One-parametric description for scalar field dark matter potentials
RReceived 1 October 2020; Revised 12 October 2020; Accepted 25 October 2020DOI: 10.1002/asna.202113942
ARTICLE TYPE
One-parametric description for scalar field dark matter potentials
Francisco X. Linares Cedeño* | L. Arturo Ureña-López** Instituto de Física y Matemáticas,Universidad Michoacana de San Nicolás deHidalgo, Edificio C-3, Ciudad Universitaria,CP. 58040, Morelia, Michoacán, México Mesoamerican Centre for TheoreticalPhysics, Universidad Autónoma deChiapas, Carretera Zapata Km 4, Real delBosque (Terán), 29040, Tuxla Gutierrez,Chiapas, México Departamento de Física, DCI, CampusLeón, Universidad de Guanajuato, 37150,León, Guanajuato, México
Correspondence * [email protected],** [email protected]
We study the cosmological evolution for a scalar field dark matter model, by con-sidering a parameterization of the evolution equations that allow us to unify in asingle parameter a family of potentials: quadratic (free case), trigonometric (Axion-like case), and hyperbolic. After exploring the cosmological dynamics of this model,we perform a statistical analysis to study the viability of such model in comparisonwith the standard Cold Dark Matter model. We found that the free case is preferredover the other scalar field potentials, but in any case all of them are disfavored by thecosmological observations with respect to the standard model.
KEYWORDS:
Scalar field, Dark matter, Cosmology
Dark matter constitutes one of the open problems in mod-ern cosmology, and in physics in general. Among the sev-eral proposals to address this problem, there are the
ScalarField Dark Matter (SFDM) models. Initial contributions weremade by considering the study of these models on galacticand cosmological scales Guzman, Matos, & Villegas-Brena(2001); Matos & Guzman (2000); Matos & Ureña Lopez(2000, 2001); Matos & Ureña López (2002), where numeri-cal solutions were obtained for the linear density fluctuations,and it was observed that a cut-off scale due to the mass ofthe scalar field naturally arises. Such cut-off has importantrepercussions on the process of structure formation, as havebeen studied since then (for recent results see Chen, Schive,& Chiueh (2017); Diez-Tejedor, Gonzalez-Morales, & Pro-fumo (2014); González-Morales, Marsh, Peñarrubia, & Ureña-López (2017); D. J. Marsh & Niemeyer (2019); D. J. E. Marsh& Pop (2015); Martínez-Medina, Robles, & Matos (2015)).On the other hand, the cosmological evolution of SFDMhas been systematically studied in Cedeño, González-Morales,& Ureña-López (2017); Diez-Tejedor & Marsh (2017);Hlozek, Grin, Marsh, & Ferreira (2015); Hlozek, Marsh, &Grin (2018); Linares Cedeño, González-Morales, & UreñaLópez (2021); D. J. Marsh (2016); D. J. Marsh & Ferreira (2010); D. J. Marsh, Tarrant, Copeland, & Ferreira (2012);D. J. E. Marsh & Silk (2014); Ureña López (2019); Ureña-López & González-Morales (2016). Particularly, in Ureña-López & González-Morales (2016) the dynamics of SFDMwith a quadratic potential was analyzed by rewriting thescalar field equations as a dynamical system and, by usingan amended version of the Boltzmann code
CLASS
Blas,Lesgourgues, & Tram (2011); Lesgourgues (2011), the Cos-mic Microwave Background (CMB) anisotropies as well asthe matter power spectrum (MPS) were obtained. Later, inCedeño et al. (2017) the analysis was generalized by con-sidering a Axion-like potential, i.e., a trigonometric cosine.Such potential is characteristic of QCD Axions Peccei &Quinn (1977a, 1977b); Weinberg (1978); Wilczek (1978) andAxion-like particles Ahn (2016); Cicoli (2014); Cicoli, Good-sell, & Ringwald (2012); Ringwald (2014); Svrcek & Wit-ten (2006); Witten (1984). When considering a trigonometricpotential for the SFDM, a bump in the MPS at small scales(large 𝑘 ’s) is obtained. We shall mention that, whereas thefiducial mass is given by 𝑚 𝜙 ≃ 10 −22 eV, Lyman- 𝛼 observa-tions seems to indicate that 𝑚 𝜙 ≥ −21 eV Armengaud,Palanque-Delabrouille, Yèche, Marsh, & Baur (2017); Iršič,Viel, Haehnelt, Bolton, & Becker (2017); Kobayashi, Murgia,De Simone, Iršič, & Viel (2017). Nonetheless, such constrainthave been obtained for a SFDM with quadratic potential, and a r X i v : . [ a s t r o - ph . C O ] F e b Francisco X. Linares Cedeño and L. Arturo Ureña-López the bump induced by the trigonometric potential helps to alle-viate this tension in the value of the scalar field mass, as wasreported by Cedeño et al. (2017), and later by Leong, Schive,Zhang, & Chiueh (2019).The main goal of this paper, is to extend to a more gener-alized SFDM evolution through the inclusion of a hyperbolicpotential, as those explored by Matos & Ureña Lopez (2000,2001, 2004); Sahni & Starobinsky (2000); Sahni & Wang(2000), and more recently by one of the authors Ureña-López(2019). As we will see, a proper change of variables will allowus to write the scalar field potential as 𝑉 ( 𝜙 ) = ⎧⎪⎨⎪⎩ 𝑚 𝜙 𝑓 [ 𝜙 ∕ 𝑓 ) ] , if 𝜆 > 𝑚 𝜙 𝜙 , if 𝜆 = 0 𝑚 𝜙 𝑓 [ 𝜙 ∕ 𝑓 ) ] , if 𝜆 < (1)where 𝑚 𝜙 is the mass of the scalar field and 𝑓 the so-calleddecay constant. As we shall see below, we will be able to com-pare the different predictions of the different families of SFDMpotentials by changing the value of a single parameter, 𝜆 , whichis associated to the decay constant 𝑓 , as we will show later.The structure of this work is as follows. In Sections 2 and 3we describe the background and linear perturbations equationsfor the SFDM, respectively. In Section 4 we calculate theBayesian evidence of the SFDM models with respect to thestandard Cold Dark Matter (CDM) model, to quantify the pref-erence of cosmological observations for one model or another.Finally, in Section 5 we discuss the main results and futureperspectives of this work. Let us consider a spatially-flat Friedmann-Robertson-Walker(FRW) line element, 𝑑𝑠 = − 𝑑𝑡 + 𝑎 ( 𝑡 ) [ 𝑑𝑟 + 𝑟 ( 𝑑𝜃 + sin 𝜃𝑑𝜙 )] , (2)where 𝑎 ( 𝑡 ) is the scale factor. The background equations forordinary matter, as well as for SFDM 𝜙 endowed with apotential 𝑉 ( 𝜙 ) are given by 𝐻 = 𝜅 (∑ 𝑗 𝜌 𝑗 + 𝜌 𝜙 ) , (3a) ̇𝐻 = − 𝜅 [∑ 𝑗 ( 𝜌 𝑗 + 𝑝 𝑗 ) + ( 𝜌 𝜙 + 𝑝 𝜙 ) ] , (3b) ̇𝜌 𝑗 = −3 𝐻 ( 𝜌 𝑗 + 𝑝 𝑗 ) , ̈𝜙 = −3 𝐻 ̇𝜙 − 𝑑𝑉 ( 𝜙 ) 𝑑𝜙 , (3c)where 𝜅 = 8 𝜋𝐺 . The dot denotes derivative with respectto cosmic time 𝑡 , and 𝐻 = ̇𝑎 ∕ 𝑎 is the Hubble parame-ter. By introducing the following change of variables Cedeñoet al. (2017); Copeland, Liddle, & Wands (1998); LinaresCedeño et al. (2021); Roy, Gonzalez-Morales, & Urena-Lopez (2018); Ureña-López & González-Morales (2016); Ureña-López (2016), Ω 𝜙 sin( 𝜃 ∕2) = 𝜅 ̇𝜙 √ 𝐻 , Ω 𝜙 cos( 𝜃 ∕2) = 𝜅𝑉 √ 𝐻 , (4a) 𝑦 ≡ − 2 √ 𝐻 𝜕 𝜙 𝑉 , 𝑦 ≡ − 4 √ 𝐻𝜅 𝜕 𝜙 𝑉 , (4b)it can be shown that the Klein-Gordon equation is written as 𝜃 ′ = −3 sin 𝜃 + 𝑦 , (5a) Ω ′ 𝜙 = 