume is L = • String cores are identified using the procedure de- 120 with periodic boundary conditions. This implies the scribed in [33] (Appendix A.2). This involves find- simulation contains (120)3 Hubble volumes at η = 1. Our ing plaquettes that are being pierced by strings. starting time is η = 0.1. Note that the comoving spatial The cell at the low-index corner of a pierced pla- difference ∆x0 = L/N0 between lattice points is such quette is tagged. that our initial state for Ψ is smooth during the initial stages of the PQ phase transition (i.e., the structure in • As strings decay the resulting radiation can produce large gradients in the field. To ensure suf- Ψ is resolved by multiple grid sites). 2 2 ficient resolution we tag cells with ∆x ψ1,2 > The EOM in (3) is solved using the strong-stability pre- ` ∇ serving Runge-Kutta (SSPRK3) method. This method 0.04. The precise numerical value is of phenomeno- is of third-order and as such one order higher than logical origin and has proven to work well in our the often used leapfrog integration scheme. We find simulation setup. that this method provides the best trade-off between numerical stability and computational costs includ- 1 ing memory consumption when compared against a log mr/H ≈ 2.6, 3.9, 5.3, 6.7 2 second- and fourth-order Runge-Kutta method. At log mr/H ≈ 8.7 9

• String radiation into radial modes is highly sup- C. Construction of the Axion Energy Density pressed at late times yet it can cause numerical Spectrum instabilities if not sufficiently resolved. To avoid a numerical breakdown we tag cells at the coarse In order to compute the axion energy density spec- 3 2 2 2 level where ∆x0η (ψ + ψ ) > 4 is fulfilled. trum, we consider the screened time-derivative of the ax- ∇ 1 2 ion field, which is defined by Strings are not stationary and thus the grid layout has a˙ scr(x) = f(x)˙a(x). (7) to be adjusted periodically. As this is computationally ` In this definition, we include a function that screens out expensive we re-grid level ` only every ∆η` = 0.2/2 . f However, we ensure that within this time interval even the locations of strings, which appear as discontinuities the fastest moving strings with v = c are always at least in the axion field and its derivative. We consider three a full string width away from any coarse-fine boundary. choices of the screening function: 2 This is done by leaving a buffer zone of 11 grid sites f(x) = [1 + r(x)/fa] (8) around each tagged cell that is refined as well. f(x) = 1 + r(x)/fa (9) The simulation was performed on NERSC’s Cori XC40 system using 1024 KNL nodes (Intel Xeon Phi Processor f(x) = 1 (no mask). (10) 7250) with, in total, 69,632 physical CPU cores and over In this work our fiducial results use (8) such that 98 TB DDR4 RAM. It ran for about 74 hours ( 5.2 a˙ (x) = ψ (x)ψ˙ (x) ψ˙ (x)ψ (x). The screening in (9) ∼ scr 1 2 1 2 Million CPU hours) in a hybrid OpenMP/MPI mode. reproduces that of [27]− while (10) corresponds to no string screening. Because 1 + r(x)/fa 1 at locations far away from string cores, screening as in≈ (8) and (9) only modify B. The string length per Hubble ξ the axion time derivative in the direct vicinity of strings. As shown in Supp. Fig. S7, the results presented in this work are relatively insensitive to the choice of screening We compute the string length per Hubble ξ, defined function, which can be understood from the fact that we in the main Article, using the algorithm from [33] that study the emission at spatial scales well beyond the string involves counting string-pierced plaquettes; our measured width. values for ξ are illustrated in Fig.2. We then fit the model The axion energy density spectrum within our simulation can then be computed as in [27] by ˜ c−2 c−1 ξ = 2 + + c0 + c1 log mr/H (5) 2 log mr/H log mr/H ∂ρa k 2 = | | dΩk a˜˙ scr(k) , (11) ∂k (2πL)3 | | to this data, though this fit is made complicated by Z where a˜˙ (k) is the Fourier transform of the fielda ˙ . We the fact that it is difficult to estimate statistical uncer- scr scr compute this energy density spectrum with the HACC tainties from our ξ measurements. We thus determine SWFFT algorithm [47] applied to the axion time deriva- these uncertainties in a data-driven way. Given that tive computed at the coarsest level of spatial resolution. we expect the uncertainties to be statistical in nature, After we have computed dρ /dk using the fast Fourier and thus determined by the finite simulation volume, a transform (FFT), we then bin our FFT data in 1774 we assign uncertainties to each measurement such that equal-sized bins between k = 0 and the maximum k, the uncertainty at a given logi log mr/H(ti) value is max −3 log /4 ≡ −3 log /4 corresponding to k /2π = 1024√3/L. This binned σ = σ e i . Here, the factor e i is the time- com ξi 0 spectrum is then used in our subsequent analysis. dependence of the square-root of the number of Hub- ble patches per simulation box, which is a proxy for the square-root of the number of independent string segments D. Measuring the string tension in the simulation volume. We then treat σ0 as a nuisance parameter that we profile over during the fit. In particular, the likelihood is We compute the effective string tension realized in our simulation following the procedure described in [24, 27]. We first compute the average energy density within our ˜ 2 (ξi−ξi) entire simulation volume using exp 2σ2 − ξi (6) ξ [ξ; ξ, c, σ0 ] = , 2 2 L M { } 2 2 2 fa i 2πσ ρtot = ∂Φ + λ Φ . (12) Y ξi h| | | | − 2 i q We then compute the average axion and radial mode en- with and ˜ denoting the data and model prediction, ξi ξi ergy densities by respectively, at the time labeled by logi. Note that we 2 denote the model by ξ with model parameter vector ρa a˙ , c = inM addition to . The uncertainties ≈ h i (13) c−2, c−1, c0, c1 σ0 1 2 1 2 λ 2 2 { } ρr r˙ + ( r) + (r + 2rfa) . in Fig.2 arise from the best-fit σ0. ≈ h2 2 ∇ 4 i 10

