Can Dark Matter Be a Scalar Field? Portal John March-Russell, Stephen M

Journal of Cosmology and Astroparticle Physics

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This content was downloaded from IP address 186.217.236.64 on 17/06/2019 at 19:16 JCAP08(2016)046 eV 1 − 0 H , 12 . hysics 0 eV at 99.7% P ≥ 33 − m te to explain the rk matter experi- the Hubble Space 10 le × ic t 56 . apeva, SP, Brazil 1 uta SP,guetá, Brazil ar Planck prior over the dark ≥ In practice, it confirms free ld with quadratic potential, er mass as á, 10.1088/1475-7516/2016/08/046 m and ement with previous studies in smouth, not found evidences for a upper , , doi: a field self interaction. [email protected] , strop A eV previously estimated for some models. , this leads to 22 1 − − 10 ∼ Mpc 1 − m J.L.G. Malatrasi [email protected] b , 8 km s . 1 ± c osmology and and osmology C = 73 0 H S.H. Pereira, 1504.04037 a dark matter theory, supernova type Ia - standard candles, da

In this paper we study a real scalar ﬁeld as a possible candida [email protected] rnal of rnal = 1). For the recent value of the Hubble constant indicated by

~ ou An IOP and SISSA journal An IOP and = 2016 IOP Publishing Ltd and Sissa Medialab srl Campus Experimental de Itapeva — R. Geraldo Alckmin, 519, It Universidade Estadual Paulista de “Júlio Mesquita Filho” [email protected] Universidade Estadual Paulista de “Júlio Mesquita Filho” Departamento de e F´ısica Campus Qu´ımica, deAv. Guaratinguet Dr. Ariberto Pereira da Cunha, 333, 12516-410 — Guaratin Institute of Cosmology andBurnaby Gravitation, Road, University PO1 of 3FX, Port Portsmouth,E-mail: U.K. c b c a c F. Andrade-Oliveira Received April 28, 2016 Revised July 22, 2016 Accepted August 9, 2016 Published August 22, 2016 Abstract. J.F. Jesus, Can dark matter be a scalar field? J

dark matter in thewe have universe. used In Unionmatter the 2.1 density SN context parameter Ia of to observational a set data free a jointly scalar lower with limit fie a on the dark matt ( Telescope, namely c.l. Such value is much smaller than Nevertheless, it is still inlimit agreement on with the them scalar oncereal we field scalar have dark field matter asthe mass a context from viable of SN candidate density Ia for perturbations, analysis. dark which matter include in scalar agre Keywords: ments ArXiv ePrint: JCAP08(2016)046 – ]. 1 2 5 6 9 6 10 31 , 30 ] is one of tter can be 23 ] it has been – 35 15 – 33 e solutions for the M model, where CDM he cosmological constant k matter is still a mystery upernovae observations [ g others candidates for DM t term, the latter being the Scalar fields are also used to At low temperatures, scalar axies and small and large low the universe, where about 5% op in the perturbative regime. of the universe and the former ded to include the dark matter cability in cosmology to explain d to a dark energy (DE) exotic tructures. In [ n by Press and Madsen [ symmetry breaking and possibly dard cosmology. The pressureless een investigated in the last decades ter (SFDM) model [ ] it is proposed a cosmological scalar 27 ]. One of the first applications of a complex 29 , – 1 – 28 seconds after the big bang. This candidate is until 30 − ]. The axion is essentially a scalar field with mass of 14 , 13 ] for a ten-point test that a new particle has to pass in order 12 ] it was studied structure formation by assuming that dark ma 32 ] for a review). 5 – eV, which has its origin at 10 ] for a review and [ ] for a review and references therein). In [ 1 5 11 26 − – , 24 10 The idea of an accelerating universe is indicated by type Ia S The first candidate to DM as a scalar field is the axion, one of th ], in agreement with ΛCDM, but the nature and origin of the dar (see [ described by a reala scalar field. phase It transition wasperturbations also of of investigated this SFDM the leads scalar to field formation of in gravitational the s early Universe. 9 stands for Cold Darkmain Matter candidate and to Λ explainhaving is the a the central current role cosmological phase for constan matter of the represents structure acceleration about formation in 30%is the of stan baryonic the matter total andstands material for content 25% of the is 70% nonbaryonicfluid remaining dark (see part, [ matter sometimes (DM). also T associate A The motion equation 1 Introduction Nowadays the most accepted model in cosmology is known as ΛCD 3 Alternative derivation — change4 of variables Constraining dark matter mass 5 Concluding remarks Contents 1 Introduction 2 The dynamics of the model Most recently, in [ the most studied models inthe quantum different field processes theory, in and the its evolution appli (see of [ the universe has b field harmonic oscillator model,drive in the agreement inflationary to phase ΛCDM of model. the universe [ to be considered a viable DM candidate). about 10 scalar field for structure formation of the universe was give shown that such modelsurface agree brightness with (LSB) rotation galaxies. curvestemperature of The dwarf and model gal finite was also temperature exten corrections up to one-lo Charge-Parity problem in QCD [ particles in the universe, the so called scalar field dark mat now one of the most accepted candidates to DM particles. Amon JCAP08(2016)046 ar (2.6) (2.3) (2.5) (2.4) (2.1) s n-Robertson-Walker stands for the Ricci R eV. As we shall see, our is Newton’s gravitational , 32 G φ y of the resulting system of − ction V and deduce the scalar L atter contributions (radiation, . r in the universe. In section II, g onstrain the free parameters of eld are obtained by varying the ] the quantum creation of scalar ] points to a mass of the scalar he context of finite temperature and φ − µν n based on a change of variables, nal field, erating model of the universe. + 2Λ) (2.2) , 38 37 T √ , , x ) R + µν 4 ( φ , 23 d g ( sm , µν − V Z T πGT mat 22 √ − S ] points to 10 + x = leads to the equations of motion for the real 4 φ = 8 + 27 d µ sm φ ] it was studied the Sextans dwarf spheroidal R µν L mat Z φ∂ – 2 – g grav 37 µ µν δg δS g − S ∂ 2 1 πG g 1 1 2 , given by √ = φ x − 2 16 − 4 = L , which leads to S √ d ]. In [ − φ µν µν L 36 ≡ g Z = R = µν with respect to T grav ) stands for the matter content of the universe, namely, , in addition to the Lagrangian density of a real massive scal S S mat sm 2.1 S L eV, while a recent work [ 22 − ) is the potential. , the energy-momentum tensor of the matter fields, is defined a φ ( µν minimally coupled to gravity, T V The second term in ( It is important to notice that some models [ In this work we study a real scalar field evolving in a Friedman Variation of the action φ constant. curvature scalar, Λ is the cosmological constant parameter field is the standard Einstein-Hilbert action for the gravitatio Gravitational constraints imposed toSFDM dark has matter halos been in investigated t in [ which is composed of thebaryons, Lagrangian neutrinos density of etc.), the standard m model is in agreement to the last one. galaxy, embedded into a scalar field dark matter halo. In [ field of about 10 particles was studied in the context of an alternative accel where background as a possible candidatewe to develop explain the the dynamics darkequations. of matte In the section model III,more and we suitable analyse make for an the numericalthe alternative stabilit integration. model derivatio by In using observational sectionfield SN IV, Ia motion data. we equation We in c conclude the in appendix. se 2 The dynamics ofWe are the interested model in the study of the action of the form scalar field. The equationsaction of with respect motion for to the the metric gravitational fi where where JCAP08(2016)046 y and (2.9) (2.7) (2.8) (2.11) (2.12) (2.17) (2.16) (2.13) (2.18) (2.10) (2.14) (2.15) sm p φ. α = 1). φ∂ ~ α ∂ ] to a similar set = p . c 21 e equation can be , pressure / Λ ) , and the scalar field ρ sm φ H Λ ρ µ √ ρ m tensor is ∂ units aryonic component with 3 ( κ . √ ) , ≡ = 0), we have φ µ ≡ ) ( ensity . φ φ i V ( ∂ µν ), we define the following dimen- V − g , l , . 2 b ) using the definitions − ˙ ) sm . ρ , φ ) ) , H 2.15 p ations adopted in [ Λ , u φ 2 1 φ φ √ 2.7 ρ ( ( α ) b − 3 = 0 = 1 φ )–( ρ = V V κ + ( ν φ∂ ) reduces to √ 2 , φ b u l + α + − V ρ dφ µ dV ∂ 2 2.12 2 2 ≡ + ˙ u ˙ ˙ − φ = 0 φ φ 2.16 + ) 2 + 1 2 2 1 b φ µν b φ ˙ sm φ g α 2 ρ – 3 – 2 p , b ( + 2 1 = κ H Hρ , p , φ∂ 2 2 V ) + φ φ − 3 − α u κ H p ρ φ √ ∂ ( φ sm + 3 = + 3 = 0 + 2 1 3 ν = ρ = V κ b 2 ¨ ˙ Λ φ 2 ˙ √ ρ ˙ H φ x ≡ φ∂ ρ + ω µ H =( , where φ 2 ≡ ∂ φ ˙ φ ρ is the Hubble parameter. sm µν 1 2 φ = T ω , u = . The equations of state of those components are respectivel φ µν ˙ a/a φ ˙ = T φ φ H ρ , p ρ φ ≡ ) 6 p κ φ √ H ( V ≡ , and + , a cosmological constant term with energy density x and . Therefore, we have b Λ φ ρ ρ sm µ α u − πG 8 φ∂ = α ≡ ∂ Λ 2 1 2 p κ The Friedmann equations obtained from this approach are (in Consequently, the Friedmann constraint ( In particular, assuming that the field is homogeneous ( Here we are assuming that the standard matter term in the abov On the other hand, the scalar field part of the energy-momentu We restrict our treatment to a universe composed of a single b In order to study the system of equations ( ≡ φ = 0, ρ b with the Friedmann constraint four-velocity where of equations): sionless variables (accordingly to the variable transform approximated by a perfect fluid characterized by its energy d which can be rewritten in the form of a perfect fluid ( energy density dark matter component p JCAP08(2016)046 ) as (2.26) (2.22) (2.23) (2.24) (2.25) (2.21) (2.19) (2.20) 2.15 , is given by )–( a M 2.12       = ln 2 is the mass of the x N m u 0 0 for the dark matter x + 3 , where − 2 b 3 2 m> Mδ~v 2 ) from a dynamical system x = 2, where of the phase space are: ′ / ntial energy contribution and ~ v ing number 2 0 0 0 0 0 } + 3 2.23 δ c φ cal constant dominated universe. . 2 zes a totally baryonic dominated 2 ) b )–( , s 2 en by the equations ( 2 3 m c b at the fixed points above are: x, 3 2 ) and consider a linear perturbation of , l + Π c 2.19 M )= 2 3 2 − φ , b x 2 ), the case (I) represents a totally kinetic ( c + b, x (2 bl V u, bs bx bu 3 2 , u 3 3 1) su c 2.18 Π 0, leading to a mass + 3 x = 2 3 m/H , − − 2 { – 4 – x, u, b, l, s b l, s, 2 x ≡ ˙ + 2 9 s> H (Π 3 Π Π H s =( 2 3 2 2 2 3 3 − sx x ~v = = = = = s ≡− 0 3 0 0 3 ′ ′ ′ ′ ′ + 3 − l b s Π x u 2 . b 2 3 ′ 3 2 ~ v 2 b 3 2 ; } s + 0 2 + xl xb xs ; x , l, 6 6 ; ; 6 } xu 0 } } 2 3 . Thus, the linearized system reduces to ; ). The eigenvalues of the matrix . 6 0 0 , } , } 2 , , δ~v 3 l 2 3 3 + 9 0 0 ], we define the vector , , s 2.24 , + − , , − 3 1 2 3 40 ~v 1 0 , , 1       , , , 3 0 ± 3 2 √ 0 → , , , = 39 , 2 3 − 0 ± 0 ~v 1 , , , , , M 3 0 0 0 6 { { { { { {± According to the Friedmann constraint ( Specializing to a quadratic scalar potential In order to study the stability of the system ( The equilibrium points or fixed points (I) (I) (II) (II) where a prime denotes derivatives with respect to the e-fold (IV) (III) particles from ( the form scalar field, the background evolution of the universe is giv and represents the Jacobian of energy scalar dominated universe,universe, the the case case (II)cosmological (III) constant characteri and represents the caseThe a (IV) case mixture (IV) a is totally of the cosmologi only scalar one pote that admits approach [ and JCAP08(2016)046 ] xt . 22 φ = 0, (3.3) (3.4) (3.1) (3.2) takes x 1 1 when − 0 , and the 2 H u 10 → − + 10 φ 2 ω eV if we assume x ∼ 33 = m − 2 r 10 ry to integrate the dy- ], when the real parts of . . Taking the limit ∼ , indicating that the large 40 u } 0 r us is the case (IV), where , ), we have H , s ≪ → ∞ ates. The resulting system of f the eigenvalues negative and 39 constrain the SFDM mass we ull, nothing can be said about 1 l constant dominated universe ms are present, the fixed point s is useful for dynamical system as the center manifold theory. x ∼ , as already presented in ref. [ , ues of the two first eigenvalues. 2.11 when the real parts are negative le to avoid such parametrization. u stress that this choice of variables 0 valuation for , such that m , nstant and a potential part of are smooth even for large values of φ 0 r, , and Ω 2 Π 0 = 0. This case is also interesting since u x 3 2 , these high frequency solutions require { p = 1 and . , + m m l } ) ≡ θ 2 θ. 3 2 θ )+ ( x r θ − 2 ( cos sin 2 , = 2 r r sin s φ – 5 – cos 4 3 2 = = r − 3 + y x 9 − s and Ω √ 2, which corresponds to = = 1 2 φ = 0, leading to / ′ ′ 0, leading to a non-null mass to the dark matter particle. 2 s r θ − ˙ φ 2 3 s > ≫ − , ) ). Nonetheless, we will do an alternative treatment in the ne 2 φ s ( 4 V 2.23 ; − } . For huge values of the mass )–( 9 2 3 1 . Thus, the new variables are: √ − x u 2 1 , 2.19 1 3 − + − 2 3 , tan 0 − , a reasonable choice of variables is , , ≡ m 0 0 or equivalently θ , , 0 0 { { In order to study the above system quantitatively we should t Given that the behaviour of Ω For the above set of eigenvalues, the most interesting one fo As it is well known from the dynamical system approach [ The limit of large mass can also be studied. From ( 1. The case (III) is also interesting, since it also has some o → ∞ ∼ (IV) (III) namical system ( In some sense weequations are is changing now: from cartesian to polar coordin s mass limit corresponds to the ΛCDM model, with at the fixed point, it is possible to estimate the mass as Although such case corresponds to a completely cosmologica others null, but it corresponds to “angle” high computational efforts (as an illustration, one single e the fixed point of interest of the dynamical system is m more than 24 hours with an Intel i5 processor), being desirab the mass section, aiming the numerical integration. 3 Alternative derivation —The change of system variables ofanalysis, differential as equations we derived had inshould discussed solve last these in equations section numerically. the However,may last we generate must high section. frequency solutions In in the order variables to the eigenvalues are positive thethe fixed fixed point point is isis unstable, stable. a and When saddle positive point.the and Finally, stability, negative when and real some more ter of accurate the methods eigenvalues should are be n the applied, stability is notBeside completely that discarded such due case to admits the null val and shown in figure it admits the possibility of coexistence of cosmological co JCAP08(2016)046 5 and = 0. (3.8) (3.6) (3.7) (3.9) (3.5) 0 r 4.5 u s 4 = 0: and 7 N 3.5 ations for higher = 10 3 µ z 2.5 EOS assumes a nicely = 0. nstrain the set of free , in such a way that it φ 0 π 2 ω u ). Hence, even imposing to fix it at the value given 0 + ,µ 1 1.5 the 0 and θ MB observations, we end up b 7 Ω ,µ,H 1 → 0 b − θ Ω = 10 0 0.5 , φ is the most well constrained to date, µ 0 Ω 0 0 ,θ b 0 − ) (3.10) φ θ 1 , we have used the normalization condition 1e+07 9e+06 8e+06 7e+06 6e+06 5e+06 4e+06 3e+06 2e+06 1e+06 b,

s p ) in function of redshift for m 2 π H , 1) 0 , = (Ω b 2 π ≡ − – 6 – Ω s 5 = cos(2 l, s − µ φ p (Π x,u,b,l Π Π , ], reducing the number of free parameters to 4. ω 3 3 2 2 2 3 ∈ 0 4.5 l 0 41 ,θ θ = = = 0 4 φ ′ ′ ′ l b . Thus, we seek to solve this system of equations over the s Ω ) with the corresponding initial conditions at b 049 [ ). π . 2 in function of redshift for p 3.5 s = 0 . As can be seen, the equations are now even simpler on u 3 = 0 2 b b 0 ~r r, θ, b, l, s z 2.5 x on figure )+ =( θ 7 ( 2 Evolution of variables ( ~r 2 with a period a) θ cos = 1 and we have defined the parametrization of the mass: , we may see that none of the “polar” variables present oscill = 10 1.5 2 2 Λ µ r 1 + Ω becomes a free parameter, . Panel . Evolution of variable φ 0 0.5 θ b) In figure It is also interesting to notice that in these new variables, + Ω 0 0 1 -1 m 0.5 -0.5 , and the system is invariant under the transformation spatial flatness, aswith predicted 5 by free inflation parameters. and Among indicated these by parameters, C Ω where with compatible limits fromby BBN Planck and and CMB. WMAP: We therefore Ω choose Figure 1 4 Constraining dark matterIn mass order to obtain limits to the dark matter mass, we have to co mass values ( parameters of the SFDM model, namely, where Π = 2 θ Ω simple expression: Panel is periodic over vector of variables JCAP08(2016)046 5 = 0. , we (4.3) (4.2) (4.4) (4.1) s µ 0 tudes, 4.5 can be θ = z 4 through a and E ], it was not L = 30). ]. 7 d 3.5 p 42 43 − 3 is the luminosity = 10 n = 0. We have plotted µ L z = d 0 2.5 t a redshift θ ng more dependent on ν onsidered the measure- 2 and , and given that 7 urve fitting [ 1.5 ), for a spatially flat Universe, N + 25 eedom ( the oscillatory behaviour of the ample and has passed recently z = 10 1 ( ta are independent of the back- L ssumptions. However, even using µ d H s 0.5 θ + µ s 0 D. , N ), it is necessary to define 0 = 5 log µ ) c z − can be written in terms of a dimensionless z H ( 1e+07 9e+06 8e+06 7e+06 6e+06 5e+06 4e+06 3e+06 2e+06 1e+06 e

( 1

