Ultralight DM bosons with an Axion-like potential: scale-dependent constraints revisited
Francisco X. Linares Cedeño, Alma X. González-Morales, L. Arturo Ureña-López
PPrepared for submission to JCAP
Ultralight DM bosons with anAxion-like potential:scale-dependent constraintsrevisited.
Francisco X. Linares Cedeño, a, Alma X. González-Morales b,c and L. Arturo Ureña-López, b a Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo,Edificio C-3, Ciudad Universitaria, CP. 58040 Morelia, Michoacán, México. b Departamento de Física, DCI, Campus León, Universidad de Guanajuato, 37150, León,Guanajuato, México. c Consejo Nacional de Ciencia y Tecnología, Av. Insurgentes Sur 1582. Colonia CréditoConstructor, Del. Benito Juárez C.P. 03940, México D.F. MéxicoE-mail: [email protected], lurena@fisica.ugto.mx,alma.gonzalez@fisica.ugto.mx
Abstract.
A scalar field φ endowed with a trigonometric potential has been proposed to playthe role of Dark Matter. A deep study of the cosmological evolution of linear perturbations,and its comparison to the Cold Dark Matter (CDM) and
Fuzzy Dark Matter (FDM) cases(scalar field with quadratic potential), reveals an enhancement in the amplitude of the masspower spectrum for large wave numbers due to the non–linearity of the axion–like potential.For the first time, we study the scale–dependence on physical quantities such as the growthfactor D k , the velocity growth factor f k , and f k σ . We found that for z < , all thesequantities recover the CDM evolution, whereas for high redshift there is a clear distinctionbetween each model (FDM case, and axion–like potential) depending on the wavenumber k and on the decay parameter of the axion-like potential as well. A semi–analytical HaloMass Function is also revisted, finding a suppression of the number of low mass halos, as inthe FDM case, but with a small increment in the amplitude of the variance and halo massfunction due to the nonlinearity of the axion–like potential. Finally, we present constraintson the axion mass m φ ≥ − eV and the axion decay parameter is not constrained withinthe prior ≤ λ ≤ by using data of the Planck Collaboration 2015. Corresponding author. a r X i v : . [ a s t r o - ph . C O ] J un ontents D and velocity growth factor f with scale-dependence 154.2 Semi-analytical Halo Mass Function 19 One of the open problems of modern physics concerns the existence of Dark Matter (DM). Atpresent we have several observations indicating that such a component of matter exists [1–5], and that it is most likely the main agent driving the formation of structure. The mostsuccessful model describing this unknown component of matter is called
Cold Dark Matter (CDM), and consists of a pressureless fluid of particles that interacts mostly gravitationallywith other components of matter [6, 7]. Although the CDM model is so far in good agreementwith most of the cosmological observations, DM nature is still unknown. It is well known thatthere are some differences at small scales between astrophysical observations and numericalsimulations based on CDM [8–23]. These differences may be due to the lack of informationabout astrophysical processes of galactic substructures and baryonic physics, but they couldalso be pointing out to characteristics and still unknown properties of the DM field. Hence,if it is the case that DM is the main responsible for the process of structure and substructureformation, then it is important to explore and analyze other DM candidates that could offer– 1 – better description of the structures at such small scales. With many different models in theliterature, it is important that a given model under study predicts observables accurately sothat comparison against observations are meaningful.In this context, models of Scalar Field Dark Matter (SFDM) have gained great relevancein modern cosmology by becoming a promising candidate to describe the DM as well, maybeeven better, than CDM. While the implementation of scalar fields in cosmology has histori-cally its origins in inflationary models of the early Universe[24, 25], and also to describe theaccelerated expansion of the Universe at late times [26–30], scalar fields also possess interestingproperties to work as DM models.Within this frame of scalar fields, a compelling DM candidate that has been vastlyinvestigated, and which is the one of interest in this work, is the
Axion , a scalar field originallyproposed to solve the Strong CP problem in QCD [31–34], and which origin can be given withina more fundamental theory such as String Theory [35–40]. Now, several models involvingaxions and axion-like particles have surged as possible source for DM [41–51], and severalexperiments such as ADMX [52], SOLAX[53], DAMA [54], COSME [55], CAST [56] aretrying to hunt directly this elusive kind of particles (other possible ways of detection can beseen in [57–65]). The lighter axion in QCD has masses of around µ eV, while for axion–likeparticles the mass lies within the range of − eV > m φ > − eV, which is the reason whythe latter are also known in the literature as Ultralight Axions . An important feature of thisDM candidate is that it can give rise to
Bose-Einstein condensates through a phase transition[66–73], and it can form caustics as well [74–81]. Thus, axions and axion-like particles arevery well motivated DM candidates from the theoretical point of view.Axion models in which the scalar field potential includes only the quadratic term, usu-ally referred as free case or fuzzy dark matter (FDM), have been extensively studied in theliterature [82–89]. We will refer to it as the FDM case from now on. However, such modelsdo not capture all the implications that arise when including a full axion potential.In this work we will focus on a model that incorporates a trigonometric potential thatis typical in axion studies, defined by V ( φ ) = m a f a [1 + cos ( φ/f a )] , (1.1)Here, m a is the axion mass, f a is the axion decay constant, and the two together m a f A make up the height of the potential. In typical axion models, there is a relationship betweenthe mass and the decay constant in which they are inversely proportional to each other, inparticular for axions coming from M-Theory and Type IIB string theory [36, 90, 91], wherethe decay constant is of the order of GeV .The choice of the potential in Eq. (1.1) codifies the shift symmetry of the axion field, andour main aim is to analyze in detail the cosmological implications arising from the nonlinearityof such potential. Previous works for this have shown some semi-analytical treatment [88, 92],while a first attempt to a full analysis was presented in [93]. The effects on the CMB andMPS of such anharmonic potential, but considering different exponents [1 − cos( φ/f )] n with n = 1 , , , have been studied in [94]. However, when the cosmological evolution of the scalarfield is that of dark matter (for n = 1 ), the predictions are basically the same as those ofFDM. As we will show in the present work, when considering extreme values of the decayconstant with the potential (1.1), it is possible to quantify deviations from the FDM case,regarding the structure formation at linear regime, like the enhancement of the mass powerspectrum (MPS) at small scales reported in [92, 93], as well as to analyze implications forother observables. – 2 –n outline of this work is as follows. In Section 2 we study the cosmological back-ground evolution and linear perturbations regime, by means of establishing new variablesand a dynamical system that lead us to a generalization of the fluid equations. Using anamended version of the Boltzmann code class [95], we track the evolution and growth ofthe perturbations. At the end of this section we develop a detailed analysis of the tachyonicinstability suffered by the density perturbations, and we show that only a set of wavenumberscorresponding to small scales are affected by such instability.The matter and temperature power spectra that arise from the axion model are presentedin Section 3, and we use them to impose some bounds on the free parameters of the model:the axion mass m a and the decay parameter f a mentioned above. We also make a qualitativeassessment of how the Lyman- α
1D mass power spectrum could constrain the parameters ofour model with Ly- α . We observe that while the FDM model with masses m a ≤ − eVis ruled out, it is possible for the axion field to pass the constraints if endowed with thetrigonometric potential (1.1).Motivated by the characteristic cut-off that this model presents in the mass power spec-trum (MPS), in Section 4 we define both the growth factor D k and the velocity growth factor f k , not only as a function of the scale factor but also with their dependence on the length scale.We also build the combination [ f k σ ] ( z ) , and no major difference with respect to the CDMprediction were found. We then analyze the semi-analytical Halo Mass Function (HMF),which, like in the case of the MPS, it shows an enhancement in its amplitude when consider-ing the potential (1.1). Finally, in Section 5, we give some conclusions and perspectives forsome future work. In this section we show the dynamical equations for the evolution of both, background andlinear perturbations of the axion model (1.1). Following previous work [93, 96], we rewritethese equations as a dynamical system and then, by a polar change of variables, we obtaina set of first order differential equations which is more appropriate for numerical studies ofultra–light axions than using directly the field equations.