In computing ρa and ρr, we mask regions of the simu- where ∂ρi/∂k is the axion energy density spectrum at ηi. lation volume that are at the highest level of reﬁnement At each ηi, we consider a power-law model of the form to exclude string contributions. Note that in computing k k −q both ρtot and ρr, we neglect the small contribution of the f ; A, q = A (20) thermal mass in (1). The string energy density is then H { } H straightforwardly obtained from and adopt the parametrized form

ρs = ρtot ρa ρr. (14) −p − − k k σ ; B, p, C = B + C (21) Using the string energy density, we may determine the H { } H effective tension by to describe the combined statistical and systematic un- 2 µdata = t ρs/ξ , (15) certainty in the data. We then analyze the data at each ηi with the Gaussian likelhood i, which is of the form with the subscript “data” denoting the measured value, L 2 which can be compared to the theoretically expected 1 1 di,j fj i [di; i] = exp − (22) string tension at large values of log mr/H: L M √ −2 σ j 2πσj " j # Y 2 mr µth πfa log . (16) where d is the value of the numerically computed in- ≈ H i,j stantaneous emission spectrum at the jth value of k/H This comparison is illustrated in Supp. Fig. S2 for times computed at time ηi. The model predictions for the mean between log mr/H = 8 and log mr/H = 9. and the error at the jth value of k/H are specified by the Importantly, we only want to compare the leading log model parameters i = Ai, qi,Bi, pi,Ci for each time M { } behavior of µdata and µth. Moreover, the addition of a ηi. The values of k/H and associated data that enter refinement level at log mr/H 8.7 changes the effective the likelihood are restricted to satisfy k/H > xIR and UV cutoff in the numerical calculation,≈ leading to a dis- −1 k/H < xUVmr/H. continuity in the measured effective tension. To analyze In performing the analysis, we only analyze emission the effective tension, we thus adopt a simple logarithmic spectra which contain at least 10 bins between kIR ≡ growth model for the effective tension HxIR and kUV mr/xUV. We make the fiducial analysis ≡ choices of using the screening function of (8), kIR = 50H 2 log + log 8 7 µ1fa mr/H µb, mr/H . and kUV = mr/16, and using a finite difference in time- µ = 2 ≤ (17) (µ1fa log mr/H + µa, else , spacings corresponding to ∆ log mr/H = 0.25. The im- pact of varying these fiducial choices, which is marginal, which allows for a different constant offset before (µb) is illustrated in the Supp. Figs. S4, S5, S6, and S7. and after (µa) the addition of the refinement level but Using the likelihood in (22), we determine the maxi- enforces uniform logarithmic growth of the string ten- mum likelihood estimateq î for the emission index at each sion. We use a Gaussian likelihood with data-driven un- ηi. Since the likelihoods are quadratic to very good ap- certainty on the µdata values σµ; we treat σµ as a nuisance proximation, we also determine Gaussian uncertainties 2 2 2 parameter in addition to µa,b. Profiling over the nuisance σq onq î at each ηi by 1/σ = ∂ /∂ log i evaluated i qi − qi L parameters we determine µ1 = 3.7 0.5, which should at the likelihood-maximizing model parameters. After be compared to the theoretically expected± value = . µ1 π obtainingq î and σqi at each ηi, we join the results to study the possible evolution of q. We use a Gaussian likelihood E. Instantaneous Emission Analysis 2 (ˆqi−q˜i) exp 2(σ2+σ2 ) − qi Here we describe the method by which we fit a power- q [q; q, σ] = (23) L M h 2 2 i i 2π(σ + σ ) law model to the instantaneous emission spectrum. Up Y qi to an overall normalization, the instantaneous emission q where ˜ is the model prediction at time specified by spectrum is given by qi ηi parameters q. We include an additional error term M k 1 ∂ 3 ∂ρa σ as a nuisance parameter which is added in quadra- F 3 R . (18) ture with the data-driven to address possible system- H ∝ R ∂t ∂k σqi atic effects. In this work, we consider two possibilities In our simulation framework, time evolution is performed for the evolution of q, the first that q grows linearly as in terms of η and hence the instantaneous emission Fi at q(log mr/H) = q1(log mr/H) + q0 and the second that q conformal time ηi is calculated by numerical finite differ- is constant such that q(log mr/H) = c0. As in our analy- ence as sis of the individual instantaneous emission spectra, the ∂ρ maximum likelihood estimates and uncertainties of the k 1 η3 i+1 η3 ∂ρi i+1 ∂k i ∂k parameters σ, q0, and q1 can be determined via standard Fi 4 − , (19) H ∝ η ηi+1 ηi frequentist techniques. i − ! 11

