The impact of the locally measured Hubble parameter on the mass of Sterile neutrino
MMNRAS , 1–8 (2019) Preprint 18 October 2019 Compiled using MNRAS L A TEX style file v3.0
The impact of the locally measured Hubble parameter onthe mass of Sterile neutrino
M. Ebadinejad, (cid:63) Astronomy Centre, Department of Physics and Astronomy, University of Sussex, East Sussex, Brighton BN1 9QH, UK
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
We present a precise analysis to test hypothetical models involving sterile neutrinos be-yond the standard flat-ΛCDM cosmology with the CMB observations from the
Planck mission and BAO measurements. This analysis shows that adding the locally measuredHubble parameter H = 73 . ± .
75 km s − Mpc − to the data removes the needfor the informative physical m thermalsterile prior in CMB constraints of m effν,sterile . Underthe constraints from the data containing the locally measured H we obtain an upperlimit m effν,sterile < .
306 eV scale mass for the massive sterile neutrino, and an upperlimit Σ m ν < .
214 eV scale mass for the three degenerate massive neutrino (95 percent confidence level). We also obtain the value σ = 0 . +0 . − . (95 per cent confidencelevel), which is in compatibility with the constraints from Planck σ level. We find that introducing parameter m effν,sterile to the model of cosmologyreduces the σ value and moves it closer to the obtained value for this parameter fromthe KiDS-450 analysis. Our results show that the locally measured Hubble parametercan increase constraints on σ values. Key words: astroparticle physics – neutrinos – cosmic background radiation – earlyUniverse – cosmology: observations
If they exist, Sterile neutrinos would make revolutionarychanges in our understanding of physics from the smallest tothe largest scales. The assumption that the sterile species arethermalized along with the three flavours of massive (active)neutrinos in the early Universe created immense motivationamongst particle physicists and cosmologists to search forthis hypothetical particle. They interact only via the grav-itational force, rather than the fundamental forces of theStandard Model (SM) of particle physics. Hence, their exis-tence would introduce physics beyond the SM.In particle physics, from the Z-Boson experiment in theLarge Electron-Positron Collider (LEP), the number of ac-tive (light) neutrinos which are sensitive to the weak interac-tions of the SM is three, corresponding to the three flavoursof neutrino (Particle Data Group et al. 2004). However, it isalso possible to have the number of mass states in neutrinosgreater than the number of active flavour states. In this case,the results of the LEP experiment are justified, and the ex-tra neutrino states must be considered as a sterile neutrino.The anomalies in Short Baseline (SBL) neutrino oscillation (cid:63)
E-mail: [email protected] (ME) measurements, for instance the results of the Liquid Scintil-lator Neutrino Detector (LSND) experiment, provide inde-pendent evidence of neutrino conversions at a greater masssplitting difference of masses squared. In this case a fourthneutrino, a light sterile with a mass of ∼ (1eV), is required(Aguilar et al. 2001; Palazzo 2013). Also the results of theMiniBooNE experiment indicate the conversions of candi-date flavour neutrinos to a sterile at SBLs (Aguilar-Arevaloet al. 2007). This experiment in Fermilab reports the resultsfrom an analysis of electron-neutrino ( ν e ) appearance datain SBLs, assuming a mass oscillation which corresponds toa sterile neutrino (Aguilar-Arevalo et al. 2018).Cosmology is significantly affected by sterile neutrinosfrom the radiation-dominated era in the early Universe tothe late Universe with matter and radiation energy den-sity (Abazajian 2017). The light sterile neutrinos with eV-scale mass, and the relic of massive active neutrinos, con-tribute to the total radiation energy density of the Uni-verse as the characteristic dark radiation. Where the HubbleLaw dominates the expansion, they fix the expansion ratein the radiation-dominated era of the Universe (Lesgour-gues & Pastor 2006). They affect cosmic microwave back-ground (CMB) anisotropies (Hu & Dodelson 2002; Ma &Bertschinger 1995). The anisotropies of the CMB contain c (cid:13) a r X i v : . [ a s t r o - ph . C O ] O c t M. Ebadinejad. information on the mass and energy content of the Universeand the origin of cosmic neutrinos from the early Universe(Abazajian et al. 2015). Hence, cosmological observations ofthe CMB can potentially provide us with evidence for theexistence of sterile species, independent of the laboratoryexperiments in particle physics (Abazajian 2017). CMB ob-servations are useful to probe the predictions beyond theflat Lambda-Cold Dark Matter (ΛCDM) model of cosmol-ogy (Walcher 2004; Lesgourgues et al. 