3( 𝑤 𝑡𝑜𝑡 − 𝑤 𝜙 )Ω 𝜙 , (5b) 𝑦 ′1 = 32 ( 𝑤 𝑡𝑜𝑡 ) 𝑦 + Ω 𝜙 sin( 𝜃 ∕2) 𝑦 , (5c)where a prime indicates derivatives with respect to the e-foldnumber 𝑁 = ln( 𝑎 ∕ 𝑎 𝑖 ) .Until this point, the dynamical equations are valid for anydark matter potential. Once the potential is specified, it can beshown that 𝑦 can be written in terms of the other variables, i.e., 𝑦 = 𝑦 ( 𝜃 , Ω 𝜙 , 𝑦 ) . We will be interested in trigonometric andhyperbolic functions of the scalar field, whereas the quadraticpotencial will be a particular case. For the foregoing potentialswe define the parameter 𝜆 ≡ ∓3∕ 𝜅 𝑓 , and then the variable 𝑦 is written as: 𝑦 = 𝜆 Ω 𝜙 cos( 𝜃 ∕2) . (6)In this way, when we consider positive values of 𝜆 we will bedealing with a scalar field endowed with a trigonometric cosinepotential, and for 𝜆 < the potential will be the hyperboliccosine. Notice that 𝜆 = 0 corresponds to the limit 𝑓 → ∞ ,for which we recover the standard Fuzzy Dark Matter scenariowhere the SFDM potential is a quadratic function 𝑉 ∝ 𝜙 .To study the behaviour of the cosmological evolution forboth background and linear perturbations for the potentials (1),we choose fixed values for the potential parameter given by 𝜆 = { , , −3 .
225 × 10 } for the quadratic ( 𝜆 = 0 ), trigono-metric ( 𝜆 > ), and hyperbolic ( 𝜆 < ) potential, respectively.Likewise, all the numerical solutions we show in what followswere obtained for the fiducial mass 𝑚 𝜙 = 10 −22 eV. The cosmo-logical evolution of the SFDM energy density 𝜌 𝜙 is shown inFigure 1 , where it can be seen that it is until some moment ofthe cosmological evolution that 𝜌 𝜙 eventually evolves as CDM(black) for the three cases, quadratic (yellow), trigonometric(blue), and hyperbolic (red) potential.The characteristic delay in the CDM–like evolution of 𝜌 𝜙 due to the nonlinearities of the trigonometric potential can beappreciated in Figure 1 , as was reported in detail in Cedeñoet al. (2017); Linares Cedeño et al. (2021). Whereas 𝜌 𝜙 forthe quadratic and trigonometric potentials remains constant atearly times, in the hyperbolic case the energy density evolvesas a radiation component. rancisco X. Linares Cedeño and L. Arturo Ureña-López z m = 10 eV CDM= 0> 0< 0
FIGURE 1
Scalar field energy density 𝜌 𝜙 as function of theredshift 𝑧 for the quadratic (yellow line), trigonometric (blueline), and hyperbolic potential (red line). The evolution ofCDM in the case of the standard Λ CDM model is shown forcomparison (black line).It must be noticed that the SFDM equation of state (EoS),given by 𝑤 𝜙 = − cos 𝜃 , oscillates very rapidly in the range [−1 ∶ 1] , and these oscillations can be averaged so that theeffective EoS on cosmological times is ⟨ 𝑤 𝜙 ⟩ = 0 . This aver-aging procedure is done numerically as explained in Ureña-López & González-Morales (2016), which does not alter thetrue evolution of the SFDM energy density (see also Cook-meyer, Grin, & Smith (2020) for a comparison between cut-offprocedures used in the literature.). When considering linear perturbations, the line element in thesynchronous gauge is given by 𝑑𝑠 = − 𝑑𝑡 + 𝑎 ( 𝑡 )( 𝛿 𝑖𝑗 + ℎ 𝑖𝑗 ) 𝑑𝑥 𝑖 𝑑𝑥 𝑗 , 𝜙 ( ⃗𝑥, 𝑡 ) = 𝜙 ( 𝑡 ) + 𝜑 ( ⃗𝑥, 𝑡 ) , (7)where ℎ 𝑖𝑗 and 𝜑 are the metric and scalar field perturbations,respectively. The linearized Klein-Gordon equation for thescalar field perturbation is written (in Fourier space) as Fer-reira & Joyce (1997, 1998); Perrotta & Baccigalupi (1999);Ratra (1991) ̈𝜑 ( ⃗𝑘, 𝑡 ) = −3 𝐻 ̇𝜑 ( ⃗𝑘, 𝑡 ) − [ 𝑘 𝑎 + 𝜕 𝑉 ( 𝜙 ) 𝜕𝜙 ] 𝜑 ( ⃗𝑘, 𝑡 ) − 12 ̇𝜙 ̇̄ℎ . (8)In a similar way to the procedure we have done for theKlein-Gordon equation (3c), we propose the following changeof variables for the scalar field perturbation 𝜑 and its deriva-tive ̇𝜑 Cedeño et al. (2017); Ureña-López & González-Morales (2016) −Ω 𝜙 𝑒 𝛼 cos( 𝜗 ∕2) = √ 𝜅 ̇𝜑𝐻 , −Ω 𝜙 𝑒 𝛼 sin( 𝜗 ∕2) = 𝜅𝑦 𝜑 √ . (9)If we introduce the following transformation 𝛿 = − 𝑒 𝛼 sin[( 𝜃 − 𝜗 )∕2] , 𝛿 = − 𝑒 𝛼 cos[( 𝜃 − 𝜗 )∕2] , (10)we obtain 𝛿 ′0 = [ −3 sin 𝜃 − 𝑘 𝑘 𝐽 (1 − cos 𝜃 ) ] 𝛿 + 𝑘 𝑘 𝐽 sin 𝜃𝛿 − ̄ℎ ′ 𝜃 ) , (11a) 𝛿 ′1 = [ −3 cos 𝜃 − 𝑘 𝑒𝑓𝑓 𝑘 𝐽 sin 𝜃 ] 𝛿 + 𝑘 𝑒𝑓𝑓 𝑘 𝐽 (1 + cos 𝜃 ) 𝛿 − ̄ℎ ′ 𝜃 , (11b)where we have defined the effective Jeans wavenumber as 𝑘 𝑒𝑓𝑓 ≡ 𝑘 − 𝜆𝑎 𝐻 Ω 𝜙 ∕2 . (12)The expression (12) encodes the effects that the SFDMpotential can have on the evolution of linear density perturba-tions. The system of equations (5) and (11) have been studiedfor the cases 𝜆 = 0 (quadratic potential) Ureña-López &González-Morales (2016) and 𝜆 > (trigonometric potential)Cedeño et al. (2017). In the former, the mass of the scalar field 𝑚 𝜙 induces a cut-off scale in the MPS at small scales. In the lat-ter, the non-linear effects of the trigonometric potential inducea cut-off as well, but with a bump at such small scales. Thecase 𝜆 < (cosh potential) has been studied in Matos & UreñaLopez (2000); Ureña-López (2019), and the resultant MPS isthe same as in the case of the quadratic potential for the samevalue of the SFDM mass 𝑚 𝜙 .The numerical solutions obtained from the linear pertur-bation equations allow us to build cosmological observables,such as the temperature anisotropies of the Cosmic MicrowaveBackground (CMB), and the aforementioned MPS. For this weuse the Boltzmann code CLASS
Blas et al. (2011); Lesgourgues(2011), which was amended to solve the SFDM equations ofmotion together with all other relevant cosmological equations.More details about the amendments can be found in Cedeñoet al. (2017); Linares Cedeño et al. (2021); Ureña-López &González-Morales (2016).We show in the top panel of Figure 2 the CMB temperaturefor each of the SFDM models, as well as that for the stan-dard CDM model. It can be seen that there are not notoriousdifferences in the anisotropies spectra between the cosmolog-ical models. Likewise, the MPS is shown in bottom panel ofFigure 2 . The characteristic cut–off at large wavenumbers ispresent for all the potentials. However, for the trigonometriccase there is also a bump at the cut–off scale, which have been
Francisco X. Linares Cedeño and L. Arturo Ureña-López already analyzed in Cedeño et al. (2017). Notice that the MPS,as is also the case for the CMB spectrum in Figure 2 , forthe quadratic and hyperbolic potentials are very similar, whichindicates the existence of a degeneration between these twomodels. l l ( l + ) C l / ( K ) CDM= 0> 0< 0PlanckSPTACT k ( h / Mpc )10 k P ( k ) m = 10 eV CDM= 0> 0< 0Lyman-