F. DM abundance calculation produce less axions by number density and thus be less important for the final DM abundance. Still, in order Here we describe the calculation of the DM abundance to be conservative we estimate the maximum amount of string DM that may be produced by the string network by as- from the quantity na , which is described in the main Article. Define Λ 400 MeV; then the temperature- suming that at T∗ all of the energy density in ρs is trans- dependent axion mass≡ is well characterized by a power- ferred instantaneously to axions with spectrum F (k/H). law [48]: This provides a contribution to the axion energy den- decay str str sity Ωstr Ωa /2, with Ωa being the contribution 4 ≈ α Λ to the DM abundance from axions produced prior to T∗. 2( ) = a Λ (24) ma T 2 n ,T , We allow for this possibility when determining that the fa (T/Λ) maximum allowed axion mass is 180 µeV, but we do not with αa and n dimensionless constants. The most re- include this contribution when estimating the minimum cent lattice simulations agree with the dilute instanton allowed axion mass of 40 µeV. −7 gas approximation and support αa = (4.6 0.9) 10 In this work we assume that the radial mass, mr, is ± × 10 12 for n 8.16 [49], which are the values we assume in of order the decay constant fa 10 10 GeV. How- ≈ ∼ − this work (note that these uncertainties are sub-dominant ever, one possibility is that mr fa, as may happen in to those from the axion production from strings from e.g. supersymmetric theories where mr is related to the our simulations). We also approximate the temperature- supersymmetry breaking scale; in this case, mr & TeV dependent number of relativistic degrees of freedom as is possible [51]. If ma TeV, then log(mr/H) 50, 0 γ 0 ∼ ∼ g∗(T ) g (T/MeV) , with g 50.8 and γ 0.053, which is large enough such that our conclusion that ≈ ∗ ∗ ≈ ≈ which has been shown to match the full result for g (T ) ma (40, 180) µeV produces the correct DM abundance ∗ ∈ up to a few percent over the temperature range 800 < is still valid in this scenario. T < 1800 MeV relevant for this calculation [50]. We also Note that in these estimates we must perform the fit of assume that the numbers of radiation and entropy de- the model δ √ξ to the H/k −1 data illustrated in Fig.4. grees of freedom are the same over the temperature range The H/k −×1 data doh not havei easily estimated uncer- of interest, since the difference between these is also at taintiesh andi so, as we have illustrated multiple times al- the level of a few percent [49] and thus a sub-dominant ready, we determine these uncertainties in a data-driven source of uncertainty. way by assigning the uncertainties of all data points a The temperature T∗ is defined as the tempera- value σ, which we profile over when determining δ. The ture at which 3H(T∗) = ma(T∗); using H(T∗) = uncertainties in Fig.4 reflect the best-fit value of σ. 2 Lastly, the derivation above assumes that number- π g∗(T )/90T∗ /Mpl, with Mpl the reduced Planck mass, we may solve explicitly for T∗. We assume that the changing processes are not important in the QCD p string network evolves uninterrupted up to T∗ but that phase transition since a/fa . 1. Note that | | 2 for it quickly evaporates and is not a significance the formula in the main Article for (a/fa) T < T∗ h i source of axions (but see below). In this approximation for the string-induced axion radiation arises from 2 2 2 axion number density is conserved for T < T∗, so that the relation (a/fa) = (1/fa ) dk dρa/dk(1/k ), with t h i we may write the axion DM abundance today as in (2). dρ /dk = dt0Γ0/H0(R0/R)3F (kR/R0H0) and primes a R str (6+n+γ)/(4+n+γ) 1.17 denoting quantities evaluated at t0. Note that Ωa fa fa , which is the ∝ QCD ≈ R same scaling as for Ωa , since for both contributions the fa-scaling has the same origin (see