The Einstein-Klein-Gordon equations for a minimally-coupled scalar field φ endowed with ageneric potential V ( φ ) , in a Friedmann-Robertson-Walker spacetime with null spatial curva-ture are given by H = κ (cid:88) j ρ j + ρ φ , ˙ H = − κ (cid:88) j ( ρ j + p j ) + ( ρ φ + p φ ) , (2.1a) ˙ ρ j = − H ( ρ j + p j ) , ¨ φ = − H ˙ φ − dV ( φ ) dφ , (2.1b)where κ = 8 πG , a dot denotes derivative with respect to cosmic time t , and H is the Hubbleparameter. Also, the scalar field energy density ρ φ and pressure p φ are given by the canonicalexpressions: ρ φ = 12 ˙ φ + V ( φ ) , p φ = 12 ˙ φ − V ( φ ) . (2.2)– 3 –n order to transform the Klein-Gordon (KG) equation (2.1b), we define a new set ofpolar variables based on previous works [97–99], Ω / φ sin( θ/ ≡ κ ˙ φ √ H , Ω / φ cos( θ/ ≡ κV / √ H , y ≡ − √ H ∂ φ V / . (2.3)with which the KG equation can be written, for the particular case of potential (1.1), as adynamical system in the form: θ (cid:48) = − θ + y , (2.4a) y (cid:48) = 32 (1 + w tot ) y + λ φ sin θ , (2.4b) Ω (cid:48) φ = 3( w tot − w φ )Ω φ . (2.4c)Here a prime denotes derivative with respect to the number of e -foldings N ≡ ln( a/a i ) ,with a the scale factor of the Universe and a i its initial value. The decay constant appearsexplicitly in the newly defined (dimensionless) parameter λ = 3 /κ f a , and then the FDMcase with λ = 0 (studied in Ref. [98]) is obtained in the limit f a → ∞ . In contrast, we seethat the mass parameter m a does not appear at all in the new equations of motion. Followingthe classification suggested in [99], the decay constant is an active parameter, whereas themass is a passive one that does not have any influence in the evolution of the field φ . Theequation of state (EoS) for the axion field is directly related to the dynamical variable θ as, w φ ≡ p φ ρ φ = x − y x + y = − cos θ . (2.5)Eq. (2.4) is a compact representation of the KG equation, and they reveal that the truevariables driving the scalar field dynamics are { θ, y , Ω φ } . They also show that the effect ofthe trigonometric potential of Eq. (1.1) is encoded in one free parameter given by λ , and thenit will be possible to analyze in one stroke the cosmological properties of both, the axion field( λ > ) and the FDM case ( λ = 0 ), see [93, 98]. For a correct numerical implementation of the equations of motion (2.4) within a cosmologicalsetting, it is necessary to estimate the right initial conditions of the dynamical variables atvery early times. As done in Ref. [98] for the FDM case, in this section we find semi-analyticalsolutions for the radiation dominated era and extrapolate them to the present time.Assuming that all quantities are small and positive, i.e. ( θ, y , Ω φ ) (cid:28) , Eq. (2.4) takesthe form (at linear order), θ (cid:48) (cid:39) − θ + y , y (cid:48) (cid:39) y , Ω (cid:48) φ (cid:39) φ , (2.6)whose analytical solutions are θ = (1 / y + C ( a/a i ) − , y = y i ( a/a i ) , Ω φ = Ω φi ( a/a i ) , (2.7)where a subscript i denotes the corresponding initial value for each variable. The solu-tions (2.7) are the same as those of the quadratic potential studied in [98], basically becausethe second term on the rhs of Eq. (2.4b) is of second order, which means that at early timesthe influence of λ in the solutions should be negligible.– 4 –ssuming that the axion field starts to behave as CDM at a = a osc , when it starts tooscillate rapidly around the minimum of the potential and the EoS first passes through thevalue w φ = 0 (corresponding to θ = π/ ), it can be shown that the estimated initial conditionsare obtained from the following equations, θ i = 25 mH i , y i = 5 θ i , Ω φi = a i a Ω φ Ω r , a = π θ − i a i (cid:112) π / , (2.8)where H i and a i are the initial values of the Hubble parameter and the scale factor, and Ω r ( Ω φ ) is the present radiation (axion) density parameter (see [98] for more details).We now find a next-to-leading order solution for the initial conditions that takes intoaccount the presence of λ , and for that we follow an iterative method. Let us consider thefirst order solutions (2.7), substitute them in Eq. (2.4b) and solve for a new solution of y .We find that y = 5 θ i ( a/a i ) + λ φi θ i ( a/a i ) . (2.9)If we now use the foregoing solution and plug it into the right hand side of Eq. (2.4a), we findthat a corrected solution for θ is θ = θ i ( a/a i ) (cid:20) − λ
72 Ω φi + λ
72 Ω φi ( a/a i ) (cid:21) , (2.10)whereas the solution for Ω φ remains the same. From the combination of the above equations,we obtain from the matching condition at a = a osc that a (cid:18) λ
72 Ω φ Ω r a osc (cid:19) = π θ − i a i (cid:112) π / . (2.11a)Notice that for λ = 0 we recover, as expected, the required matching equation for the quadraticpotential, see the last equation in (2.8). Instead of Eq. (2.8), we will use the new set ofEqs. (2.9), (2.10) and (2.11a) to calculate the initial conditions of the dynamical variables. Asshown in the appendix D.5, the iterative integration method could be used again to generatea higher-order equation to determine a osc , but we will restrict ourselves to Eq. (2.11a) as itis enough for the purposes of this paper.In contrast to the FDM case, there is an additional trigonometric constraint that is char-acteristic of the axion potential, and that can be obtained directly from the definitions (2.3), m H i = y i + 2 λ Ω φi . (2.11b)Although we use it only as an additional constraint for the initial conditions, it should beemphasized that Eq. (2.11b) is of general applicability at all times. Again, for the case λ = 0 we recover the usual expression of the FDM case, namely y i = 2 m/H i . Hence, the initialconditions in the general case are obtained from the combined solution of Eqs. (2.11a), (2.11b)and y i = 5 θ i (cid:18) λ
40 Ω φi (cid:19) , Ω φi = a i a Ω φ Ω r . (2.11c)The initial conditions are further adjusted by means of the shooting procedure imple-mented in class to give the right current values of the physical parameters. The values of– 5 – osc for different values of λ , as obtained from the numerical solutions, are shown in Table 1,where it can be seen that the onset of the scalar field oscillations suffers a delay as λ in-creases. The most extreme value that we will consider is λ = 10 , as for larger values isdifficult to calculate the initial conditions because of the exponential sensitivity that appearsin the estimation of a osc . λ log( a osc ) -6.159 -6.159 -6.159 -6.143 -6.048 -5.838 δθ — ◦ ◦ ◦ ◦ . ◦ Table 1 : Numerical values for the onset of oscillations of the axion field for each value of λ . For λ = 0 , , , oscillations start at the same time, whereas for λ = 10 , , , wenotice that oscillations start later as λ increases. In the last row we show the initial fielddisplacement from the top of the axion potential, for a comparison with the Extreme AxionWave Dark Matter model [92], see Appendix C for details.A comparison of the evolution of the CDM and axion densities is shown in Figure 1.While the CDM density redshifts as a − , we see that the axion energy density remains constantbefore the start of the field oscillations at a = a osc (cid:39) − , but afterwards the two densitiesevolve together. As also shown in the inset, the onset of the field oscillations depends onthe value of the decay constant through the parameter λ , and in general the oscillations aredelayed as the value of λ increases, which is consistent with the numerical results shownin Table 1. We can also notice that the transition of the axion energy density to the CDMbehavior happens more abruptly for larger values of λ , which is one of the consequences of theexponential sensitivity of the numerical solutions on the initial conditions that we discussedabove. Now, we consider linear perturbations around the background values of the FRW line element(in the synchronous gauge) as well as for the scalar field in the following form: ds = − dt + a ( t )( δ ij + h ij ) dx i dx j , φ ( x, t ) = φ ( t ) + ϕ ( x, t ) , (2.12)where h ij and ϕ are the metric and scalar field perturbations respectively. The linearized KGequation, in Fourier space and for a general potential, reads [100–103]: ¨ ϕ = − H ˙ ϕ − (cid:18) k a + ∂ V ( φ ) ∂φ (cid:19) ϕ −
12 ˙ φ ˙¯ h , (2.13)where ¯ h = ¯ h jj is the trace of scalar metric perturbations, and k is the comoving wavenumber.Although a functional dependence of the scalar field perturbation is not explicitly shown,note that Eq. (2.13) is written for a Fourier mode ϕ ( k, t ) . After a change of variables to thenew quantities δ and δ (see Appendix A for details), Eq.