Constraints on Dark Matter annihilation and Its Equation of State after Planck Data
aa r X i v : . [ a s t r o - ph . C O ] D ec August 9, 2018 20:57 WSPC/INSTRUCTION FILE rdm
Modern Physics Letters Ac (cid:13)
World Scientific Publishing Company
Constraints on Dark Matter annihilation and Its Equation of Stateafter
Planck
Data
Lixin Xu ∗ Institute of Theoretical Physics, School of Physics & Optoelectronic Technology, DalianUniversity of Technology, Dalian, 116024, P. R. China
Received (Day Month Year)Revised (Day Month Year)In this paper, the annihilation of dark matter f d ǫ with nonzero equation of state w dm was studied by using the currently available cosmic observations which include the geo-metric and dynamic measurements. The constrained results show they are anti-correlatedand are w dm = 0 . +0 . − . and f d ǫ = 1 . +0 . − . respectively in 1 σ regions.With the including of possible annihilation of dark matter, no significant deviation fromΛCDM model was found in the 1 σ region. Keywords : Dark matter; Cosmology; Equation of statePACS Nos.: 98.80.Cq, 95.35.+d, 98.70.Vc
1. Introduction
The nature of dark matter (DM) is still one of the biggest puzzle in particle physicsand cosmology. Its existence was confirmed by the observations from galaxy rota-tion curves, gravitational lensing, the large scale structure formation and cosmicmicrowave background (CMB) under the assumption that the Einstein’s gravitytheory is corrected. Actually, the cold dark matter plus a cosmological constant Λ,the so-called ΛCDM model, can almost agree with the most recent cosmic observa-tions which include the type-Ia supernovae, the baryon acoustic oscillation (BAO)and CMB successfully at large scales. However, it has several potential problemson smaller scales.
How to explain the discrepancies on large and small scale iscurrently still under debate. The warm dark matter has been proclaimed as a po-tential solution to the small scale difficulties of cold dark matter.
3, 7–11
Since thehot dark matter was ruled out due to the difficulty in forming the observed largescale structure, the remained focus point is whether the DM is cold or warm. Ifone takes the dark matter as a perfect fluid, its properties are characterized by itsequation of state (EoS) w dm and effective sound speed c s,eff in its rest frame. Andtheir values should be determined by the cosmic observations. A significant nonzero ∗ [email protected] 1 ugust 9, 2018 20:57 WSPC/INSTRUCTION FILE rdm Lixin Xu value of w dm indicates the dark matter is warm rather than cold. The sound speeddetermines the sound horizon of the fluid via the equation l s = c s,eff /H . The fluidcan be smooth (or cluster) below (or above) the sound horizon l s . If the sound speedis smaller, the perturbation of the fluid can be detectable on the relative large scale.In turn, the clustering fluid can influence the growth of density perturbations ofmatter, large scale structure and evolving gravitational potential which generatesthe integrated Sachs-Wolfe effects. Since the dark matter is responsible to formingthe large scale structure of our Universe, we assume the effective speed of sound c s,eff = 0 in this work. Of course, one can extend to the case of nonzero c s,eff easily. Different values of w dm would change the background evolution history ofour Universe. That means that the critical epoch, for example the equality timeof matter and radiation, would be moved earlier or later; and the peaks of CMBtemperature power spectrum would be increased or decreased; and the large scalestructure formation history would be changed too. The comoving sound horizon r s at the recombination epoch was also changed, as a result the observations ofBAO, the standard ruler, can be used to determine the values of w dm . And theluminosity-redshift relation can also be used to constrain its values by treating Iasupernovae (SN) as standard candles due to the possible modification to the back-ground evolution history trough the Hubble expansion rate. Then one can use theobserved SN and BAO data sets to constrain the EoS of DM and fix the backgroundevolution history from the geometric side. For the SN data points as ”standard can-dles”, the luminosity distances will be employed. In this paper, we keep to use theSNLS3 which consists of 472 SN calibrated by SiFTO and SALT2, for the detailsplease see. Although the photometric calibration of the SNLS and the SDSS Su-pernova Surveys were improved, they are still unavailable publicly. For the BAOdata points as ”standard ruler”, we use the measured ratio of D V /r s , where r s isthe comoving sound horizon scale at the recombination epoch, D V is the ”volumedistance” which is defined as D V ( z ) = [(1 + z ) D A ( z ) cz/H ( z )] / where D A is theangular diameter distance.Due to the different properties of dark matter as indicated by different valuesof w dm , here we have set the effective sound speed c s,eff to zero, the formation ofthe large scale structure and distribution of galaxies would be efficient indicatorsof this difference. Then the information from the large scale structure would beimportant to pin down the properties of dark matter from the dynamic side as acompensation to the geometric measurements such as SN and BAO. This is mainlybecause the fact that different cosmology models may have the same backgroundevolution but the dynamic evolution would be different. That means that dynamicevolution is necessary to break the degeneracies between model parameters. Thanksto the measurements of WiggleZ Dark Energy Survey, a total 238 ,
000 galaxies inthe redshift range z < . < z < .