e.g. [26]). While we do not simulate the QCD phase transition in G. Semi-analytic analysis of string evolution this work, it is important to keep in mind that the string network does evolve non-trivially during the QCD epoch. In the main Article we pointed to an argument related As illustrated in the simulations in [26], the string net- to the logarithmically-increasing string tension for why work collapses rapidly after T∗. In particular, the string ξ may be expected to increase logarithmically in time as network in [26] was completely gone at temperatures of well. Here, we expand upon that argument as well as order T T∗/1.5. In our approximation where the string give an argument for why q = 1 may be expected for ∼ network evolves uninterrupted until T∗ the string network the spectrum. As the string network evolves in the scal- 2 has energy ρs = 4H (T∗)µ(T∗)ξ∗ at T∗. Between T∗ and ing regime axions are produced at a rate Γa 2Hρs, 2 ≈ 1/1.5T∗ all of that energy is transferred to axion radia- where ρs = ξµ/t is the energy density in strings, and ∼ tion. However, it is likely that the spectrum of radiation µ µ0 log(mr/H) is the string tension, to leading order ≈ 2 during this collapse is shifted to the UV compared to the in large log. Recall that µ0 = πfa . The tension µ has function F (k/H) from before the mass turns on, since af- a logarithmic divergence that is regulated in the IR by −1 ter T∗ the axion mass ma(T ) is much larger than Hubble the scale of string curvature H because of energy as- and thus provides an IR cut-off for the radiation spec- sociated with the axion field∼ configuration, which wraps trum that is further in the UV compared to that for the around the string. Physically we may imagine that the axion-string network prior to the QCD phase transition. long strings are composed of a random walk of smaller Since the spectrum is shifted towards the UV, it should segments that we refer to as correlation lengths, which 12 may evolve dynamically and straighten on timescales of by using order H−1. Denote the number density of correlation lengths as n . Then, we may relate Γ = n dE /dt, dρ dn` ξsub Hµ c a c c ρ = d` = d`µ` = Dµ`max = , (26) where dE /dt is the power transferred to axions by the d` d` t2 c Z Z straightening correlation lengths. Previous studies of 3 leading to D ξsub H/t with ξsub H representing the collapsing closed string loops and straightening string ≈ kinks have shown that the loops and kinks lose energy total string length in sub-horizon sized string loops. 2 We are interested in the spectrum of axions emitted by as dE/dt = αfa , with α (1 10), regardless of the loop and− kink sizes [14, 20∼, O52, 53−]. We assume that the string network. A string loop of length ` emits ax- 2 ions dominantly at the fundamental frequency k 1/`. the correlation lengths radiate as dEc/dt = αfa for ∼ some α 1 100. Solving the energy balance− equation Meanwhile, the string loop radiates energy at a rate ∼ − dE = αf 2. We can now combine all of this knowledge then leads to a time-dependent correlation length Lc dt a ≈ with (25− ) to find α/[πH log(mr/H)] for large log. Let us now assume that there are Nstr strings in total per Hubble patch, with dn αcf 2 each string∼ composed of a random walk of smaller correla- F [k/H] αf 2 ` = a . (27) ∝ a dk k tion lengths. This then implies that at large log(mr/H) the parameter ξ scales as ξ Nstr(π/4α) log(mr/H), We thus find that given the attractive behavior seen in ≈ which reproduces the observed scaling for Nstr few, Supp. Fig. S3, that the instantaneous spectrum of axions consistent with the simulation data as illustrated∼ in emitted by the network should be approaching q = 1. As Fig.2, and α (10). a side-note, given this understanding of the string loop One of the∼ most O important results of this Article is distribution, we can easily derive the