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Neutrino properties and Cosmology
Available online at www.sciencedirect.com
Nuclear Physics B (Proc. Suppl.) 235–236 (2013) 321–328 www.elsevier.com/locate/npbps
Neutrino properties and Cosmology L. Verdea a ICREA
& ICC-UB/IEEC, Marti i Franques 1, 08028 Barcelona, Spain
Abstract I will present some recent work on constraints on neutrino properties from cosmology, touching upon present and possibly future constraints on neutrino masses, number of families and mass hierarchy. Keywords: cosmology, large-scale structure
1. Introduction In the past few years there have been new developments in the effort of constraining neutrino properties with cosmology, both concerning new data sets and new theoretical results. On the data side, the Cosmic Microwave Background (CMB) angular power spectrum damping tail has been accurately measured by ground-based experiments like The South pole Telescope (SPT,[1]) and the Atacama Cosmology Telescope (ACT, [2]). Large scale structure surveys like the Sloan III BOSS and WiggleZ [3, 4] have measured the baryon acoustic oscillations (BAO) at redshift z > 0. There have new been direct measurements of the Universe expansion rate and history [5, 6]. Finally, future, massive large-scale strutter survey have been presented and approved (e.g., the Euclid mission [7]). On the theory side, better modeling of non-linearities via N-body simulations (and perturbation theory) have become available. Also, a new possibility has emerged of constraining the mass hierarchy with cosmology. In what follows we will use Σ to denote the total neutrino mass (i.e., the sum of the masses of the different families), Nν the number of neutrino species (or families), m the lightest mass state and M the heaviest mass state. Since oscillations indicate that one mass splitting is much smaller than the other one we will make the approximation that the small mass splitting can be neglected. For cosmological applications this is an extremely good approximation. Before we start, one remark is in order. In cosmol0920-5632/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nuclphysbps.2013.04.028
ogy many (if not all) parameters constraints are modeldependent, and the constraints discussed here are no exception. In cosmology we do observations (although we call them experiments), we only have one observable Universe and we can’t perturb it slightly and observe what happens. From these observations we want to infer the properties of the (fictitious) “ensemble” of all possible observable Universes as this is a good proxy for the properties of the whole Universe (beyond the observable one). So, interpretations of observations are inevitably made in the framework of the standard big bang model. In most cases a reference model is chosen, typically a minimal ΛCDM, where the Universe is spatially flat, dark energy is a cosmological constant and the primordial power spectrum of fluctuations is a power law. In this “reference” or “baseline” model, neutrinos are massless and the model has 6 cosmological (interesting) parameters which are to be constrained by data. Then typically one parameter at the time is being added and constrained (e.g., Σ, Nν etc.).
2. Neutrino masses Neutrinos with total mass Σ 1eV become nonrelativistic before recombination, leaving an imprint (and thus being constrained by) the primary CMB. If the total mass is instead Σ 1eV they become nonrelativistic after recombination. They still alter the matter radiation equality (compared to a “reference” model
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Figure 1: Left: Figure from [15]; total neutrino mass constraint forecast from the ground-based Large Synoptic Survey telescope (LSST) survey. The curves correspond to the normal and inverted hierarchies as constrained by oscillations experiments. The horizontal bands are the 1 and 2 σ errors on the sum of the masses (mtot is Σ). The forecast uses the weak gravitational lensing signal and assumes a prior on cosmological parameters corresponding to the statistical power of Planck mission’s data. Right: total neutrino mass forecast for the space-based Euclid mission. The signal used is galaxy clustering and Planck prior is also assumed. The calculations were presented in [16]. Stars indicated the fiducial models considered, shaded (blue) region and (blue) vertical lines indicate the 1 and 2 σ errors around two fiducial models (Σ = 0.05, 0.2eV). Note how in both cases, the error depend on the fiducial model chosen. For low Σ a detection is achieved with Euclid if the background cosmology is assumed to be a minimal ΛCDM. For higher Σ more flexible “baseline” models can be assumed where extra parameters such as curvature, equation of state parameter for dark energy and Nν can also be simultaneously constrained.
with massless neutrinos) but in the CMB this effect can be canceled by degeneracies with other parameters. After recombination however, finite neutrino masses suppress the matter power spectrum on scales smaller than the free-streaming length. In fact massive neutrinos free stream out of the potential wells smoothing small-scale perturbations. The effect is in principle large: the linear theory prediction is a 40% suppression at k > 0.2h/Mpc for Σ = 1eV compared to a model with massless neutrinos, this suppression is 15% at k > 0.2h/Mpc for Σ = 0.3eV. The latest Wilkinson Microwave Anisotropy Probe (WMAP7) results for the sum of neutrino masses is Σ < 1.3 eV at 95% confidence level [8]. This constraint has not improved much since the initial data release despite longer integration time of the experiment and subsequent improvement of the data (as expected from the above considerations). Even the measurements of the CMB damping tail does not improve the constraint much as seen in the analysis e.g., of STP data [9]: the collaboration needed to add low redshift universe information (secondary anisotropies, constraints on the amplitude of present-day clustering of matter, the σ8 parameter) to improve the constraint further. The above considerations indicate that large-scale structure measurements (secondary CMB anisotropies)
are needed to further constrain neutrino masses. In the past couple of years several groups have produced updated constraints on Σ using galaxy clustering in combination with CMB e.g., [10, 11, 12, 13]. At the time of this conference the most recent result of these comes from the latest data release of the BOSS survey [14]. All these measurements converge to Σ < 0.3eV at 95% confidence. This limit is valid for a 7 parameters model (a minimal ΛCDM +Σ), and gets degraded when more freedom is given to the model in terms of extra cosmological parameters. This constraint is worst that what one would have naively expected: Σ = 0.3 eV implies a 15% suppression of the linear matter power spectrum at small scales, and the galaxy power spectrum at these scales is measured to much better that 15% precision. However a number of factors enter in interpreting the galaxy power spectrum in terms of cosmological parameters: the relation between the dark matter and galaxy clustering properties become highly uncertain on these scales, there are degeneracies with other cosmological parameter that must be marginalized over, and there are effects of non-linearities which contribute to increase the error-bars. Future surveys however might have enough statistical power to detect the effect of neutrino masses in the sky. Fig.1 shows forecasts for two surveys set-ups. On
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the left the ground-based Large Synoptic Survey telescope (LSST) survey, the forecast uses the weak gravitational lensing signal. For the background cosmology and for breaking degeneracies, a prior based on the expected performance of the Planck satellite mapping the CMB primary anisotropy is assumed. Figure from [15]. On the right the space-based Euclid mission. The signal used is galaxy clustering and Planck prior is also assumed. The calculations were presented in [16]. It is important to note that errors depend on the fiducial model and on assumptions about the background cosmology one is willing to make. For low Σ fiducial values a detection can be achieved if the background cosmology is assumed to be a minimal ΛCDM. For higher Σ more flexible “baseline” models can be assumed where extra parameters such as curvature, equation of state parameter for dark energy can also be simultaneously constrained and still a detection can be achieved. Cosmological constraints on neutrino masses are already competitive compared with forthcoming particle physics experiments and forecasts from future surveys indicate that cosmology should be able not only to place strong upper limits on Σ but also to detect a non-zero neutrino mass by “seeing” the effect of the neutrino mass on the large-scale structure. However it is important to recall as discussed above that in cosmology, this is done as part as parameter-fitting in the context of a cosmological model. In this sense the constraints will be “model dependent”. All forecasts that I mentioned (and most of the forecasts present in the literature) are carried out under two important assumptions a) that systematic effects are under control and b) that linear theory predictions for the growth of structures apply on the scales of interest. Regarding systematic effects, as famously stated by Rumsfeld, “there are known unknowns and unknowns unknowns”. Known unknowns are for example real world effects in a survey or galaxy bias. The approach would be to model these and marginalize over them as much as possible. Different approaches also are affected by different systematics, for example weak gravitational lensing is not affected by galaxy bias as galaxy surveys are. “Unknown unknowns” are more difficult to deal with. It would be very useful to have an internal consistency check that allows one to cross check that a result of Σ > 0 is with confidence a neutrino mass detection. More on this later. What about non-linearities? On small scales, where most of the signal is, non-linearities become important. In the last few years there has been progress in modeling non-linearities of the cosmological structures in the presence of massive neutrinos both using perturba-
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tion theory e.g., [19, 20] and using N-body simulations [21, 22, 23] and also using hybrid approaches where the neutrinos are evolved analytically using perturbation theory and the dark matter is evolved numerically [24]. On the N-body side, when simulating the effect of neutrinos on non-linear growth of structure there are different options. The initial investigations simulated just the effect of a total mass, i.e. assuming degenerate hierarchy. Several different approaches have been used in the literature, using particles, grids or hybrid methods. Very recently also the hierarchy has been simulated[17]. All approaches converge and agree in their predictions in the interesting regime. In all cases non-linearities enhance the suppression of power effect predicted by linear theory. Figure 2 (from [17]) shows this. The three panels correspond to Σ = 0.3eV degenerate hierarchy, and Σ = 0.1eV normal and inverted hierarchy. Solid lines are linear theory predictions and points are the simulation results. Different colors (shades) correspond to different redshifts. At scales 0.1 < k < 1 nonlinearities enhance the effect of Σ on the power spectrum, making the predictions based on liner theory conservative in this respect. However we should bear in mind that these figures show the effect for the dark matter in real space. Only gravitational lensing probes directly the dark matter; galaxy surveys probe the galaxy field in redshift space. 3. Number of effective species The expansion rate in the early Universe is given by H(t) 8πG/3(ργ + ρν where ργ ) is the density of photons and the density of neutrinos is taken to be ρν ∝ T 4 Neff . The standard value (corresponding to Nν = 3) is Neff = 3.04 but any thermal background of light articles or anything affecting the expansion rate will be captured in Neff , not just neutrinos. This goes under the name of “dark radiation”, being ργ the radiation that is not dark. Of course, one obvious route is to look at Big bang nucleosynthesis, as the synthesis of light elements is extremely sensitive to the early Universe expansion rate. The difficult lies in inferring the primordial abundance from observations in the “late” Universe. The other route is to look at the CMB. Any change from Neff = 3.04 affects the matter-radiation equality and so the sound horizon at decoupling. A degeneracy with Ωm h2 and H0 follows. There are however other subtle effects due to anisotropic stress and the effect of matter-radiation equality on diffusion damping, which imprint small changes in the CMB damping tail. In the past couple of years a lot of attention has been devoted to whether cosmology indicates that Neff > 3.04
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Figure 2: Figure from [17]. Effect of non-linearities in the matter power spectrum in the presence of massive neutrinos. The three panels show the % effect of a non-zero neutrino mass on the matter power spectrum. Solid thin lines are the liner theory predictions (at z = 2, 1, 0 from top to bottom) and symbols are the (non-linear) N-body simulations results. Note how non-linearities enhance the effect compared to liner theory. Dotted lines are semi-analytic fits using ”halofit” by [18].
Figure 3: Posterior contours (1 and 2 σ joint confidence level) in the H0 -Neff plane. Left: WMAP7 data alone. Note the degeneracy that is artificially cut by the (top hat) prior imposed. Right: in red the result from combining WMAP7 data with the Baryon Acoustic Oscillations measurements; light (blue) shaded region shows the H0 constraint from [5]. Should the H0 measurement be overestimated by about one sigma the best fit for Neff from the combination CMB+H0 would be at 3 rather than around 4.
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and maybe indicates extra (sterile) neutrinos. A thorough review of the literature can be found in [25] and in particular their table 3. The numbers in the Neff column are indeed somewhat skewed towards Neff > 3.04 indicating that cosmological analyses consistently give best fit values > 3.04 consistent therefore with dark radiation. One however should keep in mind that a) these determinations are not independent as the WMAP data and the H0 determination are in common and b) it is always barely 2 σ, maybe except when the ACT+BAO data are included. What may be going on? Before jumping to the conclusion that cosmology indicates the presence of dark radiation and/or sterile neutrinos it is worth considering in detail how these constraints are obtained. First, it is important to recall that these determinations are obtained by first exploring a multi-dimensional parameter space and then marginalizing the corresponding posterior to derive a 1-dimensional constraint. In many cases, to shrink error-bars, many different data-sets are combined; should any two data sets be slightly in tension, a shift of the recovered parameter(s) would follow. Recall that statistically one out of three times the best fit value is more than one sigma away from the true value; thus combining three or more data-sets one should expect some “tension”. Also, most of these analyses are done in the Bayesian framework, where the posterior is used to do parameter inference. The posterior is the likelihood (which is given by the data) multiplied by the prior (which is chosen by the human). In a high-dimensional space where the data might poorly constrain one (or more) degeneracy directions, the prior can have a big effect. Figure 3 illustrates the first point. The solid (black) dark lines show the 1 and 2 σ confidence region of the WMAP posterior. The (blue) shaded band shows the H0 measurement by [5]. Note the degeneracy present when considering only WMAP data, degeneracy that is artificially cut by the (top hat) prior imposed. On the right panel, the light (blue) shaded region shows the H0 constraint from [5]. Should the H0 measurement be overestimated by about one sigma, Neff from the combination CMB+H0 would be around 3 rather than around 4. An independent determination of the low redshift expansion rate, H(z) for z < 1 has recently been presented in [6, 26]. This determination, in combination with WMAP CMB data , yields lower values for the best fit Neff : Neff = 3.38 ± 0.5 for WMAP+H(z) data and Neff = 3.37 ± 0.34 for WMAP+H(z)+SPT data. Regarding the effect of the prior on parameter inference, while a discussion is still on-going in the literature, it is instructive to look at a quantity that by con-
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Figure 4: Constraints on Neff derived from the profile likelihood ratio. The error bars are likely underestimated, but the results are, by construction, prior independent. Here ΔNeff = Neff − 3.04. Solid bars are approximately 1σ and dotted 2 σ. As we can see all dataset combinations are consistent with the standard Neff value at the 2σ level.
struction is prior-independent, like the profile likelihood ratio. Profile likelihood ratios can be constructed for exactly the same data sets combinations for which the standard Bayesian analysis has been carried out. Parameter inference in the Bayesian sense with the profile likelihood ratio is not straightforward: error-bars are likely to be under-estimated. Nevertheless if a priorindependent measurement, with error-bars likely to be under-estimated, shows no evidence for deviations from the standard Neff value, one should be very careful before taking seriously the corresponding prior-dependent evidence. A summary of the profile likelihood ratioderived constraints on Neff can be found in [27] and is summarized in Fig.4. In summary: cosmology is consistent with 3 (but also with 4) neutrinos at 2σ level. To satisfy cosmological bounds these must be light neutrinos (< 0.5eV) e.g.[28]. More wiggle room can be gained by going beyond the baseline minimal ΛCDM model, the error-bars gets larger, but this is a keen to adding epicycles to one’s theory. Of course there are always radical options like e.g., avoiding thermalization. Future (planned or proposed) surveys, like e.g., the Euclid mission, should shed light on this issue as forecasted error-bars for Neff are 1 [16]. On the other hand is is safe to say that the cosmic neutrino background has been detected at 4σ as shown in Fig. 5.
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Figure 5: Cumulative PDF for Neff . Thin dark lines: WMAP7 alone. Thick lighter lines: WMAP7+SPT. Solid: CDF = (1-CDF). The horizontal dotted lines shows the 99.7% or 0.003 level.
Neff 0
P(Nˆ eff )d Nˆ eff . Dashed
Figure 6: Figure from [29]. Left: the “response” of the linear matter power spectrum to a change in Δ. