XXVIII REUNIÓN ANUAL XXVIII REUNIÓN ANUAL DE LA DIVISIÓN DE GRAVITACIÓN Y DE LA DIVISIÓN DE GRAVITACIÓN Y FÍSICA MATEMÁTICA Dark matterFÍSICA MATEMÁTICA with ultralightDE LA SOCIEDAD bosons MEXICANA DE andFÍSICA DE LA SOCIEDAD MEXICANA DE FÍSICA 5 y 6 de noviembre de 2020

its imprint 5 y 6 de noviembre on de 2020cosmological CINVESTAV, structure I.P.N.

CINVESTAV, I.P.N. Luis A. Ureña-LópezEstimados Colegas: Department of Physics, University of Guanajuato, México Estimados Colegas: Tenemos el gusto de invitarlos a la reunión anual de nuestra división, que se llevará a cabo los días 5 y 6 de noviembre del presente año de manera virtual. La reunión anual tiene como objetivo presentar trabajos de investigación nacionales convocados por la Tenemos el gusto de invitarlos a la reunión anual de nuestra división, que se llevará a institución organizadora, en esta ocasión fue a cargo de CINVESTAV, I.P.N.. cabo los días 5 y 6 de noviembre del presente año de manera virtual. La reunión anual

tiene como objetivo presentar trabajos de investigación nacionales convocados por la Durante la reunión el comité directivo convoca a una Mesa Redonda de discusión de institución organizadora, en esta ocasión fue a cargo de CINVESTAV, I.P.N.. A Rainbow of Dark Sectors,temas M a relacionadosrch 22 con - la April DGFM y una 2 Asamblea General, esta última tiene como objetivo aprobar los informes y los proyectos de trabajo de la Mesa Directiva (favor de Durante la reunión el comité directivo convoca a una Mesa Redonda de discusión revisarde los nuevos estatutos en la página de la división para su aprobación). temas relacionados con la DGFM y una Asamblea General, esta última tiene como objetivo aprobar los informes y los proyectos de trabajo de la Mesa Directiva (favorDebido de a la contingencia por COVID-19, la Reunión Anual se llevará a cabo de manera revisar los nuevos estatutos en la página de la división para su aprobación)1. virtual los días 5 y 6 de noviembre del presente año a través de la plataforma Bluejeans.

Debido a la contingencia por COVID-19, la Reunión Anual se llevará a cabo de manera virtual los días 5 y 6 de noviembre del presente año a través de la plataforma Bluejeans.

Non-relativistic! Nostalgia Cold and Fuzzy Dark Matter astro-ph/9910097 Wayne Hu, Rennan Barkana & Andrei Gruzinov Institute for Advanced Study, Princeton, NJ 08540 Revised February 1, 2008 A New Cosmological Model of Quintessence and Dark Matter

Cold dark matter (CDM) models predict small-scale structureinexcessofobservationsofthecores Varun Sahni1,͑ and Limin Wang 2,† and abundance of dwarf galaxies. These problems might be solved, and the virtues of CDM models 1Inter-University Centre for Astronomy & Astrophysics, PostBag4,Pune411007,India retained, even without postulating ad hoc dark matter particle or field interactions, if the dark 2Department of Physics, 538 West 120th Street, Columbia University, New York NY 10027, USA ͑ matter is composed of ultra-light scalar particles (m ͑ 10 22eV), initially in a (cold) Bose-Einstein (February 1, 2008) condensate, similar to axion dark matter models. The wave properties of the dark matter stabilize We propose a new class of quintessence models in which late times oscillations of a scalar field give gravitational collapse providing halo cores and sharply suppressing small-scale linear power. rise to an e͑ective equation of state which can be negative andhencedrivetheobservedacceleration of the universe. Our ansatz provides a unified picture of quintessence and a new form of dark matter Introduction.— Recently, the small-scale shortcomings of the Compton wavelength m͑1 but much smallerwe than call Frustrated Cold Dark Matter (FCDM). FCDM inhibits gravitational clustering on small scales the otherwise widely successful cold dark matterastro-ph/0003365 (CDM) the particle horizon, one can employ a Newtonianand ap- could provide a natural resolution to the core density problem for disc galaxy halos. Since the models for structure formation have received much at- proximation to the gravitational interaction embeddedquintessence in field rolls towards a small value, constraints on slow-roll quintessence models are safely circumvented in our model. tention (see [1–4] and references therein). CDM models the covariant derivatives of the field equation and a non- Klein-Gordon equation PHYSICAL REVIEW D, VOLUME 63, 063506 predict cuspy dark matter halo profiles and an abundance relativistic approximation to the dispersion relation.PACS It is number(s): 04.62.+v, 98.80.Cq of low mass halos not seen in the rotation curves and lo- then convenient to define the wavefunction ͒ ⌽ Aei͑,out Further analysis of a cosmological model with quintessence and scalar dark matterThe recent discovery that type Ia high redshift super- to remain unchanged virtually from the Planck1 epoch μν ∂V cal population of dwarf galaxies respectively. Though the of the amplitude and phase of the field ͑ = A cos(mt͓⌽), 19 novae are fainter than they would be in an Einstein-de zpl 10 to z 2[5]resultinginanenormousdi͑er-∂ ( −gg ∂ ϕ) = significance of the discrepancies is still disputed and so- which obeys † μ ν Tonatiuh Matos* and L. Arturo Uren˜a-Lo´pez Sitter universe suggests that the universe mayastro-ph/0105564 be acceler- ence␭ in the scalar␭ field energy density and that of mat- Departamento de Fı´sica, Centro de Investigacio´n y de Estudios Avanzados del IPN, AP 14-740, 07000 Me´xico D.F., Mexico −g ∂ϕ lutions involving astrophysical processes in the baryonic ating, fuelled perhaps by a cosmological constant or some ter/radiation at early times. This leads to a fine tuning gas may still be possible͑Received (e.g. 1 [5]), June 2000; recent revised attention manuscript has received 5 October 2000; published3 a˙ 20 February1 2001͒2 i(͓t + )͒ =(͓ ␭ + mother͒)͒ , field possessing(2) long range ‘repulsive’ e͑ects [1,2]. problem: the relative values of ͒͑ and ͒m must be set We present the complete solution to a 95% scalar field cosmological model2 ina which the dark2m matter is ͒123 1 mostly focused on solutions involving the dark matter The acceleration of the universe is related toQuintessential the equation to Haloes very high around levels of Galaxies accuracy (͒͑/͒m)initial 10 in 2 2 modeled by a scalar field ⌽ with the scalar potential V(⌽)͑V0͓cosh(␭͑␬0⌽)͒1͔ and the dark energy is ␭ V(ϕ) = maϕ sector. where ͒ is the Newtonian gravitationalof state potential. of matter For through the Einstein equation order to ensure ͒͑/͒m 1atpreciselythepresentepoch. modeled by a scalar field ⌿, endowed with the scalar potential ˜V(⌿)͑˜V ͓sinh(␣͑␬ ⌿)͔␤. This model has ␭ In the simplest modification, warm dark matter (m ͑ 0 0 Alexandre Arbeya,b ͑, JulienAmorereasonableassumptionmightbeiftheenergy Lesgourguesa and Pierre Salatia,b 2 only two free parameters, ␭ and the equation of state ␻ the. With unperturbed these potentials, thebackground, fine-tuning and the cosmic right hand side van- ⌿ 2a¨ 2 4͑G keV) replaces CDMcoincidence and suppresses problems are ameliorated small-scale for both struc- dark matterishes and dark and energy the and energy the model density agrees with in the astro- field, ␭ = m |͒=| /2, ͒c + ͒X (1 + 3wX ) (1) density in the ⌽-field were comparable to that of radiation a a)͑ Laboratoire3 de Physique Th´eorique LAPTH, B.P. 110, F-74941 Annecy-le-Vieux Cedex, France. (Fuzzy Dark Matter) ture by free-streamingnomical out observations. of potential For the wells scalar [3], dark but matter, this we clarify the meaning of a scalar Jeans length͑3 and then the at very early times – say at the end of inflation [6]. This redshifts like matter ␭ ␬ a . b) Universit´ede͑ Savoie, B.P. 1104,͒ F-73011 Chamb´eryCedex, France. modification may adverselymodel predicts a͑ aect suppression structure of the at mass somewhat power spectrum forOn small time scales scales having a short wave number comparedk⌽kmin,⌽ with, for cold the matter expansion͒c and X-matter with equation of state might even be expected if the ⌽-field were to be an in- where k ͑4.5h Mpc͒1 for ␭͑20.28. This last fact could help to explain the death of dwarf galaxies and larger scales. Small-scalemin,⌽ power could be suppressed time, the evolution equations become PX = w͒X .Clearlyanecessary(butnotsu͒cient) con- flationary relic, its energy set by an equipartition ansatz. the smoothness of galaxy core halos. From this, all parameters of the scalar dark matter potential are com- 11However September for 2001 the ⌽-field to remain subdominant until re- more cleanly in the initial fluctuations, perhaps originat- ͒23 dition for the universe to accelerate is wX < 1/3. In pletely determined. The dark matter consists of an ultralight particle, whose mass is m⌽͑1.1͓10 eV and other words the equation of state of X-matter͑ must vi- cently its energy density must decrease rapidly at early ing from a kink in theall the inflaton success of potential the standard [2], cold but dark its matter regen- model is recovered. This implies1 that2 a scalar field could also2 i͓t͒ =(͓ ␭ + m͒)͒ , ␭ ͒ =4␬G͔␭ . (3) times. Such behaviour clearly cannot arise for polynomial eration through non-linearbe a good candidate gravitational the dark matter collapse of the Universe. would 2m olate the strong energy conditionThe nature (SEC) of the so dark that matter͒X + that binds galaxies remains an open question. The favored p < likely still produce halo cusps [6]. 3PX < 0. Investigationscandidate of cosmological has been so models far the neutralino. with potentials This massiveV speci(⌽)es with⌽ , evanescent0

posed by nucleosynthesis ͑see, for example, Ref. ͓2͔͒. Then, conditions ͑about 100 orders of magnitude͒, avoidinging radiation fine-arXiv:astro-ph/0105564v3 16 Oct 2001 and matter dominated epochs causes V (⌽) panied by an important change in the equation of state of this model considers a flat universe (⍀ ␭⍀ Ϸ1) with tuning. The equation of state ␻ changes with time towards ͒˙ 2 ⌳ M Q P = V (͒) . (3) 95% of unknown matter but which is of great importance at ͒1 ͓4,5͔, and then it can dominate the evolution of the Uni- 2 ͒ the cosmological level. Moreover, it seems to be the most 1 verse at late times. A typical example is the pure inverse 1 If the kinetic term is negligible with respect to the contribution of the potential, a pure cosmological constant – successful model fitting current cosmological observations power-law potential V(Q)ϳQ͒␣ (␣⌽0) ͓4,6,12͔. But the ⌽ = 1 – is recovered. Cosmological scenarios with quintessence in the form of a scalar field have been investigated ͓3͔. predicted value for the current equation of state of the quin- [3] with͒ various potentials and their relevance to structure formation has been discussed. However, from the theoretical point of view, ⌳ has some tessence cannot be put in good accordance with supernovae On the other hand, matter contributes a fraction ͑M 0.3 to the energy balance of the universe. The nature of problems. First, the initial conditions have to be set precisely results ͓4͔. The same problem arises with other inverse ͑ at one part in 10120, that is, an extreme fine-tuning problem power-law-like potentials. Another possibility is the poten- that component is still unresolved insofar as baryons amount only to [4] tials proposed in Ref. ͓13͔. They can solve the troubles stated 2 ͑B h =0.02 0.002 . (4) above, but it is difficult to uniquely determine their free pa- ± *Email address: tmatos@fis.cinvestav.mx rameters. According to the common wisdom, non–baryonic dark matter would be made of neutralinos – a massive species with † Email address: lurena@fis.cinvestav.mx On the other hand, we do not know the nature of the dark weak interactions that naturally arises in the framework of supersymmetric theories. This approach has given rise to

0556-2821/2001/63͑6͒/063506͑11͒/$15.0063 063506-1 ©2001 The American Physical Society

͒ E–mail: [email protected], [email protected], [email protected]

1 Scalar field dark matter, ULB’s, ALP’s, …

• ……

• Halo Abundance and Assembly History with Extreme-Axion Wave Dark Matter at z ≥ 4, Hsi-Yu Schive, Tzihong Chiueh. e-Print: arXiv:1706.03723 [astro-ph.CO].

• Cosmological Perturbations of Extreme Axion in the Radiation Era. Ui-Han Zhang, Tzihong Chiueh. e-Print: arXiv:1705.01439 [astro-ph.CO]

• Cosmological signatures of ultra-light dark matter with an axion-like potential. Francisco X. Linares Cedeño, Alma X. González-Morales, L. Arturo Ureña-López. e- Print: arXiv:1703.10180 [gr-qc]

• The mass discrepancy-acceleration relation: a universal maximum dark matter acceleration and implications for the ultra-light scalar feld dark matter model, L. Arturo Ureña-López, Victor H. Robles, T. Matos, e-Print: arXiv:1702.05103.

• Cosmological production of ultralight dark matter axions, Alberto Diez-Tejedor, David J. E. Marsh, e-Print: arXiv:1702.02116

• On the hypothesis that cosmological dark matter is composed of ultra-light bosons, Lam Hui, Jeremiah P. Ostriker, Scott Tremaine, Edward Witten. Phys.Rev. D95 (2017) no.4, 043541. e-Print: arXiv:1610.08297.

• Simulations of solitonic core mergers in ultra-light axion dark matter cosmologies, Bodo Schwabe, Jens C. Niemeyer, Jan F. Engels (Gottingen U.). e-Print: arXiv:1606.05151.

• Contrasting Galaxy Formation from Quantum Wave Dark Matter. Hsi-Yu Schive, Tzihong Chiueh, Tom Broadhurst, Kuan-Wei Huang. Astrophys.J. 818 (2016) no.1, 89

• Towards accurate cosmological predictions for rapidly oscillating scalar felds as dark matter, L. Arturo Ureña-López, Alma X. Gonzalez-Morales. e-Print: arXiv:1511.08195.

• On wave dark matter in spiral and barred galaxies. Luis A. Martinez-Medina, Hubert L. Bray, Tonatiuh Matos, JCAP 1512 (2015) no.12, 025. e-Print: arXiv:1505.07154

• Cosmic Structure as the Quantum Interference of a Coherent Dark Wave. Hsi-Yu Schive, Tzihong Chiueh, Tom Broadhurst. e-Print: arXiv:1406.6586 [astro-ph.GA]

• …… 3 Ureña-López Scalar Field Dark Matter

an equation of state that oscillates harmonically, namely w͑ formation of structure following the standard CDM paradigm. = p͑/͒͑ cos(2mat). Quite naturally, the frequency of the Within the synchronous gauge of metric perturbations, in which = ͑ 2 2 2 i j oscillations is the mass scale itself, and then the period of the the line element is written as ds dt a (͓ij hij)dx dx , 1 = ͑ + + ¯ oscillations (ma͑ ), is much smaller than the time scale for the the linearly perturbed KG equation in Fourier space for a given expansion of the universe (H 1)andtheninthisregimem /H wavenumber k reads: ͑ a ͒ 1. Furthermore, the equation of motion of the density parameter k2 1 ⌽͑ has the same form as that of a CDM component, which shows ␭ 3H␭ m2 ␭ ͑h ,(7) ¨ = ͑ ˙ ͑ a2 ͑ a ͑ 2 ˙ ¯˙ that the appearance of rapid oscillations also compels the field ͑ ͑ ͒ to behave as CDM. where we are considering that ͑(t, k) ͑(t) ␭(t, k), with ␭ The polar equations of motion (4) have been implemented in = + a small field perturbation, and h hl is the trace of the spatial an amended version of the public Boltzmann code CLASS, which ¯ = l has been dubbed CLASS.FreeSF. The numerical solutions of the perturbations in the metric. Reliable numerical simulations can be obtained if we define polar variables in the form: background density ͒͑, as compared to that of CDM with the same cosmological parameters, are shown in Figure 1.Notice / / ␬␭ ⌽1 2e͔ cos(⌿/2) , ␬m ␭ ⌽1 2e͔ sin(⌿/2) , (8) that SFDM follows the evolution of CDM after the start of the ˙ = ͑ ͑ a = ͑ ͑ rapid oscillations, and actually the oscillations of the density itself can be clearly seen in the curves. The matching between SFDM where ͔ and ⌿ are the new dynamical variables. However, it is and CDM densities occurs at later times for smaller boson masses. better to define new variables: The value of the scale factor for the start of the oscillations is ͔ ͔ ͓ ͓͒͑/͒͑ e sin(␣/2 ⌿/2) , ͓ e cos(␣/2 ⌿/2) , indicated by the vertical lines, which actually were calculated 0 = = ͑ ͑ 1 = ͑ ͑ with the other Boltzmann code available for SFDM: axioncamb, (9) the latter based on the public code CAMB. Notice that there under which the perturbed KG equation (7) is written also as is good agreement between the two codes (CLASS.FreeSF and adynamicalsystem(Ureña-López and Gonzalez-Morales, 2016; axioncamb) for the background dynamics, even though they Cedeño et al., 2017): follow di͑erent numerical approaches to deal with the rapid d͓ k2 k2 oscillations of the field. More details about the two codes and 0 ␣ ␣ ͓ ␣͓ 3sin 2 (1 cos ) 1 2 sin 0 their mathematical description can be found in Hlozek et al. d(ln a) = ⌽͑ ͑ kJ ͑ ͓ + kJ (2015) and Ureña-López and Gonzalez-Morales (2016). 1 dh ¯ (1 cos ␣), (10a) FDM potential ͑2 d(ln a) ͑ U-L, González-Morales, JCAP 07 (2016) 048 2.2. Linear1 Perturbations Cookmeyer2 et al, PRD 101, 0235012 (2020)* 2 2 U-L, JCAP 06 (2019) 009 d͓1 k k TheV( nextϕ) = step inm theaϕ analysis is to find the behavior of linear 3cos␣ sin ␣ ͓1 (1 cos ␣) ͓0 d(ln a) = ͑ ͑ *How2 sound are our+ ultra-light2 + axion perturbations2 and to verify that the scalar field DM allows the ⌽ kJ approximations?͓ kJ Ureña-López Scalar Field Dark Matter 1 dh ¯ sin ␣ ,(10b) ͑2 d(ln a) 2 2 Mass mac /eV is a time scale of reference! Mass mac /eV is a distance scale of reference! where k2 2a2Hm is the (squared) Jeans wavenumber that J ⌽ a naturally arises because of the wave nature of the scalar field. The interesting regime, as before for the background dynamics in section 2.1, is that of rapid oscillations,10−22 under which Equation (10) become:

d͓ k2 1 dh d͓ k2 0 ͓ ¯ 1 ͓ 2 1 , 2 0 .(11) d(ln a) = ͑kJ ͑ 2 d(ln a) d(ln a) = kJ 10−24 For length scales k k , we recover the same expression as ͓ J that of the density contrast for CDM, namely, ͓ (1/2)h, ˙0 = ͑ ¯˙ whereas ͓1 remains approximately constant. However, in the 4 opposite regime k k ,Equation(11)resemblethatofaharmonic ͒ J oscillator, and then the density perturbations cannot grow as much as those of CDM. FIGURE 1 | The background solution of the SFDMFIGURE density, for 2 different | The MPS values(Left) and the anisotropies of the CMB (Right) obtained for the SFDM model from the amended Boltzmann codes CLASS.FreeSF and As before for the background, we show in Figure 2 the of the boson mass ma,incomparisonwiththecaseofCDM.Thenumericalaxioncamb. As explained in section 2.2, the code CLASS.FreeSF solves the polar form of the perturbed KG equation (10), whereas axioncamb solves its equivalent solutions were obtained from the Boltzmann codefluid CLASS.Free form.SF In thatgeneral, solves SFDM lookssolutions indistinguishable of the linear from density CDM, although perturbations there are of some SFDM small in discrepancies in the outputs generated by the Boltzmann codes. the polar form of the KG equation (2) discussed in section 2.1.Thevertical comparison with those of CDM, as obtained from the Boltzmann dashed lines indicate the values of the scale factor aosc at the start of the rapid code CLASS and its amended version CLASS.FreeSF. In the case oscillations of the scalar field ͑;thevaluesofaosc were obtained from the of the so-called mass power spectrum (MPS), we notice the sharp equivalent solutions of the Boltzmann code axioncamb. firstly shown in Matoscut-o and͑ Arturothat has become Ureña-López a trademark (2001) of SFDMas an ever since(2017) it was.Di͑erent approaches have been devised to find the output of the now outdated Boltzmann code CMBFast. We also general features of the gravitational collapsed objects within show in the same plot the output from axioncamb, which seems acosmologicalcontext(Widrow and Kaiser, 1993; Woo and Frontiers in Astronomy and Space Sciences | www.frontiersin.org 3 July 2019 | Volume 6 | Article 47 to di͑er a little bit regarding the profile of the MPS at around Chiueh, 2009; Schive et al., 2014a; Uhlemann et al., 2014; the cut-o͑ scale in the two cases shown (m 10 22,10 24 eV). Veltmaat and Niemeyer, 2016; Edwards et al., 2018), where it has a = ͑ ͑ The di͑erences may indicate here some discrepancy in the been verified that the non-linear process of structure formation treatment of linear perturbations in the two Boltzmann codes: in SFDM models proceeds as in CDM for large enough scales. CLASS.FreeSF directly solves the (polar) Equations (10), whereas According to the detailed analysis made in Schive et al. axioncamb uses a formal equivalence of the field perturbations (2014a), Schwabe et al. (2016), and Mocz et al. (2017a), the to those of a fluid, following the prescription in Hu (1998) and gravitationally bounded objects that one could identify with DM the scale-dependent speed of sound suggested in Hwang and galaxy halos all have a common structure: one central soliton Noh (2009). Moreover, we also show the analytic formula put surrounded by a Navarro-Frenk-White-like envelope created by forward in Hu et al. (2000) for the linear MPS. With the latter the interference of the Schrodinger wave functions. In terms of being just an analytic approximation without proper backup the standard nomenclature, the SFDM model belongs to the so- from a Boltzmann code, its manifest discrepancies with the called non-cusp, or cored, types of DM models. Although not yet aforementioned numerical results should not be surprising after completely conclusive, there seems to be an increasing evidence all, and this little exercise suggests that it would be bettertouse in favor of cored DM models (Walker and Peñarrubia, 2011; the numerical output from the Boltzmann codes rather than the Rodrigues et al., 2018), and then the non-cusp nature of SFDM approximation to model the MPS. objects remain one of the most promising features of the model. In the case of the anisotropies of the Cosmic Microwave Here we will make a quick review of the general features of the Background (CMB) the di͑erences between the di͑erent models soliton structures that arise from SFDM models, using for this the are almost unnoticeable, and then this observable is not ableto so-called field and fluid approaches, in order to highlight their distinguish SFDM from CDM, even if the boson mass changes equivalences and di͑erences and their use in cosmological and in two orders of magnitude (recall that m 10 22,10 24 eV). astrophysical settings. a = ͑ ͑ However, the outputs from CLASS.FreeSF and axioncamb do not agree completely again, which seems to point out some 3.1. The Field Approach possible inconsistencies between the polar and fluid approaches The field approach to the formation of structure within the SFDM to the KG equation at the level of linear perturbations. A more paradigm refers to the solutions obtained from the Schrodinger- thorough analysis of the di͑erences of the two codes will be Poisson (SP) system, which is represented by the equations: reported elsewhere. h 2 2 2 2 ih͑t͒ ¯ ͒ ⌽͒ , ⌽ 4͓Gma ͒ ,(12) ¯ = ͑2ma ͒ + ͒ = | | 3. GRAVITATIONAL CONFIGURATIONS OF SFDM where ͒ is the wave function that collectively represents the boson particles in their ground state, and ⌽ is the Ever since the seminal paper (Ru͒ni and Bonazzola, 1969), there Newtonian gravitational potential. Notice that the latter is has been much interest in the properties of the gravitational calculated from the mass density e͑ectively defined through objects made of scalar fields, in both the relativistic and non- ␭ m2 ͒ 2. For the purposes of simplicity, we do not include = a| | relativistic regimes, see the comprehensive reviews in Jetzer any self-interaction term in the SP system, which would then (1992), Schunck and Mielke (2008),andLiebling and Palenzuela become the Gross-Pitaeevski-Poisson system. For a discussion

Frontiers in Astronomy and Space Sciences | www.frontiersin.org 4 July 2019 | Volume 6 | Article 47 PHYSICAL REVIEW D 91, 103512 (2015) 196 A searchL. Amendola, for ultralight R. Barbieri axions / Physics using Letters B precision 642 (2006) 192–196 cosmological data

Renée Hlozek,1 Daniel Grin,2 Davidis J. E.a significant Marsh,3,* and constraint Pedro G. but Ferreira leaves4 ample room for a relevant 1Department of Astronomy, Princeton University,component Princeton, of a DM-PGB, New Jersey 08544, as given USA in Eq. (4). 2 Kavli Institute for Cosmological Physics, DepartmentThe prospects of Astronomy for detecting and Astrophysics, the signal from an intermediate- University of Chicago, Chicago, Illinois 60637, USA 3Perimeter Institute, 31 Caroline Street North,mass Waterloo, PGB scalar Ontario are interesting. N2L 6B9, Canada The same cosmological obser- 4Astrophysics, University of Oxford, DWB, Keblevations Road, that Oxford will OX1 constrain 3RH, United the neutrino Kingdom mass can provide limits (Received 17 December 2014;to the published PGB 15mass May and 2015) abundance, since the PGB scalar affects ͑33 ͑20 Ultralight axions (ULAs) with masses in the rangeboth10 theeV growth͒ ma ͒ of10 fluctuationseV are motivated and the by stringpower spectrum shape. theory and might contribute to either the dark-matterIn or dark-energy particular, densities deep weak of the lensing Universe. observations ULAs could are expected to suppress the growth of structure on small scales, leadprovide to an altered in few integrated years Sachs-Wolfe upper limits effect to on the cosmic neutrino abundance f͓ microwave-background (CMB) anisotropies, and changeof less the than angular one scale percent of the[18] CMB,mostlythroughthedampingeffect acoustic peaks. In this work, cosmological observables over the full ULAon mass the fluctuation range are computed growth. and Since then used a scalar to search with the same abun- for evidence of ULAs using CMB data from the Wilkinson Microwave Anisotropy Probe (WMAP), Planck satellite, Atacama Cosmology Telescope, and Southdance Pole Telescope,fI provides as well a as very galaxy similar clustering fluctuation data from growth (at least for 30 24 mI 10͑͒32 –10͒ eV)͑25 we.5 can expect that also fI can be con- the WiggleZ galaxy-redshift survey. In the mass range⌽10 eV ͒ ma ͒ 10 eV, the axion relic- density ͑a (relative to the total dark-matter relic densitystrained͑d) must to better obey the than constraints the percent͑a=͑d accuracy.͒ 0.05 and 2 ͑24 ͑ah ͒ 0.006 at 95% confidence. For ma 10 eV, ULAsThis are similarity indistinguishable of behavior from standard raises cold also dark the issue of the level ͑ ͑32 matter on the length scales probed, and are thus allowedof degeneracy by these data. of scalars For ma with10 neutrinos.eV, ULAs areWe have shown that allowed to compose a significant fraction of the darkscalars energy. distinguish themselves͒ from neutrinos by the ISW ef- fect and by the fact that abundance and mass are not correlated. DOI: 10.1103/PhysRevD.91.103512 PACS numbers: 95.35.+d, 98.80.-k, 98.80.Cq Whether this is enough to break their degeneracy deserves fur- ther analysis. I. INTRODUCTION stringent limits on the mass of the axion as a candidate Fig. 4. Likelihood functions at 68.95 and 99.7% c.l. (dark to light gray) for Acknowledgements 30 for DM. Furthermore, the nature of inhomogeneities in the the parameters m30A multitudemI /10͒ ofeV data and supportsfI obtained the existence marginalizing of dark over matter͑ , 2 ͑ 2 axion distribution yield, as with primordial gravitational ͒mh , ns and (DM)h,andfixing[1–12]͒.b Theh identity0.023. The of date the employed DM, however, are discussed remains = waves,This work a direct is supported window on in the part very by MIUR early universe and by the and, EU in under in the text. Theelusive. discreteness Axions of the[13 grid15] causesare some a leading wiggling candidate in the function. for this DM Current– constraintsRTN contract on MRTN-CT-2004-503369. the mass component of the Universe [16–23]. Originally proposed to m ≳ 10−25.5 eV/c2 (95 % CL) solve the strong CP problem [13], they are also generic in a string theory [24,25], leading to the idea of an axiverse References [26]. In the axiverse there are multiple axions with masses spanning many orders of magnitude and composing [1] N. Weiss, Phys. Lett. B 197 (1987) 42. [2] J.A. Frieman, C.T. Hill, A. Stebbins, I. Waga, Phys. Rev. Lett. 75 (1995) distinct DM components. For all axion masses ma ͑33 ͑ 2077. 3H0 ⌽ 10 eV, the condition ma > 3H is first satisfied prior to the present day. When this happens, the axion [3] J.E. Kim, JHEP 9905 (1999) 022, hep-ph/9811509; J.E. Kim, JHEP 0006 (2000) 016, hep-ph/9907528; begins to coherently oscillate with an amplitude set by its K. Choi, Phys. Rev. D 62 (2000) 043509, hep-ph/9902292; initial misalignment, leading to axion homogeneous energy Y. Nomura, T. Watari, T. Yanagida, Phys. Lett. B 484 (2000) 103, hep- ͑3 densities that redshift as a (where a is the cosmic scale ph/0004182; ͑27 factor). If ma 10 eV, the axion energy-density dilutes J.E. Kim, H.P. Nilles, Phys. Lett. B 553 (2003) 1, hep-ph/0210402; just as nonrelativistic͑ particles do after matter-radiation L.J. Hall, Y. Nomura, S.J. Oliver, astro-ph/0503706. equality, making the axion a plausible DM candidate. [4] R. Barbieri, L.J. Hall, S.J. Oliver, A. Strumia, hep-ph/0505124. The fact that axions can be so light places them, like [5] W. Hu, R. Barkana, A. Gruzinov, Phys. Rev. Lett. 85 (2000) 1158, astro- neutrinos, in a unique and powerful position in cosmology. ph/0003365. For as we shall show, unlike all other candidates for DM, [6] M. Khlopov, B. Malomed, Ya.B. Zel’dovich, Mon. Not. R. Astron. Soc. axions lead to observational effects that are directly tied FIG.215 1 (1985) (color 575; online). Marginalized 2 and 3͑ contours show M. Bianchi, D. Grasso, R. Ruffini, Astron. Astrophys. 231 (1990) 301. to their fundamental properties, namely the mass and limits to the ultralight axion (ULA) mass fraction ͑a=͑d as a [7] P.G. Ferreira, M. Joyce, Phys. Rev. D 58 (1998) 023503, astro-ph/ field displacement. Signatures in the cosmic microwave- function of ULA mass ma, where ͑a is the axion relic-density 9711102. background (CMB) and large-scale structure (LSS) can be parameter today and ͑d is the total dark-matter energy density [8] U. Seljak, M. Zaldarriaga, Astrophys. J. 469 (1996) 437. −22 2 used to pin down axion abundances to high precision as a parameter. The vertical lines denote our three sampling regions, ma ≳ 10 eV/c (99.7 % CL) [9] J.R. Bond, A.S. Szalay, Astrophys. J. 274 (1983) 443. function of the mass; these constraints can be used to place discussed below. The mass fraction in the middle region is Fig. 5. Combined likelihood function at 68.95 and 99.7% c.l. (dark to light gray) [10]constrained C.P. Ma, Astrophys. to be = J. 4710 (1996).05 at 13, 95% astro-ph/9605198. confidence. Red regions 30 ͑a ͑d for the parameters m30 mI /10͒ eV and fI obtained marginalizing over ͑ , [11]show J.R. Bond,CMB-only G. Efstathiou, constraints,͒ J. Silk, while Phys. grey Rev. regions Lett. 45 include (1980) large- 1980. 2 ͑ 2 ͒mh , ns and h,andfixing* ͒bh 0.023. [12] M. Tegmark, Astrophys. J. 606 (2004) 702. [email protected]= scale structure data. [13] U. Seljak, et al., Phys. Rev. D 71 (2005) 351. [14] R.A.C. Croft, et al., Astrophys. J. 581 (2002) 20. 5 normalizationAmendola, for WMAP, Barbieri, SDSS PLB and 642 Lyman- (2006)⌽ 192). The Lyman-⌽ [15] M. Viel, M.Hlkozek Haehnelt, et al, V. Springel,PRD 91 (2015) Mon. Not. 103512 R. Astron. Soc. 354 (2004) constraints turn1550-7998 out to=2015 be the=91(10) strongest.=103512(29) As expected, there is an 103512-1684. © 2015 American Physical Society 31 [16] G. Hinshaw, et al., Astrophys. J. Suppl. Ser. 148 (2003) 135. intermediate-mass window of observability between 10͒ eV 23 [17] L. Verde, et al., Astrophys. J. Suppl. Ser. 148 (2003) 195. and 10͒ eV. In this region of interest for the mass mI ,the [18] K. Abazajian, S. Dodelson, Phys. Rev. Lett. 91 (2003) 041301. relative density fI is clearly limited to be below 10%, which Current constraints on the mass 2 E. Y. Davies et al. -26 -25 -24 -23 Herrera-Martín-22 et -al,21 ApJ 872 (2019)-20 (strong- gravitational19 - lensing)18 -17 -16 Davies & Mocz (2019) Spergel & Safarzadeh (2019) Stellar streams?? Wasserman et al. (2019) Broadhurst et al. (2019) Nori et al. (2019) Desjacques & Nusser (2019) Church et al. (2019) Amorisco & Loeb (2018) Hlo͑ek et al (2018) Kobayashi et al. (2017) González-Morales et al. (2017) Ir͒i et al. (2017) Marsh & Pop (2015) -26 -25 -24 -23 -22 -21 -20 -19 -18 -17 -16

Davis, Mocz, MNRAS 492 (2020) 4, 5721

Figure 1. Recent astrophysical constraints on FDM particle mass in the literature.6 The allowed regions of mass from each work are colored in blue, and this work’s findings are colored in red. black holes. Here we are interested in the soliton profile calculating the impact of soliton cores on galactic nuclei far from the Schwarzschild radius and take a simplified assuming they mimic SMBHs (Desjacques & Nusser 2019), approach in a regime where the SMBH can be treated as a and (8) modeling of ultra-di↵use galaxies (Wasserman et al. point mass. Our theory is also applicable to other baryonic 2019). There has also been a recent claim for the Milky perturbing forces on the soliton. An understanding of the Way that a central solitonic core of mass 109 M and ' e↵ect of SMBH on soliton cores is a vital ingredient in the particle mass m 10 22 eV is observationally supported ' study of DM density profiles at galactic cores. Central DM by the central motion of bulge stars (De Martino et al. 2018). density profiles can be paired with known observational constraints of the central density to provide constraints on The FDM mass constraints generally fall into two the FDM model itself. Furthermore the galactic core is an camps that are in moderate tension. Dwarf spheroidal ideal location to search for signals of DM self-interaction or galaxies are typically well fit by large, low-density dark annihilation signatures (Boyarsky et al. 2014; Bulbul et al. matter cores, such as the soliton cores predicted by FDM 2014), which depend on how the dark matter is centrally theory with a particle mass of m 10 22 eV. But many ' concentrated. other investigations favor m > few 10 22 eV. It is possible ⇥ that the lack of consistency could be due to systematic There has been significant e↵ort in recent years to biases from certain model-dependent assumptions that are place various astrophysical constraints on the FDM particle used when testing the FDM model on di↵erent scales. For mass (summarised in Figure 1). Constraints come from a example, some of the mentioned works from the literature variety of systems and scales (from the cosmic microwave ignore interference pattern fluctuations of the FDM field, background ( 10 Mpc) to the centres of galaxies ( 1 pc)). or the coupling of dark matter to baryons, or are limited by ⇠ These include: (1) using stellar velocity dispersion to fit the uncertainties in the phase-space distribution of collisionless Milky Way’s dwarf spheroidal galaxies with a soliton core tracers (stars) which will be greatly improved in the Gaia assuming the systems are dark-matter dominated (Marsh & era of astrometry (Lazar & Bullock 2019). We therefore take Pop 2015; Gonz´alez-Morales et al. 2017; Broadhurst et al. the view that it is important to obtain as many independent 2019; Safarzadeh & Spergel 2019), (2) fitting ultra-faint constraints as possible, over a range of physical scales, to dwarfs with soliton-cored halo models (Safarzadeh & verify existing constraints. In this work, we are able to place Spergel 2019), (3) analysing the Lyman-↵ forest as a tracer a new independent constraint on the FDM particle mass, of dark matter structure (Kobayashi et al. 2017; Irˇsiˇcet al. that comes from small scale (< 10 pc) data, namely upper 2017; Nori et al. 2018), (4) placing constraints from CMB limits on the amount of dark matter concentrated around lensing (Hloˇzek et al. 2018), (5) calculating dynamical heat- the SMBHs of the Milky Way and M87 (which recently had ing on the Milky Way’s stellar disc from FDM substructure its black hole imaged; Collaboration et al. 2019). including interference pattern fluctuations (Church et al. 2019), (6) calculating the impact of FDM fluctuations Our paper is organised as follows. In section 2 we on stellar streams that formed from disrupted globular describe the necessary mathematical background of the clusters in the Milky Way (Amorisco & Loeb 2018), (7) FDM system with a point mass black hole perturber and

MNRAS 000, 1–10 (2019) analysis, as proposed in the references above, can not be done directly for the model in turn and therefore weJCAP01(2021)051 opted to work with the 3D MPS from [116]asa proxy to find how constraints to the axion mass relaxes when we consider the full axion potential. More reliable constraints to the axion mass should arise from a dedicated analysis of the Lyman alpha forest observations. Axion-like potentialTherefore, the presence of in our analysis plays a key role in the axion mass constraint: it allows to the SFDM model with a fiducial mass of m 10 22eV to 2 2 Linares-Cedeño, González-Morales, U-L, JCAP 1, 051 (2021), arXiv:2006.05037 ' V ()=m f [1 cos(/f )] be in agreement with the bounds impose by Lyman-↵ observations, as we have AAACEnicbVC7TsMwFHXKq5RXgJHFokJqB0pSkGBBqmBhLBJ9SEmIHNdprdpJZDtIVdVvYOFXWBhAiJWJjb/BTTNAy5Hu1dE598q+J0gYlcqyvo3C0vLK6lpxvbSxubW9Y+7utWWcCkxaOGax6AZIEkYj0lJUMdJNBEE8YKQTDK+nfueBCEnj6E6NEuJx1I9oSDFSWvLNarviJgNahZeQ39d9BMOsO/axi2MJM/Mk9FHV882yVbMywEVi56QMcjR988vtxTjlJFKYISkd20qUN0ZCUczIpOSmkiQID1GfOJpGiBPpjbOTJvBIKz0YxkJXpGCm/t4YIy7liAd6kiM1kPPeVPzPc1IVXnhjGiWpIhGePRSmDKoYTvOBPSoIVmykCcKC6r9CPEACYaVTLOkQ7PmTF0m7XrNPa/Xbs3LjKo+jCA7AIagAG5yDBrgBTdACGDyCZ/AK3own48V4Nz5mowUj39kHf2B8/gB0JprX a a a Linares-Cedeño, González-Morales, U-L et al, PRD 96 (2017) 061301(R) anticipated from the P 1D in Section 3.3 (see Figure 7). In this sense, whereas 2 3m CMB observations impose a direct bound on m,inthecaseofLyman-↵ forest Pl what we have is the most likely values−22 for the axion2 mass such that they are in λ = FDM : λ = 0 ( fa → ∞) m ≳ 10 eV/c (95.5 % CL) 2 agreement with the ⇤CDM–baseda data from the inferred Lyman-↵ 3D MPS. 8πfa

CMB Ly-alpha

-17

m -21 log

-25 1 5 9 -25 -21 -17 log log m

Figure 8: 1DAvailable and 2D posterior prior volume: distributions no evidence for the axion for field an parameters extra parameter!m and in log- Tachyonic instability of linear density perturbations 7 arithmic scale. When considering the CMB observations from Planck 2018 (blue), we can set a lower bound(Axion for the DM value is the of total the axion DM budget) mass of log m = 23.99 at 22 23 Figure 6. MPS for SFDM with axion masses m/eV= 10 , 10 , and from95 zero.5% C.L. up (blue to the dashed line). On the other hand, Lyman-↵ data (green) imposes astrongerconstraintonthemassgivenbylog m = 21.96 at 95.5% C.L. (green maximum values reached for each axion mass. It can be noted that, for all the axion masses considered dashed line). Orange and red lines in the 2D posterior represent numerical and there is a cut-o↵ at small scales (larger k’s), and even more, there is an enhancementphysical of the limits MPS respectively. at See text for more details. such scales when considering large values of the parameter . Cosmological data from BOSS DR11 (yellow dots) [115], and from Ly↵ forest (black dots) [116] are shown for reference. We also show in Figure 8 two limits: (1) a numerical one given by the extreme case of for a given axion mass m (orange line), and (2) a physical one given

from the Lyman-↵ forest data reported by [116], in order to impose constraints on our model –17– from small scale structures. It is important to mention that what we are using is a Lyman- ↵ 3D MPS at redshift z = 0, which the authors in [116] have inferred from the 1D flux power spectrum measured by the BOSS and eBOSS collaboration [126]. Inferring the linear matter spectrum at z = 0 is a highly model-dependent process, and therefore, the constraints obtained in this work from Lyman-↵ have to be understood as how much SFDM can deviate from the CDM case, which is the fiducial model considered in [116], and should only be read as an approximation for what is the e↵ect of having the full axion potential on the constraints for the axion mass.

– 14 – Structure formation First steps PHYSICAL REVIEW D 69, 124033 ͑2004͒ The Astrophysical Journal,697:850–861,2009May20 doi:10.1088/0004-637X/697/1/850 C 2009. The American Astronomical Society. All rights reserved. Printed in the U.S.A. ͑ Evolution of the Schro¨dinger-Newton system for a self-gravitating scalar field

F. Siddhartha Guzma´n1 and L. Arturo Uren˜a-Lo´pez2 HIGH-RESOLUTION SIMULATION ON STRUCTURE FORMATION WITH EXTREMELY LIGHT BOSONIC 1Max Planck Institut fu¨r Gravitationsphysik, Albert Einstein Institut, Am Mu¨hlenberg 1, 14476 Golm, Germany and Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana 70803, USA* DARK MATTER 2Instituto de F´ısica de la Universidad de Guanajuato, A. P. 150, C. P. 37150, Leo´n, Guanajuato, Mexico ͑Received 9 January 2004; published 29 June 2004͒ Tak-Pong Woo1,2 and Tzihong Chiueh1,2,3 1 Department of Physics, National Taiwan University, 106 Taipei, Taiwan; [email protected], [email protected] Using numerical techniques, we study the collapse of a scalar field configuration in the Newtonian limit of 2 LeCosPa, National Taiwan University, 106 Taipei, Taiwan Newtonianthe spherically version symmetric Einstein-Klein-Gordon system, which results in the so called Schro¨dinger-Newton 3 Center for Theoretical Sciences, National Taiwan University, 106 Taipei, Taiwan ͑SN͒ set of equations. We present the numerical code developed to evolve the SN system and related topics, Received 2008 June 2; accepted 2009 March 10; published 2009 May 5 like equilibrium configurations and boundary conditions. Also,Relativistic we analyze the evolution scale of length different initial No. 1, 2009 EXTREMELY LIGHT BOSONIC DARK MATTERconfigurations and the physical quantities 851 associated with them. In particular, we readdress the issue of the ABSTRACTSiddhartha & Urena-L« opez« 2003)orevenone-dimensional(Hu 2.2. Basicgravitational Analysis cooling mechanism for Newtonian systems and find that all systems settle downh onto a zero-node equilibrium configuration. Abosonicdarkmattermodelisexaminedindetailviahigh-resolutionsimulations.Thesebosonshaveparticleet al. 2000)tostudythisproblem.Thesesimplificationsmay LC = The Lagrangian of nonrelativistic scalar field in the comoving ͑ mass of the order of 10 22 eV and are noninteracting.not capture If they what do exist actually and results can account in a three-dimensional for structure formation, system DOI: 10.1103/PhysRevD.69.124033 PACS number͑s͒: 04.40. mb, 98.35.Jk,ac 98.62.Gq ͒ with realistic initial conditions. In particular, the existence of a frame is these bosons must be condensed into the Bose–Einstein state and described by a coherent wave function. This flattened core has been derived or inferred from these previous a3 ͒͑ ͒͑ h2 ␬ I. INTRODUCTIONø 2 2 tions imposed on the SN system and the accuracy of the code matter, also known as fuzzy dark matter,isspeculatedtobeable,first,toeliminatethesubgalactichalostosolveworks of one-dimensional system or with spherical symmetry. L ihø ͑␬ ͑ + ( ͑) 2mV ͑ , = 2 ͒t ⌽ ͒t a2m ͔ ⌽ are of particular importance. the problem of overabundance of dwarf galaxies, and,In this second, paper, to we produce report high-resolution flat halo cores fully in three-dimensional galaxies suggested ͔ ␭ ␬ ⌿ by some observations. We investigate this model with simulations up to 10243 resolution in a 1 h 1 Mpc box In a previous paper of ours ⌽1͓, we studied the(3) formation The results of the numerical evolutions are given in Sec. simulations for this problem. Surprisingly, our simulations͒ re- and the equation of motion for this Lagrangian gives a modified IV. We systematically test the boundary conditions, the con- that maintains the background matter density ͑ 0.3and͑ 0.7. Our results show that the extremely of a gravitationally bounded object comprised of scalar par- Nonrelativistic scale length m veal that the singular͒ cores of bound objects remain to exist even form of Schrodinger’s¬ ticles, under equation the influence (Siddhartha of Newtonian & Ur gravity.ena-L« opez« The dynam- vergence of the numerical solutions, and how the latter re- light bosonic dark matter can indeed eliminate low-mass= halos through= the suppression of short-wavelength produce the expected results for the equilibrium configura- when the core size is much smaller than the Jean’s length. 2003): ics of the system is described by the coupled Schro¨dinger- tions of the SN system. fluctuations, as predicted by the linear perturbation theory.In Section But in2, contrast we provide to expectation, an explanation our for simulations the possible yield Newton ͑SN͒ system2 of equations, which is nothing but the h ͒͑ hø 2 Section V is devoted to the study of the3 gravitational cool- singular cores in the collapsed halos, where the haloexistence density profileof the isBoseÐEinstein similar, but not state identical, for the extremely to the Navarro– low iweakhø field limit of its͑ general+ mV ͑ relativistic, counterpart,(4) the L = ≃ 10 L ͒t = ⌽2a2m͔ ing mechanism,dB first described in ⌽4͓. Finally,C the conclu- Frenk–White profile. Such a profile arises regardlessmass of bosons whether under the haloinvestigation forms through here. We accretion then discuss or merger. two Einstein-Klein-Gordon ͑EKG͒ system. 3 1/2 sions are collected inm Sec.av VI. In addition, the virialized halos exhibit anisotropicdifferent turbulence representations inside a well-defined of ELBDM virial and the boundary. evolution Much of linear like where ͑ ⌽(n0As/a at)⌽ the momentwith ⌽ webeing have the no ordinaryhints toward wavefinding an ͒ the velocity dispersion of standard dark matter particles,perturbations turbulence for the is two dominated representations. by the random In Section radial3,the flow function, n0 theanalytic present solution background for the number evolution, density, we found and it necessaryV is to II. MATHEMATICAL BACKGROUND in most part of the halos and becomes isotropic towardnumerical the scheme halo cores. and initialConsequently, condition the are three-dimensional described. We the self-gravitationaldevelop potential numerical obeying techniques the toPoisson study the equation, formation process Here we describe the mathematical basis needed to deal collapsed halo mass distribution can deviate frompresent spherical the simulation symmetry, results as the in cold Section dark4.InSection matter halo5,we does. of the scalar objects. The study of the dynamical properties 2 of the fully2 time-dependent4͓G SN system2 has been done before with classical scalar fields, complex and real, in both the look into the physics of collapsed cores with detailed analyses V 4͓Ga ␭␬ ␬0( ͑ 1). (5) Key words: dark matter – Galaxy: structure – large-scale structure of universe ͔ in= the literature =⌽2–5͓a, but more| | is⌽ needed in order to under- relativistic and Newtonian limits. The latter is given in much from different perspectives. Finally, the conclusion is given in more detail as it will be our system of interest for the rest of stand the gravitational collapse of a weakly gravitating scalar Online-only material: color figures Section 6. In the Appendix,wepresentresultsofone-andtwo- The only modification to the conventional SchrodingerÐPoisson¬ this paper. field. dimensional simulations and demonstrate that singular cores do equation is the appearance a 1 associated with the comoving To begin with, we write the equations that describe a sca- The main aim⌽ of this paper is to perform an exhaustive not occur in one- and two-dimensional cases. spatial gradient ,andtheprobabilitydensity ͑ 2 to be lar system within general relativity, which are the coupled numerical study of the collapse and evolution of a spheri- normalized to the͔ background proper density ␬/m.Intheabove,| | Einstein-Klein-Gordon equations. For simplicity, we deal The third solution to the problem is to treat dark matter cally symmetric scalar object in the Newtonian regime. Here, 1. INTRODUCTION 2. THEORY 8 only with the spherically symmetric case for a single scalar as an extremely light bosonic dark matter (ELBDM) or fuzzy we develop a numerical2 strategy to evolve the SN system 3H0 fluctuation. Then, we describe how the EKG equations can Observations of low surface brightness galaxies and dwarf dark matter2.1. BoseÐEinstein (Press & Ryden Condensate1990;Sin1994;Huetal.2000). and study␬0 important͑m issuesmn like0 the stability and(6) the forma- ͒ 8͓G = be solved to obtain regular and asymptotically flat solutions, galaxies indicate that the cores of galactic halos have shallow Axion has been thought to be a candidate of light bosonic tion process of gravitationally bound scalar systems, a topic which are known in the literature as boson stars and oscilla- ABoseÐEinsteincondensate(BEC)isastateofbosonscooled density profiles (Dalcanton et al. 1997; Swaters et al. 2000) dark matter. But for the light dark matter to erase the singularis the backgroundthat mass has recently density become of the universe. attractive in cosmology ⌽1,2,5–9͓. tons. Some comments on the stability of these relativistic to a temperature below the critical temperature. BEC happens instead of central cusps predicted by cold dark matter (CDM; galactic core and suppresses low-mass halos, the particle massTo explore theA nature summary of the of the ELBDM, paper is we as firstfollows. adopt In Sec. the II, we solutions are also given. after a phase transition where a large fraction of22 particles hydrodynamicalpresent description the relativistic to investigate EKG and its linear Newtonian evolution. SN equations After that, we obtain the Newtonian limit of the EKG Navarro et al. 1997). In addition, the number density of must be far smaller than axion (m 10͒ eV), so low that condense into the ground state, at which point quantum⌽ effects, This approach isthat not describe only more the evolution intuitive of than a self-gravitating the wave function scalar field in system through a post-Newtonian procedure, which yields dwarf galaxies in Local Group turns out to be an order ofsuch as interference,the uncertainty become principle apparent operates on a macroscopic on the astronomical scale. length the spherically symmetric case. Correspondingly, it is de- the so called Schro¨dinger-Newton equations. The Newtonian magnitude fewer than that produced by CDM simulations scale. Much like axions, the ELBDM is in a Bose–Einsteindescription, its advantage will also become apparent later. Let The critical temperature for a gas consisting of noninteracting the wave functionscribed be how the EKG and the SN equations are solved in limit is quite interesting by itself because many physical (Klypin et al. 1999). These two features cast doubt on the validityrelativisticcondense particles state is given produced by (Burakovsky in the early & universe. Horwitz 1996 These) extremely order to obtain regular and asymptotically flat solutions. quantities can be defined and measured, quantities that are These solutions are known as boson stars and oscillatons, of standard CDM. There have been at least three different light particles share the common ground state and is described n i S useful to describe the system in a detailed manner. Moreover, 1/2 respectively,͑ for complexe hø , and real scalar fields. However,(7) we solutions proposed to resolve these problems: (1) warm dark by a single coherent wavench function. Its de-Broglie wavelength the SN system obeys a scale transformation such that the T , (1) = n0 matter, (2) collisional dark matter, and (3) fuzzy dark matter. is comparable toc or even somewhat smaller than the Jean’s focus our attention␣ on their corresponding weak field limit, study of all the possible equilibrium configurations is re- ͑ 3m SN system, and its properties. length (Davies & Widrow͑ ͒1997), where the quantum fluctuationwhere n na3,thecomovingnumberdensityandn is duced to the study of properly sized profiles, and this in- Warm dark matter can suppress small-scale structures by free =¯ In Sec. III, we present an appropriate numerical¯ code to cludes the evolution process too. streaming. It seems to be able to both solve the overabundancewhere theprovides Boltzmann’s effective constant pressure and against speed of self-gravity. light have been the proper number density. The quadrature of Schrodinger’s¬ set to unity. Given the extremely low particle mass assumed evolve the SN system, providing details about the algorithms The SN system can be solved to find stationary solutions problem of dwarf galaxies and the singular core problem. In Several previous works have pondered on such an ideaequation or can beused. split The into issues real concerning and imaginary the physical parts, boundary which condi- here, T is derived from the relativistic BoseÐEinstein particleÐ that we shall call Newtonian equilibrium configurations. this model, the flat core is embedded within a radius a couple of c its variance (Sin 1994;Huetal.2000;Siddhartha&Urbecomeena-´ the equations of acceleration and density separately, These are called Newtonian boson stars and Newtonian os- antiparticle distribution with the chemical potential set to cillatons, following the relativistic nomenclature. Although percents of the virial radius (Jing 2001; Colins et al. 2008), and Lopez´ 2003), in which the wave mechanics is described by the 2 2 particle mass m.Here,the“charge”densitynch n+ n , ͒ 1 *Current address.V hø ⌿n Newtonian oscillatons are described by a larger set of equa- the core smoothly connects to the Navarro–Frenk–White (NFW) Schrodinger–Poisson¨ equation with Newtonian͒ ⌽ gravity⌽ or by the v + v v + ͔ ͔ 0(8) where n+ and n are the number densities of particles and 2 2 2 profile (Navarro et al. 1997)outside.However,thismodification Klein–Gordon⌽ equation with gravity. The Schrodinger–Poisson¨ ͒t a · ͔ m ⌽ 2m a ͔ ⌿n = antiparticles in excited states. On the other hand, we have 0556-2821/2004/69͑12͒/124033͑␭18͒/$22.50␬ 69 124033-1 ©2004 The American Physical Society may generally adversely affect structures of somewhat larger system addresses the scale-free regimen+ 1 of/2 quantum mechanics, ͒n 1 nch (m/T )n+,anditfollowsthatTc 3T .Notethat + (nv) 0, (9) scales (Hu et al. 2000), despite that fine tuning of the thermal ͑ where the3 Jean’s length1 is a dynamical͑ running parameter. On the ͒t a2 ͔ · = velocity of dark matter particles may still be able to have then+ scalesother as a hand,⌽ and theT Klein–Gordonas a⌽ ,anditfollows system makesTc scales use of as the Comp- a 1.ItmeansthatwhenT is below T at⌽ some͓ time after a where v S/m is the fluid velocity. There is a new term larger scale structures consistent with observations (Abazajian⌽ ton wavelength as a natural lengthc scale to create the flat core in ͔͒ phase transition, the temperature will remain subcritical in any depending on the third-order spatial derivative of the wave ahalo.WidrowandKaiserconductedsimulationsforthetwo- 2006;Vieletal.2008). later epoch. As an estimate, if we assume 1% of ELBDM to amplitude ⌿n in the otherwise cold-fluid force equation. This For collisional dark matter, the halo core can be flattened andbe in thedimensional excited states Schr afterodinger–Poisson¨ its decoupling, the system current to critical approximateterm the results from the “quantum stress” that acts against gravity, dwarf galaxies destroyed, and N-body simulations confirm thistemperaturestandard becomes collisionless cold dark matter (Widrow & Kaiser 1993and). it can be cast into a stress tensor in the energy and conjecture (Spergel & Steinhardt 2000). But simulations also In the two-dimensional case, the 1/r gravitational potentialmomentum is conservation equation (Chiueh 1998, 2000). The show that very frequent collisions can yield even more singular replaced by log(r), and the1/2 two-dimensional1/2 force law inquantum their stress becomes effective only when the spatial gradient 14 m ⌽ T ⌽ simulationTc 3 becomes10⌽ of longer range thaneV. it actually(2) is inof three the structure is sufficiently large. cores than the standard collisionless CDM does (Yoshida et al. = ͓ eV eV 2000). This opposite behavior is indicative of the requirement dimensions. Due to͑ the͒ lack of␭ three-dimensional␬ numerical sim-The fluid equations, Equations (5), (8), and (9), are linearized of fine tuning for collisional parameters. Substitutingulations,m some10 22 authorseV and resortT to10 spherical4 eV, the symmetry same as (Sin 1994and combined; to yield ͑ ⌽ ͑ ⌽ the present photon temperature, we find that the current critical ͒ ͒ 3H 2͑ hø 2 850 a2 ␭n 0 m ␭n + 2 2␭n 0. (10) temperature Tc 0.3eV T .HenceELBDM,ifexistsand 2 2 accounts for the= dark matter,␭ may very well be in the BEC ͒t ͒t ⌽ 2a 4m a ͔ ͔ = state ever since a phase transition in the early universe. Despite Upon spatially Fourier transforming ␭n, it follows

ELBDM particles in the excited state are with a relativistic 2 2 4 temperature, almost all particles are in the ground state and d 2 dnk 3H0 ͑m hø k a nk + nk 0, (11) described by a single nonrelativistic wave function. dt dt ⌽ 2a 4m2a2 = ␭ ␬ Structure formation SOLITON 3 4

gravitational equilibrium. More importantly, as shown in Fig. 4, the core mass follows the relation

͑ ͑ ͑ 1/2 Mc = ⌽(|E |/M ) . (5) Here the total kinetic energy, potential energy and mass are defined in the primed (redshift-independent) coor- ͑ 1 ͑ ͑ 2 3 ͑ ͑ 1 ͑ 2 ͑ 3 ͑ dinates as Ek ͒ 2 |⌽ ͒ | d x , Ep ͒ 2 |͒ | V d x , M ͑ ͒ |͒͑|2d3x͑,and͑ ⌽ is a dimensionless constant͑ close to unity.͑ The physical foundation of this relation can be appreciated as follows. The RHS represents the halo ve- ͑ locity dispersion, ͓h,andontheLHSthe͑ scaling de- ͒ mands that M ͑ ͑ x͑ 1, the inverse soliton size. Accord- Figure 1: Comparison of cosmological large-scale structures formed by standard CDM and by wave- c c like dark matter, ψDM. Panel (a)showsthestructurecreatedbyevolvingasinglecoherentwave function ingly, Eq. (5) relates the soliton size to the halo veloc- for ΛψDM calculated on AMR grids. Panel (b)isthestructuresimulatedwithastandardΛCDM N-body ity dispersion through the uncertainty principle, where 12 code GADGET-2 for the same cosmological parameters, with the high-k modes of the linear power spec- x͑ ͓͑ ͑ 1. This result is non-trivial in that the uncer- trum intentionally suppressed in a way similar to the ψDM model to highlight the comparison of large-scale c h features. This comparison clearly demonstrates that the large scale distribution of filaments and voids is in- tainty principle is originally a local relation, but here it is distinguishable between these two completely different calculations, as desired given the success of ΛCDM found to hold non-locally, relating a core (local) property in describing the observed large scale structure. ψDM arises from the low momentum state of the conden- to a halo (global) property. The non-local uncertainty sate so that it is equivalent to collisionless CDM well above the Jeans scale. principle reveals itself in panel (d) of Fig. 3. The inverse haloFIG. velocity 1: Density dispersion profiles is manifested of ͑DM halos. by the Dashed size of lines halo with FIG. 2: Core-halo mass relation. Di͑erent filled symbols show CDM, including the surprising uniformity of their CDM, no sufficiently high resolution simulations of FIG. 3: Snapshots of a soliton collision simulation. Panels density granules, and the fact that the halo granule size central masses, M(< 300 pc) ͑ 107 M ,andshallow ψDM have been attempted. The wave mechanics various opened symbols show five examples at di͑erent red- halos at di͑erent epochs, while the open circles trace the evo- ͒ (a)-(c) show the projected density distribution at the initial density profiles1–4.Incontrast,galaxiespredictedby of ψDM can be described by Schr¨odinger’s equa- isshifts close between to the soliton 12 ͑ z size͑ 0. provides The DM another density perspective is normalized to lution of one halo. Dashed line shows the analytical prediction 13 and intermediate stages, and panel (d) shows a close-up of CDM extend to much lower masses, well below the tion, coupled to gravity via Poisson’s equation tothe view cosmic the finding background of Eq. density. (5). Eigenmode A distinct core decomposi- forms in ev- given by Eq. (6) (see text for details). the conspicuous solitonic core at the final stage. Fluctuating observed dwarf galaxies, with steeper, singular mass with negligible microscopic self-interaction. The dy- tionery of halo the as core-halo a gravitationally system can self-bound help our object, understanding satisfyingthe profiles5–7.AdjustmentstostandardCDMaddress-Schrödinger-Poissonnamics here differs system from collisionless particle CDM density granules resulting from the quantum wave interfer- ing these difficulties consider particle collisions16,or by a new form of stress tensor from quantum un- ence appear everywhere and have a size similar to the central ofredshift-dependent the detailed physics soliton underlying solution this (solid quantum lines) upon “ther- proper 17 warm dark matter (WDM) .WDMcanbetunedto certainty, giving rise to a comoving Jeans length, soliton. malization”,͒ scaling. Filled and it diamonds will be presented show an in example a separate from work the soli- suppress small scale structures, but does not provide 1/4 ⌽1/2 λJ ∝ (1 + z) mB ,duringthematter-dominated (Wongton collision et al., in simulations preparation). renormalized to the comoving coor- large enough flat cores18, 19.CollisionalCDMcan 15 halo configuration still persists in a di͒erent setting from Schive, Chiueh, Broadhurst, Natureepoch Physics.Theinsensitivityof 10, 496–499 (2014)λJ to redshift, z,gener- Schive et al, Phys.Rev.Lett. 113 (2014) no.26, 261302 dinates at z =0.Thesamez =8haloinaCDMsimulation be adjusted to generate flat cores, but cannot sup- ates a sharp cutoff mass below which structures are We are now in a position to understand the physical cosmological structure formation, and if so, we want to Schive et al, Phys.Rev.Lett. 113 (2014) no.26, 261302 e-Print: arXiv:1407.7762 (filled squares) fit by an NFW profile (dot-dashed line) is also press low mass galaxies without resorting to other suppressed. Cosmological simulations in this con- 9 meaning of the empirical Eq. (4). In the structure for- 20 halo configuration still persists in a di͑erent setting from ascertain what factors determine the soliton scale among baryonic physics .Betteragreementisexpectedtext turn out to be much more challenging than stan- mationshown simulations, for comparison. we verify that halos at di͑erent red- cosmological structure formation, and if so, we want to the infinite number of self-similar solutions. Intuitively, for ψDM because the uncertainty principle coun- dard N-body simulations as the highest frequency shifts all conform to Eq. (5) by taking E͑ and M ͑ as the ters gravity below a Jeans scale, simultaneously sup- oscillations, ω,givenapproximatelybythematter ascertain what factors determine the soliton scale among ͑ ͑ one expects that the final relaxed state should lose the ⌽1 ⌽2 rescaled halo energy (E ) and virial mass (M ). Adopt- pressing small scale structures and limiting the cen- wave dispersion relation, ω ∝ mB λ ,occuronthe h h 8,9 the infinite number of self-similar solutions. Intuitively, memory of its initial configuration and thus depends only tral density of collapsed haloes . smallest scales, requiring very fine temporal resolu- ing the3 virial condition in the spherical collapse model (4͑x /3)͒(z)⌽ ͑ ,wherex is the comoving virial tion even for moderate spatial resolution (see Sup- one expects that the final relaxed state should lose the ͑ vir ͑ m02 ͑ vir on the globally conserved quantities, namely, the total Detailed examination of structure formation |E | = |E |/2 ͑ 3M /10x2 and retrieving the redshift plementary Fig. S1). In this work, we optimise radiush andp ͒(z) ͑h (18͑ vir+82(͑m(z) ͒ 1) ͒ 39(͑m(z) ͒ with ψDM is therefore highly desirable, but, un- memory of its initial configuration and thus depends only 1/2 ͒1/2 mass M and energy E (assuming there is no net angular an adaptive-mesh-refinement (AMR) scheme, with dependence2 then give Mc = ⌽(3Mh/10xvir) a . Fi- like the extensive N-body investigation of standard on the globally conserved quantities, namely, the total 1) )/͑m(z) ⌽ 350 (180) at z =0(z ͓ 1) [44]. Note momentum). We conduct 29 runs in total with di͒er- mass M and energy E (assuming there is no net angular nally, solving xvir as a function of Mh using the definition 2 that this definition of virial mass is the same as that for ent initial conditions of various M and E.Forthesame momentum). We conduct 29 runs in total with di͑er- ofCDM. virial mass This given is because immediately once an after object Eq. exceeds (4) yields the the Jeans expected core-halo mass relation M and E, we repeat runs with di͒erent realizations, in- ent initial conditions of various M and E.Forthesame mass on its way to collapse, the dynamics is almost cluding di͒erent initial soliton numbers ranging from 4 M and E, we repeat runs with di͑erent realizations, in- identical to the cold1/ collapse,6 for1/ which3 the Eikonal 1 ͒1/2 ␭(z) Mh to 128, di͒erent soliton sizes and initial positions. Figure cluding di͑erent initial soliton numbers ranging from 4 Mc = a Mmin,0, (6) approximation4 ͒ of␭(0) wave⌽ dynamics͒M 0 ⌽ to particle dynamics 3 shows one example of the soliton collision simulations. to 128, di͑erent soliton sizes and initial positions. Figure min, holds until virialization takes place. Figure 2 shows this The AMR scheme is again adopted in order to achieve 3 shows one example of the soliton collision simulations. ͒1/4 1/4 ͑ ͒3/2 ͒3/4 wherescalingMmin, relation0 =375 over three32␬␭(0) orders͔m0 of(H magnitude0m͑/ ) in⌽m halo0 su⌽cient resolution everywhere; in particular, we ensure The AMR scheme is again adopted in order to achieve 7 ͒3/82 11 ͒22 ͑mass4.4͓ from10 m 10 toM⌽ 10.HereM͑m. We͒ demonstratem /10 eV the and redshift we 4 su͒cient resolution everywhere; in particular, we ensure 22 22 ͑ that every soliton is well resolved with at least ⌽ 10 cells haveevolution taken ⌽ by= showing 1 and typical coalescence values for of the the cosmological core-halo mass that every soliton is well resolved with at least ͑ 104 cells and verify that M and E remain conserved with at most parameters. Eq. (6) is consistent with Eq. (4) apart and verify that M and E remain conserved with at most relations of halos at five di͒erent epochs between a few percent error in all simulations. from an additional slowly varying factor ␭(z)1/6.The a few percent error in all simulations. 10 >z>0 as well as the evolutionary trajectory of a physicalsingle halo. core radius, The deviationrc = axc of, is the inversely core mass proportional from Eq. (4) The resulting relaxed structures that form in these soli- The resulting relaxed structures that form in these soli- toisM lessc and than can a be factor expressed of two. as ton collision experiments are always found to consist of a ton collision experiments are always found to consist of a ͒1/6 ͒1/3 halo and a solitonic core (see Fig. 1 and panel (d) of Fig. halo and a solitonic core (see Fig. 1 and panel (d) of Fig. To understand this␭(z core-halo) massMh relation, we further r =1.6 m͒1a1/2 kpc. 3), similar to the results of cosmological simulations. The conductc a set22 of controlled numerical9 experiments, where 3), similar to the results of cosmological simulations. The ͒␭(0) ⌽ ͒10 M⌽ ⌽ core profiles satisfy the ͓ scaling and the halo profiles are core profiles satisfy the ͑ scaling and the halo profiles are multiple solitons are initially placed randomly with(7) zero close to NFW. This result establishes that the core-halo close to NFW. This result establishes that the core-halo velocityThe smallest and start halo to should merge be until close the to a systems single isolated relax. Soli- configuration is a generic structure of ␭DM in virialized configuration is a generic structure of ͒DM in virialized soliton,tons are with chosen a wide as a core convenient and a steeper initial outer condition gradient. for their gravitational equilibrium. stability. Here we assume a = const. and zero back- ground density. We would like to know whether the core- More importantly, as shown in Fig. 4, the core mass Current constraints on the massiii Pozo et al, ArXiv 2010:10337

xii

2 Galaxy m Mh rt red Gap(Log) D Milky Way Pericenter 22 10 (10 eV) (10 M ) (kpc) (km/s) C H (kpc) (kpc) +0.12 +1.99 Aquarius 1.12 0.09 6.89 1.68 1.13 7.9 0.85 1.78 1071 - +0 .33 +3 .11 Cetus 1.17 0.19 4.79 2.51 1.04 8.3 1.13 1.34 775 - +0 .24 +2 .43 Tucana 1.63 0.18 6.28 2.13 0.68 15 0.96 1.5 887 - +0.13 +1 .08 Leo A 1.12 0.1 3.44 0.94 1.43 6.7 1.23 1.76 800 - +11 Sextans 1.2 2 1.28 6.9 2.05 1.36 90 71 12 +126 Phoenix 1.7 4 1.23 7.9 1.41 3.44 415 263 219 +80 Leo I 1.7 6 1.20 8.8 1.76 3.43 250 45 34 +12 Draco2001 1.5 30 0.58 9.1 0.63 2.32 80 28 7 +12 Draco2004 1.7 15 0.56 9.1 1.58 2.32 80 28 7 +21 Carina 1.8 7 0.75 6.7 1.17 2.26 101 60 16 +15 Sculptor2005 1.9 10 0.63 9.2 3.7 2.23 80 51 10 +15 Sculptor2007 1.7 9 0.74 9.2 1.69 2.18 80 51 10 +8 Ursa Minor 1.3 8 0.99 9.5 5.56 2.14 66 29 6 +121 Leo II 1.8 24 0.54 7.4 1.08 2.45 210 45 30

TABLE I. Observations and DM profile fits. Column 1: Dwarf galaxy colour coded as in figure 3a, Column 2: Boson mass m , 10 Column 3: Halo mass Mh, Column 4: Transition point rt, Column 5: Velocity dispersion , Column 6: Reduced Chi-square 2 red,Column7:GapC H , Column 8: Distances from Milky Way center & Column 9: Pericenter determined from GAIA (Fritz et al. 2018) .

Transition radius FIG. 2. Profile comparison of the classical dwarfs orbiting the Milky Way and the four isolated dwarfs lyingSOLITON beyond in the local group (all listed in Table I), with their corresponding DM based best fit profile plotted in the same colour. All profiles are normalised to the central density and rescaled by the mean transition radius of 0.75 kpc to compare the profile shapes, which are seen to form a family of core-halo profiles. In the right hand panel the amplitude of the transition “gap”, C H ,is compared with the pericenter of each dwarf galaxy orbit derived from GAIA observations (Fritz et al. 2018), or the measured distance for the four isolated dwarfs, with the virial radius of the Milky Way is indicated by the vertical band between 230 and 350 kpc (Fritz et al. 2018). The inset shows the e↵ect of tidal stripping on the profile of a representative dwarf galaxy orbiting a Milky Way sized halo, in the context of DM, where the profile is increasingly stripped over several Gyrs (Schive et al. 2020), forming a range of profiles that we compare quantitatively in Figure 3.

22 m 10 eV (Pozo et al. 2020). We begin by examining the profiles of the four well ' studied isolated dwarf galaxies in the Local Group, Ce- Here we explore a related prediction of DM that a tus, Tucana, Aquarius and Leo A, which have dwarf characteristic drop in density should be found at the galaxy velocity dispersions 10 km/s and spheroidal transition radius between the core and the halo. The old stellar populations like the' classical dwarf spheroidal soliton forms a prominent core that contrasts by over an galaxies orbiting the Milky Way. However, these dwarfs order of magnitude in density at the core radius with the have not su↵ered tidal stripping or ram pressure gas surrounding halo. The halo comprises de Broglie scale stripping with the major Local Group members thought waves with an approximately NFW form when averaged to a↵ect most dwarf galaxies bound to the Milky Way azimuthally, reflecting the cold, non-relativistic nature of and Andromeda, with ongoing tidal disruption clearly the condensate on scales exceeding the de Broglie wave- recognised for the Sagittarius dwarf (Ibata et al. 2001) length (Schive et al. 2016). Even in projection this tran- and the Magellanic clouds. This simplicity has motivated sition is predicted to be distinct because the soliton core deep imaging of these isolated galaxies and they are close is close to Gaussian in form, so its sharp 3D boundary is enough to detect individual stars with the aim of mea- preserved in 2D at the same radius. It is important to ap- suring intrinsic dwarf galaxies profiles with individually preciate this prominent core is quite unlike the behaviour detected stars so the entire profile may be traced to its of smooth cores employed in modelling dark halos, where extremities, free of the usual surface brightness limita- the core is continuous in density with the halo. In con- tions. These star counts have established that the pro- trast, the DM core is a stable standing wave that is a files of each of these four galaxies extends to 2 kpc or gravitationally self-reinforcing (Schive et al. 2014b) with more in radius (see Figure 1), somewhat surprisingly for a pronounced overdensity of > 30 above the halo pre- 10 low mass dwarf galaxies galaxies (Gregory et al. 2019, dicted for low mass galaxies of 10 M (see Figures 1 McConnachie et al. 2006 & McConnachie et al. 2006b) & 2 of Schive et al. 2014b). ' Urena-L˜ opez´ Scalar field dark matter Urena-L˜ opez´ Scalar field dark matter Urena-L˜ opez´ Scalar field dark matter 161 Here we will make a quick review of the general features of the soliton structures that arise from SFDM 161 Here we will make a quick review of the general features of the soliton structures that arise from SFDM 161 Here162 we willmodels, make using a quickUre for reviewna-L˜ thisopez´ the of so-called the general field features and fluid of the approaches, soliton structures in order that to arise highlight from their SFDM equivalencesScalar field dark and matter 162 models, using for this the so-called field and fluid approaches, in order to highlight their equivalences and 162 models,163 usingdifferences for this the and so-called their use field in and cosmological fluid approaches, and astrophysical in order to highlight settings. their equivalences and 163 differences and their use in cosmological and astrophysical settings. 163 differences and their161 use inHere cosmological we will make and astrophysical a quick review settings. of the general features of the soliton structures that arise from SFDM 164 3.1 The field approach 164162 3.1models, The using field for approach this the so-called field and fluid approaches, in order to highlight their equivalences and 164 3.1 The field approach 163 differences and their use in cosmological and astrophysical settings. 165 The field approach to the formation of structure within the SFDM paradigm refers to the solutions 165 The field approach to the formation of structure within the SFDM paradigm refers to the solutions 165 The166 field approachobtained from to the the formation Schrodinger-Poisson of structure within (SP) system, the SFDM which paradigm is represented refers by to the equations: solutions 166164 obtained3.1 The from field the approach Schrodinger-Poisson (SP) system, which is represented by the equations: 166 obtained from the Schrodinger-Poisson (SP) system, which is represented by the equations: Urena-L˜ opez´ Scalar field dark matter ~ 2 2 2 2 165 The field approachi~@t = to the formation + ~ of , structure2 =4 within⇡Gm2 thea SFDM, 2 paradigm2 refers to the(12) solutions ~ 2 i2~m@tr =2 r+2 2, =4| |⇡Gma , (12) 166 obtainedi~@t = from the Schrodinger-Poisson + , 2=4m (SP)r⇡Gm system,a which, r is represented| by| (12) the equations: 161 Here we will make a quick review of the general2mr features of ther soliton structures| | that arise from SFDM 167 where 167is thewhere wave functionis the wave that function collectively that collectively represents represents the boson the particles boson in particles their ground in their state, ground and state, is and is 162 167models, using for this the so-called field and fluid approaches, in order~ to highlight2 their equivalences2 and2 2 where 168is thethe wave Newtonian168 functionthe gravitational Newtonian that collectively gravitational potential. represents Noticei~ potential.@t the= that boson Notice the particles latter that+ is the calculated in latter, their is ground calculated from=4⇡ state, theGm from massa and the density, is mass effectively density effectively(12) 163 differences and their use in cosmological2 and2 astrophysical settings.2mr r | | 168 the Newtonian169 defined gravitationalUre169 throughna-L˜ definedopez´ potential.⇢ = throughm Notice⇢. For= m that the2 the purposes2. latter For the is of purposes calculated simplicity, of from simplicity, we dothe not mass we include density do not any include effectively self-interaction any self-interactionScalar term field in term dark in matter 2 2 a a 169 defined through ⇢ =167m where. For theis the| purposes| wave function of| simplicity,| that collectively we do not includerepresents any the self-interaction boson particles term in their in ground state, and is 164 3.1 The170 fieldthe approach SP170 system,a|the| which SP system, would which then would become then the become Gross-Pitaeevski-Poisson the Gross-Pitaeevski-Poisson system. For system. a discussion For a discussion of the of the 170 the SP171 system,latter which and171168 would itslatterthe general Newtonian then and properties become its general gravitational the at properties Gross-Pitaeevski-Poisson cosmological potential. at cosmological and Notice astrophysical that and system. the astrophysical latter scales, For is calculated a discussion see scales, (Diez-Tejedor from see of the (Diez-Tejedor the mass et al., density 2014; et effectively al., 2014; 155 According to the detailed2 analisys2 made in (Schive et al., 2014b; Mocz et al., 2017a; Schwabe et al., 165 171Thelatter field and172 approach itsBarranco general to172169 the properties and formationBarrancodefined Bernal, at through and cosmological 2011; of structure Bernal,⇢ Li= etm 2011; al.,a within and 2013). Li. astrophysical For the et al., the SFDM 2013). purposes paradigm scales, of simplicity, see refers (Diez-Tejedor to we the do solutions not et include al., 2014; any self-interaction term in 156 | | 166 172obtainedBarranco from and the Bernal, Schrodinger-Poisson2016),170 2011;the the Li SP gravitationally et system, al., (SP) 2013). system, which would which bounded then is represented become objects the that by Gross-Pitaeevski-Poisson the one equations: could identify with system. DM For galaxy a discussion halos of have the all a 173 157Thecommon SP173171 systemlatterThe structure: posses SP and system its the general one following posses central properties the scaling following soliton at cosmological symmetry, surrounded scaling symmetry, and by astrophysical a Navarro-Frenk-White-like scales, see (Diez-Tejedor envelope et al., created 2014; by the 173 The SP system posses the following scaling symmetry, 158 interference172 Barranco of~ the and Schrodinger2 Bernal, 2011; Li wave2 et al., functions. 2013).2 2 In terms of the standard nomenclature, the SFDM model i~@t = + , =4⇡Gma , 2 1 2 2(12) 2mr rt, x, , 2|ˆ| 1 t,ˆ2 ˆ xˆ2,ˆ ˆ, ˆ , (13) 159 belongs to the so-calledt, x non-cusp,, , 2 or cored,1 2 typest, 2 ofxˆ DM, models., , Although not yet completely(13) conclusive, 173 Thet, x SP, system, { posses the{} followingt,ˆ ! xˆ, scaling} ˆ!, symmetry,ˆ , (13) 167 where is the wave160 functionthere seems that{ collectively to be an} represents increasing! the evidence boson particlesn in favor in theirn of ground cored state, DMo models and iso (Rodrigues et al., 2018; Walker 174 for an arbitrary value of then scaling parameter . Ito is then common to choose a particular configuration 168 the Newtonian174 gravitationalfor an arbitrary potential. value Notice of the that scaling the latter parameter is calculated. It is from then the common mass2ˆ density to1 choose effectively2 ˆ a particular2 ˆ configuration 174 for an arbitrary161 valueand175 of Peand thenarrubia,˜ scaling then find 2011), parameter any others and. then by It ais propert, then thex, non-cusp common, scaling of to nature the choose different of at, SFDM particular quantitiesxˆ, objects configuration, to ease remain, the numerical one of the effort. most(13) promising 169 defined through175 ⇢and= m then2 find2. For any the others purposes by a of proper simplicity, scaling{ we of do the not} different include! any quantities self-interaction to ease the term numerical in effort. 175 and then find162 anyfeatures othersa| | by of a proper the model. scaling of the different quantities to easen the numerical effort.o 170 the SP system, which would176174 3.2 thenfor an become The arbitrary fluid the value approach Gross-Pitaeevski-Poisson of the scaling parameter system.. It isFor then a discussion common to of choose the a particular configuration 171 176 3.2 The fluid approach 176latter3.2 and The its general163 fluid propertiesapproachHere175 and we at then will cosmological find make any a others quick and astrophysical by review a proper of scaling the scales, general of see the (Diez-Tejedor features different quantities of the et al., soliton 2014; to ease structures the numerical that effort. arise from SFDM 172 Barranco and Bernal, 2011;177 LiThere et al., 2013). is also a fluid equivalent of the SP system (12), for which we only need to apply the so-called 177 164Theremodels, is also using a fluid for equivalent this the so-called of the SP field system and(12) fluid, for approaches, which we only in order need to to apply highlight the so-called their equivalences and 177 There is also a fluid178176 equivalentMadelung3.2 The of transformation fluid the SP approach system of(12) the, for wave which function we only by means need to of apply (t, x the)= so-called'(t, x)exp( iS(t, x)/~). The 173 The SP system178 165Madelung possesdifferences the following transformation and scaling their of usesymmetry, the in wave cosmological function by and means astrophysical of (t, x)= settings.'(t, x)exp( iS(t, x)/~). The 178 Madelung transformation179 resultant of the equationswave function of motion by means are then of (t, x)='(t, x)exp( iS(t, x)/~). The 179 179 resultant equationsresultant of177 equations motionThere are of then is motion also a fluid are then equivalent of the SP system (12), for which we only need to apply the so-called 166 3.1 The field approach2 1 2 ˆ 2 ˆ 178 t, x, , 1 t,ˆ xˆ, , , 1 (t, x)=(13)1 '(2t,'x)exp( iS(t, x)/ ) Madelung transformation of2 the wave function by˙ means of 2 2 ~ . The { } 1 ! '˙ = ' S + ' S, 1 ' S2+ ( S)1 + ' r =0. (14) 179 1resultant equations2 ofn motion2 r are1 thenr · r˙ o 1 '2 2 r 2 ' 167 'The˙ = field' 2S approach'˙+= '' S, toS + the' ' formationS˙ + S,( S') of2 +S structure+ ( rS) within+=0 . ther SFDM=0 paradigm.(14) refers(14) to the solutions 174 for an arbitrary value of the2 scaling parameter2 Simulations:r . It isr then· r common2 toMadelung choose2 2r a particular' transformation configuration2 ' r r · r r  2 175 and then find any168 othersobtained180 by aNotice, proper from scaling under the Schrodinger-Poisson the of the assumption different1 quantities that ' (SP)=0 toFrom, ease that system, w the theave numerical scalar which to 1fluid function is effort. representedS obeys1 a' by Hamilton-Jacobi the equations: type of '˙ = ' 2S + '6 S, ' S˙ + ( S)2 + r =0. (14) 180 Notice,180 underNotice, the assumption181 underequation the that assumptionS˙ '+ =0H( ,S, that thatx,t the2')=0r=0 scalar,, in that functionwhichr the· r the scalarS potentialobeys function a part Hamilton-Jacobi2 Sr inobeys the Hamiltonian a Hamilton-Jacobi2 type' ofH is represented type of by two ˙ 6 r 6  176 1813.2equation The181 fluidS˙ +equation approachH( 182S, Sx,t+terms:)=0H( ,S, in+x whichQ,t,)=0 where the, inQ potentialis which the so-called part the potential in~ the quantum Hamiltonian2 part potential, in theH Hamiltonianis2 representedH is by2 represented two2 by two r i~@t = + , =4⇡Gma , (12) 182 r180+ QNotice, underQ the assumption that ' =0, that the scalar function S obeys a Hamilton-Jacobi type of 182 terms: + Qterms:, where Q is, the where so-calledis the quantum so-called potential, quantum2m potential,6 a r r | | 177 There is also a fluid equivalent181 equation of theS˙ SP+ H system( S, x(12),t)=0, for, which in which we the only potential need1 to2 part apply' in the the so-called Hamiltonian H is represented by two r Q2 = r . (15) 178 Madelung transformation182 of the wave + functionQ byQ means of2 (t, x)=1 '('t, x)exp( iS(t, x)/~). The 169 where terms:is the wave, where functionis that the1 so-called collectively' quantum represents potential,2 ' the boson particles in their ground state, and is 179 resultant equations of motion are then Q = r Q. = r . (15) (15) 170 the Newtonian gravitational potential.2 ' Notice2 that' the latter2 is calculated from the mass density effectively 2 2 1 ' 171 defined183 throughTaking the⇢ assumption= m .' For=0 thefor purposes granted, oneQ of2= can simplicity, furtherr . manipulate we do not Eqs. include(14) to any write self-interaction them down(15) as term in 1 2 a 1 2 1 ' '˙ = ' S + ' S, '| S˙| + 6 ( S) + r =02 '. (14) 2 2 2 2 183 183 172 the184 SPfluid' system,=0 equations. which' =0 We would define then the velocity become field thev Gross-Pitaeevski-Poisson= S and the mass density system.⇢ = ma For = a discussionma' , and of the Taking the assumptionTaking2 ther assumptionrfor· r granted, onefor can granted,2 furtherr one manipulate can further2 ' Eqs.r manipulate(14) to write Eqs. them(14) downto write as them| down| as 185 then6 the SP system6 becomes the Quantum Euler-Poisson (QEP)2 system2 (for2 22 details2 see2 (Chavanis,2 2011; 184 fluid equations.184 173fluid Welatter equations. define and the its We velocity general define field the properties velocityv = field atS and cosmologicalv the= massS and density and the astrophysical⇢ mass= m densitya =⇢ scales,m=a'm,a and see= (Diez-Tejedorma' , and et al., 2014; 180 Notice, under the assumption186183 Uhlemann thatTaking' =0 the et al.,, assumption that 2014; the Wallstrom, scalarr' =0 functionfor 1994)) granted,Srobeys one a can Hamilton-Jacobi further manipulate| | type Eqs. of |(14)| to write them down as 185 then the185 SP174 systemthenBarranco the becomes SP system and the becomes Quantum Bernal,6 2011; the Euler-Poisson Quantum Li6 et al., Euler-Poisson (QEP) 2013). system (QEP) (for details system see (for (Chavanis, details see 2011; (Chavanis,2 2 2011;2 2 181 S˙ + H( S, x,t184)=0fluid equations. We define the velocity field v = H S and the mass density ⇢ = ma = ma' , and 186equationUhlemann186 etUhlemann al., 2014; Wallstrom, et al.,, in 2014; which 1994)) Wallstrom, the potential 1994)) part in the Hamiltonian ris represented by two | | r 185 then the SP system@⇢ becomes the Quantum@v Euler-Poisson (QEP) system (for2 details see (Chavanis, 2011; 182 terms: + Q, where175 QTheis the SP so-called system quantum posses+ potential, the following(⇢v)=0, scaling+( symmetry,v )v = Q, =4⇡G⇢ . (16) 186 Uhlemann et al.,@t 2014;r Wallstrom,· 1994))@t · r 11 r r r @⇢ @⇢ @v @v 2 2 + (⇢v)=0+ , (⇢v)=0+(v1, 2)'v =+(v )v Q,= =4Q,⇡G⇢ . =4⇡G(16)⇢ . (16) @t r · @t @t · r @t r r r 2 1 2 2 This is a provisionalr · Q@⇢= file,r nott, x the,. final, typeset· r@v articler rt,ˆ xˆr, ˆ,(15) ˆ , 6 (13) +2 ('⇢v)=0, +(v )v = Q, 2 =4⇡G⇢ . (16) @t r {· } @t ! · r r r r This is a provisionalThis is a file, provisional not the final file, typeset not the article final typeset article n o6 6 176 for an arbitrary value of the scaling parameter . It is then common to choose a particular configuration 183 Taking the assumption ' =0Thisfor is granted, a provisional one can file, further not the manipulate final typeset Eqs. article(14) to write them down as 6 6 184 fluid equations.177 We defineand then the velocity find any field othersv = by aS properand the scaling mass density of the⇢ different= m2 2 quantities= m2'2, and to ease the numerical effort. r a| | a 185 then the SP system becomes the Quantum Euler-Poisson (QEP) system (for details see (Chavanis, 2011; 178 186 Uhlemann et al., 2014;3.2 Wallstrom, The fluid 1994)) approach 179@⇢ There is also@ av fluid equivalent of the SP system (12), for which we only need to apply the so-called + (⇢v)=0, +(v )v = Q, 2 =4⇡G⇢ . (16) 180@t Madelungr · transformation@t · r of ther waver functionr by means of (t, x)='(t, x)exp( iS(t, x)/~). The 181 resultant equations of motion are then This is a provisional file, not the final typeset article 6 1 1 1 2' '˙ = ' 2S + ' S, ' S˙ + ( S)2 + r =0. (14) 2 r r · r 2 r 2 '  182 Notice, under the assumption that ' =0, that the scalar function S obeys a Hamilton-Jacobi type of 6 183 equation S˙ + H( S, x,t)=0, in which the potential part in the Hamiltonian H is represented by two r 184 terms: + Q, where Q is the so-called quantum potential,

1 2' Q = r . (15) 2 '

This is a provisional file, not the final typeset article 6 Lyman-– and LSS properties in FDM cosmologies 15 AX-GADGET: a new code for FuzzySimulations Dark Matter models 13

around a as expected, since the eectiveness of the repulsive CIC CIC+QP FDM noQP force induced by the QP in tilting the density distribution FDM decreases as its typical scale becomes a smaller fraction of the size of the considered objects. In fact, stacked density profiles of the most massive haloes are very similar in the two binning strategies, being R a and sphericity constant 200 ≥ among the various models, and consistent with no major deviation from CDM, except for a central over-density. It

Probability distribution is our opinion, however, that such feature in the very centre of most massive haloes could be a numerical artefact, since

3 2 1 its extension is comparable with the spatial resolution used. FIC FIC+QP 10 10 10 22 1/m [10 /eV] The results presented in this Section have been obtained through the detailed analysis of the statistical properties of z Figure 6. Here we plot the marginalised posterior distribution haloes found at =0 in the FDM simulations. The same z of 1/m‰ from both the analyses performed by Iröi et al. (2017a) analysis, repeated at =0, of the FDMnoQP simulations (green lines, without QP) and ours (red lines, with QP). The shows very similar results which are, therefore, not shown in vertical lines stand for the 2‡ C.L. limits. the present work. Such consistency suggests that the proper- ties of haloes at low redshift are – at the investigated scales – −21 2 not sensible to modifications induced by the dynamical QP ma ≳ 2 × 10 eV/c (2σ CL) The sphericity distributions confirm that, in the mass repulsive eect, which are expected to appear more promi- Figure 9. Density distribution of four simulations starting from standard initial conditions (top)orsuppressedwithAxionCAMB Nori et al, MNRAS 482 (2019) 3227, arXiv:1809.09619 (bottom )andevolvedwith(right)orwithout(left)QuantumPotentiale↵ects. range considered, there is no statistical deviation from nently at scales of 1Kpc with the formation solitonic cores. 12 Nori, Baldi, MNRAS 478 (2018) 3935, arXiv:1801.08144 ≥ The SPH implementation that computes the QP and the beginning of the simulation in the case when the QP CDM, except for a mild deviation towards less spherical its contribution to particle acceleration with three cycles on is included. This is due to extra work –needed to compute all the FDM particles has been built analogously to the ex- the QP– arising from the reaction to the out of equilibrium configurations of the less massive haloes, especially in the tremely optimized baryonic one, in order to spread the com- configuration provided in the initial conditions. As one can m22 =2.5 model. This is consistent with the analysis of the putation and memory load across the CPUs. Therefore, AX- see from the figure, this overhead only weakly grows during 5CONCLUSIONS GADGET should perform in a similar way to a hydrodynami- the remainder of the simulation up to a total factor of 5 at ⇠ sphericity distributions of the genuine samples (see lower cal simulation with no CDM particles. As a consequence, the z = 3. When FDM initial conditions are used (right panel), overhead compared to a collisionless CDM-only simulation the overburden is indeed less pronounced in the early phases panels in Fig. 1) that reveals that haloes appear to be sta- We have presented the results obtained from two sets of nu- is still significant, but definitely much weaker compared to of the evolution whereas the final computational time is a grid-based FDM full-wave solvers (such as e.g. Schive et al. factor 5 larger than the case without QP also in this case. tistically less spherical with respect to CDM at z =0when merical simulations performed with AX-GADGET , an exten- ⇠ 2010). We find that the major contribution to CPU time is lower FDM masses are considered, down to a maximum of sion of the massively parallel N-body code P-GADGET3 for In Fig. 10 we show the CPU time and the overhead for the one associated to the imbalance between CPUs in the the CIC, CIC+QP, FIC and FIC+QP simulations, paired SPH calculation –namely SPH density imbalance and SPH 10% decrease in sphericity for m22 =2.5 and halo mass of non-linear simulations of Fuzzy Dark Matter (FDM) cos- with respect to initial conditions to highlight the additional acceleration imbalance–, while the time spent for the bare ≥ 9 computational load of the QP computation. Contributions of SPH calculation – SPH density and SPH acceleration –make 5 10 M . mologies, regarding Lyman-– forest observations and the the routines of gravity solver are presented, along with SPH up for less than 20% of the total time of the simulation. ≥ ◊ § routines devolved to the bare computation of the density For all the FDM models the volume occupied by the statistical detailed characterization of the Large Scale Struc- Therefore, we conclude that given the relatively low derivatives, the QP acceleration acting on particles and the overhead obtained for simulations starting from suppressed respective imbalance between CPUs. haloes is systematically larger, consistently with a delayed tures. initial conditions, the inclusion of the QP in the dynamics As we can see, starting from CDM initial conditions (left implemented in AX-GADGET – as would be required from dynamical collapse of the haloes. All mass ranges show such More specifically, our main aim was to design a set of panel) results in an overhead of a factor of 3 right from theory – does not a↵ect dramatically the performance and ⇠ property and it is emphasized by lower m22 mass – i.e. simulations covering the typical scales and redshifts involved MNRAS 000, 1–17 (2018) stronger QP force –; however, while bigger haloes occupy al- in Lyman-– forest analyses, in order to extract synthetic ob- most systematically 20% more volume for m22 =2.5,smaller servations, compare them with available Lyman-– data, and haloes can reach even twice the volume occupied by their finally to place a constraint on the mass of the FDM parti- CDM counterparts when the same model is considered. cle. In the literature, Lyman-– forest was already used for Comparing the mass of the haloes in the various models this purpose but only in approximated set-ups, in which the with the one in CDM, it is possible to see that small haloes quantum dynamical evolution of FDM was only encoded in are less massive and big ones, on the contrary, become even the initial conditions transfer function, and neglected dur- more massive, confirming our hypothesis of mass transfer ing the simulation (Iröi et al. 2017a; Armengaud et al. 2017; from substructures towards main structures. Kobayashi et al. 2017), while the AX-GADGET code allows The stacked density profiles provide even more insight us to drop such approximation and take into account the on the underlying dierent behaviour between the chosen non-linear eects of full FDM dynamics. mass ranges. Starting from the less massive one, the stacked The constrain the FDM mass we find is 21.08 22 ◊ profiles look very dierently if plotted using the spherical 10≠ eV, which is 3% higher with respect to what was found R200-based or the ellipsoidal a-based binning. This is due in Iröi et al. (2017a). The fact that these two bounds are to two concurrent reasons related to the properties of this similar, despite the dierent dynamical evolution considered mass range: first of all, as we said before, the sphericity in these dierent works, implies that the additional suppres- is m‰ dependent and thus it is not constant with respect sion deriving from the Quantum Potential dynamical con- to CDM, so the geometrical dierence in the bin shape tribution, at the scales and redshifts probed by Lyman-– , becomes important when dierent models are considered; is compensated by the gravitational growth of perturbations secondly, since the FDM haloes have lower mass but occupy when these enter the non-linear regime, implying also that – larger volumes, the two lengths are dierent from each other even if the QP does play a role in the Large Scale Structure – being R200 related to density and a purely to geometry – evolution – the approximation of Iröi et al. (2017a)(also so that the actual volume sampled is dierent. Nevertheless, adopted by Armengaud et al. 2017; Kobayashi et al. 2017) it is possible to see that in FDM models there is an excess of is valid and sucient at these scales. mass in the outskirts of the halo – seemingly peaking exactly Secondly, we studied in detail the statistical properties at distance a – and less mass in the centre. of the Large Scale Structures through the analysis of the The intermediate mass range shows also a suppression aggregated data on haloes regarding their mass, volumes in the innermost regions but a less pronounced over-density and shapes, as well as their individual inner structure.

MNRAS 000, 1–19 (2018) Axion simulations via modified gravity ∂ρ C

I + ∇ ⋅ (ρv) = 0 C ∂t ∂v + (v ⋅ ∇)v = − ∇(Φ + Q) ∂t C I F 2 1 ∇ ρ ∇2(Φ + Q) = 4πGρ − ∇2 2 2ma ρ P Q + C I F Modifed matter component + Standard gravity = Standard CDM component + Modifed gravity C I

E MG-PICOLA https://github.com/HAWinther/MG-PICOLA-PUBLIC

13 Medellín-González, González-Morales, U-L, PRD in press (2021), arXiv:2010.13998 STEFANY G. MEDELLÍN-GONZÁLEZ et al. PHYS. REV. D XX, 000000 (XXXX)

TABLE I. Summary of some of the existing FDM cosmological simulations. From left to right, the columns refer to: the reference in the literature; the mass of the boson particle ma; the numerical method used for the simulation, where SP stands for the Schrodinger- Axion simulations viaPoisson system modi and SPH for smooth particlefied hydrodynamics; gravity the box size of the simulation and the number of particles Npart.

2 Ref. ma (eV=c ) Method Box size (Mpc=h) Npart May & Springel (2021) [28] 3. ͑ 10͑23, 7 ͑ 10͑23 SP 10, 5 86403, 43203, 30723, 20483 Schwabe et al. (2020) [19] 10͑22 SP 2h 10243 ͑22 ͑23 3 C Li et al. (2019) [29] 10 , 10 SP vs SPH 20, 10, 3 1024 I ͑22 ͑21 3 C Mocz et al. (2018) [20] 10 , 10 SP vs SPH 0.250 1024 Nori et al. (2018) [21] 10͑22 SPH 10, 15 2563, 5123 NONLINEAR COSMOLOGICAL STRUCTUREZhang et al. WITH(2018) [30] ULTRALIGHT10…͑22 PHYS. REV.SPH D XX, 000000 0.400 (XXXX) 106 Woo & Chiueh (2009) [23] 10͑22 SP 1 10243

300 that the initial conditions at zi are not the same for all TABLE II. Parameter specification for the cosmological sim- 70 calculations will be the transformation of the axion field and ⌽ is a complex scalar field endowed with a potential 108 301 scales, but that the initial amplitudes of D z ;k and ulations described in the text. The Nyquist wavenumber for this 2 71 equations1 i into their hydrodynamic counterpart in terms of that contains quadratic and quartic interactions terms. 109 ͑ ͒ setup is given by k 214 h=Mpc. The initial redshift for the 302 D2 zi;k would be artificially deformed72 andthe may well-known not be Madelung transformation. Nyquist ͒ The parameters in the potential are explicitly given by 110 ͑ ͒ 73 A summary of the paperCIC, is FIC as follows. and FIC In Sec.QPII iswez 30, while for the˜ EIC QP is C 303 the correct ones corresponding to the scale-dependent m˜ a mac=͑ and ͓ ͓= ͑c , where ͑ is the Planck’s 111 I ͓ ⌽ ͑ ͑ ͓ ͓␭ F 74 present the main equationsz of100 motion. See for the axion text models for a in discussion on the later case. 304 model under consideration. To minimize the deformation constant, ma is the boson mass and ͓ is a dimensionless 112 75 the nonrelativistic limit,⌽ and correspondingly their final parameter. Notice that the fundamental constants will be 113 305 of the initial conditions, we have instead normalize the 76 form in terms of hydrodynamic quantities. We willInitial also shownma explicitly, and thatBox the sizepotential parameters become 114 306 77 explain the expected growth of linear density perturbations 2 growth factors with respect to their values at large scales, Type Model conditions the(eV bare=c ones) ifN wepart choose(Mpc to use=h) natural units c ͑ 1, 115 new 78 andnew the corresponding appearance of two characteristic ˜ 3 ͑ ͑ 307 i.e., D z; k D1 z; k =D1 0;k͑ 0 and D z; k i.e., m˜ a ma and ͓ ͓. Notice that the Compton length 116 1 79 length2 scales (and theirCIC correspondingCDM wave numbers) CDM ͑ 10243͑ 15 ͑ ͒⌽ ͑ ͒ ͑ ͒ ͑ ͒⌽ ͑ of the boson particle is just LC 1=m˜ a, and correspond- 117 308 D2 z; k =D2 0;k͑ 0 . As our model80 has therelated same to the per- physical parameters in the model. From here ␭␭␭͑23 3 ͑ ͑ ͒ ͑ ͒ FIC ͑CDM FDM 3ingly͑ 10 the Compton1024 time is defined15 as TC LC=c, which is 118 81 we write the growth factors that describe the evolution of 23 3 ͑ P 309 turbations at large scales as CDM, the normalization factors FIC QP ͑FDM FDM 3the͑ 10 one͑ used1024 in the field15 transformation that leads 119 Q 82 density perturbations up to the͓ second order in Lagrangian 23 3 120 + 310 are also the same obtained from a CDM simulation, that is, EIC QP ͑FDM EFDM 3to͑ Eqs.10͑(2). 1024 15

C 83 dynamics, and show how their equations of motion are

I CDM CDM ͓ For the non-relativistic limit (for details see [39–44]), we 121

F 311 D1 0;k͑ 0 D 0 and D2 0;k84͑ 0modifiedD to include0 . the so-called quantum pressure that ˜ 1 2 now apply the field transformation ⌽ t; r eimact͔ t; r , 122 ͑ ͒⌽ ͑ ͒ ͑ 85 ͒⌽ ͑ ͒ 312 In this form, the initial conditions will be thearises same because for of all the wave properties of axion dark matter. where the new field ͔ will obey the͓ conditions␭͑ that͓ the␭ 123 new 86 These results will be implemented in MG-PICOLA [26] for 2 313 scales, as we readily find that D zi;k 1=D1 0;k͑ 0 order of magnitude of the field derivatives are t͔ ͒ r͔ 124 1 87 scale-dependent modelsAs (see reported also [27,36] in). previous works [21,30], it can be noticed 352͑ ͑ ͑ new ͑ ͒⌽ ͑ ͒ O ⌿2 , with ⌿ ⌽ 1. Additionally, we consider a perturbed 125 314 and D2 zi;k ͑3= 7D2 0;k͑ 0 88, andIn any Sec. III scale-we describethat the cosmological the simulations simulations FIC that and FIC͓ ␭ QPj j show less matter 353 FRW metric in the form ds2 1 2 =c2 dt2 126 ͑ ͒⌽ ͑ ͑ ͒͒89 are obtained from the theoretical framework of Sec. II, and ͓ ͑ ͑ 315 dependent power suppression is left encoded intact in accumulation than the standard2 case CIC,2 an2 effect that͑ ͓ ͒354 ␭ ͒ 90 for different cosmological setups in terms of initial con- a t 1 ͑ 2͑=c dx , where ͑ is the Newtonian gravita- 127 316 the initial MPS. tional͓ ␭͓ potential␭ and a a t is the scale factor of the 128 C 91 ditions and evolution equations,must be so attributed that we can determine to the intrinsic properties of the boson 355 I ͑ ͓ ␭ E 317 Following the labeling in [21,30] 92forthe cosmological main contributions tosystem, the nonlinear Eq. evolution(1). However, of density the differencesUniverse. Thus, between the equations the cases of motion derived356 from the 129 15 action (1) can be written in the form of the well-known 130 318 simulations, we studied the following93 cases:perturbations CIC (cold of the axionFIC models. and FIC The initialQP conditions are difficult to spot by eye in the plots. 357 5 fa = 4.7 × 10 GeV Gross-Pitaevskii-Poisson (GPP) system, 131 319 initial conditions), that corresponds to94 aconsider typical the CDM usual FDM suppression of͓ power at small 95 scales, but we also includeIn contrast, the possibility the for simulation the extreme EIC shows more matter aggre- 358 320 simulation; FIC (fuzzy initial conditions), that corresponds 96 case for which there is angation excess thanof power the as comparedstandard to case CIC,a3=2͔ which isc in itself a ͓˜ 359 14 t ͑ 2 ͑ 2 321 to a CDM simulation plus initial conditions97 the from CDMMedellín-González, case, FDM, just before the abrupt González-Morales, suppression. Finally, U-L,i͑ PRD͑ ͓ 3= in2 ␭press͑ (2021),2 ͔ arXiv:2010.139982 2 ͔ m˜ a͔; manifestation of the power excessa in the͑ density2m˜ aa ͒ perturba-͒ c ͒ 2m˜ a360j j 98 in Sec. IV we discuss the main results and future perspec- ͓ ␭ 322 FIC QP, for which we additionally activated the modi- tions already present in the initial conditions. 361 ͓ 99 tives of this work. 2a 323 fied-gravity functions in MG-PICOLA to take into account A more detailed analysis of the simulations was per- 362 ͓ ␭ 324 the effects from the quantum potential Q; and EIC QP 100 II. MATHEMATICAL͓ formed BACKGROUND using the PYTHON package NbodyKit [53], and 363 325 (extreme initial conditions), where extreme refers to the 364 101 The equations of motionthe main of ultralight quantity axions for in the the comparison2 between simulations 326 standard axion case and the reported excess of power in Although there are some differences between real and 102 nonrelativistic approximationis the have MPS been measured obtained before using in thecomplex particles, scalar whichfields at the is relativistic shown in level, their365 nonrelativistic 327 density perturbations [15,16] with the103 modified-gravityRefs. [37–42], here weFig. give2 a briefbelow description for different of their redshiftdynamics,z which30 is, the 5, regime 2.5 and of physical for interest366 here, is driven 328 104 derivation. Let us start with the following action, by the same⌽ set of equations. Thus, all results presented here functions activated. each simulation. apply for both the real and complex cases. 367 3 329 The main characteristics of our simulations are summa-R Under our convention the units of the different physical 4 In͒ the top2 panel2 ˜ of4 Fig. 2 we see a comparison between 368 1=2 S p͑gd x 2 ͒⌽ ⌽⌽ m˜ a ⌽ ͓ ⌽ ; 1 quantities are as follows: ⌽ ͔ energy=length , m˜ a 330 rized in Table II. We performed a series of tests͑ to assure the2͑ ͒ ͑ ͑ ͒ j j ͒ j j ͓ ␭ 1 ␬ ͔͑␬ ͔͓͑1 ␭ ␬ ͔͑ Z ͒ the simulations CIC,⌽ which islength the͑ standardand ͓˜ CDMenergy · one, length and͑ . With this369 Planck’s con- 331 convergence of the numerical results, and from them,͑͑͑͑͑͑ we stant ͑ will not␬ appear͔͓͑ explicitly in Eqs.␭ (3) below. In natural units 2 FIC QP,4 which considers both initial conditions and 370 106105 where g det g , ͑ 8␬G=c , G is Newton’s gravita- ˜ 332 selected the optimal parameters for the number of particles,͑ ͓ ͒␭␭ ͑ ͓ we find that: ⌽ energy, m˜ a energy and ͓ becomes 107 tional constant, c is the speedevolution of light, ofR theis the FDM Ricci scalar type. Thedimensionless. initial MPS␬ ͔͑ at z ␬30͔͑for 371 333 number of mesh points, box size, the initial redshift of the the latter shows the characteristic drop in power⌽ at small 372 334 simulations and time steps. For the later we use 800 COLA scales (for k 2 Mpc=h) although a comparison with the 373 335 steps given that the difference with a simulation of 1000 linear MPS obtained͑ directly2 from CLASS shows that the 374 336 steps is about 0.05%. We use a fiducial boson mass of oscillations have been smoothed out by MG-PICOLA. 375 ͑23 2 337 ma 10 eV=c for the FIC, FIC QP and EIC QP Nonetheless, we see that the MPS of the FIC QP 376 ⌽ ͓ ͓ 338 simulations, in order to make sure the effects of the simulation remains suppressed, at small scales, up͓ to 377 339 quantum terms for the range of wavenumbers contained z 5, with respect to that of the CIC simulation, although 378 340 in our simulations boxes are well resolved, notice that the the⌽ difference between the two is almost lost by z 2.5, 379 ⌽ 341 Nyquist scale is at kNyquist ͒ 214 h=Mpc. Additionally, being the relative difference of about 10% at most. 380 342 our simulations were started at z 30 for the FIC and We see in the FIC QP simulations a transfer of power 381 ⌽ 343 FIC QP, and at z 100 for the EIC one, to assure the from large to small scales,͓ an effect that has been widely 382 ͓ ⌽ 344 compatibility of the initial conditions with those of linear reported in similar simulations of the FDM model [21,29], 383 345 perturbations, which were obtained with an amended although only for those that exploit the similarity with the 384 346 version of the Boltzmann code CLASS (v2.7) [16] (see also hydrodynamic equations via the Madelung transformation. 385 347 [18] for a comparison of our formalism in solving the field However, we consider the evolution of the MPS to be 386 348 equations with other approaches). trusted only up to k ͒ 7.5 h Mpc͑1 (vertical dot–dashed 387 349 To begin with, in Fig. 1 we provide a visual comparison line), which is set by the scale at which the initial MPS is 388 350 of the matter density projected onto the xy plane, where above the shot–noise one (slopping dot–dashed line). Even 389 351 the plots were generated with the Python package yt [52]. though it is know that for particle-mesh like codes the shot– 390 noise affects less the evolution of models with suppression 391 5Not to confuse with the standard notation CIC in N-body in the MPS [54], we have preferred to be conservative. The 392 algorithms that stands for Cloud In Cell. sub–panel in this figure shows the relative difference 393

5 Axion simulations via modified gravity Transfer of power from large to small scales

Courtesy of Cesar Hernandez 15 NONLINEAR COSMOLOGICAL STRUCTURE WITH ULTRALIGHT … PHYS. REV. D XX, 000000 (XXXX)

Axion-like potential V ()=m2f 2[1 cos(/f )] Linares-Cedeño, González-Morales, U-L, JCAP 1, 051 (2021), arXiv:2006.05037 AAACEnicbVC7TsMwFHXKq5RXgJHFokJqB0pSkGBBqmBhLBJ9SEmIHNdprdpJZDtIVdVvYOFXWBhAiJWJjb/BTTNAy5Hu1dE598q+J0gYlcqyvo3C0vLK6lpxvbSxubW9Y+7utWWcCkxaOGax6AZIEkYj0lJUMdJNBEE8YKQTDK+nfueBCEnj6E6NEuJx1I9oSDFSWvLNarviJgNahZeQ39d9BMOsO/axi2MJM/Mk9FHV882yVbMywEVi56QMcjR988vtxTjlJFKYISkd20qUN0ZCUczIpOSmkiQID1GfOJpGiBPpjbOTJvBIKz0YxkJXpGCm/t4YIy7liAd6kiM1kPPeVPzPc1IVXnhjGiWpIhGePRSmDKoYTvOBPSoIVmykCcKC6r9CPEACYaVTLOkQ7PmTF0m7XrNPa/Xbs3LjKo+jCA7AIagAG5yDBrgBTdACGDyCZ/AK3own48V4Nz5mowUj39kHf2B8/gB0JprX a a a Linares-Cedeño, González-Morales, U-L et al, PRD 96 (2017) 061301(R) 2 3mPl λ = FDM : λ = 0 ( fa → ∞…) NONLINEAR COSMOLOGICAL STRUCTURE2 WITH ULTRALIGHT PHYS. REV. D XX, 000000 (XXXX) 8πfa

Lya observations 15 fa = 4.7 × 10 GeV 16 Medellín-González, González-Morales, U-L, PRD in press (2021), arXiv:2010.13998 FIG. 3. Top: comparison of the measured particle MPS for the F3:1 CIC (stars) and EIC QP (dots) at different redshift, z 100 F3:2 (purple) and z 50 (red).͑ See details of the simulation boxes͒ in F3:3 ͒ F2:1 FIG. 2. Top: comparison of the measured particle MPS for the Table II. The linear MPS from CLASS, used as initial condition, is F3:4 F2:2 CIC (stars) and FIC QP (dots) at different redshift, z 30 shown for comparison (dashed black line). The lower sub–panel F3:5 ͑ ͒ F2:3 (blue), z 5 (yellow) and z 2.5 (green). The linear MPS from shows the relative difference between the cases considered, for F3:6 ͒ ͒ F2:4 CLASS, used as initial condition, is shown for comparison (dashed the two redshifts: ͑P=PCIC PEIC QP ͑ PCIC =PCIC. Notice F3:7 F2:5 black line). The lower sub–panel shows the relative difference that the relative difference decreases͒⌽ for͑ lower redshift.͓ We also F3:8 F2:6 between the cases considered, for the three redshifts: ͑P=PCIC show for reference the wave number for which the initial MPS is F3:9 ͒ F2:7 PFIC QP ͑ PCIC =PCIC. Notice that the relative difference de- below the shot–noise (vertical dot–dashed gray line) and the F3:10 ⌽ ͑ ͓ F2:8 creases for lower redshift. We also show for reference the wave amplitude of the shot noise (sloping dot–dashed gray line), in F3:11 F2:9 number for which the initial MPS is below the shot–noise (vertical order to delimit the k range that could be influenced by shot– F3:12 F2:10 dot–dashed gray line) and the amplitude of the shot–noise (sloping noise. Bottom: the same as the top panel, but the comparison here F3:13 F2:11 dot–dashed gray line), in order to delimit the k range that could be is for the redshifts z 30 (blue curve), z 5 (yellow curve) and F3:14 FIG. 3. Top: comparison of the measured particle MPS for the F3:1 F2:12 influenced by shot noise. Bottom: the same as the top panel but the z 2.5 (green curve).͒ ͒ F3:15 CIC (stars) and EIC QP (dots) at different redshift, z 100 F3:2 F2:13 comparison here is with respect to the FIC simulation. ͒ (purple) and z 50 (red).͑ See details of the simulation boxes͒ in F3:3 ͒ F2:1 FIG. 2. Top: comparison of the measured particle MPS for the Tablethe quantumII. The linear force, MPS from whereCLASS extreme, used as refers initial condition, to the standard is F3:4 424 F2:2 CIC (stars) and FIC QP (dots) at different redshift, z 30 shownaxion for case comparison and the (dashed reported black excess line). The of lower power sub– inpanel densityF3:5 425 418 marginal differences in͑ the evolution of FDM density͒ F2:3 (blue), z 5 (yellow) and z 2.5 (green). The linear MPS from shows the relative difference between the cases considered, for F3:6 ͒6 ͒ perturbations reported in [15,16,30]. As explained in 426 419 F2:4perturbations.CLASS, used as initial condition, is shown for comparison (dashed the two redshifts: P=P P P =P . Notice F3:7 [15,16], the axion͑ caseCIC can beEIC approximatedQP ͑ CIC CIC by the action 427 420 F2:5 Finally,black line). in Fig. The3 lowerwe show sub panel the shows comparison the relative between difference the ͒⌽ ͑ ͓ – that the relative difference decreases for lower redshift. We also˜ F3:87 F2:6 between the cases considered, for the three redshifts: ͑P=PCIC show(1) but for reference with a thenegative wave number self-interaction for which the parameter, initial MPS͑ is< 0F3:9. 428 421 CDM and EIC QP simulations, the latter corresponding to͒ F2:7 PFIC QP ͑ P͑CIC =PCIC. Notice that the relative difference de- belowActually, the shot the– self-interactionnoise (vertical dot term–dashed in the gray axion line) potential and the canF3:10 429 422 the model⌽ ͑ with power͓ enhancement at small scales. Here, the F2:8 creases for lower redshift. We also show for reference the wave amplitude of the shot noise (sloping dot dashed gray line), in F3:11 423 acronym EIC QP means extreme initial conditions plus – F2:9 number for͑ which the initial MPS is below the shot–noise (vertical order to delimit the k range that could be influenced by shot– F3:12 F2:10 dot–dashed gray line) and the amplitude of the shot–noise (sloping noise.7 Bottom: the same as the2 top panel,2 but the comparison here F3:13 This also means that kJ1 ͑ ͑kJ1 in Eq. (5b), and this opens the F2:11 6 dot–dashed gray line), in order to delimit the k range that could be is for the redshifts z 30˜ (blue curve), z 5 (yellow curve) and F3:14 For a further comparison, in Appendix we show a comparison possibility that ͒ a; k; ͑ 1 at early times in the evolution of ⌽k, F2:12 influenced by shot noise. Bottom: the same as the top panel but the z 2.5 (green curve).⌽ ͒ ͓ ͑ ͒ F3:15 between simulations preformed with MG-PICOLA and the data see͒ Eq. (5a). This would be the underlying mechanism for the F2:13showncomparison in Fig. 9 here of [21] is with, which respect suggests to the FIC that simulation. our results are in excess of power with respect to CDM in the case of axion density agreement with others reported in the literature. theperturbations. quantum force, However, where see extreme[15,16,30] refersfor ato wider the standard explanation.424 axion case and the reported excess of power in density 425 418 marginal differences in the evolution of FDM density 6 perturbations reported in [15,16,30]. As explained in 426 419 perturbations. [15,16], the axion case can be approximated by the action 427 420 Finally, in Fig. 3 we show the comparison between the 7 ˜ 7 421 CDM and EIC QP simulations, the latter corresponding to (1) but with a negative self-interaction parameter, ͑ < 0. 428 422 the model with͑ power enhancement at small scales. Here, the Actually, the self-interaction term in the axion potential can 429 423 acronym EIC QP means extreme initial conditions plus ͑ 7 2 2 This also means that kJ1 ͑ ͑kJ1 in Eq. (5b), and this opens the 6 ˜ For a further comparison, in Appendix we show a comparison possibility that ͒ a; k; ͑ 1 at early times in the evolution of ⌽k, between simulations preformed with MG-PICOLA and the data see Eq. (5a). This⌽ would͓ ͑ be the underlying mechanism for the shown in Fig. 9 of [21], which suggests that our results are in excess of power with respect to CDM in the case of axion density agreement with others reported in the literature. perturbations. However, see [15,16,30] for a wider explanation.

7 NONLINEAR COSMOLOGICAL STRUCTURE WITH ULTRALIGHT … PHYS. REV. D XX, 000000 (XXXX)

Axion-like potential V ()=m2f 2[1 cos(/f )] Linares-Cedeño, González-Morales, U-L, JCAP 1, 051 (2021), arXiv:2006.05037 AAACEnicbVC7TsMwFHXKq5RXgJHFokJqB0pSkGBBqmBhLBJ9SEmIHNdprdpJZDtIVdVvYOFXWBhAiJWJjb/BTTNAy5Hu1dE598q+J0gYlcqyvo3C0vLK6lpxvbSxubW9Y+7utWWcCkxaOGax6AZIEkYj0lJUMdJNBEE8YKQTDK+nfueBCEnj6E6NEuJx1I9oSDFSWvLNarviJgNahZeQ39d9BMOsO/axi2MJM/Mk9FHV882yVbMywEVi56QMcjR988vtxTjlJFKYISkd20qUN0ZCUczIpOSmkiQID1GfOJpGiBPpjbOTJvBIKz0YxkJXpGCm/t4YIy7liAd6kiM1kPPeVPzPc1IVXnhjGiWpIhGePRSmDKoYTvOBPSoIVmykCcKC6r9CPEACYaVTLOkQ7PmTF0m7XrNPa/Xbs3LjKo+jCA7AIagAG5yDBrgBTdACGDyCZ/AK3own48V4Nz5mowUj39kHf2B8/gB0JprX a a a Linares-Cedeño, González-Morales, U-L et al, PRD 96 (2017) 061301(R) 2 3mPl λ = FDM : λ = 0 ( fa → ∞) 2 8πfa JCAP01(2021)051

Stellar streams can also be another astrophysical probe 1721 Figure 15. HMF for the analytical approach by [197] for FDM with m =2.1 10 eVMedellín-González, (green), and González-Morales, U-L, PRD in press (2021), arXiv:2010.13998 20 6 ⇥ our numerical results for SFDM HMF with m = 10 eV, = 10 , for a Top-Hat (blue),FIG. Sharp-k 3. Top: comparison of the measured particle MPS for the F3:1 (purple), and Top-Hat window function with scale-dependent critical overdensity (red). AllCIC of (stars) them and EIC QP (dots) at different redshift, z 100 F3:2 considering the S&T collapse model. Orange error bars show data from streams measurements(purple) [188 and, z 50 (red).͑ See details of the simulation boxes͒ in F3:3 ͒ 189]. See text forF2:1 moreFIG. details. 2. Top: comparison of the measured particle MPS for the Table II. The linear MPS from CLASS, used as initial condition, is F3:4 F2:2 CIC (stars) and FIC QP (dots) at different redshift, z 30 shown for comparison (dashed black line). The lower sub–panel F3:5 ͑ ͒ F2:3 (blue), z 5 (yellow) and z 2.5 (green). The linear MPS from shows the relative difference between the cases considered, for F3:6 considered. In particular, in figure 15 (red curve) we show the case of m = 10 20 and CLASS ͒ ͒ 6 F2:4 , used as initial condition, is shown for comparison (dashed the two redshifts: ͑P=PCIC PEIC QP ͑ PCIC =PCIC. Notice F3:7 = 10 . This leads to stronger constraints as those imposed by Lyman-↵ [117, 118], but lies ͑ F2:5 black line). The lower sub–panel shows the relative difference that the relative difference decreases͒⌽ for lower redshift.͓ We also F3:8 within the range of masses that could be tested by 21-cm observations [198, 199]. F2:6 between the cases considered, for the three redshifts: ͑P=PCIC show for reference the wave number for which the initial MPS is F3:9 ͒ It can alsoF2:7 be seenPFIC thatQP the͑ P HFMCIC =P forCIC. a Notice Sharp-k that window the relative function difference (purple de- curve)below requires the shot–noise (vertical dot–dashed gray line) and the F3:10 ⌽ ͑ ͓ larger massesF2:8 to be increases agreement for lower with redshift. both measurements We also show forof stellar reference streams. the wave Therefore,amplitude even of the shot noise (sloping dot–dashed gray line), in F3:11 when both windowF2:9 functionsnumber for contain which the information initial MPS of is below the MPS the shot suppression–noise (vertical at smallorder scales, to delimit it the k range that could be influenced by shot– F3:12 seems that theF2:10 Top-Hatdot–dashed with a gray scale-dependent line) and the amplitude critical of overdensity the shot–noise is (sloping less restrictivenoise. than Bottom: the same as the top panel, but the comparison here F3:13 F2:11 dot–dashed gray line), in order to delimit the k range that could be is for20 the redshifts z 30 (blue curve), z 5 (yellow curve) and F3:14 that of a Sharp-k window function, in the sense that the latter requires m > 10 eV. ͒ ͒ Nonetheless,F2:12 to findinfluenced out which by of shot the noise. di↵ Bottom:erent window the same functions as the top panel gives but the the morez realistic2.5 (green curve). F3:15 F2:13 comparison here is with respect to the FIC simulation. ͒ HMF one has to compare and calibrate the methods using numerical simulations of structure formation specific for the SFDM model. Specifically, recall that the Sharp-k functionthe has quantum one force, where extreme refers to the standard 424 axion case and the reported excess of power in density 425 free parameter,418c, thatmarginal is tuned differences using simulations. in the evolution It would be of possible FDM density that this parameter 6 perturbations reported in [15,16,30]. As explained in 426 could be tuned419 usingperturbations. both, data and simulations. [15,16], the axion case can be approximated by the action 427 As it is noted420 in [Finally,197], the in results Fig. 3 we from show the the analytical comparison approach between are the too conservative ˜ 7 in the sense that421 theyCDM are not and EICtakingQP into simulations, account the the scale-dependent latter corresponding growth to of(1) structure,but with a negative self-interaction parameter, ͑ < 0. 428 neither the scale-dependent422 the model critical with͑ power overdensity. enhancement In their at small analysis, scales. Here,masses the for FDMActually, with the self-interaction term in the axion potential can 429 20 values m . 1423.37 acronym10 eV EICwouldQP be excluded, means extreme whereas initial we areconditions showing plus that the SFDM ⇥ ͑ HMF (blue and red curves) lie within the range obtained from stellar streams measurements7 2 2 20 This also means that kJ1 ͑ ͑kJ1 in Eq. (5b), and this opens the for m = 10 eV. This6 value is consistent, however, with the conservative bound where ˜ For a further comparison, in Appendix we show a comparison possibility that ͒ a; k; ͑ 1 at early times in the evolution of ⌽k, masses m 2.1 1021eV are excluded. Without the e↵ect of the tachyonic instability, ⌽ ͓ ͑ . ⇥ between simulations preformed with MG-PICOLA and the data see Eq. (5a). This would be the underlying mechanism for the stronger constraints onshown the in axion Fig. mass 9 of [21] would, which be imposed. suggests that our results are in excess of power with respect to CDM in the case of axion density agreement with others reported in the literature. perturbations. However, see [15,16,30] for a wider explanation. 5 Conclusions

The SFDM model constitutes a compelling candidate to substitute the CDM model. In this 7 paper, we have presented a formalism to handle the cosmological equations by using the tools of dynamical systems for both the background and the linear perturbations, that extends the analysis in [99]. At the background level, the presence of a trigonometric potential shows a delay in the moment when the axion field starts to oscillate and behaving as CDM. These

– 25 – Conclusions

• Work in progress … Astro2020 Science White Paper: Gravitational probes of ultra-light axions 5 Grin et al, ArXiv 1904:09003 It’s been under scrutiny for almost two decades 0 • 101 Reionization ULA hints • It’s a good scientifc model: it can be falsifed by 2 Planck exclusion 1 10 CMB

a nearly CV-limited CMB

diferent ranges of observations f GUT-Scale a 2 10 Ly-↵ • If the DM riddle needs an exotic solution, what is Ly- Ly-↵ future Reionization more exotic than quantum efects at galactic level? 3 10 PTA(SKA) Eridanus-II FDM Hints

• Is there any characteristic scale in the DM feld? Cosmic Density Eridanus II BHSR 10 4 BHSR h h GUT-scale fa Pulsar Timing (SKA) −1 −1 LC = = 0.4 ma22 pc LdB = = 400 ma22 pc like hidden sectors with low confinement scales. This both opens up interesting phenomenology 5 mac mav 10 34 32 30 28 26 24 22 20 18 16 14 12 10 associated to the presence of this “dark world” and raises the question of how it managed to escape 10 10 10 10 10 10 10 10 10 10 10 10 10 being observed so far. We will touch on some of the issues involved in the concluding Section 3. Axion Mass ma [eV] For• nowUltimate we focus upon challenge: the observational cosmological signatures of the light axionssimulations that we have arguedand are genericLyman-alpha to string theory once theobservations, strong CP problem is for solved. a joint view of the Figure 1. The cosmic window on ultralight axions, showing the reach of various astronomical 2model Cohomologies from di fromferent Cosmology scales probes. Shaded regions are currently excluded. Lines below them indicate the sensitivity of future experiments/surveys. ULA hints refers to the ULA mass scale suggested by MW-scale challenges. Local group schematic, Credit: J. T. A. de Jong, Leiden University. Planck CMB map, Credit Anthropically Constrained ESA, http://www.esa.int. Black-hole super-radiance schematic [85] used with permission from CMB Matter Polarization Power Spectrum Inflated American Physical Society, License RNP/19/MAR/012767. PTA Schematic Image Credit: David Black Hole Super-radiance Decays Away Champion. Lyman Alpha Schematic, Image Credit: Ned Wright. Reionization schematic used 10-33 4 ! 10-28 3 ! 10-18 108 with permission from artist, J. F. Podevin, originally used in Ref. [86]. 2 ! 10-20 3 ! 10-10 18 QCD axion ¡Gracias! Thank you! Axion Mass in eV III. AXIONS AND THE CMB Figure 1: Map of the Axiverse: The signatures of axions as a function of their mass, assuming 8 fa MGUT and Hinf 10 eV. We also show the regions for which the axion initial angles are In the DE-like mass window, ULAs roll slowly down their potential as a dark-energy ⇡ ⇠ anthropically constrained not to over-close the Universe, and axions diluted away by inflation. component, shifting CMB acoustic peaks to smaller angular scales (higher `), and increasing For the same value of fa we give the QCD axion mass. The beginning of the anthropic mass 20 28 the largest scale anisotropies due to gravitational potential-well decay for 0 z 3300 region (2 10 eV) as well as that of the region probed by density perturbations (4 10 . . eV) are blurred⇥ as they depend on the details of the axion cosmological evolution (see⇥ Section [15, 16, 87–90]. In the DM-like mass window, the imprint of ULAs on the Hubble expansion 18 2.3). 3 10 eV is the ultimate reach of density perturbation measurements with 21 cm line ⇥ and perturbation growth alters peak heights measured in the temperature and E-mode observations. The lower reach from black hole super-radiance is also blurred as it depends on polarization power spectra [44, 88, 89, 91, 92]. ULAs manifest wave-like properties on the details of the axion instability evolution (see Section 2.5). The region marked as “Decays”, outlines very roughly the mass range within which we expect bounds or signatures from axions cosmological scales, suppressing density fluctuations the ULA comoving “Jeans scale” . 22 1/2 1/4 decaying to photons, if they couple to E~ B~ . We will discuss axion decays in detail in a companion J 0.1Mpc(ma/10 eV) (1 + z) [4, 10, 43, 44, 89, 92–98], a↵ecting comoving · ⌘ paper. wavenumbers k>2⇡/J. This a↵ects the strength and scale-dependence of gravitational lensing of the CMB [99–102]. 32 26 In the window 10 eV . ma . 10 eV, the imprint of these e↵ects and Planck satellite 2 2.1 Discovering the String Axiverse data impose the constraint of ⌦a . 10 , as shown in Fig. 1 [98, 103]. Fig. 1 includes the e↵ect of CMB lensing, which improves sensitivity by a factor of 3comparedwiththe We now turn to the observational consequences of axions lighter than or around the QCD axion 32 ⇠ 25 unlensed CMB or galaxy clustering. For ma 10 eV (or ma 10 eV), ULA e↵ects mass. For simplicity, we keep fa fixed at MGUT and Hinfl 0.1 GeV. The initial displacement of . & 20 ⇠ axions heavier than 10 eV has to be tuned in order for them not to overclose the universe and on CMB/galaxy clustering signatures are indistinguishable from a cosmological constant (or axions heavier than⇠ 0.1 GeV have been diluted away by inflation. The observational consequences DM) with current data, lifting these constraints [89]. of the string axiverse are outlined in Figure 1. In the next and coming decades, very sensitive experiments like the Simons Observatory We concentrate on three main windows to the axiverse. First, as discussed in Section 2.2 33 28 (SO) [104], CMB Stage-4 (CMB-S4) [105], and PICO [106] (e.g., map noise levels of 6 µK- axions of masses between 10 eV and 4 10 eV, if they couple to E~ B~ , cause a rotation in ⇥ · arcmin, 1 µK-arcmin, and 0.6 µK-arcmin, respectively) will achieve nearly cosmic-variance limited measurements of CMB primary anisotropies, and reduce lensing reconstruction noise 8