energy density and the result that q 1, to within 5%. In order to fur- spectrum of gravity waves emitted by the network using ≈ ∼ 2 ther support this result, we show visually that the string dEGW /dt = αGW Gµ [40]. − distribution is approaching an attractive solution that Finally, we conclude by giving analytic arguments for supports q = 1. String loops can be characterized by why Supp. Fig. S3 takes the form that it does. Namely the parameter n`, which is the number of string loops that at larger lengths, ξ` `, and at smaller lengths 2 ∝ with size smaller than ` at a time t, as well as by ξ`, ξ` ` . At small lengths, the string loops shrink due to ∝ 2 which is the total length of string loops with size smaller the emission of axions giving `(t) = `0 αf (t t0)/µ − a − than `. In Supp. Fig. S3 we illustrate ξ`/ξ∞ versus the with `0 being the initial loop size at a time t0. If string length ` at various times, with ξ∞ = ξ. Visually, it is loops are formed at a constant rate with a fixed length clear that as time progresses, the string loop distribution `0, then dn`/d` dn`0 /dt = constant. Multiplying by ` ∝ 2 approaches an attractor solution, whose validity is ex- and integrating, one finds that at small lengths ξ` ` , ∝ tending over an increasingly large range of lengths. This in rough agreement with Supp. Fig. S3. sort of attractor solution for the loop distribution was Larger string loops shrink by intersecting the long, rel- also found for the fat string approximation in Ref. [24] atively straight, and infinite strings that carry most of but here we are able to show that this also holds in the the string length. The two strings will intersect at a rate physical case. Given the importance of this distribution, Γint given roughly by the average string speed over the we numerically fit a power law model to the data using average distance between strings. Upon intersecting the the same procedure described in Methods Sec.E. The infinite string, the string loop loses a random amount of treatment of uncertainties and definition of the Gaussian its string length. If the locations of the intersections are likelihood is analogous to that used for the instantaneous random, the probability distribution for the final length emission spectrum, with a power-law model of the form of the string loop, `, is proportional to its length. Thus m 2 ξ` = D` . We perform the fit at various times ηi to an initial string loop of length `0 has dP/d` = 2`/`0. obtain corresponding indices mi. The fitting range is Putting this intuition into equation form, we find H`/π (8H/mr, 1/2) to ensure we are within the at- ∈ ∞ tractor regime. We only include string loop distributions dn` dn` dP dn`0 = Γint + d`0 Γint . (28) with at least 8 data points within the fitting range and dtd` − d` ` d` d`0 Z log m /H 4. The results for m are then joint using r i The first term on the right hand side gives the loss of a Gaussian≥ likelihood identical to that for q in (23) as- loops due to intersections while the second term gives suming m is time-independent. We find m = 0.97 0.03 their production from larger loops of size . Solving for with the fit illustrated in Supp. Fig. S3. ± `0 the equilibrium distribution, we find that dn /d` 1/`. Let us now show that m = 1 implies q = 1. From a ` As before, multiplying by ` and integrating, one∝ finds m = 1 length distribution, we can calculate the number that ξ ` giving q = 1. Meanwhile, the infinite strings of strings loops with length between ` and ` + d` to be ` are mostly∝ straight except for some highly curved regions that they obtain from intersections with smaller string dξ` dn` = ` = D, (25) loops. As a result, it is natural to expect that the infinite d` d` strings radiate axions with the same q = 1 frequency for some constant D. We can determine the constant D spectrum as the string loops. 13