(2.13) is described by the following– 6 – igure 1 : Evolution of CDM and SFDM energy density for a fixed axion mass of m φ =10 − eV, and different values of the decay parameter λ of the potential (1.1). Initially theamplitude of the axion energy density is less than that of the CDM, but once the axion fieldstarts to oscillate (around a = 10 − ), it evolves just as the CDM case. Inset: It can benoticed that larger values of parameter λ delay the scalar field oscillations, then the axionfield evolves as CDM. Vertical lines indicate the onset of oscillations log( a osc ) for each valueof λ , see also Table 1.system of first order differential equations, δ (cid:48) = (cid:20) − θ − k k J (1 − cos θ ) (cid:21) δ + k k J sin θδ − ¯ h (cid:48) − cos θ ) , (2.14a) δ (cid:48) = (cid:20) − θ − k k J sin θ + Ω / φ sin (cid:18) θ (cid:19) y y (cid:21) δ + (cid:20) k k J (1 + cos θ ) − Ω / φ cos (cid:18) θ (cid:19) y y (cid:21) δ − ¯ h (cid:48) θ , (2.14b)where we defined the (squared) Jeans wavenumber as k J = H a y , which is the same defini-tion used for the case of a quadratic potential [98, 104]. Eq. (2.14) are written in general forany scalar field model, and one only requires to specify the functional form of the ratio y /y for a given potential, see for instance [99].The density and pressure contrasts δ φ , δp φ , and velocity divergence θ φ , are given by thestandard definitions [102, 103, 105], and in terms of the new perturbation variables they take– 7 –he form: δ φ = ˙ φ ˙ ϕ + ∂ φ V ϕ ˙ φ / V ( φ ) = δ , δ p φ = ˙ φ ˙ ϕ − ∂ φ V ϕ ˙ φ / V ( φ ) = sin θδ − cos θδ , (2.15a) ( ρ φ + p φ ) θ φ = ( k /a ) ˙ φϕ = k am ρ φ [(1 − cos θ ) δ − sin θδ ] . (2.15b)It is important to mention that we have gained physical interpretation for the new dynamicalvariable δ : it plays the role of the scalar field density contrast, δ φ , according to the firstexpression in Eq. (2.15a). This implies that Eq. (2.14a) is the closest we can get of a fluidequation for the scalar field perturbations. The interpretation of δ remains elusive, and itremind us of the difficulties to match Eq. (2.13) to a fluid even in the generalized case [105].For the particular case of the axion field endowed with the potential (1.1), the expres-sions (2.14) now reads δ (cid:48) = (cid:20) − θ − k k J (1 − cos θ ) (cid:21) δ + k k J sin θδ − ¯ h (cid:48) − cos θ ) , (2.16a) δ (cid:48) = (cid:34) − θ − k eff k J sin θ (cid:35) δ + k eff k J (1 + cos θ ) δ − ¯ h (cid:48) θ , (2.16b)where we have defined an effective wavenumber of the perturbations as k eff ≡ k − λa H Ω φ / .The equations of linear perturbations for the standard FDM case are again obtained when λ = 0 , for which k eff = k is just the standard Laplacian term in Fourier space. Because now y = 2 m φ /H , the Jeans wavenumber k J is then the only characteristic scale in the evolutionof linear perturbations, and the responsible for the appearance of a sharp cut-off in their masspower spectrum: linear perturbations are heavily suppressed for wavenumbers k > k J . TheJeans wavenumber is always proportional to the geometric mean of the Hubble parameter H and the boson mass m , namely k J = a √ Hm , which shows that the cut-off in the MPS issensitive to both the parameters of the axion model and to the background expansion. Moredetails about the cut-off of linear perturbations in the FDM case λ = 0 can be found in [98]. One of the main effects on linear perturbations of axion fields (for λ > ) is the appearance ofan enhancement in the growth of the density contrast δ , that was first discussed in [92, 93,106] and thereby dubbed as a tachyonic instability . Such instability provokes the appearanceof a bump in the MPS of the perturbations that is well localized in wavenumbers around theJeans one k J .To have a qualitative understanding of the tachyonic instability, we follow and extendthe procedure already outlined in [93]. Let us write Eqs. (2.16) on rapid oscillations regime,under which all trigonometric terms are time-averaged to zero, (cid:104) sin θ (cid:105) = (cid:104) cos θ (cid:105) = 0 . Hence,we find δ (cid:48) = − k k J δ − ¯ h (cid:48) , δ (cid:48) = k eff k J δ . (2.17a) Recently, the authors in [107] made a comparison of the different approximations one can find in theliterature to follow the cosmological evolution of ultra-light bosons. Such approximations, which correspondto diverse choices in cycle-averaging procedures, are necessary to deal with the rapid oscillations of theaxion field at late times, see the original field equations (2.1) and (2.13). It was there concluded that ourapproximation method, which has been used previously in Refs [93, 96], is the closest, compared to others, tothe exact solution of the field equations of motion. – 8 –f we neglect, for simplicity, the time variation of both k J and k eff , the foregoing equa-tions can be combined into the form of a forced harmonic oscillator for the density contrast,namely, δ (cid:48)(cid:48) + ω δ = − ¯ h (cid:48)(cid:48) , ω ≡ k k eff k J . (2.17b)From the above we see that the tachyonic instability requires of two conditions. Firstly, thestart of rapid oscillations of the field around the minimum of its potential, and secondly,a negative squared amplitude of the angular frequency, ω < , in Eq. (2.17b). The lattercondition is possible because the effective wavenumber k eff can be either positive or negative,although it depends on a non-simple combination of the cosmological quantities a , H and Ω φ .The tachyonic instability and the conditions for its appearance are illustrated in Figure 2.In the top panel, we show the relative difference between the axion density contrast withrespect to CDM ∆ δ ≡ ( δ φ − δ CDM ) /δ CDM , and in the lower panel the evolution of theangular frequency ω . The axion mass and the wavenumber that were chosen have the values m φ = 10 − eV and k = 8 h/ Mpc respectively, while the decay parameter was chosen as λ = 0 (FDM case, f → ∞ ) and λ = 10 (extreme case, f (cid:39) − m Pl ). The light gray regionindicates the period of time when the tachyonic condition ω < occurs in the case λ = 10 . Figure 2 : (Left) Evolution of the relative difference between density contrasts ∆ δ (top) andfrequency ω (bottom) for a fixed axion mass of m = 10 − eV and wavenumber k = 8 h/ Mpc ,evaluated at λ = 0 (solid blue line) and λ = 10 (solid red line). The light gray regionindicates the duration of the tachyonic instability, while the vertical dotted blue and redlines show the onset of the oscillation of the scalar field for λ = 0 and λ = 10 respectively.(Right) Frequency ω as function of the wavenumber k for an axion with m φ = 10 − eV and λ = 10 , and for three different times of the scalar field evolution: at the onset of oscillation a osc = 1 . × − (green line), at a threshold value a th = 4 . × − (blue line), and at theend of the tachyonic instability a end = 2 . × − (purple line). The region within the twovertical dashed green lines indicates the range of wavenumbers that will suffer the tachyoniceffect. The black horizontal solid line stands for ω = 1 . See text for more details.In the FDM case we see that the angular frequency is always positive and less thanunity, < ω = k /k J < , and then Eq. (2.17b) is just the equation of motion of a forcedoscillator and the tachyonic instability never happens. The density contrast, δ , for thechosen wavenumber, can not catch up completely with the CDM solution after the onset– 9 –f rapid oscillations (at around a (cid:39) − . ), but nonetheless keeps a constant ratio withrespect to CDM at late times (this constant ratio can be explained in terms of the growthfactor, see Sec. 3 below). The result is that the amplitude of the MPS at this wavelength issuppressed respect to the CDM one. The particular value k = 8 h/ Mpc was chosen becauseit corresponds, approximately, to the cut-off scale in the FDM case for m φ = 10 − eV.In contrast, for the value λ = 10 we see that the onset of oscillations (at a osc =1 . × − ) occurs after the appearance of the tachyonic instability (at a (cid:39) − . ). At thesame time, and after the onset of oscillations, the amplitude of the axion density contrast δ grows quickly reaching larger values than that of CDM. This growth persists until justafter the tachyonic instability disappears (when once again ω > ) at around a (cid:39) − . .After this, the density contrast δ then evolves like in the FDM case and keeps a constantamplitude with respect that of CDM at late time (see also Sec. 3 below). As a result, theMPS at k = 8 h/ Mpc is now enhanced with respect to that of CDM. The tachyonic effectand its duration is scale dependent, as we show in Figure 2, where we plot the frequency | ω | as function of the wavenumber k , for an axion with m φ = 10 − eV and λ = 10 , andthree fixed times. The green curve corresponds to the time at the onset of the axion fieldoscillations (at a = a osc ), and then we expect the tachyonic instability to start happening forthose wavenumbers for which ω < − , that is, for those in the range < k/ ( h/ Mpc) < .There is a characteristic time, labeled as the threshold for tachyonic instability at a = a th (blue curve), for which just a small range of wavenumbers around k = 8 h/ Mpc barely complywith the condition ω = − . Moreover, we also see that wavenumbers k > h/ Mpc haveleft the tachyonic regime by this time, as ω > for them. Finally, the end of the tachyonicinstability at a = a end is also shown. The value of a end is somewhat arbitrary, but we havechosen it such that even the smallest of the wavenumbers in the initial range of tachyonicinstability is no longer stimulated, and then the downturn of the (purple) curve occurs at k ∼ h/ Mpc .Summarizing, we find that for large scales k (cid:46) h/ Mpc the tachyonic instability ispractically non-existent, and also for them the condition < ω (cid:28) is accomplished atall times. The evolution of the density contrast for these scales is governed by the equation δ (cid:48)(cid:48) (cid:39) − (1 / h (cid:48)(cid:48) (see Eq. (2.13)), and we obtain for them the same solution as for CDM linearperturbations, that is δ (cid:39) − (1 / h . Likewise, small scales k (cid:38) h/ Mpc are also alwaysfree from tachyonic instabilities as for them ω > at all times. The latter condition meansthat they do not longer grow with the CDM solution, but now they must be suppressed asin the standard FDM case. Therefore, wavenumbers within the range (cid:46) k/ ( h/ Mpc) (cid:46) will present an enhancement in their density contrast amplitude, as was shown in Figure 2for the case k = 8 h/ Mpc.The range of wavenumbers k that suffer a tachyonic instability is mainly determinedby the axion mass m φ . The arguments above show that the instability appears around thewavenumber that marks the cut-off of the corresponding FDM case. As shown in the example,the range of wavenumbers that suffers a tachyonic instability in the case λ (cid:54) = 0 shift to larger(smaller) values for larger (smaller) axion masses. The solutions of Eq. (2.16) are useful to build up cosmological observables such as the CMBanisotropies and the MPS, which can then be contrasted with observations. In this section– 10 –e first present a qualitative comparison with the observables, and then present the detailsand results from a parameter estimation procedure.