2013). There is the as-sumption that sterile neutrinos are partially or completelythermalized at the decoupling time in the early Universeand they affect the CMB (Abazajian et al. 2015; Gariazzo2016). The constraints on the linear matter power spec-trum of thermalized sterile neutrinos at eV-scale mass andthe constraints on massive active neutrinos are similar asboth are measured in large scale structure (LSS) observa-tions when they are non-relativistic (Abazajian 2017). In asimilar way to active neutrinos, the eV-scale mass thermal-ized sterile neutrinos in the Big Bang nucleosynthesis (BBN)epoch contribute to the total radiation energy density whenthey are relativistic. They significantly alter the effectivenumber of relativistic species ( N eff ) in the early Universe.This is while massive neutrinos later influence matter den-sity. Hence, BBN and LSS can constrain a partially or fullythermalized sterile neutrino (Steigman et al. 1977; Sasankanet al. 2017; Mathews et al. 2018). The N eff is a quantitythat precisely describes relativistic species, such as massiveactive neutrinos as well as eV-scale mass light sterile neutri-nos at the BBN epoch with high relativistic energy density;when the abundant light species constrain the thermalizedsterile neutrinos in the early Universe. But N eff does not ac-curately describe the non-relativistic sterile neutrinos overthe period of structure formation (Cyburt et al. 2016). Inthe case of KeV-scale massive sterile neutrinos-as a possibledark matter particle candidates-they do not affect BBN dueto their abundance, which is very small in relative density,compared to radiation in the radiation-dominated epoch ofBBN. Therefore, in this work we show that data from ob-servation of the CMB, will only provide constraints on lightsterile neutrinos. However, we also assume the presence ofmassive sterile neutrinos, via partial or complete thermaliza-tion in the early Universe (Barger et al. 1998). The presenceof a massive sterile neutrino in cosmological models pro-vides a potential solution to the neutrino oscillation anoma-lies (Kopp et al. 2013). This follows recent research whichis motivated by models including massive sterile neutrinos(Hamann & Hasenkamp 2013; Battye & Moss 2014; Wymanet al. 2014). Apart from ordinary massive neutrinos, thepresence of additional massive particles, such as massivesterile neutrinos, would resolve the tension between Planck
CMB observations and other astrophysical data by leadingto a lower value in the amplitude of mass fluctuation ( σ ),(Planck Collaboration et al. 2016b). In this paper, we in-vestigate the constraints on massive sterile neutrinos by the Planck observations and other data in an extended model ofstandard ΛCDM. We also look to the constraints on σ anddiscuss whether a possible massive sterile neutrino wouldsignificantly alter this parameter.The existence of a sterile neutrino would alter the cur-rent favoured SM of cosmology: flat-ΛCDM. CMB observa-tions are based on cosmological parameters which are de-fined at low redshift (low z) and depend on the SM. It is useful to measure some of these parameters locally, to de-termine whether they are independent of the cosmologicalmodel. The measurement of quantities such as the Hubbleconstant in the local Universe helps to test the consistency ofthe current flat-ΛCDM model and its constraints on cosmol-ogy, as the observations are independent of the model itself(Verde et al. 2013). Hence, addition of the directly measuredHubble parameter at low redshift to CMB cosmological dataat high redshift helps one to investigate whether the exten-sion of the SM of cosmology by a new parameter is validated.The proposed way to do the test is by extending the mainΛCDM model using one or more extra parameters, and con-straining these parameters with current observations.The ΛCDM model successfully fits various observations.In this paper, we use the recent locally measured Hubbleparameter in addition to the Planck
CMB observations andgalaxy BAO data, to test the impact from the introducingsterile and massive neutrinos parameters to the standardΛCDM, on the late cosmological parameters, H , σ , Ω m .We focus on testing the impact on the extended parame-ters m effν,sterile massive sterile neutrino, Σ m ν three degener-ate massive neutrino, and the parameter σ , from the locallymeasured H ; this is what we usually can not do in standardΛCDM due to its tension with the local Universe. In this work the latest observed Cosmic Microwave Back-ground, CMB anisotropy-temperature data by the
Planck mission in 2015 (Planck Collaboration et al. 2016b) is used.We consider the likelihood at 0 ≤ l ≤ < l <
30 which is the low- l part of the TTpower spectrum with the range including polarization, andPlik, the CMB TT likelihood in the range of 30 < l < H =73 . ± .