3, 0 . < z < .
5, 0 . < z < . . < z < .
9. Thecorresponding power spectrum in the four redshift bins was measured; for details,please see. We estimate the nonlinear growth from a given linear growth theoryugust 9, 2018 20:57 WSPC/INSTRUCTION FILE rdm
Constraints on Dark Matter annihilation and Its Equation of State after Planck Data power spectrum based on the principles of the halo model; actually the HALOFITformula will be used in this paper. At the scale of halo, the growth of halos dependson the local physics, and not on the details of precollapse matter and the large scaledistribution of matter. Thus, in the nonlinear regime the growth depends only on thenonlinear scale, the slope and curvature of the power spectrum. In this work, weloosen the constraint to a zero equation of state and investigate the simplest modelfor dark matter, i.e. the one with a constant w dm . Therefore, the main informationis stored in the matter power spectrum. Also we assume the HALOFIT formula isstill suitable for this case, though the formula would be modified due to the free-streaming of warm dark matter. When the measurements from WiggleZ are used,the BAO data points from WiggleZ are not included. Because they come from thesame galaxy sample as the P ( k ) measurement.The initial conditions of the cosmological perturbations were fixed by the cosmicobservations from CMB. Here the firs release of Planck data was employed due to theimprovement of the quality of the cosmological data. It allows us to give a tighterconstraint to the cosmological parameter space when one uses the full informationof CMB from the recently released
Planck data sets which include the high-l TTlikelihood (
CAMSpec ) up to a maximum multipole number of l max = 2500 from l = 50, the low-l TT likelihood ( lowl ) up to l = 49 and the low-l TE, EE, BBlikelihood up to l = 32 from WMAP9, the data sets are available on line. On the particle physics side, the weakly interacting massive particle (WIMP)is a well motivated candidate. The direct detection experiments, such as DAMA, CoGeNT and CRESST, have suggested a low dark matter mass; for recentreviews, please see. The annihilation of the dark matter will release energy in theform of standard model particles. And the released energy would be absorbed bythe surrounding gas, causing the gas to heat and ionize. Then the ionized fractionof electron χ ion would be changed with respect to the redshift z . The excess freeelectrons interact with the CMB photons through the Thomson scattering, resultinga damping of the CMB power spectrum on small scales and a boost in the EEpolarization power spectrum on large scales. Then through the measurements ofCMB power spectrum, the properties of the dark matter can be detected indirectly.Several authors have studied the effects to the CMB power spectrum due to thedecay and annihilation of the dark matter. In the literature, the equation of state of the dark matter was constrained fromthe different sides, for the recent results, please see Ref. and references therein.However, we should notice that the evolution of the energy density of the darkmatter with respect to the redshift z is changed from (1 + z ) to (1 + z ) w dm ) ,when the equation of state of the dark matter w dm is not zero in the constant case.Then the equation of state is degenerated with annihilation of the dark matterugust 9, 2018 20:57 WSPC/INSTRUCTION FILE rdm Lixin Xu particles through injected energy per unit volume at redshift z dEdV dt (cid:12)(cid:12)(cid:12)(cid:12) χ ¯ χ = 2 M χ c h σv i N χ N ¯ χ ≈ . × − [1 + z ] w dm ) eVs − cm − × (cid:20) M χ c (cid:21) − (cid:20) Ω χ h . (cid:21) (cid:20) h σv i × − cm / s (cid:21) , (1)which is defined as the energy liberated by the dark matter annihilations, where M χ ≡ M ¯ χ is the mass of the dark matter particle and its antiparticle; h σv i isthe thermally averaged product of the cross section and relative velocity of theannihilating the dark matter particles; and N χ ≡ N ¯ χ = N χ, [1 + z ] w dm ) is thenumber density of the dark matter particles and their antiparticles, here N χ, ≈ . × − cm − h Ω χ h . i h M χ c i − is the present value of N χ . And this releasedenergy will be deposited into the intergalactic medium (IGM), going into heating,and ionizations or excitations of atoms with a fraction f d ( z ), i.e. dE d dV dt (cid:12)(cid:12)(cid:12)(cid:12) χ ¯ χ = f d ( z ) dEdV dt (cid:12)(cid:12)(cid:12)(cid:12) χ ¯ χ = f d ( z ) ǫ N H [1 + z ] w dm ) eVs − (2)with the dimensionless parameter ǫ = 1 . × − (cid:20) M χ c (cid:21) − (cid:20) Ω χ h . (cid:21) (cid:20) h σv i × − cm / s (cid:21) . (3)Here N H ≈ . × − cm − [1 + z ] is the number density of hydrogen nuclei in theUniverse. In this work, we take f d as a constant in the range [0 , CosmoRec to include a extra model parameter w dm .By a combination of CMB, SDSS BAO, SN and WiggleZ, the EoS of dark matterand f d ǫ will be constrained. The plan of this paper is as follows: in section 2, wepresent the main background evolution and perturbation equations for dark matterwith an arbitrary EoS. In section 3, the constrained results are presented via theMarkov Chain Monte Carlo (MCMC) method. Section 4 is the conclusion.
2. Background and Perturbation equations
For a nonzero value of the EoS of DM, the Friedmann equation for a spatially flatFRW universe reads H = H h Ω r a − + Ω b a − + Ω dm a − w dm ) + Ω Λ i , (4)where Ω i = ρ i / M pl H are the present dimensionless energy densities for the radia-tion, the baryon, the dark matter and the cosmological constant respectively, whereΩ dm + Ω r + Ω b + Ω Λ = 1 is respected for a spatially flat universe.ugust 9, 2018 20:57 WSPC/INSTRUCTION FILE rdm Constraints on Dark Matter annihilation and Its Equation of State after Planck Data In the synchronous gauge the perturbation equations of density contrast andvelocity divergence for the dark matter are written as
47, 49 ˙ δ dm = − (1 + w dm )( θ dm + ˙ h w dm w dm δ dm − H ( c s,eff − c s,ad ) (cid:20) δ dm + 3 H (1 + w dm ) θ dm k (cid:21) , (5)˙ θ dm = −H (1 − c s,eff ) θ dm + c s,eff w dm k δ dm − k σ dm . (6)following the notations of Ma and Bertschinger, where c s,ad is the adiabatic soundspeed of the dark matter c s,ad = ˙ p dm ˙ ρ dm = w dm − ˙ w dm H (1 + w dm ) . (7)In this work, we assume the shear perturbation σ dm = 0 and the adiabatic ini-tial conditions. And for simplicity, we only consider a constant w dm in this work,although it is easy to be extended to the non-constant case.
3. Constrained Results
To constrain the equation of state and annihilation of the dark matter from thecurrently available cosmic observations, on can use the Markov chain Monte Carlo(MCMC) method which is efficient in the case of more parameters. We modifiedthe publicly available cosmoMC package to include the perturbation evolutionsof the dark matter with a general form of the equation of state according to theEq. (5) and Eq. (6). The recombination was also modified to include the effectdue to the possible nonzero equation of state of the dark matter. The followingeight-dimensional parameter space was adopted P ≡ { ω b , ω c , Θ S , τ, f d ǫ , w dm , n s , log[10 A s ] } (8)which priors are summarized in Table 1. The new Hubble constant H = 72 . ± . − Mpc − was also adopted. The pivot scale of the initial scalar powerspectrum k s = 0 . − is used in this paper.Eight chains were run on the Computing Cluster for Cosmos for every cosmo-logical models with different values of w dm and f d ǫ with priors which are gatheredin the second column of Table 1. The chains are stopped when the Gelman & Rubin R − R − ∼ .
02 which guarantees the accurate confidence limits.The constrained results are summarized in Table 1. The result is compatible toour previous result obtained in Ref. Correspondingly, the one-dimensional andtwo-dimensional contours for ǫ f d and w dm was plotted in Fig. 1.To show the effects of w dm and f d ǫ to the ionized fraction of electron χ e ( z ),we fix the other relevant cosmological parameters to their best fit values as given inugust 9, 2018 20:57 WSPC/INSTRUCTION FILE rdm Lixin Xu
Table 1. The mean values with 1 , , σ errors and the best fit values ofmodel parameters, where SNLS3, BAO, Planck +WMAP9 and WiggleZ mea-surements of matter power spectrum are used.Prameters Priors Mean with errors Best fitΩ b h [0 . , .
1] 0 . +0 . . . − . − . − . . c h [0 . , .
99] 0 . +0 . . . − . − . − . . θ MC [0 . ,
10] 1 . +0 . . . − . − . − . . τ [0 . , .
8] 0 . +0 . . . − . − . − . . f d ǫ [0 ,
10] 1 . +0 . . . − . − . − . . w m [ − . , .
2] 0 . +0 . . . − . − . − . − . n s [0 . , .
5] 0 . +0 . . . − . − . − . . A s ) [2 . ,
4] 3 . +0 . . . − . − . − . . Λ - 0 . +0 . . . − . − . − . . m - 0 . +0 . . . − . − . − . . σ - 0 . +0 . . . − . − . − . . z re - 10 . +1 . . . − . − . − . . H - 68 . +0 . . . − . − . − . . Y P - 0 . +0 . . . − . − . − . . A s e − τ - 1 . +0 . . . − . − . − . . / Gyr - 13 . +0 . . . − . − . − . . Table 1 and vary the values of w dm and f d ǫ around their best fit values. Their effectson the ionization due to different values of w dm and f d ǫ are plotted in Figure 2. Asshown in this figure, large values of w dm will lower the ionized fraction of electron.And the annihilation of the dark matter will increase the ionized fraction of electronsas expected. Just due to the annihilation of the dark matter, the ionized fraction ofelectron is increased instead of downing to almost zero. Therefore, it can be easilyunderstand that the comic observations of CMB data can be used to constrain theannihilation of the dark matter.
4. Conclusion
In this paper, the degeneracy between the equation of state and annihilation ofthe dark matter was studied by using the currently available cosmic observationswhich include SNLS, SDSS and BAO to fix the background evolution and theCMB of the first 15.5 months data from
Planck to fix the initial conditions ofthe perturbations and the WiggleZ measurement of power spectrum to fix the largestructure formation. We have found the latest data sets provide that constraint w dm = 0 . +0 . . . − . − . − . and f d ǫ = 1 . +0 . . . − . − . − . in 3 σ regions. The current data show also the anti-correlation between w dm and f d ǫ .With the including of possible annihilation of DM, no significant deviation fromΛCDM model is found in the 1 σ region.ugust 9, 2018 20:57 WSPC/INSTRUCTION FILE rdm Constraints on Dark Matter annihilation and Its Equation of State after Planck Data −0.002 0.000 0.002 w m ǫ f d −0.0020.0000.002 w m Fig. 1. The one-dimensional marginalizeddistribution on individual parameters and two-dimensional contours with 68% C.L., 95% C.L.for ΛwDM model by using CMB+BAO+SN+WiggleZ data points.
Acknowledgments
L. Xu thanks the invitation of Prof. M. Yu. Khlopov to give contribution to thisspecial issue of MPLA. L. Xu’s work is supported in part by NSFC under the GrantsNo. 11275035 and ”the Fundamental Research Funds for the Central Universities”under the Grants No. DUT13LK01.
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