) , s , θ + µ + θ L SN H − z E d ) data, with 34 measurements [ ) in function of redshift for z M ( ≡ – 7 – 5 )= − H r,b,l N dz ( dD = (1+ SN ′ 4.5 L m D d ) in diﬀerent redshifts. These kind of observational data 4 z )= ) in function of redshift for ( ~p | H 3.5 z s,θ ( ]: 3 SN 44 µ z are respectively the apparent and absolute supernova magni 2.5 by: 2 Evolution of variables ( SN . Changing to the independent variable ) is the set of free parameters of the model and D ) 0 z 0 M ( a) H 1.5 H ,µ,H and ≡ 1 0 ) ,θ z . Panel SN 0 ( 0.5 . Evolution of variables ( φ m E l r b b) here in order for both variables stand on same axis scale. (Ω Since we have no analytic expression for On its turn, the comoving distance can be related to Next, we considered a SNe Ia data sample, which, although bei The parameters dependent distance modulus for a supernova a As a ﬁrst attempt to constrain these free parameters we have c 0 µ 0 1 0.2 0.1 0.8 0.7 0.6 0.5 0.4 0.3 0.9 ≡

+ r, b, l b, r, ~p comoving distance differential equation. The luminosity distance distance in units of Megaparsecs. where by the following relation [ where arrive to: Figure 2 the fiducial cosmological model,through more it refined consists and of model a independent large methods data of light s c θ enough to constrain the 4model free along parameters, with mainly the because relatively of small number of degrees of fr are quite reliableground because cosmological in model, general justthe such relying current observational on most da astrophysical complete a compilation of computed through the expression ment of the Hubble parameter Panel JCAP08(2016)046 , , ) ]. 3 = 17 34, . 0 43 . 3.7 φ (4.6) (4.5) (4.8) (4.7) 0 levels. )–( 2 < 30, 6 . χ 3.3 comprises 0 . 0 φ 0 φ c = 2 Ω H 2 -Ω . < 2 χ µ based on Planck o is given by: 10 0 : ∗ 19 φ 6 (flat prior on this o,i ∗ . . 974. We have found µ . ˜ 976. The best fit is 0 L . N M − dM 25 + 5 log )] < i = 0 i z ) rk matter mass, we have 0 ( ≡ σ = 0 2 ν φ L set from Suzuki et al. [ 2ln ∗ ,µ χ 2 ν D Ω 0 83, respectively. The best fit = 580 for the Union 2.1 data − . d another for high mass. The χ M dependence by rewriting the in the range of low DM mass < sly with the system ( ,θ n 2 the plane log = 0 0 0 ] 5 log[ φ φ 2 04 H ∗ n . Ω χ statistics with 897, . , , we use a Gaussian prior of Ω M 2 ∗ =1 SN,o,i n i σ χ (0) = 0. In order to constrain the free µ M 17 and 11 . ( P )+ D 2 2 i − z 2 = 561 ( χ = σ ) A , we use a prior over Ω B as 0.94, 0.51 and 0.13 at the same L 0 1 2 ~p | 30, 6 φ C i . , the degeneracy changes to prevent a mass D µ − − 2 min z , 2 011, at 1 ( . χ C o,i 0 10 = 2 – 8 – of the Planck analysis. The result is shown on µ SN 2 = is the corrected distance modulus for a given SNe exp ± − µ & , or, at least, give it an inferior limit. As Planck χ σ [ )] 2 i ˜ m N χ z µ 2 i ) = 5 log ( . The result of this analysis can be seen in figure 854, which corresponds to σ n 265 =1 . z 0 L . i ∞ X ( at ∆ θ SN,o,i D + −∞ µ = 18 38 SN = 0 . . Z to the quite wide range 0 0 was estimated from a ), µ 5 log[ 0 = 560 0 − 2 SN ~p φ χ )= 38+0 17 4.1 dm . . =1 . The degeneracy remains, however, over the dark matter mass n i is now better constrained in the interval 0 0 2 min . Then, we marginalize the likelihood over ,µ − χ 0 0 0 P 086 061 . . φ 38+0 15 . . H 0 ,θ 0 = 0 − +0 − is dimensionless luminosity distance and φ B 065+0 045 22 . . , (Ω D . 0 ) 2 i ˜ 68, with inferior limits given by 1.12, 0.64 and 0.12 at ∆ 1 L . σ − z its corresponding individual uncertainty and over the parameter = 0 ) is given by ( i 041+0 027 =1 . . ~p ˜ 10). For higher masses, 0 σ n i L | 0 φ i = 19 +0 − z . P ( µ = (1+ ] yields the limit Ω 022, which is approximately 2 is a normalization constant. The corresponding ˜ . µ = 252 L 0 SN . 41 being . 83, respectively. . µ A N D . i 4 ± . We have found, from this analysis, z was 19.95 and we may infer lower limits to As usual on this analysis, we marginalize over the Since we are mainly interested on the constraints over the da As one may see, Ω As one may notice, there is a strong degeneracy for Ω In this case, restricting Ω In order to alleviate the degeneracy over Ω 1 = 0 σ µ − 0 265 φ . and 11 at 3 distance modulus: Ω compilation. where Ia at which can be seen as a sixth equation to be solved simultaneou parameters of the model we considered the Union 2.1 SN Ia data where whose likelihood now presents twobest maxima, one fit for is low mass an where with the initial condition for the comoving distance, The best-fit set of parameters where we have plotted the statistical confidence contours on figure marginalized all the dependence over where 0 (10 the best fit Ω determination. for interval), we have found a (2013) [ results in order to constrain the mass JCAP08(2016)046 = ce m . The 0 φ ior over Ω ns correspond to 68.3%, eV for a Hubble constant of er as a scalar field, where we gle parameter free scalar field. 33 − 10 d and blue regions, respectively). ely). × 038) . 0 12, corresponds to a dark matter mass . ± – 9 – = 0 557 . µ = (1 m , as indicated by the most recent analysis from the Hubble Spa 1 − Mpc 1 − ]. 8kms . 45 1 . Confidence contours of SFDM from Union 2.1 SN data + Planck pr . Confidence contours of SFDM from Union 2.1 SN data. The regio , which corresponds to ± 1 − 0 The lowest limit on this context, H = 73 0 12 . 5 Concluding remarks In this work, wehave investigated analysed the the hypothesis dynamics of of a the quadratic dark potential matt for a sin Figure 4 Figure 3 95.4% and 99.7% c.l. (green, red and blue regions, respectiv 0 regions correspond to 68.3%, 95.4% and 99.7% c.l. (green, re H Telescope [ JCAP08(2016)046 φ ρ dark (A.3) (A.1) (A.2) ], search for par- 46 eV), which opens the 32 on of the universe, since e in order to obtain an . Using the fact that motion equation. While − N m this analysis in a future 10 ], an upper bound limit for btained from the continuity small. t against different potential t. , AO was supported by CNPq ∼ 04297/2015-1, and to UNESP 47 .g DAMA [ for dynamical interpretations. y low inferior limit to the dark , a observations. Our constraints Λ m thout Borders. JLGM is grateful ctuations would get washed out cture formation, hot dark matter ρ ation, while observations indicate ] ( = 0 eV), we have shown that any dark + n. ) . 27 2 , which is more suitable for numerical on which could eventually furnish an N 22 2 φ ) a 3 ( − ′ H − N = 0 ( e V ′ 0 b = ln + ρ dφ dV ′ N πGφ φ ), we can write the Hubble parameter as + 4 )+ φ ˙ ( φ φ – 10 – − ( V H 3 + 3 V eV. Although such lower limit is much smaller than + ′ ] (of about 10 33 + 3 2 H H ) 37 − ¨ πG φ , N 8 10 ( ′ 15–120 GeV). More recently [ + 23 = × φ ′′ , 2 ∼ denotes derivation with respect to 2 φ 56 22 H . H 1 2 H 1 ≥ 197TeV was established, based on relic abundance of thermal )= and m ∼ N φ ( φ ρ ): 2.13 Studies of the density perturbation evolution should be mad A very small mass may be in conflict with the structure formati Writing it in terms of the e-folds number, Experiments involving dark matter particles detections, e integration, we have: where the primes at can be written as the constraints obtained in [ Considering the SN Ia observationalmatter data, mass, we namely, have found a ver matter particles annihilating via a long-range interactio upper limit on thework. scalar field Another darkdependencies matter forthcoming of mass. study the We is scalar mayupper field, trying perfor limit including to to self-interacti dark test matter this mass. resul Acknowledgments SHP is grateful to CNPq— for Campus the financial Experimental support, de grantsBrazil No. Itapeva-SP, through a for 3 fellowship the within the hospitality.to programme Prof. F Science wi C. A. O. Matos for financial support, PAIA gran Carlão The motion equation Another way to investigateit the may SFDM be model numericallyThe intensive, is motion this equation through for approach the aequation can field scalar as be field ( useful is well known and can be o ticles with mass indark the matter range mass ( of question if the mass of the scalar fieldit dark consists matter of ultra could hot beis dark so matter. disfavoured In since the inby framework of free-streaming this stru leading case to theearly an galaxy galaxy-size earlier formation. density superclusters flu form matter mass greater than this oneare is also also much compatible similar with to SN I the value recently obtained in [ JCAP08(2016)046 and (A.4) (A.8) (A.5) (A.9) (A.6) (A.7) Ω : 1 ] ≥ (2006) 2105 z (2007) 98 (2004) 194 e at D 15 , 34 659 = 0 , Measurements of , we arrive at an equation Φ , 2 , astro-ph/9812133 . [ µ tten Λ0 + 2 ]. ameter Observational evidence from . ′ ify the equations, we apply the Astrophys. J. 2 constant + Ω Φ Braz. J. Phys. , Int. J. Mod. Phys. ]. Φ , N , 2 2 3 ]. 2 SPIRE ) Λ0 . (1999) 565 E − µ ]. IN 2 0 e 2 N 0 ( SPIRE ] [ Φ ′ b 2 φ 2 + 517 IN + 6Ω Φ SPIRE µ 2 1 2 ′ ] [ 3 IN SPIRE . N 2 πG + Ω 3 + Φ 8 ] [ − IN 0 2 [ − ) 2 m 1 ′ 0 e The cosmological constant H = r 2 0 N and define the dimensionless quantities Φ b – 11 – φ ≡ ≡ 2 Φ( = 2 3Ω 2 µ Φ H φ 0 collaboration, S. Perlmutter et al., µ 2 3 πGρ Astrophys. J. φ 2 1 + 8 , Ω m ′ (1992) 499 ). This equation can be solved to find the free scalar field 1 2 astro-ph/0207347 = = Φ [ 2 2 30 φ ]. ]. ]. Reconstructing dark energy A.6 The cosmological constant and dark energy Φ collaboration, A.G. Riess et al., hep-th/0212290 astro-ph/9805201 )= Ω [ 2 [ 0 φ µ ( H 3 H V SPIRE SPIRE SPIRE 2 IN IN IN 1 (2003) 559 E ] [ ] [ ] [ 2 Cosmological constant: the weight of the vacuum New Hubble space telescope discoveries of type Ia supernova 75 (2003) 235 + is given by ( (1998) 1009 ′′ 2 Alternative dark energy models: an overview Φ 380 0 high redshift supernovae 116 H H ]. 42 ≡ =constant. Inserting this result into the motion equation, 2 Λ SPIRE from ρ E IN astro-ph/0611572 astro-ph/0402109 astro-ph/0610026 Λ supernovae for an accelerating universe and a cosmological [ [ [ Supernova Search Team Astron. J. Rev. Mod. Phys. Ann. Rev. Astron. Astrophys. Phys. Rept. Supernova Cosmology Project [ narrowing constraints on the early behavior of dark energy [5] V. Sahni and A. Starobinsky, [6] [2] P.J.E. Peebles and B. Ratra, [3] S.M. Carroll, W.H. Press and E.L. Turner, [4] J.A.S. Lima, [7] [1] T. Padmanabhan, [8] A.G. 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