75 km s − Mpc − , by Riess et al. (2016) in orderto find out the impact of this observable on sterile neutrinosin the analysis. Thus, we have two sets of observational datain this paper: CMB + BAO and CMB + BAO + H . We use the software called camb , Code for Anisotropies inthe Microwave Background by Lewis (2013). This is a codefor cosmological calculations as well as calculating CMB andmatter power spectra. In the process of modelling and anal-ysis, the constraints on the cosmological parameters basedon the ΛCDM model are produced using camb
Boltzmanncode (Lewis et al. 2000). These parameters are: ω b = Ω b h , MNRAS000
Boltzmanncode (Lewis et al. 2000). These parameters are: ω b = Ω b h , MNRAS000 , 1–8 (2019) he locally measured Hubble parameter ω c = Ω c h , 100Θ MC , τ , A s and n s , which are, respectively,baryon matter density, present cold dark matter density, theratio between the acoustic horizon and the angular diameterdistance at the time of decoupling, Thomson scattering op-tical depth, amplitude of first curvature perturbation powerspectrum and the spectral index of the first curvature per-turbation. However, these constraints are not reported inthis work. But the constraints on the derived parametersof Ω m , σ and H ; the total matter density, the mass fluc-tuation amplitude, and the Hubble parameter, respectively,are reported. In addition, all the temperature nuisance pa-rameters which are used by Planck likelihood, are allowedto vary in this analysis. We adopt the same priors from thebaseline
Planck
CosmoMC software, which is aMarkov-Chain Monte-Carlo (MCMC) engine for exploringcosmological parameter space for plotting and presenting theresults (Lewis & Bridle 2002).We compare the sets of data with the theoretical mod-els to test our speculation on the models in the ΛCDMparadigm of cosmology. In the modelling we consider thelight and massive sterile neutrinos as two cases to be addedinto the cosmological models; the parameter N eff for the lightsterile neutrino as it contributes to the dark radiation, andthe additional parameter, m effν,sterile , for the massive sterileneutrino. Under the assumption that the sterile neutrinosare thermalized along with the active neutrinos in the earlyUniverse, the physical mass of sterile neutrino, m thermalsterile ,which is thermally distributed, is measured via the equationbelow (Planck Collaboration et al. 2016b). m thermalsterile = ( N eff − . − / m effν,sterile , (1)Where in a universe with an extra relativistic particlesuch as a possible sterile neutrino, we must consider thevalue greater than 3.046 for N eff avoiding a negative value for m thermalsterile . We use the arranged data sets from the previoussubsection to constrain the parameters specified by light andmassive sterile neutrinos, in the extended models. Thus, wecompare the models ΛCDM, ΛCDM + N eff and ΛCDM + N eff + m effν,sterile under the same data sets. We discuss the implications of the fitting results for sterileneutrinos, by looking to the constraints from our data setson the extended parameters, N eff and m effν,sterile as well as,the constraints on the derived parameters σ , Ω m and H .The impact of the additional data, the locally measured H ,on the parameters, is investigated especially. We include theparameter specified for the massive active neutrino in someof the models in order to check the constraint from the dataon the massive neutrino, showing whether the models as-sociated with this parameter could better fit the data. Wekeep the total mass of the three degenerate massive neutri-nos Σ m ν = 0 .
06 eV scale mass under the assumption of theminimal-mass normal hierarchy.
As explained in Section 1, light sterile neutrinos serve ascandidates for the dark radiation. N eff = 3 is for the threeSM neutrinos that were thermalized and decoupled well be-fore the electron-positron annihilation in the early Universe.In standard cosmology it is predicted that N eff = 3.046 forthe three active neutrinos. This is because of the transferof a small amount of entropy due to the non-instantaneousdecoupling of neutrinos at electron-positron annihilation(Mangano et al. 2005). In the search for a relic of an ex-tra relativistic species, for example a light sterile neutrino,we presume that an additional light particle is produced be-fore re-combination, with an energy density that scales withthe expansion exactly like that of active neutrinos. Here welook for a larger value for this parameter in our fitting re-sults that may correspond to the presence of an extra lightspecies. Hence, a measurement of extra relativistic degreesof freedom in the early Universe, ∆ N eff = N eff - 3.046 > N eff to vary increases the uncertainty in the H constraint from Planck
CMB observations, alleviating the tension with thelocal measurement of H (Bernal et al. 2016; Planck Collab-oration et al. 2016b; Riess et al. 2016). In this work, Table 1shows that, under the constraints from CMB + BAO + H data, for the ΛCDM and the ΛCDM + N eff models we ob-tain the values of H = 68.51 ± − Mpc − , and H = 70.7 ± − Mpc − , respectively. Here, the derivedvalue of H for the ΛCDM model might not be reliable dueto the Hubble tension which is caused by a tension metricfor locally measured value of H against CMB + BAO data.The difference between value of local H = 73 . ± .
75 kms − Mpc − and the derived value of H for the ΛCDM + N eff model, is 1.08 σ . It indicates that here the addition ofthe parameter N eff can relieve the tension between this dataand the local H , showing a compatibility with the localmeasurement; hence an extra relativistic species such as apossible light sterile neutrino in our model of cosmology, cansurely improve cosmological fit to the data, as is indicatedby the results of total χ in Table 1. This result is consistentwith the obtained results of the similar works, for example(Planck Collaboration et al. 2016b). We also obtain a dis-crepancy at 2.82 σ level from the current Planck H , inthe standard ΛCDM. One thing we perceive here is that thefavoured flat-ΛCDM model of cosmology for large scales,which is well described by CMB observations, does not fitboth measurements well, or that the systematic errors couldbe the cause of the tension between the local and the largescale measurements (Verde et al. 2013; Bernal et al. 2016).The fitting results also show that, the addition of BAO toCMB data leads to only a small reduction in the value oftotal χ for the models, as BAO contains only a small partof the whole observational data we use and it contributesless than the CMB in fitting the models to the data combi-nations; but the results in Table 1 also show that the BAOdata helps to give a better constraint on the cosmological pa-rameters, as it produces smaller errors associated with theobtained results. MNRAS , 1–8 (2019)
M. Ebadinejad.
Table 1.
Fitting results for the cosmological models including a massive sterile neutrino m effν,sterile , light sterile neutrino N eff and massiveactive neutrino Σ m ν with 1 ± σ errors. We quote the 95 per cent confidence level upper limits for the parameters that cannot be wellconstrained. Model Data H N eff m effν,sterile Σ m ν Ω m σ Total χ ΛCDM CMB 67.33 ± ± ± . +2 . − . . +0 . − . – 0.06 0.311 ± . +0 . − . ± ± ± ± ± ± N eff CMB+BAO 69.1 ± ± ± ± N eff CMB+BAOH0 70.7 ± ± ± ± N eff m effν,sterile CMB+BAO 69 . +0 . − . < . < .
468 0.06 0.303 ± . +0 . − . N eff m effν,sterile CMB+BAOH0 70.8 ± ± < .
306 0.06 0.296 ± . +0 . − . N eff Σ m ν CMB+BAO 69.1 ± ± < .
240 0.302 ± ± N eff Σ m ν CMB+BAO H . +1 . − . . +0 . − . – < .
214 0.295 ± ± N eff m effν,sterile Σ m ν CMB+BAO 69 . +0 . − . < . < . < .
216 0.303 ± . +0 . − . N eff m effν,sterile Σ m ν CMB+BAO H . +1 . − . . +0 . − . < . < .
213 0.297 ± . +0 . − . Ω m H ΛCDMΛCDM+ N eff + m eff ν, sterile (a) Ω m H ΛCDMΛCDM+ N eff + m eff ν, sterile (b) Figure 1.
The two-dimensional marginalized contours (68 and95 per cent confidence level) in the Ω m − H plane, from theconstraints of the CMB + BAO (a) and the CMB + BAO + H (b) data combinations in the ΛCDM and ΛCDM + N eff + m effν,sterile models. The constraints from CMB + BAO were released in
Planck papers. In this work we use the updated BAO in thedata set.
The obtained values of constraints from the data combina-tions on the parameter N eff , infer the values of the extrarelativistic degrees of freedom in the early Universe, givenan extended ΛCDM model. In the same way as baseline Planck ± .