FIGURE 2 (Top) Temperature anisotropies of the CMB. Fora scalar field mass given by 𝑚 𝜙 = 10 −22 eV, the differencesbetween SFDM and CDM are negligible. The data points arefrom the Planck, SPT and ACT collaborations. (Bottom) MPSfor each of the scalar field dark matter models. Whereas for thetrigonometric potential there is a bump at large wavenumbersbesides the characteristic cut–off, the quadratic and hyperbolicpotentials are very similar in their prediction of the structureformation. The data points are from Chabanier et al. (2019)and are shown for reference. The standard procedure in cosmology for the comparison oftheoretical models with data is to use the Bayes theorem andperform large numerical Monte Carlo runs to calculate the pos-terior distributions of the model parameters, with this one canestimate the values of the latter that correspond to the maxi-mum likelihood. This kind of studies can be found in Cedeñoet al. (2017); Hlozek et al. (2015) for the quadratic andtrigonometric potentials using available cosmological data.Such study does not exist yet for the cosh potential, but given itssimilarities with the quadratic one, one can safely assume thatthe resultant constraints in the parameteres will be equally sim-ilar. For brevity, we will follow here a different approach andcalculate the so-called Bayesian evidence for each one of themodels, which will allow us to compare the models one witheach other and see whether available data show any preferencefor any of them.Bayes theorem states that, given some observed data 𝐷 ,the probability of a model 𝑀 described in terms of a set ofparameters Θ , is given by the Posterior : (Θ ∣ 𝐷, 𝑀 ) = ( 𝐷 ∣ Θ , 𝑀 )Π(Θ ∣ 𝑀 ) ( 𝐷 ∣ 𝑀 ) , (13)where is the Likelihood function, Π represents the set ofpriors, which contain the a priori information about the param-eters of the model, and is the so-called Evidence, to whichwe pay particular attention .The evidence normalises the area under the posterior ,and is given by ( 𝐷 ∣ 𝑀 ) = ∫ 𝑑 Θ ( 𝐷 | Θ , 𝑀 )Π(Θ | 𝑀 ) . (14)When comparing two different models 𝑀 and 𝑀 using theBayes’ theorem (13), the ratio of posteriors of the two models and will be proportional to the ratio of their evidences.This leads to the definition of the Bayes Factor 𝐵 , which inlogarithmic scale is written as log 𝐵 ≡ log [ ( 𝐷 ∣ 𝑀 ) ] − log [ ( 𝐷 ∣ 𝑀 ) ] (15)If log 𝐵 is larger (smaller) than unity, the data favours model 𝑀 ( 𝑀 ). To assess the strength of the evidence contained inthe data, Jeffreys (1961) introduces an empirical scale, whichquantifies the strength of evidence for a corresponding range ofthe Bayes factor. We follow the convention of Kass & Raftery(1995); Raftery (1996) in presenting a factor of two with thenatural logarithm of the Bayes factor.To calculate the evidence, we first generate a series ofMCMC with the cosmological estimator parameters MONTEPYTHON , developed by Audren, Lesgourgues, Benabed, & For a comprehensive review of Bayesian model selection, we refer the readerto Trotta (2008). rancisco X. Linares Cedeño and L. Arturo Ureña-López Prunet (2013), and using data from Planck 2018 Collaboration,see Aghanim et al. (2020a, 2020b) . We then follow a recentproposal by Heavens, Fantaye, Mootoovaloo, et al. (2017), inthat the unnormalized posterior ̃ (Θ ∣ 𝐷, 𝑀 ) is proportionalto the number density 𝑛 (Θ ∣ 𝐷, 𝑀 ) . Since the number densityis given by 𝑛 (Θ ∣ 𝐷, 𝑀 ) = 𝑁 (Θ ∣ 𝐷, 𝑀 ) =