DATA AVAILABILITY

All data products of this work may be made available by the corresponding authors upon request. Supplemen- tary animations are available at https://bit.ly/amr_ axion.

CODE AVAILABILITY

The AMReX code framework used in this work is publicly available at https://amrex-codes.github.io/. Addi- tional code may be made available upon request.

ACKNOWLEDGMENTS

We thank Marco Gorghetto, David Marsh, Javier Re- dondo, Alejandro Vaquero, Ofri Telem, and Giovanni Villadoro for fruitful discussions. M.B. was supported by the DOE under Award Number DESC0007968. J.F. and B.R.S. were supported in part by the DOE Early Career Grant DESC0019225. This research used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Sci- ence User Facility located at Lawrence Berkeley National Laboratory, and the Lawrencium computational cluster provided by the IT Division at the Lawrence Berkeley National Laboratory, both operated under Contract No. DE-AC02-05CH11231. This research was supported by the Exascale Computing Project (17-SC-20-SC), a col- laborative effort of the U.S. Department of Energy Office of Science and the National Nuclear Security Adminis- tration. 14

Supplementary Figures and Tables for Dark Matter from Axion Strings with Adaptive Mesh Reﬁnement

Malte Buschmann, Joshua W. Foster, Anson Hook, Adam Peterson, Don E. Willcox, Weiqun Zhang, and Benjamin R. Safdi

This work 2 This work (low resolution) String width Γ [# grid sites] Gorghetto et al. 2020 0 0 1 2 3 4 5 6 7 8 9 log(mr/H)

Figure S1. String width Γ in units of number of grid sites as a function of simulation time parameterized by log mr/H. Each sudden increase in string width is due to the transformation ∆x → ∆x/2 when an extra level is added. The first four refinement levels are added when Γ/∆x = 4 leading to a sudden increase to Γ/∆x = 8 each time. To test the effect of limited resolution we performed a low-resolution simulation (dashed red) that is identical to the main result (solid black) up to log mr/H ≈ 5.3 but does not add any extra refinement levels afterwards. We compare this to the approximate resolution of simulations on a static lattice (blue) presented in [27]. Simulations on a static lattice over-resolve string cores at early times while under-resolving them at late times. The gray horizontal line corresponds to the often used resolution criteria mr∆x . 1.

28 Fit π log(mr/H) Data

26 2 a µ/f 24

8.2 8.4 8.6 8.8 9.0 log (mr/H)

Figure S2. The string tension realized in our simulation as a function of time (see Methods Sec.D for details of how this is computed). The theoretical string tension as a function of log mr/H is shown in grey, to leading order in large log, while the string tensions measured in our simulation with data driven errors are shown in dark blue. Only the leading log growth of the data is expected to match the theoretical expectation; we find consistency between the linear growth in log mr/H of the theoretical and measured string tensions. Note that an additional refinement level is added during the red band, at log mr/H ≈ 8.7, leading to a change in the overall offset of the µ data. 15

100 1.2

1 10− 1.1 2 10− ∞ 3 1.0 m /ξ 10− = 8 ` /H ξ mr log 4 10− 0.9 7 6 5 10− 5 0.8 4 6 10− 3 2 1 0 1 2 3 10− 10− 10− 10 10 10 10 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 H`/π log mr/H

Figure S3. (Left) The total string length in string loops smaller than `, ξ`, versus ` for various values of log mr/H. It is clear that as time progresses the string loop distribution is approaching an attractor solution. We perform a power-law ﬁt of the m form ξ` = D` to the string loop distribution within the attractor regime (dashed lines). (Right) Distribution of the index m. The result is joint assuming no time dependence ﬁnding m = 0.97 ± 0.03 (gray band). As argued in the text, this attractor solution leads to an axion spectrum with q = 1.

mr mr/4 log H = 8.75 This Work mr w/o Systematic log H = 8.25 mr/6 mr/8

mr/12

mr/16

mr/24 UV Cutoﬀ Variations mr/28

mr/32 1.0 1.1 1.2 1.3 0.3 0.2 0.1 0.0 0.1 0.2 − − − q q1 [Linear Fit]

Figure S4. (Left) A comparison of the best-fit values for the index q of string emission at selected times for several choices of the UV cutoff of the fit range. (Right) Same as left but for q1. All index evolution results for these variations are presented in detail in Tab. S2. Note that the “w/o Systematic” data points do not include the systematic nuisance parameter σ as given in the likelihood in (23).

log mr = 8.75 This Work 30H H mr w/o Systematic log H = 8.25 50H

75H IR Cutoﬀ Variations 100H

1.00 1.02 1.04 1.06 1.08 1.10 1.12 0.6 0.4 0.2 0.0 − − − q q1 [Linear Fit]

Figure S5. As in Fig. S4, but comparing choices of the IR cutoﬀ. All index evolution results for these variations are presented in detail in Tab. S1. 16

This Work w/o Systematic 0.5 Variations

r 0.25 H m

∆ log log(m /H) = 8.0 0.125 r log(mr/H) = 7.5

0.6 0.8 1.0 1.2 1.4 0.10 0.05 0.00 0.05 − − q q1 [Linear Fit]

Figure S6. As in Fig. S4, but comparing choices of ∆ log(mr/H) used in calculating the instantaneous emission spectrum. All index evolution results for these variations are presented in detail in Tab. S3.

mr mr log H = 8.75 log H = 8.25 This Work w/o Systematic No Mask

(1 + r/fa) Mask

Masking Variations 2 (1 + r/fa) Mask

1.00 1.02 1.04 1.06 1.08 1.10 0.10 0.05 0.00 0.05 − − q q1 [Linear Fit]

Figure S7. As in Fig. S4, but comparing choices of the string masking function. All index evolution results for these variations are presented in detail in Tab. S4.

Coeﬃcient xIR = 30 xIR = 50 xIR = 75 xIR = 100

q1 0.07 ± 0.07 −0.04 ± 0.08 −0.06 ± 0.13 −0.32 ± 0.26

q0 0.41 ± 0.58 1.36 ± 0.69 1.5 ± 1.12 3.68 ± 2.21 const. q0 0.98 ± 0.04 1.02 ± 0.04 1.0 ± 0.05 1.02 ± 0.07

Table S1. Results of the fits to the spectral evolution holding all our fiducial analysis choices fixed but for various IR cutoffs xIR. We provide the fits and uncertainties for the q1 and q0 in the linearly growing index model and the best fit constant for const. q0 in the constant index model. Our fiducial choice of xIR = 50 is shown in bold.

Coeﬃcient xUV = 4 xUV = 6 xUV = 8 xUV = 12 xUV = 16 xUV = 24 xUV = 28 xUV = 32

q1 0.17 ± 0.05 0.08 ± 0.04 0.04 ± 0.05 −0.01 ± 0.06 −0.04 ± 0.08 0.08 ± 0.09 0.08 ± 0.12 −0.2 ± 0.2

q0 −0.22 ± 0.37 0.39 ± 0.32 0.7 ± 0.41 1.09 ± 0.51 1.36 ± 0.69 0.36 ± 0.78 0.34 ± 1.05 2.74 ± 1.68 const. q0 1.12 ± 0.05 1.06 ± 0.03 1.03 ± 0.03 1.03 ± 0.03 1.02 ± 0.04 1.05 ± 0.04 1.05 ± 0.04 1.03 ± 0.05

Table S2. As in Tab. S1, but for varying UV cutoff xUV with all other parameters fixed to their fiducial values. Our fiducial choice of xUV = 16 is shown in bold. 17

1 10 kUV = mr/8 kUV = mr/6 kUV = mr/4

) [arb.] 10 k/H (

F 1 10− mr log H = 8.75 0 200 400 600 800 1000 1200 1400 1600 k/H 2 kUV = mr/8 kUV = mr/6 kUV = mr/4

q 1

0 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 mr log H Figure S8. (Top) Example fits to the instantaneous emission spectrum at log(mr/H) = 8.75 for our three largest choices of the UV cutoff for the fitting range. The data indicated in grey corresponds to the range k ∈ (30H, mr/8), with associated fit in black. In red, which show the fit obtained with the range k ∈ (30H, mr/6), which includes the grey and additionally light-red data. In dark blue, we show the fit obtained with the range k ∈ (30H, mr/4), which includes the grey, light-red, and light-blue data. Error bars have been obtained in a data-driven way from the fits using the procedure described in Methods Sec.E. Visible mismodeling at large k/H biases the fitted power-law towards artificially larger q. This can be contrasted with the results shown in Fig.3, where a more conservative choice of UV cutoff does not result in apparent mismodeling at an identical time. (Bottom) The time evolution of the emission spectrum index for these large choices of UV cutoff for the fitting range. A clear trend of increasing q is obtained for the largest UV cutoff, suggesting that choices of large UV cutoff may result in unphysical growth in the fitted spectral index. Evidence for the linear growth of q in log(mr/H) was claimed in [27] based on analysis performed with the fitting range k ∈ (30H, mr/4).

