Probing Dark Energy with Neutrino Number
aa r X i v : . [ a s t r o - ph . C O ] O c t KIAS-P14057
Probing Dark Energy with Neutrino Number
Seokcheon Lee ∗ School of Physics, Korea Institute for Advanced Study, Heogiro 85, Seoul 130-722, Korea
From measurements of the cosmic microwave background (CMB), the effective number of neutrinois found to be close to the standard model value N eff = 3 .
046 for the ΛCDM cosmology. One canobtain the same CMB angular power spectrum as that of ΛCDM for the different value of N eff byusing the different dark energy model ( i.e. for the different value of w). This degeneracy between N eff and w in CMB can be broken from future galaxy survey using the matter power spectrum. PACS numbers: 04.20.Jb, 95.36.+x, 98.65.-r, 98.80.-k.
1. INTRODUCTION
The existence of relic neutrinos is a generic feature of the hot big bang model. This cosmic neutrinohas been indirectly measured from the analysis of the cosmic microwave background (CMB) angular powerspectrum, as well as primordial abundances of light elements and other cosmological observables [1–4].Measurements of the Planck satellite CMB alone have led to a constraint on the effective number of neutrinospecies, N eff = 3 . ± .
34 which is consistent with the stand cosmological model prediction N eff = 3 . etc. ), then one can vary N eff to include theminto the analysis [8]. Both cosmological and particle physics observational evidences for the existence ofextra neutrino species are still in debate [6, 9–16].Big bang nucleosynthesis (BBN) has emerged as one of the foundation of the hot big bang theory,compounding the Hubble expansion and the CMB [17, 18]. Compared to other elements in the earlyuniverse (D, He, and Li), the abundance of helium, He is insensitive to the matter density of theUniverse, because all neutrons are tied up in helium. Instead, an increase in the number of neutrino, N eff causes the faster expansion rate of the Universe, therefore more neutrons will survive until nucleosynthesiswhich leads to an increase in the Helium abundance, Y P . This proportional direction of the degeneracybetween N eff and Y P has the orthogonal direction when one considers the equal ratio of the acoustic scaleto the diffusion scale. Y P should be decreased as N eff increases [19]. This fact provides strong constraintson both parameters. One needs to observe He from recombination in extremely low-metallicity regionsto be found in extragalactic HII regions. Y P is obtained from the extrapolation to zero metallicity but isaffected by systematic uncertainties. Izotov et al. use both near-infrared spectroscopic observations andoptical range ones of high-intensity HeI emission line in 45 low-metallicity HII regions to get [20, 21]Y P = 0 . ± . . (1-1)The primordial abundance of He could be appreciated to the zero-metallicity in terms of an extrapolationby a model of chemical evolution of galaxies. An alternative low value using a Monte Carlo Markov Chaintechnique is reported by Aver et al. [22] Y P = 0 . ± . . (1-2) ∗ [email protected] There have been great works on the effects of relativistic species quoted as an effective number of neutrinospecies, N eff on CMB and large scale structure (LSS) [19, 23, 24]. However, we focus on the degeneracybetween the N eff and the equation of state (eos) of dark energy, w. This degeneracy can be confused withother degeneracies between N eff and the Hubble parameter, h.In the next section, we investigate the degeneracy between N eff and w on CMB angular power spectrum.We extend this degeneracy on the matter power spectrum in Section 3. In Section 4, we also investigatethe degeneracies between w (or N eff ) and h. We draw our conclusions in Section 5.