The CMB power spectrum for both CDM and SFDM, for a couple of values of the axionmass, is shown in Figure 3, where we have included data from the Planck Collaboration. For a fiducial axion mass of m φ = 10 − eV (left panel) we observe that, regardless of thevalue of λ , the axion field reproduces the CMB spectrum as good as CDM. In fact, the majordiscrepancy between both cases is of ∼ . for large multipoles. In contrast, for an axionmass of m φ = 10 − eV (right panel), we clearly note that the CMB spectrum does not fit theobservational data, with a major discrepancy of ∼ for l ∼ . Figure 3 : Temperature Power Spectrum for CDM and SFDM for two axion masses: m φ =10 − , − eV. The effect of λ is clearly noted for the latter where, for large multipoles, thedifferences are greater as the value of λ increases. See text for more details.We have also considered CMB observations for high multipoles, as can be seen in Fig-ure 4, where we have also included a wider range of values for m φ and λ . We observe in theupper panel that for large multipoles the case of an axion mass of − eV with quadratic( λ = 0 ) and trigonometric potential ( λ = 10 ) still have more amplitude than the extremecase with m φ = 10 − eV and λ = 10 . In particular, we observe that for a given axion massthe effect of consider λ > is to increase the amplitude of the CMB power spectrum incomparison with the FDM case (this can be clearly seen when m φ = 10 − eV). The lowerpanel shows that observations such as Planck, SPT and ACT, do not constrain the fiducialcase of a free axion with mass − eV and λ = 0 . On the other hand, for an axion withmass and decay parameter given by m φ = 10 − eV and λ = 6 × , respectively, we notethat experiments such as ACT rules out such combination of parameters. Thus, consideringnumerical solutions with a difference within sub-percent levels with respect to CDM, andwith lower amplitude that the minimum sensitivity of CMB experiments, the range of axionmasses with λ (cid:54) = 0 consistent with CMB observations seem to be given by m φ > − eV.This will be important in Section 3.4 when we carry out the statistical analysis. – 11 – l = l ( l + ) C l / ( K ) CDM m =10 eV, =0 m =10 eV, =10 m =10 eV, =6×10 m =10 eV, =0 m =10 eV, =10 PlanckSPTACT Multipole Moment l l ( K ) Figure 4 : CMB anisotropies for high multipoles. Data from Planck (green dots), SPT [108](red dots) and ACT [109] (blue dots) are shown to compare with our numerical solutions. Formasses lighter as m φ = 10 − eV and λ = 0 , , we found notorious discrepancies with the ob-servational data. The lower panel shows the relative differences between CDM and SFDM with (cid:8) m φ = 10 − eV , λ = 0 (cid:9) (yellow dashed line) and (cid:8) m φ = 10 − eV , λ = 6 × (cid:9) (red dottedline). The horizontal green, red and blue lines indicate the minimum sensitivity for Planck,SPT and ACT observations respectively, given by ( σ Planck , σ
SPT , σ
ACT ) = (3 . , . , . µ K . Likewise, Figure 5 shows the MPS for CDM (black line) and SFDM with masses m φ =10 − , − eV for several values of the decay parameter λ = 0 , , , , (solid gray,dashed gray, dashed blue, dotted yellow, and solid cyan line respectively), as well as forthe extreme values of λ corresponding to each value of the axion mass λ = 4 . × for m φ = 10 − eV (dashed green line), and λ = 10 , . × for m φ = 10 − eV (dashed green,and dotdashed red line respectively). For each case we observe the well-known cut-off at largewavenumbers, but this time there is also present a bump in the MPS at the cut-off scale foreach value of the axion mass. As discussed in Sec. 2, the tachyonic instability produces anenhancement in the density contrast after the onset of the oscillations of the axion field, andsuch instability is going to be present in the MPS, at least for a range of wavenumbers asexplained in [93] and in Section 2.4. It is important to note that, for the case m φ = 10 − eVwith λ = 10 (green dashed line in the bottom Figure) the bump is within the range ofwavenumbers showed in Figure 2. For a qualitative comparison, we have included data fromBOSS DR11 (yellow dots) [110], and from Ly α forest (black dots) [111]. Based on the comparison of the CMB anisotropies and MPS with available data, we see thatthe SFDM model describes such cosmological observables as good as CDM model does, aslong as the axion mass is m φ = 10 − eV. This is a lower value than that imposed by Lyman- α observations of the 1-dimensional flux power spectrum (P1D), for the axion mass endowedwith a quadratic potential (FDM case), given by m φ (cid:38) − eV [112, 113] . Note that [106] reports a different constraint with the same observations, claiming that including quantumpressure to numerical simulations of FDM leads to a lower bound of m φ = 10 − eV. – 12 – P ( k ) m = 10 eV Linear Regime Semi and Non Linear Regime
CDM= 0= 10= 10 = 10 = 10 = 4.3 × 10 Anderson et al, 2014Chabanier et al, 2019 wavenumber k ( h /Mpc) P ( k ) / P ( k ) C D M ( % ) P ( k ) m = 10 eV Linear Regime Semi and Non Linear Regime
CDM= 0= 10= 10 = 10 = 10 = 10 = 1.5 × 10 Anderson et al, 2014Chabanier et al, 2019 wavenumber k ( h /Mpc) P ( k ) / P ( k ) C D M ( % ) Figure 5 : MPS for SFDM with axion masses m φ / eV = 10 − , − , and λ from zero up tothe maximum values reached for each axion mass. It can be noted that, for all the axionmasses considered there is a cut-off at small scales (larges k ’s), and even more, there is anenhancement of the MPS at such scales when considering large values of the parameter λ .Cosmological data from BOSS DR11 (yellow dots) [110], and from Ly α forest (black dots)[111] are shown for reference. See text for more details.To qualitatively assess the constraints that the Lyman- α P1D can impose to the modelunder consideration, we compare in Figure 6 the relative difference with respect the LCDMmodel, for the 1-dimensional matter power spectrum with the precision of current P1Dmeasurements with data sets such as eBOSS [114], HIRES/MIKES[115] and XQ-100 [116](yellow, blue and red rectangle respectively). We do this for the following combinations: m φ = 4 × − eV for λ = 0 , . × (green lines), m φ = 10 − eV for λ = 0 , × (bluelines), and m φ = 10 − eV for λ = 0 , . × (red lines). We can see that combinationswith λ = 0 are excluded by the data except for the larger mass, while combinations with non– 13 –ull and larger values of the decay parameter λ might be allowed. This means that an axionfield endowed with a trigonometric potential could still be allowed by Lyman- α observations.Definite constraints, of course, might come from a full analysis of the Lyman- α P1D withcurrent and future data such as DESI [117].
Figure 6 : 1D MPS for the Axion field compared to the Λ CDM one. We show the cases m φ = 4 × − eV for λ = 0 , . × (green lines), m φ = 10 − eV for λ = 0 , × (bluelines), and m φ = 10 − eV for λ = 0 , . × (red lines). For reference we have includedcolored rectangles indicating the rough precision of current data from BOSS [114] (yellow),HIRES/MIKES [115] (blue) and XQ-100 [116] (red) to show that these experiments can beused to constraint the axion field parameters m φ and λ . In this Section we will analyze the parameter space of our model in order to find constraintsusing data from the Planck 2015 data release [118]. This is done by using the parameterestimator code
Monte Python [119], to compute the posterior distribution of several cos-mological parameter by implementing Bayes’ Theorem, which reads P (Θ | D) = Π(Θ) L (D | Θ)E(D) (3.1)where Θ stands for the parameters of the cosmological model, D is the data from cosmologicalsurveys, Π is the prior probability, the likelihood L representing the probability distributionof the data for each allowed input Θ , and the evidence E which encodes how well our originalassignments managed to predict the data, and which can be calculated as E = (cid:82) Π(Θ) L (D | Θ)dΘ .Our model is defined by two parameters, the axion mass m φ and the decay param-eter λ , and additionally by the standard cosmological parameters of Λ CDM model, thephysical baryon density parameter ω b , the (logarithmic) power spectrum scalar ampli-tude log(10 A s ) , the scalar spectral index n s , the Thomson scattering optical depth dueto reionization τ reio , and the angular size of sound horizon at decoupling θ s . Note thatwe do not include Ω c (dark matter density parameter) because that information will beprovided by our axion field. Thus, in total we have 7 cosmological parameters given by– 14 – = (cid:2) ω b , log(10 A s ) , n s , τ reio , θ s , log m φ , log λ (cid:3) , where we have defined the scalar fieldparameters m φ and λ in logarithmic scale. We are going to consider the CMB as the cos-mological observable to constraint our model, hence, we will take the data and likelihoodsfrom Planck Collaboration 2015. The initial input given to the code to run the chains issummarized in Table 2, where the initial mean value, as well as the priors and the 1- σ value,are specified for each of the parameters Θ . Particularly, the input for the axion field parame-ters m φ and λ are chosen to be consistent with the numerical solutions obtained with class .Thus, the means and priors for m φ and λ will be setted based on the cosmological evolutionof the axion field that we were able to explore numerically.Param mean prior min prior max 1- σ ω b ∗ θ s . None None × − ln A s . None None 0.0029 n s τ reio log λ log m φ -22 -26 -16 0.05 Table 2 : Initial input for the parameters Θ of our SFDM model. Whereas no priors werespecified for the standard cosmological parameter (only a lower bound prior for τ reio of 0.04),the prior for the axion field parameters were chosen according to the numerical solution weobtained from the class code.We have run the chains with the Metropolis-Hasting algorithm, with the Gelman-Rubinconvergence criterion [120] fulfilling R − < . . The minimum of the likelihood and the χ function we obtained are respectively given by − ln L min = 5636 . , χ = 1 . × . The posteriors are shown in Figure 7. While the standard cosmological parameters (cid:2) ω b , log(10 A s ) , n s , τ reio , θ s (cid:3) show their observed values at the present day, the axionfield parameters m φ and λ have a non–Gaussian posterior. However, the axion mass hasa lower bound given by log m φ = − . at . CL. This is consistent with the previousresult shown in Section 3.1, where we compare our numerical solutions with data from theCMB anisotropies (see Figure 4). Thus, whereas a restriction for the axion mass was found,it seems that the data from CMB is not able to constraint the value of the decay parameter λ . That is, for all the values of λ we were able to explore, it was possible to find consistentnumerical solutions for the rest of the cosmological parameters. D and velocity growth factor f with scale-dependence It is well known that the growth factor D for CDM model does not present explicit scaledependence, i.e., it is independent of the wavenumber k , but it is the transfer function T which carries such information. Such separation of variables on the gravitational potential Φ given by Φ( k, a ) ∝ T ( k ) D ( a ) , can be done in a standard CDM scenario, and it allows tostudy the growth of matter overdensities in the structure formation process [121–125]. Forinstance, the standard parameterization for the velocity growth factor is given by f ( z ) = – 15 – .041.041.041.041.04 θ s l n A s n s τ r e i o -5-2.3-0.4991.34 l og λ -25 -23 -21 -19 -17 log m φ ω b -25-23-21-19-17 l og m φ θ s ln A s n s τ reio -5 -2.3 -0.499 1.3 4 log λ Figure 7 : 1D and 2D posterior distributions for the axion field parameters m φ and λ (inlogarithmic scale) together with the standard cosmological parameters of CDM model. Wecan set a lower bound for the value of the axion mass of log m φ = − . at . C.L. Onthe other hand, a flat posterior is obtained for the decay parameter λ , indicating that CMBanisotropies do not constraint such parameter, at least within the prior − ≤ λ ≤ . Seetext for more details. Ω γm ( z ) [126–132], where γ is called the growth index, and Ω m is the energy density parameterfor the total matter as function of the redshift z . Such expression does not contain explicitinformation of k . However, the scale-dependence on the quantities D and f have been studiedin alternatives models of gravity [125, 133–139] mainly due to the appearance of an effectiveNewton’s constant containing explicit dependence on k . Therefore, while it is true that withinthe CDM scenario the growth factor D and its velocity f are the same for every mode k , thismay not be true in particular for models with a cut-off in the mass power spectrum, such asthose we are studying in this work.To explore possible deviations from the CDM model on such cosmological quantities, inthis section we present an approach to obtain the evolution of both, the growth factor D and– 16 –he velocity growth factor f as function of the wavenumber k for SFDM with the axion–likepotential. As starting point, let us revisit the system of equations that rules the dynamicsof the SFDM linear perturbations after the onset of rapid oscillations. From Eq. (2.16), andconsidering cos θ ∼ sin θ ∼ , we find δ (cid:48)(cid:48) + ω δ = − ¯ h (cid:48)(cid:48) k k J k (cid:48) J k J δ , (4.1)where, in contrast to Eq. (2.13), we are not neglecting the evolution of the Jeans wavenumber k J . Two main features can be seen in Eq. (4.1): 1) the solution of δ will always be coupledto δ , and 2) the solution for δ will depend on the wavenumber k .Following recent literature, where the growth factor is defined in terms of the densitycontrast [85, 125, 131, 140–146], we define a scale-dependent growth factor D as D k ( z ) ≡ δ ( z, k ) δ ( z = 0 , k ) , (4.2)so that D k ( z = 0) = 1 . The definition given in Eq. (4.2) allows us to generalize the growthfactor in such a way that it is possible to track its evolution for each wavenumber k . Thisis done in Figure 8, where we show the growth factor D k ( z ) for k = 10 − Mpc − (yellow), k = 0 . Mpc − (blue), k = 10 Mpc − (red), for SFDM with mass m φ = 10 − eV, and which isendowed with a quadratic potential (FDM case λ = 0 , dashed lines), and with a trigonometricpotential with tachyonic instability as well ( λ = 1 . × , dotted lines). The initial amplitudefor the growth factor with trigonometric potential is smaller than that of the FDM case, butaround z ∼ the growth factor with λ = 1 . × suffers the tachyonic instability and itsamplitude increases faster than the free axion case. It is important to recall that such fastincrement of the growth factor amplitude, and therefore in the density contrast, is translatedas a bump in the mass power spectrum for large k ’s, as was shown in Figure 5. Interestinglyenough, from z ∼ up to the present day all curves evolve as CDM , which implies thatfor z < the growth factor D k ( z ) in Eq. (4.2) becomes effectively scale-independent. Figure 8 : Growth factor D k ( z ) for an axion mass m φ = 10 − eV with both, quadraticpotential (dashed lines) and trigonometric potential (dotted lines). The tachyonic instabilityis manifested for the latter as a fast increment of amplitude for D k at z ∼ . Horizontaldotted gray line indicates D = 1 , where all curve converge at z ∼ .– 17 –oing further, the definition given by Eq. (4.2) enables us to write the velocity of thegrowth factor f k ( z ) as follows, f k ( z ) = d log D k ( N ) dN = − (1 + z ) d log D k ( z ) dz = − (1 + z ) d log δ ( z, k ) dz . (4.3)The dependence on k for the function shown above can be seen in Figure 9, where the colorsand the line style for each curve are the same as in Figure 8. Notice that the velocity growthfactor for k = 10 − Mpc − is the same as that of CDM and is not affected by the values of λ ;that is, at large scales we recover the same behavior of CDM. Similarly, for k = 0 .
53 Mpc − the evolution is also independent of the values of λ , although the CDM evolution is notrecovered for z (cid:38) . Thus, it is possible to distinguish between CDM and SFDM at highredshifts. The result is different for the wavenumber k = 10 Mpc − , where we can see thatthe evolution of f k is different for the two values of λ considered. However, from z ∼ tothe present day, the evolution of f k for each mode and for each value of λ is the same as thatof CDM. This means that at late times the MPS of the axion field should keep a constantratio with respect to that of CDM. Figure 9 : Velocity growth factor f k ( z ) for an axion with mass m φ = 10 − eV. Dashed(dotted) lines correspond to λ = 0 ( λ = 1 . × ) , and yellow, blue and red lines indicatewavenumbers k = 10 − , . , Mpc − respectively. For k (cid:28) Mpc − the velocity growthfactor evolves as CDM for all redshift, whereas for k > Mpc − each mode evolve indepen-dently until z ∼ , where all curve converge to the CDM case, and the velocity growth factoris the same for all wavenumbers. See text for more details.For k = 10 Mpc − , we attribute the notorious difference at z > between the FDMcase and the axion-like potential to the tachyonic instability, since this effect is manifestedat such range of scale (see Figure ?? ). Finally, since the growth factor D k and the velocitygrowth factor f k coincide with those of CDM for < z < , the combined observable f k σ at < z < (range within which we can search for observational constraints) will be insensitiveto the details of the axion case, as can be seen in Figure 10, where the overlapped curvescorrespond to the same values of wavenumbers k and decay constant λ as those in Figures 8and 9. – 18 – igure 10 : Velocity growth factor f k and variance σ combined as function of both, wavenum-ber k and redshift z . The overlapped curves have the same values of k and λ as those of theprevious Figures 8 and 9. Observational data are shown in colored squares from 2dFGRS[147], WiggleZ [148], 6dFGRS [149], VIPERS [150], SDSS DR7 Main [151], BOSS DR12[152], FastSound [153], eBOSS DR14Q [154], 2MTF [155] and SDSS-II [156].Whereas strong constraints have been imposed to the mass of the scalar field dark matter m φ (through galactic observations, mass power spectrum and CMB anisotropies), havingobservations of matter distribution at high redshifts can be useful to explore the nature ofDM, and particularly to constraint the decay parameter λ of the axion field. The cosmologicaleffects of such parameter have not be studied in great detail, and we are showing that it hasa characteristic imprint on the structure formation, at small scales (see MPS in Figure 5) aswell as at high redshifts (Figure 8 y 9). The
Halo Mass Function (HMF) encodes the comoving number density of dark matter halosas function of the halo mass, and it constitutes a representative cosmological probe of darkmatter and dark energy. It can be used for example to constraint the value of the combinedparameters σ and Ω M (the power spectrum normalization and the matter density parameterrespectively), and also to characterize the dark energy equation of state ω [157–159]. A halois an overdensity of matter, which lie on the nonlinear regime of structure formation. Tostudy such objects numerical simulations have to be carried out [160–162]. However, semi-analytical analysis can be performed as well, as have been shown in [163–166]. Particularly,the procedure to obtain the semi analytical HMF of our model will be similar to that givenby [167, 168].First, we define the window functions we are going to implement: the Top-Hat windowfunction W T H , which is a filter with spherical symmetry in real space, and the Sharp-k windowfunction W SK , defined as a Top-Hat function in Fourier space. They are given, in Fourierspace, by W T H ( kr ) = 3( kr ) [sin( kr ) − kr cos( kr )] , W SK ( kr ) = Θ(2 π − kr ) . (4.4)The Top-Hat function is useful to work with the LCDM model, while the Sharp-k functionis useful for suppressed power spectra, which is the case of the axion field. More discussionabout the choice of the window functions are given in [96, 169–171] and references therein.– 19 – igure 11 : Square root of the variance at z = 0 as function of the halo mass for Λ CDM(solid black line) and SFDM for λ = 0 (solid lines) and λ = 10 (dotted lines) with Top-Hat(blue lines) and Sharp-k (red lines) window functions respectively.One of the quantities of interest is the variance , which is calculated as σ ( r ) = (cid:90) d (cid:126)k (2 π ) P ( k ) W ( kr ) . (4.5)In Figure 11 we show the square root of the variance at redshift z = 0 for Λ CDM and theaxion field, the latter with quadratic ( λ = 0 ) and trigonometric ( λ = 10 ) potential, and m φ = 10 − eV. The variance of the axion field, for both the Top Hat and Sharp-k windowfunctions, show a constant value for small masses, in contrast to the result of the Λ CDMmodel which is always increasing. The asymptotic values for the quadratic and trigonometricpotentials are different; for the latter it can be seen that it is the tachyonic instability, andultimately the non-linearities of the trigonometric potential, that enhances the value of σ atsmall masses.To be able to study the gravitational collapse, it is necessary to take into account thescale-dependence that inherently SFDM models possess. Let us explain this as follows: differ-ent wavenumbers will grow at different rates, as was shown in Figure 8 and 9 for the growthfactor D k ( z ) and the velocity growth factor f k ( z ) respectively. Thus, the gravitational col-lapse for each mode k will be different. The threshold value at which some matter fluctuationassociated to a given mode k will collapse, is known as the critical overdensity , which withina standard CDM scenario is defined by [167, 172–174] δ crit = 1 . D CDM ( z = 0) D CDM ( z ) , (4.6)where D CDM is the growth factor for CDM D CDM = 5Ω m H (cid:90) daa H . (4.7)In the case of SFDM, we can in principle apply a similar expression, but using the growthfactor introduced in Eq. (4.2), δ crit = 1 . D k ( z = 0) D k ( z ) . (4.8)– 20 –e are interested in building up the HMF at z = 0 , and even when Eq. (4.8) contains explicitdependence on the wavenumber k , the growth factor for SFDM coincides with the CDM caseat later times, as was discussed in Section 4.1. Therefore, this approach will not be useful tostudy the effects of gravitational collapsing with scale dependence on the HMF.Notwithstanding, we can rather consider approaches as those that have been carried outon previous studies on this subject [85, 168, 175], where the authors introduce a definition ofthe growth factor in terms of several density contrasts rates. Particularly, in Eq. (10) from[168] it is shown the relative amount of growth between CDM and SFDM as D CDM ( z ) D SFDM ( M, z ) = δ CDM ( k, z ) δ SFDM ( k, z ) δ CDM ( k , z h ) δ SFDM ( k , z h ) δ SFDM ( k , z ) δ CDM ( k , z ) δ SFDM ( k, z h ) δ CDM ( k, z h ) , (4.9)where k = 0 . h/ Mpc is a pivot scale, and z h = 300 is the redshift at which the shape ofthe CDM power spectrum has frozen in. We observe that the pivot scale is small, and forsuch mode the growth factor of SFDM will evolve as CDM. Then, the second and third ratioin Eq. (4.9) are δ CDM ( k , z h ) /δ SFDM ( k , z h ) = δ CDM ( k , z ) /δ SFDM ( k , z ) (cid:39) . On the otherhand, the overall effect of the last quotient on the right hand side of Eq. (4.9) occurs for k > where δ SFDM ( k, z h ) < δ CDM ( k, z h ) , suppressing the amplitude of the growth factor for suchwavenumbers at z = z h , while for k < such quotient is equal to 1. Besides, notice that z h ∼ , which is the order of magnitude of redshift where the cosmological evolution of thegrowth factor is basically that of CDM (see Figure 8), and thus, the last term of the aboveequation can be taken as δ CDM ( k, z h ) /δ SFDM ( k, z h ) (cid:39) almost independently of the valueof the wavenumber k . Thereby, the main responsible to carry the scale dependence of thegrowth factor D will be the first term in Eq. (4.9). We concluded that the critical overdensitycan be written as δ crit ( k ) = 1 . δ CDM ( k, z ) δ ( z, k ) . (4.10)From this expression it can be seen that for small wavenumbers the CDM case is recov-ered, since k → erases the scale-dependence on δ crit . In other words, the density contrastfor the axion field will evolve as CDM for small values of k , specially at late times. On theother hand, the axion density contrast for large values of k do not grows as CDM in all itsevolution. Particularly at present day, δ has less amplitude than δ CDM , which is clearly seenin the MPS on Figure 5. Note that our definition of the critical overdensity given by theabove equation is a reduction from that used by authors in [85, 168, 175], where a particularnormalization and an analytical function based on Axion camb results are implemented for ascale/mass-dependent growth factor. Within our analysis, such scale dependence is encodedin the density contrast given by our new dynamical variable δ ( z, k ) , and which we have ob-tained numerically from class . We want to highlight that from our definition (4.10) we canrecover the results from the previous work mentioned above. For example, Figure 12 showsthe critical overdensity as function of the wavenumber, analogous to that of Figure 2 from[168], where δ crit is shown as function of the mass. Such comparison is valid for an axionmass of − eV (green line on Figure 2 from [168], and blue line in Figure 12), since in thiswork we have consider the effect of tachyonic instability in the critical overdensity as well.We observe that δ crit shows a clear scale dependence for wavenumbers k > h/ Mpc, which istranslated to small halo masses, as we shall see below.– 21 – igure 12 : Critical overdensity δ crit at redshift z = 0 as function of the wavenumber k foran axion field with mass m φ = 10 − eV. For an axion with quadratic potential (FDM case,blue line), δ crit grows for wavenumbers k > h/ Mpc, while for the trigonometric potential(red line) there is a decrease for < k Mpc /h < due to the tachyonic instability, and thengrows like the quadratic case. The horizontal black line indicates the value δ crit = 1 . .Notice that for the trigonometric potential (red line) there are wavenumbers for whichthe critical overdensity is less than in the CDM case, implying that structures associated tosuch modes will be able to grow with a threshold δ crit lower than in a standard CDM scenario,and also compared to the case of a free axion. This is why the MPS exhibits a bump at smallscales, as was shown in Figure 5.On modeling the gravitational collapse we will consider both, the Press-Schechter (P&S)and the Sheth-Tormen (S&T) formalism for spherical and ellipsoidal collapse models respec-tively [163, 165]. Such collapse models are encrypted in the following function f ( ν ) = (cid:113) νπ e − ν/ for P&S ,A (cid:113) qνπ (1 + qν ) − p e − qν/ for S&T , (4.11)where ν ≡ δ crit /σ is the peak height of perturbations, while A = 0 . , p = 0 . , q = 0 . for the S&T model in Eq. (4.11) according to [167]. Finally, the semi-analytical HMF has thefollowing expression dnd ln M = −
12 ¯ ρM f ( ν ) d ln σ d ln M . (4.12)Now we can analyze the HMF for an axion field endowed with a trigonometric potential,and compare it with the CDM prediction, as well as with the free axion case. Figure 13 showsthe semi-analytical halo mass function at redshift z = 0 and axion mass m φ = 10 − eV.We separate our analysis in three different cases depending on the window functionimplemented: Top-Hat, Sharp-k, and Top-Hat including the critical overdensity with scaledependence. For all cases we consider the collapse models given at Eq. (4.11), for the FDMcase with quadratic potential ( λ = 0 ) and an axion field with trigonometric potential ( λ =10 ). When considering the Top Hat window function W T H without a scale dependent criticaloverdensity, differences between the HMF for SFDM and CDM appear at small mass scales, ascan be seen at upper left panel in Figure 13. However, since we have used Eq. (4.8), the HMF– 22 – igure 13 : Halo Mass Function for the CDM (black line) and SFDM models with the samevalues of λ, m φ and z as Figure 11. Two different collapse models are shown, P&S (blue lines)and S&T (red lines). Upper left: HMF with a Top Hat window function. Upper right: HMFwith a Top Hat window function and scale dependent critical overdensity. Bottom: HMFwith a Sharp-k filter.do not exhibits the cut-off of the MPS when using this window function. This is because, aswe mentioned before, the dependence on k in the growth factor (4.2) is lost at late times. Onthe other hand, including a critical overdensity with dependence on scale through Eq. (4.10)(upper right panel in Figure 13), a steep cut-off appears at M ∼ M (cid:12) /h for λ = 0 and M ∼ M (cid:12) /h for λ = 10 . This result is consistent with that of [85, 168] for the particularcase in which SFDM constitutes all the DM content, i.e., when Ω φ / Ω CDM = 1 . Finally, theHMF with the Sharp-k function W SK is shown in the lower panel of Figure 13. In this case,we have use Eq. (4.8), since the cut-off at a given scale is captured by the Sharp-k windowfunction, as discussed by [96]. Note that, whereas the turn around of the halo mass functionis slightly different for λ = 0 and λ = 10 , the cut-off for both of them occurs approximatelyat the same range of mass scale (cid:46) M ( h/M (cid:12) ) (cid:46) .For all the cases studied we observe as a new general feature in the HMF, an incrementin its amplitude when considering one of the two following considerations:1.- ellipsoidal collapse S&T model (red lines in Figure 13),2.- Axion-like potential in the tachyonic instability regime (dotted lines in Figure 13).