23 at the68 per cent confidence level. Here, the CMB + BAO con-straint on N eff provides a deviation from the standard 3.046value only at the 0.75 σ level; as we discussed earlier, theaddition of the BAO data to CMB improves the fit of thedata to the models only with a small amount. Moreover, thederived value of Hubble parameter H = 69 . ± . − Mpc − is obtained under this data set for ΛCDM + N eff ,which is in discrepancy (below 1 σ error) with the derived H = 68 . ± . − Mpc − of the same analysis in thepublished Planck H data set well constrains N eff in the model, with the measured value of 3.45 ± .
20 at the68 per cent confidence level, providing a stronger preferenceat the 2.02 σ level for the extra dark radiation density. Thismight be an indication of the presence of extra relativisticspecies such as a potential light sterile neutrino. Here, the MNRAS000
20 at the68 per cent confidence level, providing a stronger preferenceat the 2.02 σ level for the extra dark radiation density. Thismight be an indication of the presence of extra relativisticspecies such as a potential light sterile neutrino. Here, the MNRAS000 , 1–8 (2019) he locally measured Hubble parameter impact of the directly measured local H on fitting the ex-tended model to this data set is significant, improving thefit by ∆ χ = 3.80 for the model. Here we have the parameter m effν,sterile which denotes themassive sterile neutrino, in addition to N eff , in the ΛCDMmodel. Fig. 1 shows how Ω m affects the constraints on the H , having a negative correlation for both models under thedata combinations. Panel(b) in the figure shows that com-bining the local H measurement with the data results inthe reduction of Ω m values and an increase in H values forboth models in comparison to the plot in panel (a). Table1 shows that the additional m effν,sterile , improves the fit ofthe model to the data by only a small amount, as we find asmall reduction on the total χ value for the model associ-ated with this parameter in comparison to the model withonly the parameter N eff . Under the constraint from both ofthe data sets we use in this analysis, for the model including m effν,sterile , we find slightly higher derived values for parame-ters H and Ω m , and lower values for σ , in comparison withthe model containing only N eff . It is notable that the mea-sured values for parameters H and σ under CMB + BAOdata are in slight disagreement with values obtained fromthe same analysis in the Planck H is to increase the measured valuesof σ in the given extended SM, where in ΛCDM + N eff + m effν,sterile the massive sterile neutrino helps to pull down the σ values but increases the derived H values. As discussedin the baseline Planck m ν as in the base ΛCDM, could potentially help to resolvethe tension between Planck observations and the late astro-physical measurements by introducing sufficient freedom toallow higher values of H and lower values of σ . Here we consider the case of the extended model which is de-scribed earlier in this subsection. The potential degeneratemass above the minimal baseline, as well as the relativisticparticle contribution from massive active neutrinos to N eff ,is considered. Hence, we simultaneously constrain both pa-rameters in the extended model. For thermally distributedsterile neutrinos, m effν,sterile is related to the physical massvia Eq.(1). To measure the constraints on m effν,sterile , we usethe same prior, m thermalsterile <
10 eV, on the physical thermalmass as the one used in the baseline
Planck N eff to avoid negative values for the physical mass m thermalsterile , asexplained for Eq.(1).Table 1 shows that, under the constraints of the CMB+ BAO data for massive sterile neutrinos, we obtain onlyan upper limit of m effν,sterile < .
468 eV scale mass at the 95per cent confidence level, which is tighter than the obtainedvalue m effν,sterile < .
571 eV scale mass in the
Planck H data westill obtain an upper limit of m effν,sterile < .
306 eV scale massat the 95 per cent confidence level, indicating that the locallymeasured H only helps to constrain the mass by tighteningthe constraints on the parameter m effν,sterile . However, theseresults are similar to the results of the baseline Planck
Chandra
X-ray observations of the Andromeda galaxy by Horiuchiet al. (2014).
It is found that neutrinos acquire mass via neutrino mass os-cillation (Fukuda et al. 1998). A normal mass hierarchy withΣ m ν ≈ Planck ob-servations. There is also possibility of a degenerate hierarchywith larger mass for active neutrinos Σ m ν (cid:38) σ . Neutrino masses be-low 1eV have only a mild effect on the CMB power spectrumas they are relativistic. But there is some sensitivity of CMBanisotropy to neutrino masses when neutrinos begin to beless relativistic at the time of recombination (Planck Col-laboration et al. 2016b). The total χ test results presentedin Table 1 show that the addition of parameter Σ m ν to theextended models, improves the fit of the model by only asmall amount for both data sets.The neutrino mass constraints from other works includethe analysis of the KIDS-450 data (Hildebrandt et al. 2017),a derived neutrino mass constraint leading to Σ m ν < . Planck
CMB results, BAOmeasurements and measured Ly α power spectrum producesan upper limit of Σ m ν < .
14 scale mass at 95 per centconfidence level Palanque-Delabrouille et al. (2015).We use the prior with the range (0.01 1) to constrain theparameter Σ m ν . Table 1 shows that under the constraints ofCMB + BAO data we derive an upper limit Σ m ν < . m ν < .
266 eV scale mass of similar analysisin baseline
Planck H measurement in the data set helps toconstrain only a tighter upper limit Σ m ν < .
214 eV scalemass at the 95 per cent confidence level. The obtained upperlimit values of Σ m ν for the model associated with massivesterile neutrino are also presented in Table 1. MNRAS , 1–8 (2019)
M. Ebadinejad. m eff ν, sterile σ CMB + BAOCMB + BAO + H0 (a) m eff ν, sterile N e ff CMB + BAOCMB + BAO + H0 (b)
Figure 2.
The constraint results for the ΛCDM + N eff + m effν,sterile model from the CMB + BAO and the CMB + BAO + H data combinations are shown in two-dimensional marginalizedposterior contours (68 and 95 per cent confidence level). Panel(a):Shows the significant impact from local H on the mass of highmass sterile neutrinos. Panel(b): The grey shading shows the ex-cluded region of parameter space by the prior m thermalsterile <
10 eV,where the neutrinos with the high mass near to the tail of prior,behave like dark matter. σ Table 1 indicates that under the CMB data combined withupdated BAO measurement in this paper, we derive largerconstraint values for σ , in comparison to the results of sim-ilar analysis in baseline Planck H has an impact on σ , increasing its de-rived constraint values for our extended models, where forinstance we obtain the highest value of 0.851 ± N eff . These results also show that forthe models included with parameters m effν,sterile or Σ m ν wederive smaller constraint values for parameter σ , where amassive sterile neutrino helps to pull down the amount ofconstraints on this parameter more than the degenerate mas-sive neutrinos. This effect is also discussed in earlier workssuch as Planck collaboration results. In plot (a) of Fig. 2,we show the impact of massive sterile neutrinos on σ underthe constraints from both data combinations in ΛCDM + N eff + m effν,sterile . The plot indicates that m effν,sterile is nega-tively correlated with σ , which implies that the higher mass,for instance the massive sterile neutrino in the cosmologicalmodel, leads to a lower σ . The obtained constraint valuesof σ in Table 1 are consistent with the observed tendencyin the plot.One issue with the constraints on m effν,sterile fromCMB observations, is the need for the informative physi- cal m thermalsterile prior. In plot (a) of Fig. 2, we assume that theobserved high value of m effν,sterile is caused by this problem.But interestingly, this plot also indicates that the additionallocally measured H value might resolve this problem by re-ducing the high values of m effν,sterile . The effect of very mas-sive neutrinos on the CMB spectrum is identical to that ofcold dark matter (Planck Collaboration et al. 2016b) whichcannot be distinguished. Hence, a prior of m thermalsterile < H ,the measured value for dark radiation is pulled away fromthe standard value N eff = 3.046. From this, we may concludethat there would be a lesser distribution probability in thehigh-mass tail. Further, it suggests that there could be noneed for a high physical-mass tail when constraining massivesterile neutrinos from the CMB under the locally measured H . It is assumed that the consideration of a low redshiftdata combination, such as the package of SZ (the PlanckSunyaev-Zeldovich), the Planck
Lensing, WL (the weak lens-ing), galaxy clustering, and the local measurement of theHubble constant H , can lead to a better constraint on σ resulting in a lower value (K¨ohlinger et al. 2017). However,the results in this work show that the inclusion of an H -measured at low redshift-in our data combination does nothelp to obtain this.Fig. 3 shows the plots of introduced parameters in var-ious neutrino extensions of the base ΛCDM model againstthe late Universe cosmological parameters. Parameter con-tours show the clear degeneracies between cosmological andthe introduced parameters; and the contribution from thelocally measured H , for instance, its significant effect on m effν,sterile , decreasing high values of mass for sterile neutrinois clear. The massive active neutrinos are degenerated with the cos-mological parameters. They slow down the growth of thematter fluctuation (Lesgourgues et al. 2013). Here we in-clude the massive active neutrino along with a massive ster-ile neutrino in the cosmological model in order to find outtheir contribution to σ . The plot in panel (a) of Fig. 4 in-dicates that, the model with a massive sterile neutrino canproduce a lower value for parameter σ , in comparison tothe model with only massive active neutrino; Table 1 showsthat under the constraint of CMB + BAO we obtain σ = 0 . +0 . − . and σ = 0.834 ± σ that covers the lower value of σ . Thus, under theconstraint from this data combination for the model withboth, m effν,sterile and Σ m ν , we produce a constraint with alower and upper limit of σ = 0 . +0 . − . at the 95 percent confidence level. These results imply that the potentialmassive sterile neutrinos in cosmology could actually slowdown the structure growth in the Universe more than the MNRAS000
Lensing, WL (the weak lens-ing), galaxy clustering, and the local measurement of theHubble constant H , can lead to a better constraint on σ resulting in a lower value (K¨ohlinger et al. 2017). However,the results in this work show that the inclusion of an H -measured at low redshift-in our data combination does nothelp to obtain this.Fig. 3 shows the plots of introduced parameters in var-ious neutrino extensions of the base ΛCDM model againstthe late Universe cosmological parameters. Parameter con-tours show the clear degeneracies between cosmological andthe introduced parameters; and the contribution from thelocally measured H , for instance, its significant effect on m effν,sterile , decreasing high values of mass for sterile neutrinois clear. The massive active neutrinos are degenerated with the cos-mological parameters. They slow down the growth of thematter fluctuation (Lesgourgues et al. 2013). Here we in-clude the massive active neutrino along with a massive ster-ile neutrino in the cosmological model in order to find outtheir contribution to σ . The plot in panel (a) of Fig. 4 in-dicates that, the model with a massive sterile neutrino canproduce a lower value for parameter σ , in comparison tothe model with only massive active neutrino; Table 1 showsthat under the constraint of CMB + BAO we obtain σ = 0 . +0 . − . and σ = 0.834 ± σ that covers the lower value of σ . Thus, under theconstraint from this data combination for the model withboth, m effν,sterile and Σ m ν , we produce a constraint with alower and upper limit of σ = 0 . +0 . − . at the 95 percent confidence level. These results imply that the potentialmassive sterile neutrinos in cosmology could actually slowdown the structure growth in the Universe more than the MNRAS000 , 1–8 (2019) he locally measured Hubble parameter H ΛCDM+ N eff + m eff ν, sterile Ω m ΛCDM+ N eff + m eff ν, sterile m eff ν, sterile σ ΛCDM+ N eff + m eff ν, sterile ΛCDM+ N eff + m eff ν, sterile ΛCDM+ N eff + m eff ν, sterile N eff ΛCDM+ N eff + m eff ν, sterile ΛCDM+ N eff + m eff ν, sterile +Σ m ν ΛCDM+ N eff + m eff ν, sterile +Σ m ν Σ m ν ΛCDM+ N eff + m eff ν, sterile +Σ m ν ΛCDM+ N eff + m eff ν, sterile +Σ m ν ΛCDM+ N eff + m eff ν, sterile +Σ m ν m eff ν, sterile ΛCDM+ N eff + m eff ν, sterile +Σ m ν ΛCDM+ N eff + m eff ν, sterile +Σ m ν ΛCDM+ N eff + m eff ν, sterile +Σ m ν N eff ΛCDM+ N eff + m eff ν, sterile +Σ m ν CMB+BAO CMB+BAO+H0
Figure 3.
68 and 95 per cent confidence level constraints from
Planck
CMB + BAO and
Planck
CMB + BAO + H on the lateUniverse cosmological parameters H , Ω m , σ , and the introduced parameters in the neutrino extensions of the base ΛCDM model. Thecomparison between marginalized contours show the parameter degeneracies and the contribution from the locally measured H . Herepriors on parameters are chosen from base Planck Ω m σ ΛCDM+ N eff +Σ m ν ΛCDM+ N eff + m eff ν, sterile +Σ m ν ΛCDM (a) σ ΛCDM+ N eff +Σ m ν ΛCDM+ N eff + m eff ν, sterile +Σ m ν ΛCDM (b)
Figure 4.