2. CMB AND N eff We briefly review the sensitivity of the CMB angular power spectrum to the cosmological parametersto investigate the degeneracy between the effective number of neutrino, N eff and the equation of thedark energy, w. Let define the present value of the energy density contrast, Ω i = ρ i /ρ cr0 . In additionto dark energy, i can be either the radiation (r) composed of the photon ( γ ) and neutrino ( ν ) or thematter (m) comprised of the cold dark matter (c) and the baryon (b). We define our fiducial modelas a flat ΛCDM with cosmological parameters values as (h , ω γ , ω ν , ω b , ω c , w , A S , n s , N eff , Y P , τ )=(0.6715,2 . × − , 1 . × − , 0.0221, 0.1203, -1, 2 . × − , 0.9616, 3.046, 0.25, 0.0927). First, the ratioof odd to even peaks depends on the balance of the gravity and the pressure in a baryon and photon fluid.Thus, if one wants to obtain the same CMB angular power spectrum for different dark energy models, thenone needs to fix the ratio of the present energy density of the baryon, Ω b0 h ≡ ω b to that of the photon,Ω γ h ≡ ω γ . Because the present value of the photon energy density is accurately measured ( i.e. thetemperature of the photon), one can fix the energy density of the baryon. Thus, we use the same values of ω b and ω γ for all models. Second, amplitudes of all peaks depend on the matter and the radiation energiesequality epoch, z eq = ω m /ω r −
1. Thus, one needs to fix the z eq for different values of N eff and this causeschanges in the dark matter energy density. ω c [ N eff ] = ω γ (1 + 0 . N eff )(1 + z eq ) − ω b , (2-1)where we use ω m = ω b + ω c and ω r = ω γ + ω ν . Also the peak location depends on the characteristic angularsize of the fluctuation in the CMB as the acoustic scale. It is determined by the sound horizon at thelast scattering, r s ( z ∗ ) and the comoving angular diameter distance, d ( c ) A ( z ∗ ). The acoustic angular size isdefined by θ s [ z ∗ , N eff , w , h] ≡ r s [z ∗ , N eff , w , h]d (c)A [z ∗ , N eff , w , h] , (2-2)where h is defined from the Hubble parameter H = 100hkm / s / Mpc. Both r s and d (c)A are a function ofthe reduced Hubble parameter, E ( z ) E [ z, N eff , w , h] = HH = 1h q ω m (1 + z) + ω r (1 + z) + (h − ω m − ω r )(1 + z) . (2-3)Even though, the expression for E ( z ) given by Eq.(2-3) is true only for the constant w, one can extendthe consideration for the time varying ones []. Because θ s ( z ∗ ) is determined from the observation, one canfind the relation between parameters ( N eff , w, and h) in order to obtain the same value of θ s ( z ∗ ). In thissection, we keep the value of h fixed and investigate the degeneracy between N eff and w. For the high l acoustic peaks, CMB anisotropies on scales smaller than the photon diffusion length are damped by thediffusion. One needs to consider the mean diffusion distance at the last scattering surface, r d ( z ∗ ) and theangular scale of the diffusion length, θ d ( z ∗ ). Increasing N eff leads to the smaller r d which would decreasethe amount of damping. This effect can be compensated by decreasing the Helium abundance Y P . In orderto obtain the same CMB angular power spectrum for different cosmological models, one needs to fix theratio between two angular scales θ s ( z ∗ ) and θ d ( z ∗ ). From this fact, one can find the relation θ s [ z, N eff , w , h] θ d ( z ∗ )[ z, N eff , w , h , Y P ] = r s ( z ∗ )[ z, N eff , w , h] r d ( z ∗ )[ z, N eff , w , h , Y P ] . (2-4) N eff w ω c ω de Y P A S (10 ) σ fσ ∆ fσ A (fid) S (10 ) σ fσ ∆ fσ N eff . w, ω c ( ω de ), Y P are obtained from Eqs. (2-2),(2-1), and (2-4), respectively. f is the present value of the growth rate of the matter perturbation and fσ is biasfree observable. ∆ fσ is difference of fσ between each model and the fiducial one. A S (10 ) ≡ A S × is an inputparameter and σ is obtained from CAMB. A (fid) S (10 ) is the A S for the fiducial model. Thus, one can find Y P [N eff ] from the Eq. (2-4) in addition to the w[ N eff ] obtained from Eq. (2-2). So farwe consider both the locations of acoustic peaks and the ratio of the peak amplitudes. Also, one needsto consider the global amplitude of the peaks. This can be adjusted either by adjusting the amplitude ofthe primordial density field A s or by matching the integrated Sach-Wolfe (ISW) effect. We investigate theboth cases.From the above consideration, we obtain various models which can produce almost same CMB angularpower spectra as that of our fiducial model. We summarize results in Table.I. We show the dependence ofw, ω c , ω de , and Y P on N eff . We also show changes in σ and f σ due to the different choice of normalizationwhich will be explained in the next section. As N eff increases, so does ω ν when ω b , ω γ , h, and z eq are fixed.This leads to increasing ω c from Eq. (2-1). When N eff varies from 2.0 to 4.0, ω c varies from 0 . . N eff increases, so does E ( z ). However, in order to keep θ s ( z ∗ ) equal for increasing N eff , oneshould increase w too. This causes the changes in w from − .