1400 β ξ × 1200

1000 1 − i

2 800 k H

h 600

400

200 q = 1.06 β = 840 70 0 ± 6.0 6.5 7.0 7.5 8.0 8.5 log mr/H

Figure S9. The inverse expectation value h(H/k)2i−1 computed using the axion spectrum F (k/H) by numerically integrating −q the spectrum to k/H = xmax = 50 and then analytically integrating the power law distribution F (x) ∝ x from xmax to the log∗ UV cut-oﬀ at k/H ∼ e for log∗ ≈ 65 (as in Fig.4). Smaller values of β correspond to larger axion ﬁeld values. Here, we illustrate the result for the maximum allowed q of 1.06, which leads to the smallest β consistent with our simulation results. 18

Fid. Fit Fid. Data Lo-res. Data Lo-res. Fit

) [arb.] 10 k/H ( F

mr log H = 7.75 60 80 100 120 140 k/H 1.5

1.0 q

0.5 Fid. Fit Fid. Data Lo-res. Fit Lo-res. Data

7.00 7.25 7.50 7.75 8.00 8.25 8.50 8.75 mr log H Figure S10. As in Fig.3, but comparing fits to the emission spectrum and index evolution for our fiducial simulation output and a lower-resolution simulation. The two simulations are identical until log mr/H ≈ 5.3 when the low-resolution simulation stops adding extra refinement levels. The low-resolution simulation is then run until log mr/H ≈ 8, when we saturate the mr∆x . 1 resolution requirement. (Top) A comparison of the emission spectra and fits for the fiducial simulation (data in grey, fit in black) and the lower-resolution simulation (data in light red, fit in maroon) at log mr/H ≈ 7.75, which is the final emission spectrum obtained in our lower-resolution simulation. The lower-resolution simulation prefers a larger power-law index in the fit, and the data-driven errors are somewhat larger than in our fiducial simulation. (Bottom) A comparison of the best-fit emission spectrum index as a function of log mr/H for the fiducial and lower-resolution simulation. Over the range of log mr/H to which we are sensitive in the lower-resolution simulation, the emission spectra realize larger indices, suggesting that resolution loss in a uniform resolution simulation that saturates the resolution criteria may lead to a systematic bias towards index growth.

Coeﬃcient ∆ log = 0.125 ∆log = 0.25 ∆ log = 0.5 ∆ log = log 2

q1 0.0 ± 0.09 −0.04 ± 0.08 −0.05 ± 0.09 −0.1 ± 0.06

q0 1.02 ± 0.72 1.36 ± 0.69 1.4 ± 0.73 1.84 ± 0.46 const. q0 1.03 ± 0.04 1.02 ± 0.04 1.02 ± 0.03 1.03 ± 0.01

Table S3. As in Tab. S1, but now holding all our fiducial analysis choices fixed, with the exception of the size of the step in log(mr/H) used in the finite difference for the calculation of the instantaneous emission spectrum. We vary between ∆ log(mr/H) ∈ {0.125,.25,.5, log(2)}, with the log(2) differences corresponding to a Hubble time spacing. Our fiducial choice of ∆ log(mr/H) = 0.25 is shown in bold.

Coeﬃcient Eq.8 Mask Eq.9 Mask Eq. 10 Mask

q1 −0.04 ± 0.08 −0.04 ± 0.08 −0.05 ± 0.08

q0 1.36 ± 0.69 1.37 ± 0.65 1.39 ± 0.7 const. q0 1.02 ± 0.04 1.02 ± 0.03 1.02 ± 0.03

Table S4. As in Tab. S1, but now holding all our fiducial analysis choices fixed, with the exception of the choice of screening mask. We vary this choice between the screening functions described in (8), (9), and (10). Our fiducial choice of screening in the form of (8) is shown in bold.