Panel(a): Two-dimensional marginalized posterior contours (68 and 95 per cent confidence level) of the constraint resultsfrom
Planck
CMB + BAO data combination for the ΛCDM + N eff + Σ m ν model (gray contour) and the ΛCDM + N eff + m effν,sterile + Σ m ν model (red contour); and from Planck
CMB data for the base ΛCDM model (blue contour), in the Ω m - σ plane. It shows thesignificant difference on Ω m constraints between the blue contour, and both, the gray and red contours, which is due to the use of theupdated BAO data in this analysis. A significant difference can be seen in σ constraints between the red contour and the gray and bluecontours, which is due to introducing m effν,sterile parameter in the model (red contour). Panel(b): One-dimensional marginalized posteriordistributions for σ . standard massive neutrinos. As a further result of this anal-ysis we report the compatibility in this parameter with theconstraints from Planck σ level.Although, in KiDS-450, the kilo Degree survey, the analy-sis of the data from the weak gravitational lensing cosmicshear power spectrum, which is based on 450 deg , resultedin the value S (cid:39) ± S ≡ σ (cid:112) Ω m / . Planck data at the3.2 σ level (K¨ohlinger et al. 2017). Our result is in tensionwith the result from the Kids-450 analysis at the (cid:39) σ level. This shows that including parameter m effν,sterile in themodel of cosmology reduces the σ value, and moves it closerto the value from the Kids-450 for this parameter. Further-more, our result disagrees-at the 1.9 σ level-with the S = 0 . +0 . − . (68 per cent confidence level) obtained from therecent cosmological analysis of cosmic shear two-point cor-relation functions (TPCFs) from Hyper Suprime-Cam Sub-aru Strategic Program (HSC SSP) first-year data, covering136.9 deg of the sky (Hamana et al. 2019). Our result alsodisagrees-at a 2.05 σ level-with the value of S = 0 . +0 . − . (68 per cent confidence level) from the results of measure-ments of cosmic shear in a galaxy survey from Dark EnergySurvey (DES) year 1 shape catalogs, over 1321 deg of thesouthern sky (Troxel et al. 2018)The Ω m constraint results in the panel (a) of the Fig. 4is in discrepancy with the published constraints in baseline Planck
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M. Ebadinejad. N eff + Σ m ν , whereas, the Ω m constraints are very similarbetween these two models in Planck results. This is due tothe updated BAO data we use in this analysis which leadsto the smaller constraints Ω m . Sterile neutrinos could be a natural extension to the SM ofcosmology to help better understand the properties of theUniverse.As we predicted in this study, the current CMB, as wellas some low redshift BAO observations, may not provideus with good information on the mass of particles such assterile neutrinos in an extended ΛCDM model; as we onlyobtain an upper limit on parameter m effν,sterile . In the caseof three degenerate massive neutrino Σ m ν , we find only anupper limit on the mass of this particle too. Addition of thelocally measured Hubble parameter H in the data helpsto constrain the mass by only tightening the constraints onthe parameters m effν,sterile and Σ m ν . Involving massive ster-ile neutrino in cosmology may help to slow down the rateof structure growth in the Universe, by leading to lower σ values, where we show the compatibility in obtained σ withthe constraints from Planck
Planck
CMB observations and the lateastrophysical measurements of H and σ ; whereas, the im-pact from standard massive neutrinos on these parametersis very small. We observe that the impact from locally mea-sured H on the value of the CMB constraints of m effν,sterile might solve the issue of the need for an informative phys-ical m thermalsterile prior. We find that the addition of the local H in the observational data, results in increasing constraintvalues for σ .A future study on the B-mode signal from CMB polar-ization observations might provide hints on the nature of allphysical properties of the Universe that influence the forma-tion of large structure, such as the mass of neutrinos or darkmatter. ACKNOWLEDGEMENTS
This paper is based on the author’s masters thesis, writtenunder the supervision of Professor Antony Lewis of the As-tronomy Centre in the Department of Physics and Astron-omy at the University of Sussex. The author wishes to thankProfessor Lewis for very helpful comments and discussionsover the period of his supervision as well as the helpful hintsto some questions that he provided over the preparation ofthis paper. Appreciation also goes to the anonymous refereefor very helpful comments that improved the manuscript andits presentation. The author wishes to thank his friends, DrRalph Eatough from Max Planck institute for Radio Astron-omy in Bonn and Dr Mark Purver, former student at theUniversity of Manchester, for proof-reading the manuscript.
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