174 to − .
864 for same ranges of N eff . Thesevalues obtained numerically from Eq. (2-2). The value of Y P is decreased as N eff increased in order tokeep the same ratio of θ s ( z ∗ ) /θ d ( z ∗ ) for different models. One can understand this from Eq.(2-4). r s ( z ∗ ) isdecreased as N eff increases. Thus, one needs to decrease r d ( z ∗ ) as N eff and this leads to decreasing Y P . A S is inferred in order to reduce the difference of the CMB angular power spectrum at high multipoles betweenmodels. Both σ and f are obtained for given cosmological models. Especially, we adopt σ values obtainedfrom CAMB. We adopt fiducial model values for the spectral index ( n s ), the optical depth ( τ ), and theHubble parameter (h), because n s changes the overall tilt of CMB power spectrum, τ affects to the relativeamplitude for l ≫
40 with respect to the lower multipole, and h also affects to the global location andamplitude. We show CMB angular power spectra of different models, D l with different normalizations inFig. 1. In the top left panel of Fig.1, we show the CMB power spectra for different models. Their differencesbetween models and the fiducial one with adopting the varying A S ((9)) given in the Table.I are depicted inthe bottom left panel. One can see the degeneracy (less than 2% for all models) in high l between models.Main differences come from ISW effect which might not be distinguished from the observation. The dashed,long-dashed, solid, dot-dashed, and dotted lines correspond N eff = (2, 2.5, 3.046, 3.5, 4.0), respectively. Inthe lower panel, we also show the power spectra difference between various models and the fiducial one,∆ D l ≡ (cid:16) D l ( N eff ) − D l ( N (fid)eff ) (cid:17) /D l ( N (fid)eff ) × N (fid)eff = 3 . l except N eff = 2. If one adopt the fiducial model A S = 2 . × − for all models, then one obtainsalmost degenerated CMB power spectra at low l . This is depicted in the top right panel. This confirmsthe fact that ISW effect for the different N eff values is negligible as shown in [19]. The differences of D l are appeared on high l . More interesting effect is the shifts on the acoustic peaks. Thus, when one claimsthe shift in the high l peaks due to the changing in the effective neutrino number, one should also considerthe degeneracy in A S . The differences become about 5 % at l ∼ D l @ Μ K D N eff = =- Y p = A s = N eff = =- Y p = A s = N eff = =- Y p = A s = N eff = =- Y p = A s = N eff = =- Y p = A s = A S D D l l D l @ Μ K D N eff = =- Y p = N eff = =- Y p = N eff = =- Y p = N eff = =- Y p = N eff = =- Y p = A S = ´ - D D l l FIG. 1: CMB angular power spectra for different models and their differences from the fiducial model with differentnormalization.
Top left ) CMB angular power spectra for N eff = 2 (dashed), 2.5 (long-dashed), 3.046 (solid), 3.5(dot-dashed), and 4 (dotted), respectively. Bottom left ) The differences of CMB power spectra between N eff = 2(2.5, 3.5, 4.0) model and the fiducial one depicted by dashed (long-dashed, dot-dashed, dotted) line. Top right )CMB angular power spectra using the same A S (10 ). Bottom right ) The differences of CMB power spectra betweenmodels with the same notation as the left panel.
3. LSS AND N eff In the previous section, we investigate the CMB angular power spectrum degeneracy between N eff andother cosmological parameters. It is natural to expect that this degeneracy might be broken in the measure-ment of the matter power spectrum, P ( k ). The most obvious effect of varying N eff appears in the turnoverscale at k ∼ .
02h Mpc − which is related to the size of the particle horizon at the matter-radiation equalityand hence is determined by ω m and ω r , k eq [ z eq , N eff , w , h] = H c E[z eq , N eff , w , h]1 + z eq . (3-1)As N eff increases, so does E ( z eq ). Thus, k eq becomes larger as N eff increases. However, k eq has yet to berobustly detected and future galaxy survey might provide this information to probe structure at the largestscales. Thus, future galaxy survey is promising to constrain the effective number of neutrino. Also, boththe slope and the amplitude of the matter power spectrum depend on the ratio ω b /ω m . As N eff increases,so does ω m with constant ω b . Thus, the slope of the matter power spectrum becomes more moderate as N eff increases. BAO phase depends on the sound horizon at baryon drag and its amplitude is also relatedto the Silk damping scale. Thus, one can find the drag epoch of each model if one obtain the accurateBAO signature around k ∼ .
1h Mpc − . This effect can provide a useful information on N eff [3]. Also, σ depends on both ω m and A s . If we keep A S fixed, then σ decreases as ω m increases. However, this is nottrue when we vary the A S . The amplitude of the linear matter power spectrum decreases as N eff increases.Also the growth rate of the matter perturbation, f depends on ω m . As ω m increases, so does f . Also onecan consider the bias free quantity f σ which also increases as N eff does [25, 26]. The difference in f σ between models are less than 10 % as shown in Table.I and thus can be distinguished from future galaxysurvey such as Euclid and LSST.These results are summarized in Table I. When A S is allowed to vary, there exits no direction of thedegeneracy between σ and N eff ( i.e. ω c ). In this case, σ becomes 0 . . , . , . , . N eff = 2 . . , . , . , . f σ varies from 0 .
406 to 0.475 for same ranges of N eff . The direction P H k L @ H M p c (cid:144) h L D - ´ - N eff = =- Y p = A s = N eff = =- Y p = A s = N eff = =- Y p = A s = N eff = =- Y p = A s = N eff = =- Y p = A s = A S D P H k L k @ h (cid:144) Mpc D P H k L @ H M p c (cid:144) h L D - ´ - N eff = =- Y p = N eff = =- Y p = N eff = =- Y p = N eff = =- Y p = N eff = =- Y p = A S = ´ - D P H k L k @ h (cid:144) Mpc D FIG. 2: Linear matter power spectra for different models and their differences from the fiducial model with differentnormalization.
Top left ) CMB angular power spectra for N eff = 2 (dashed), 2.5 (long-dashed), 3.046 (solid), 3.5(dot-dashed), and 4 (dotted), respectively. Bottom left ) The differences of CMB power spectra between N eff = 2(2.5, 3.5, 4.0) model and the fiducial one depicted by dashed (long-dashed, dot-dashed, dotted) line. Top right )CMB angular power spectra using the same A S (10 ). Bottom right ) The differences of CMB power spectra betweenmodels with the same notation as the left panel. of the degeneracy between σ and ω c becomes the same as the well from the cluster abundance counts, σ Ω γm ≃ . A S is fixed. σ ( f σ ) varies from 0 . . . . . ≤ N eff ≤ . σ and Ω m0 . We show the present linear matter power spectra of different models, P ( k ) with different normalization in Fig.2. In the top left panel of Fig.2, we show the matter power spectrafor different models. It is obvious that each model has the different turnover scale. k eq varies from 0 . . ≤ N eff ≤ .
0. The slope of P ( k ) at k < k eq is same for all models because we fix n s .The dashed, long-dashed, solid, dot-dashed, and dotted lines correspond N eff = (2, 2.5, 3.046, 3.5, 4.0),respectively. Their differences between models and the fiducial one with adopting the varying A S ((9)) aredepicted in the bottom left panel. We define ∆ P ( k ) ≡ (cid:16) P ( k, N eff ) − P ( k, N (fid)eff ) (cid:17) /P ( k, N (fid)eff ) × P ( k ) becomes 25 (18, 12, 10) % at k = 0 .
02h Mpc − when N eff = 2 (2.5, 3.5, 4). Also ∆ P ( k ) is sub-percent level at k = 0 .
1h Mpc − for all models. If one adopts the fiducial model A S = 2 . × − for allmodels, then one obtains the matter power spectra with more consistent slopes at k > k eq . This is depictedin the top right panel. The differences of D l are appeared on high l . In the bottom right panel, we showthe ∆ P ( k ) for the different models. ∆ P ( k ) becomes 25 (17, 12, 8) % at k = 0 .
02h Mpc − when N eff = 2(2.5, 3.5, 4). ∆ P ( k ) becomes 3 (3, 1.5, 1.5) % at k = 0 .
1h Mpc − for N eff = 2 (2.5, 3.5, 4) model. However,the linear matter power spectrum is not able to be used directly due to bias problem. Thus, it is better tocompare the bias free parameter f σ as we mentioned.
4. DEGENERACIES (w, h ) AND ( N eff , h ) In previous sections, we investigate the degeneracy of N eff and w from CMB and LSS. We briefly inves-tigate the degeneracy between w and h in this section. We also probe the degeneracy between N eff andh. In the first case, even though there is no change in N eff , w is degenerated with N eff and we want toinvestigate how it can be separated from the degeneracy with h. Results are summarized in Table.II. N eff = 3 .
046 w = − .
0w h σ fσ ∆ fσ N eff h Y P A S (10 ) σ fσ ∆ fσ -0.8 0.6139 0.785 0.455 2.1 2.0 0.6226 0.3049 2.12 0.7909 0.418 -6.4-0.9 0.6422 0.815 0.451 1.1 2.5 0.6464 0.2782 2.16 0.8174 0.431 -3.3-1.0 0.6715 0.840 0.446 0 3.046 0.6715 0.25 2.21 0.8452 0.446 0-1.1 0.7012 0.875 0.442 -1.1 3.5 0.6917 0.2274 2.22 0.8607 0.454 1.8-1.2 0.7320 0.905 0.437 -2.1 4.0 0.7132 0.2032 2.26 0.8819 0.466 4.3TABLE II: The degeneracy between w (or N eff ) and h. In the first case, we fix N eff = 3 .
046 and check the degeneracybetween w and h. In the second case, we fix w = − . N eff and h. A. (w, h ) First, we investigate the degeneracy in w and h from CMB with N eff = 3 . ω c . However, one needs tofix E ( z ) to produce the same acoustic angular size. If h increases, so does h × E(z) at low z . Thus, one needsto decreases w to make E ( z ) moderate. Thus, h decreases as w increases. If w varies from − . − . . . l in CMB angular power spectrum. Except this effect, models produce almost identical D l .This is shown in the top left panel of Fig.3. The dashed, long-dashed, solid, dot-dashed, and dotted linescorrespond (w , h) = (-1.2,0.7320), (-1.1,0.7012), (-1.0,0.6715), (-0.9,0.6422), and (-0.8,0.6139), respectively.We use the same normalization A S = 2 . × − for all models. The differences between models are about1 % at large angle, and they become sub-percent level at high l as shown in the bottom left panel. Wealso investigate the matter power spectra for models. Because we fix all cosmological parameters except wand h, the equality wavenumber k eq = a eq H eq /c should be same for all models. However, as one can seein the top right panel of Fig.3, there are differences in turnover scales between models. This is due to theclustering of the dark energy at large scale ( i.e. at small k ). On small scales, the dark energy is smooth andthe dark energy perturbation is damped and does not contribute the matter density perturbation. However,on large scales, the dark energy clusters and contributes to the energy density and pressure perturbations.The amplitudes of matter power spectra for different models depend on the choice of normalization. Weadopt the same primordial amplitude for all models, A S = 2 . × − . However, one can vary A S and thiscase the amplitudes of matter power spectra can be changed. When w decreases, the transfer function ofthe matter power spectrum. The effective Compton wavenumber of dark energy is approximated as [27, 28] k w = 3 Hc p (1 − w)(2 + 2w − wΩ m ( z )) . (4-1)Also one can approximate the transfer function as [29] T ( k ) ≃ , when k w D ( z eq ) ≪ (cid:16) k w D ( z eq ) (cid:17) − , when k w D ( z eq ) ≫ . (4-2)Thus, one find that as w increases, so does k w . This causes the decreasing T ( k ) as w increases. There areabout 20 % differences between models as shown in the bottom right panel of Fig.3. However, there is biasconcern with this differences and if we check the differences of f σ , then the difference between w = − . D l @ Μ K D w =- = =- = =- = =- = =- = A S = ´ - , N eff = D D l l P H k L @ H M p c (cid:144) h L D - ´ - w =- = =- = =- = =- = =- = A S = ´ - , N eff = D P H k L k @ h (cid:144) Mpc D FIG. 3: CMB angular power spectra for different models and their differences from the fiducial model with differentnormalization.
Top left ) CMB angular power spectra for N eff = 2 (dashed), 2.5 (long-dashed), 3.046 (solid), 3.5(dot-dashed), and 4 (dotted), respectively. Bottom left ) The differences of CMB power spectra between N eff = 2(2.5, 3.5, 4.0) model and the fiducial one depicted by dashed (long-dashed, dot-dashed, dotted) line. Top right )CMB angular power spectra using the same A S (10 ). Bottom right ) The differences of CMB power spectra betweenmodels with the same notation as the left panel.
B. ( N eff , h ) In this subsection, we investigate the degeneracy between N eff and h. Now, we keep all other cosmologicalparameters fixed except these two, Y P , and A S . Again, due to the change in the effective number of neutrino,one obtains the change in the ω c . This also causes the change in the Hubble parameter to match θ s . AlsoY P changes due to obtain the same θ r /θ d ratio for all models. These changes in Y P and A S are almost sameas those in the Section. II, the degeneracy between N eff and w. In stead of changing w for the varying N eff ,one can obtain almost same effect by varying h. If we fix w = − .
0, h varies from 0.6226 to 0.7132 when N eff changes from 2.0 to 4.0. This is shown in Table.II. The corresponding changes in CMB angular powerspectra are dominated in low l due to ISW effect. This is shown in the top left panel of Fig.4. The dashed,long-dashed, solid, dot-dashed, and dotted lines correspond N eff = (2, 2.5, 3.046, 3.5, 4.0), respectively.The differences of D l between models are shown in the bottom left panel of the same figure. All are aboutless than 2 % for entire region of l . The matter power spectra in these models are shown in the top rightpanel of Fig.4. As N eff increases, so does h and leads to increasing k eq (inversely decreasing T ( k )). Thus,one obtains the slight decreasing P ( k ) as N eff increases. We use the same notation for this panel as thatof the left one. ∆ f σ is -6.4 (-3.3, 1.8, 4.3) between N eff = 2 .
5. CONCLUSIONS
We investigate the cosmic microwave background degeneracy on the effective number of neutrino and theequation of state of dark energy. One of the most accurate measurement of CMB is the acoustic scale whichdepends on both N eff and w. We showed that CMB is degenerated for the different dark energy modelskeeping other cosmological parameters fixed [30]. This degeneracy might be broken when one combineCMB with LSS. Thus, one should also consider different dark energy models when one investigate the N eff from CMB and LSS. This will be a challenge for confirming the concordance model. We also investigate D l @ Μ K D N eff = = A s = N eff = = A s = N eff = = A s = N eff = = A s = N eff = = A s = =- D D l l P H k L @ H M p c (cid:144) h L D - ´ ´ - N eff = = A s = N eff = = A s = N eff = = A s = N eff = = A s = N eff = = A s = =- D P H k L k @ h (cid:144) Mpc D FIG. 4: CMB angular power spectra for different models and their differences from the fiducial model with differentnormalization.
Top left ) CMB angular power spectra for N eff = 2 (dashed), 2.5 (long-dashed), 3.046 (solid), 3.5(dot-dashed), and 4 (dotted), respectively. Bottom left ) The differences of CMB power spectra between N eff = 2(2.5, 3.5, 4.0) model and the fiducial one depicted by dashed (long-dashed, dot-dashed, dotted) line. Top right )CMB angular power spectra using the same A S (10 ). Bottom right ) The differences of CMB power spectra betweenmodels with the same notation as the left panel. the degeneracy between w (or N eff ) and h. Acknowledgments
We would like to thank for useful discussion. This work were carried out using computing resources ofKIAS Center for Advanced Computation. We also thank for the hospitality at APCTP during the programTRP.
APPENDIX
Ratio of odd-to even peaks is due to the gravity-pressure balance in fluid. Thus, we adopt ω b ω γ fromPlanck. Amplitude of all peaks (damping during MD) depends on z eq . One can find z eq from the Planckbest fit values for N eff and ω c . Thus, we fix z eq for all models. From these, one can directly relate the N eff to ω c . We limit our consideration to the flat universeΩ m0 = Ω r0 (1 + z eq ) → z eq = Ω m0 Ω r0 = Ω c0 + Ω b0 Ω γ h ( ) / N eff i → Ω c0 = (1 + z eq )Ω γ (cid:16) . N eff (cid:17) − Ω b0 (A-1) → ω c [ N eff ] = ω γ (cid:16) . N eff (cid:17) (1 + z eq ) − ω b In order to fix the location of peak, one should fix θ s ( z ∗ ) = r s ( z ∗ ) /d ( c ) A ( z ∗ ). First adopt the best fit valuefor N eff = 3 . θ s ∗ [ N eff , w ] = θ s ∗ [3 . , − . w for the different values of N eff . If one fixes z eq , ω γ , and ω b , then ω c depends on N eff and thus w depends on ω c ( i.e. N eff ). Now we consider r s and r d to make sure the ratio of θ s to θ d isconstant for the different models. r s ( z ∗ ) = c √ H Z ∞ z ∗ dz p R [ z ] E [ z ] , (A-2) d ( c ) A ( z ∗ ) = cH Z z ∗ dzE [ z ] , (A-3) r d ( z ∗ ) = vuut cπ H Z ∞ z ∗ (1 + z ) dzσ T X e n b (1 − Y P ) E [ z ] " R + (1 + R )6(1 + R ) = vuut cπ H σ T X e ω b − . Y P (1 . · − )(1 − Y P ) Z ∞ z ∗ dz (1 + z ) E [ z ] " R + (1 + R )6(1 + R ) , (A-4)where R [ z ] = 3 ρ b / ρ r = ω b ω γ (1 + z ) − , σ T is the Thomson scattering cross section, X e is the ionizationfraction, and Y P is the Helium fraction.Big Bang Nucleosynthesis prediction depends on the baryon density ω b . It is related to the baryon tophoton ratio, η ≡ n b /n γ . Relativistic neutrinos contribute to the radiation energy density of the Universe ρ r ρ r = π (cid:16) k B T γ (cid:17) (1 + z ) " (cid:16) (cid:17) / N eff (A-5)Also the critical energy density of the Universe at present is ρ cr0 ≡ H πG N = 3 · (100h km / s / Mpc) π · . × − cm g − s − G (14)N G N = 1 . × − h G (14)N G N [g / cm ] , (A-6)where we use the new value for the Newton’s gravitational constant G (14)N = 6 . × − cm g − s − [31].If one adopts the old value of G N , then one obtains the slightly different value of ρ cr0 . The photon numberdensity is given by n γ = 2 ζ (3) π k B T γ ~ c ! = 2 ζ (3) π . · − (eV / K) × . . · − (eV · s) / (2 π ) × . · (cm / s) ! T γ, . ! = 410 . T γ, . ! cm − (A-7)Also, the baryon number density is n B = ρ B m B = m H m B ρ B m H = m H m B ρ B ρ cr ρ cr m H = 11 − . Y P Ω B . · − h MeVcm − . G (14) N G N ! = 1 . · − − . Y P ω b G (14) N G N ! cm − , (A-8)where we use m H = 938 . η ≡ ( n B /n γ ) is given by η ≡ n B n γ = 1410 .