– 23 –hese two new features in the HMF can be contrasted with recent results obtainedin [176], where observational constraints on Warm Dark Matter (WDM) and Fuzzy DarkMatter (FDM) models are imposed. In particular, the HMF for FDM is modeled by imple-menting the analytical function (cid:18) dnd ln M (cid:19) SF DM = f ( M ) + f ( M ) (cid:18) dnd ln M (cid:19) CDM , (4.13)where the functions f and f are given by f ( M ) = β exp (cid:34) − (cid:18) ln MM × M (cid:12) (cid:19) /σ (cid:35) , f ( M ) = (cid:34) (cid:18) MM × M (cid:12) (cid:19) − α (cid:35) − /α , (4.14)and the CDM halo mass function is (cid:18) dnd ln M (cid:19) CDM = 3 . × − (cid:18) M . × M (cid:12) (cid:19) − . (cid:18) MM (cid:12) (cid:19) . (4.15)The different parameters used in the above expressions are α = 0 . , M /m − . =4 . , M /m − . = 2 . , β/m . = 0 . , σ = 1 . , and where m = m φ / − eV. In orderto put a limit value on the FDM mass, in that work the parameter m is varied up tothe maximum value such that the HMF is more suppressed than the excluded WDM cases.Doing so, the mass obtained in [176] is m φ = 2 . × − eV. We show the HMF for this resultin Figure 14 (green solid line), as well as the HMF for the CDM model (black solid line).We have also included the numerical results for the HMF obtained in this work, consideringonly those which are consistent with the reported values of the HMF according to stellarstreams measurements [177]. These measurements refer to a stream of stars (the GD-1 stream)that have been detected in Sloan Digital Sky Survey (SDSS) data [178]. Stellar streams areoriginated from the tidal disruption of globular clusters, forming an elongated structure that,when it is gravitationally perturbed by dark subhaloes, some gaps in the stellar distribution ofsuch elongated structure are produced. Therefore, stellar stream observations would provideinformation about the dark matter subhaloes [179–186].All our numerical results presented in Figure 13 underestimate the result of the analyticalapproach modeled by Eq. (4.13) when considering m φ = 2 . × − eV. For such mass value,and indistinctly of the collapse model, only those HMF with a Top-Hat window functionare consistent with the stellar stream measurement data. This is because the suppressionimpressed in the HMF due to the Top-Hat window function is less than that imposed by theSharp-k window function, and even less in comparison with that of a Top-Hat with criticaloverdensity with scale dependence, as we already showed in Figure 13. Therefore, if we wantto consider the HMF for the axion field with a pronounced suppression, stronger constraintshave to be imposed to our model ( m φ ≥ . × − eV). This will be also the case whenconsidering an analytical model as that presented in [186], where the bound for the mass isgiven by m φ ≥ . × − eV.We found that, for m φ = 10 − eV, it is possible to be in agreement with streams mea-surements for an axion field HMF when a Top-Hat with a scale-dependent critical overden-sity is considered. This is achieved precisely with the ellipsoidal S&T collapse model in the– 24 – M ( M / h ) d n / d l n M ( h / M p c ) m =10 eV CDM=0 (P&S - Top-Hat)=10 (P&S - Top-Hat)=0 (S&T - Top-Hat)=10 (S&T - Top-Hat)=10 (S&T - Top-Hat with crit ( k ))FDM with m =2.1×10 eV (Schutz 2020)Streams (Banik et al, 2019) Figure 14 : HMF for the analytical approach by [176] for FDM with m φ = 2 . × − eV(green solid line), and our numerical results for SFDM HMF with m φ = 10 − eV. Orangeerror bars show data from streams measurements [177, 185]. See text for more details.presence of tachyonic instability (with λ = 10 ), as it is shown in Figure 14 (dotted redline). Therefore, the two new considerations mentioned above and included in our analysis,which lead to an increment in the amplitude of the axion HMF, can play an important rolein order to guarantee consistency with stellar stream measurements for haloes with masses ∼ − M (cid:12) . However, this will be possible only for axion masses m φ ≥ − eV, whichconstitutes a stronger constraint as those imposed, for instance, by Lymann- α [112, 113],but lies within the range of masses that could be tested by 21-cm observations [187, 188].As it is also noted in [176], the results from the analytical approach are too conservative inthe sense that they are not taking into account the scale-dependent growth of structure, nei-ther the scale-dependent critical overdensity. In their analysis, masses for FDM with values m φ (cid:46) . × − eV would be excluded, whereas we are showing that the SFDM HMF witha Top-Hat window function for any of the collapse model studied (yellow and blue lines) lieswithin the range obtained from stream measurements for m φ = 10 − eV. Particularly, wehave shown that the suppression of subhaloes due to a SFDM endowed with a trigonometricpotential is in agreement with the constraints imposed by measurements of stellar stream-ing when the axion mass is m φ = 10 − eV. Without the effect of the tachyonic instability,stronger constraints on the axion mass would be imposed. Ultra–light DM bosons as SFDM model constitute a compelling candidate to substitute theCDM model. In this paper we presented a formalism to handle the cosmological equations byusing the tools of dynamical systems for both the background and the linear perturbations. Atthe background level, the presence of a trigonometric potential shows a delay in the momentwhen the axion field starts to oscillate and behaving as CDM. These values of the onset ofoscillations are shown in Table 1 in terms of the scale factor a osc . We have explored withsome depth the effect dubbed as tachyonic instability, which occurs due to the nonlinearitiesof the potential (1.1). For extreme values of λ , once the axion field starts to oscillate thedensity contrast grows with more amplitude than that of standard CDM and SFDM with a– 25 –uadratic potential (FDM). We indicated the duration of the tachyonic instability as well asthe range of wavenumbers that suffer such effect.On the other hand, we built observables such as the CMB anisotropies and the 3D and 1Dmatter power spectrum. The power spectrum of temperature fluctuations for the axion fieldshows major discrepancies for high multipoles, and a limiting case given by m φ ≥ − eVand λ ≤ × states the values of masses and decay constants that are in agreement withhigh multipoles experiment such as ACT and SPT. When considering the Planck experiment,large values of λ are allowed, as can be seen on the lower panel of Figure 4, and which isconsistent with our statistical analysis using Planck data (Figure 7). When analyzing the3D matter power spectrum, the well known cut-off at small scales is reproduced for both,FDM and the axion field. Nonetheless, for large values of λ a bump appears at the cut-off scale as consequence of the tachyonic instability. This is the imprint of the axion–likepotential (1.1) on the formation of large scale structures. As an additional analysis, wecomputed the 1D matter power spectrum, which is closely related to the flux power spectrumthat can be used to constraint DM models with Lyman- α observations. For a scalar fieldendowed with a quadratic potential (FDM), a lower bound have been imposed to the mass,ruling out masses with values m φ < . × − eV. However, we have observed that whenconsidering a trigonometric potential, mass values lower than this bound are allowed. Thus,this kind of observations and the experiments that involve them can be used to constraintboth parameters, m φ and λ .When performing the statistical analysis, we obtain a lower bound for the axion massgiven by log m φ = − . at . C.L. This result is consistent with our numerical analysisobtained for the CMB anisotropies. However, the decay parameter λ is not constrained, andall the values we explored numerically have equal probability, i.e., the parameter λ presentsa flat posterior. Thus, SFDM with quadratic and trigonometric potential are in agreementwith CMB observations, and it is the axion mass who still plays an important role in theconstrains on this type of models for such observations.Motivated by the scale-dependence of the scalar field dark matter models, we proposed agrowth factor D k and a velocity growth factor f k with explicit dependence on the wavenumber k . This was performed through the density contrast and its derivative. Effectively, there aredifferences in the evolution of D k and f k for each value of k , but all of them evolve as colddark matter from certain value of redshift ( z ∼ for D k , and z ∼ for f k ) to the presentday ( z ∼ ). Having this quantities allowed us to build the combined parameter f k σ wherethe differences are marginal when comparing with CDM.The tachyonic instability is manifested in both the variance and the halo mass function asan enhancement in the amplitude for low masses. The Top-Hat and Sharp-k window functionshave been considered, and each one affect the HMF in a different way. On one hand, the HMFwith a Top Hat window function present a decrease at small masses for SFDM in comparisonwith the CDM model, but such decrease is not the one expected from a mass power spectrumwith a cut-off. However, when considering a critical overdensity with explicit dependence onscale, the HMF exhibits a steep cut-off. On the other hand, with the Sharp-k window functionthe halo mass function for the axion field has a cut-off less pronounced than the mentionedabove, but approximately at the same mass scales. All the cases studied were performed fortwo different gravitational collapse models, the Press-Schechter and the Sheth-Tormen forspherical and ellipsoidal collapse model respectively. Both of them produce qualitatively thesame HMF, with small differences at small scales.Future astronomical observations planned by collaborations such as the Dark Energy– 26 –pectroscopy Instrument (DESI) [117] and the Large Synoptic Survey Telescope (LSST, nowVera C. Rubin Observatory) [189] will explore the Universe with major accuracy. Particularly,the LSST will be able to constraint light bosonic dark matter mass m φ ∼ − eV by probingthe MPS for halos with ∼ M (cid:12) [190]. On the other hand, the Sloan Digital Sky Survey(SDDS) can be used for searches of low-surface brightness dwarf galaxies at small scales, as isdiscussed by authors in [191]. Besides, the 21cm signal detected by EDGES [192] can be usedto study the properties of dark matter [193], in particular to probe small scale structures [187].In fact, it was recently proposed that through forthcoming experiments such as the SquareKilometre Array (SKA) [194], 21cm observations can be used to constraint the scalar fieldmass as well at wavenumbers < k < (Mpc − ) [188]. It has been recently studiedthe implications of a post-inflationary symmetry breaking of axion-like particles on the bornof first generation stars, and on small scale structures [195]. For a recent review on othergravitational probes for ultra-light axions see [196]. Therefore, the physics at these smallscales and high redshifts, will reveal more information about the properties of this model ofaxion-like dark matter. Acknowledgments
FXLC acknowledges CONACYT and the Programa para el Desarrollo Profesional Docentefor financial support. This work was partially supported by Programa para el DesarrolloProfesional Docente; Dirección de Apoyo a la Investigación y al Posgrado, Universidad deGuanajuato 099/2020; CONACyT México under Grants No. A1-S-17899, 286897, 297771;and the Instituto Avanzado de Cosmología collaboration.
A General dynamical variables for scalar field perturbations
To translate the same scheme we used for the background evolution of axion fields in Section 2,where we were able to write down a dynamical system for the KG equation, we propose thefollowing new variables for the scalar field perturbation ϕ and its derivative ˙ ϕ [96], u = (cid:114) κ ˙ ϕH = − Ω / φ e α cos( ϑ/ , v = κy ϕ √ − Ω / φ e α sin( ϑ/ , (A.1)which after substitution on the perturbed KG equation (2.13) lead to the following differentialequations ϑ (cid:48) = 3 sin ϑ + 2 ω (1 − cos ϑ ) + y − e − α h (cid:48) sin (cid:18) θ (cid:19) sin (cid:18) ϑ (cid:19) +Ω / φ (cid:20) cos (cid:18) ϑ − θ (cid:19) − cos (cid:18) θ (cid:19)(cid:21) y y , (A.2a) α (cid:48) = −
32 (cos ϑ + cos θ ) − ω sin ϑ + e − α h (cid:48) sin (cid:18) θ (cid:19) cos (cid:18) ϑ (cid:19) + Ω / φ (cid:20) sin (cid:18) θ (cid:19) + sin (cid:18) ϑ − θ (cid:19)(cid:21) y y . (A.2b)– 27 –or numerical purposes, it is convenient to use as angular variable the difference ˜ ϑ ≡ θ − ϑ ,and then from Eqs. (2.4a) and (A.2a) we obtain ˜ ϑ (cid:48) = − (cid:104) sin θ + sin (cid:16) θ − ˜ ϑ (cid:17)(cid:105) − ω (cid:104) − cos (cid:16) θ − ˜ ϑ (cid:17)(cid:105) + e − α h (cid:48) (cid:34) cos (cid:32) ˜ ϑ (cid:33) − cos (cid:32) θ − ˜ ϑ (cid:33)(cid:35) +Ω / φ (cid:20) cos (cid:18) θ − ˜ ϑ (cid:19) − cos (cid:18) θ (cid:19)(cid:21) y y , (A.3a) α (cid:48) = − (cid:104) cos (cid:16) θ − ˜ ϑ (cid:17) + cos θ (cid:105) − ω sin (cid:16) θ − ˜ ϑ (cid:17) + 12 e − α h (cid:48) (cid:34) sin (cid:32) ˜ ϑ (cid:33) + sin (cid:32) θ − ˜ ϑ (cid:33)(cid:35) + Ω / φ (cid:20) sin (cid:18) θ (cid:19) + sin (cid:18) θ − ˜ ϑ (cid:19)(cid:21) y y . (A.3b)If we further define the variables δ = − e α sin( ˜ ϑ/ and δ = − e α cos( ˜ ϑ/ , Eq. (A.3) can beproperly combined to obtain the dynamical system shown in Eq. (2.14). B Fluid interpretation of the equations of motion of SFDM density per-turbations
Here we report about the fluid interpretation of Eq. (2.16) in terms of the standard fluidvariables for linear perturbations, namely the density contrast δ = δ and the divergence ofthe velocity perturbation θ φ (see Eq. (2.15b)). Following the procedure in [107], we first con-sider the equations of density perturbations well within the regime of rapid field oscillations,Eqs. (2.17a), but written in the form, δ (cid:48) = − θ φ − ¯ h (cid:48) , θ (cid:48) φ = − a (cid:48) a θ φ + k a m φ (cid:32) − ρ φ a k f φ (cid:33) δ , (B.1)where now a prime denotes derivative with respect to τ . Notice that we have used the relation θ φ = k am φ δ , which is found from Eq. (2.15b) for rapid oscillations. A quick comparison withthe standard fluid equations for axion fields (see for instance Eqs. (13) and (14) in [107]),leads us to conclude that the averaged value of the sound speed c s of the axion field, in thenonrelativistic limit, is given by (cid:104) c s (cid:105) (cid:39) k a m φ (cid:32) − ρ φ a k f φ (cid:33) . (B.2)The standard result of the FDM case is obtained in the limit f φ → ∞ ( λ → ), namely (cid:104) c s (cid:105) (cid:39) k a m φ (eg [197]). C Extreme Axion Wave Dark Matter
The tachyonic instability of SFDM in the axion case was firstly studied in [198], from thefield perspective, and was dubbed Extreme Axion Wave Dark Matter (EA ψ DM). Assumingan axion potential in the form V ( φ ) = 2 m φ f φ sin ( φ/ f φ ) , the dynamics of the field startsclose to maximum of the potential, and then the extreme label refers to initial conditions such– 28 –hat φ i /f φ → π . For instance, some of the most extreme values considered in [92] were of theorder δθ ≡ π − φ i /f φ (cid:39) . ◦ .To find the relation between the extreme initial conditions used in [92] and our approach,we proceed as follows. Considering our convention for the axion potential (2.11b), we find forthe initial conditions that m φ H a i Ω r λ cos( φ i / f φ ) = Ω φi . (C.1)In our convention, ESFDM is achieved if φ i /f φ → , and then we see that an extreme initialcondition on the field φ i translates into an extreme initial condition on the density parameter Ω φi → . However, the latter’s value is not independent, as for any choice of the potentialparameters m φ and λ (ie f φ ), one has to fine tune Ω φi to get the right value of Ω φ at thepresent time.The above is the main reason why, in our approach, the extreme case of initial conditionsis interlinked with the (decay) parameter λ : larger values of the latter asks for smaller valuesof φ i /f φ , that is, for more extreme values in the sense that φ i /f φ → . For the fiducialmodel with m φ = 10 − , we find δθ = (174 ◦ , ◦ , ◦ , ◦ , . ◦ , . ◦ ) corresponding to λ = (10 , , , , , . × ) . Thus, our formalism allows initial conditions as extremeas those reported in [92], but covering the whole evolution of the Universe. D Higher order algebraic equation for the scale factor on the onset ofoscillations
In Section 2, we obtained an expression to determine the scale factor at the onset of oscillation a osc given by Eq. (2.11a), which was important to determine the initial conditions for theevolution of the background variables when λ > . Now we will show that it is possible toobtain a more accurate expression for a osc by means of an iterative integration of the equationsof motion at early times.Considering again a radiation domination era, let us take the solution for y given byEq. (2.9) and plug it into Eq. (2.4b), which leads to the new solution, y ( a ) = y ,i (cid:18) aa i (cid:19) + λ φ,i θ i (cid:18) aa i (cid:19) − (cid:18) λ Ω φ,i (cid:19) θ i (cid:18) aa i (cid:19) + 12 (cid:18) λ Ω φ,i (cid:19) θ i (cid:18) aa i (cid:19) . (D.1)This solution can be used in Eq. (2.4a) to obtain a new solution on θ , which can be shown tobe θ ( a ) = θ i (cid:18) aa i (cid:19) (cid:40)(cid:34) − λ Ω φ,i
72 + 1726 (cid:18) λ Ω φ,i (cid:19) (cid:35) + λ Ω φ,i (cid:18) − λ Ω φ,i (cid:19) (cid:18) aa i (cid:19) + 926 (cid:18) λ Ω φ,i (cid:19) (cid:18) aa i (cid:19) (cid:41) . (D.2)Setting the previous expression to the onset of oscillations, i.e., a = a osc and θ = π/ , weobtain a quartic order equation for a osc a (cid:34) (cid:18) λ
72 Ω φ, Ω r, (cid:19) a osc + 926 (cid:18) λ
72 Ω φ, Ω r, (cid:19) a (cid:35) = πθ − i a i (cid:112) π / , (D.3)– 29 –here we have used Eq. (2.8). Following the same iterative scheme, we can find higher ordersolutions for y and θ , which we do not show, but that lead to a fifth order equation for a osc , a (cid:34) (cid:18) λ
72 Ω φ, Ω r, (cid:19) a osc + 926 (cid:18) λ
72 Ω φ, Ω r, (cid:19) a + 27442 (cid:18) λ
72 Ω φ, Ω r, (cid:19) a (cid:35) = πθ − i a i (cid:112) π / . (D.4)We have noticed that higher order solutions follow a similar pattern as that in Eq. (D.4),which resembles that of the series expansion of the exponential series, except for the numericalcoefficients. But a close comparison between the two series shows that, (cid:18) λ
72 Ω φ, Ω r, (cid:19) a osc + 926 (cid:18) λ
72 Ω φ, Ω r, (cid:19) a + 27442 (cid:18) λ
72 Ω φ, Ω r, (cid:19) a + · · · < (cid:18) λ
72 Ω φ, Ω r, (cid:19) a osc + 12 (cid:18) λ
72 Ω φ, Ω r, (cid:19) a + 16 (cid:18) λ
72 Ω φ, Ω r, (cid:19) a + · · · = e (cid:18) λ
72 Ω φ, r, (cid:19) a osc . (D.5)Although not a formal demonstration, this exercise shows that a better estimation ofthe scale factor at the onset of the oscillations could be made from the expression a exp (cid:18) λ
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