𝑁 ̃ (Θ ∣ 𝐷, 𝑀 ) ( 𝐷 ∣ 𝑀 ) , (16)where 𝑁 is the lenght of the chain, then the evidence is givenby ( 𝐷 ∣ 𝑀 ) = 𝑎 𝑁 . (17)By determining the proportionality constant 𝑎 , it is possible tocalculate the evidence directly from the MCMC chains. Thesoftware developed for this approach is called MCE VIDENCE ,and it has been used to calculate the evidence for severalalternative cosmological models, as it is shown by Archidia-cono, Hannestad, & Lesgourgues (2020); Binnie & Pritchard(2019); Gómez-Valent, Pettorino, & Amendola (2020); Heav-ens, Fantaye, Sellentin, et al. (2017); Kouwn, Oh, & Park(2018).Figure 3 shows the Bayes factor comparing all the SFDMmodels against the standard CDM. It can be seen that the CDMmodel is preferred over the SFDM models. The horizontallines indicate the Jeffrey’s scale, from which we obtain thatthe SFDM with quadratic potential is disfavored, with positiveevidence, against the CDM model, and that the SFDM withtrigonometric and hyperbolic are also disfavored, but now withstrong and very strong evidence, respectively. l o g B WeakPositiveStrongVery Strong
CDMSFDM with = 0SFDM with > 0SFDM with < 0
FIGURE 3
Bayes factor (15) estimated for the SFDM modelsdiscussed in the text with respect to the standard CDM one.See the text for more details. In this paper we have studied three different SFDM models,taking advantage of a formalism that allows the calculation ofthe cosmological observables in a unified manner. The SFDMmodels have been considered in the literature as serious alter-natives to the standard CDM one, being all of them consistentwith current observations at hand when considered individu-ally. Here we present the first comparison of them with respectto the CDM case, using the Bayesian evidence.Our preliminary results indicate a preference of the cosmo-logical observations for the CDM model, with some positiveevidence against the free case ( 𝜆 = 0 ), and strong and verystrong evidence against the trigonometric ( 𝜆 < ) and cosh( 𝜆 > ) potentials. Part of the explanation is that SFDM mod-els introduce at least one extra parameter, which may be penal-ized by the Bayesian evidence. However, the strong reasonbehind the rejection of the SFDM models can be that currentcosmological observations do not yet indicate the existence ofcharacteristic scales related to the dark matter component, e.g.,the clear presence of a cut-off in the MPS of density perturba-tions. On the other hand, we have explored the full possibilityof varying both, the scalar field mass and the potential param-eter 𝜆 . This could lead to combinations in the parameter spacethat are ruled out by CMB observations.As future work, we will analyze these models for fixedmasses, and focusing on the potential parameter, in order toestablish a more proper comparison against CDM. Besides, itwill be useful to include other set of data, such as those fromLyman- 𝛼 forest, to be able to constraint the SFDM modelsby considering the cut-off at large wavenumbers in the MPS,which is a clear distinction of these models in comparisonwith the CDM case. Acknowledgments.-
Francisco X. Linares Cedeñoacknowledges the receipt of the grant from the AbdusSalam International Centre for Theoretical Physics, Trieste,Italy; and CONACYT and the Programa para el DesarrolloProfesional Docente for financial support. This work was par-tially supported by Programa para el Desarrollo ProfesionalDocente; Dirección de Apoyo a la Investigación y al Pos-grado, Universidad de Guanajuato under Grant No. 099/2020;CONACyT México under Grants No. A1-S-17899, 286897,297771, 304001; and the Instituto Avanzado de Cosmologíacollaboration.
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