802 1 . × − − . Y P Ω B h G (14) N G N ! T γ, . ! − = 273 . − . Y P ω b G (14) N G N ! . T γ, ! (A-9)0 [1] J. Lesgourgues and S. Pastor, New J. Phys. , 065002 (2014) [arXiv:1404.1740].[2] E. Giusarma, E. D. Valentino, M. Lattanzi, A. Melchiorri, and O. Mena, Phys. Rev. D , 043507 (2014)[arXiv:1403.4852].[3] W. Sutherland and L. Mularczyk, Mon. Not. Roy. Astron. Soc. , 3128 (2014) [arXiv:1401.3240].[4] K. N. Abazajian, and et al. , [arXiv:1309.5383].[5] Planck Collaboration : P. A. R. Ade, and et al. , [arXiv:1303.5076].[6] Z. Hou, and et al. , Astrophys. J. , 74 (2014) [arXiv:1212.6267].[7] J. Lesgourgues and S. Pastor, Phys. Rept. , 307 (2006) [aXiv:astro-ph/0603494].[8] C. Brust, D. E. Kaplan, and M. T. Walters, JHEP , 058 (2013) [arXiv:1303.5379].[9] G. Mention, M. Fechner, Th. Lasserre, Th. A. Mueller, D. Lhuillier, M. Cribier, and A. Letourneau, Phys. Rev.D , 073006 (2011) [arXiv:1101.2755].[10] W. Rodejohann and H. Zhang, Phys. Lett. B , 81 (2014) [arXiv:1407.2739].[11] S. Rajpoot, S. Sahu, and H. C. Wang, Eur. Phys. J. C , 2936 (2014) [arXiv:1310.7075].[12] MiniBooNE Collaboration : A. A. Aguilar-Arevalo, and et al. , Phys. Rev. Lett. , 181801 (2010)[arXiv:1007.1150].[13] G. Hinshaw, and et al. , Astrophys. J. Suppl. , 19 (2013) [arXiv:1212.5226].[14] J. L. Sievers, and et al. , J. Cosmol. Astropart. Phys. , 060 (2013) [arXiv:1301.0824].[15] E. Di Valentino, and et al. , Phys. Rev. D , 023501 (2013) [arXiv:1301.7343].[16] L. Verde, S. M. Feeney, D. J. Mortlock, and H. V. Peiris, J. Cosmol. Astropart. Phys. , 013 (2013)[arXiv:1307.2904].[17] A. Coc, J.-P. Uzan, and E. Vangioni, [arXiv:1307.6955].[18] K. A. Olive, Nucl. Phys. Proc. Suppl. , 79 (2000) [arXiv:astro-ph/9903309].[19] Z. Hou, R. Keisler, L. Knox, M. Millea, and C. Reichardt, Phys. Rev. D , 083008 (2013) [arXiv:1104.2333].[20] Y. I. Izotov, T. X. Thuan, and N. G. Guseva, accepted in Mon. Not. Roy. Astron. Soc.[arXiv:1408.6953].[21] Y. I. Izotov, G. Stasinska, and N. G. Guseva, Astron. Astrophys. , 33 (2013) [arXiv:1308.2100].[22] E. Aver, K. A. Olive, R. L. Porter, and E. D. Skillman, J. Cosmol. Astropart. Phys. , 017 (2013)[arXiv:1309.0047].[23] A. Font-Ribera, P. McDonald, N. Mostek, B. A. Reid, H-J. Seo, and A. Slosar J. Cosmol. Astropart. Phys. ,023 (2014) [arXiv:1308.4164].[24] D. Valentino, Eleonora; Melchiorri, Alessandro; Mena, Olga J. Cosmol. Astropart. Phys. , 018 (2013)[arXiv:1304.5981].[25] S. Lee, [arXiv:1205.6304].[26] Y.-S. Song and W. J. Percival, J. Cosmol. Astropart. Phys. , 004 (2009) [arXiv:0807.0810].[27] S. Lee and K.-W. Ng, [arXiv:1010.2291].[28] C.-P. Ma, R. R. Caldwell, P. Bode, and L. Wang, Astrophys. J. , L1 (1999) [arXiv:astroph/ 9906174].[29] J. A. Peacock, [arXiv:astro-ph/0309240].[30] S. Lee, [arXiv:1409.1355].[31] G. Rosi, et.al. , Nature510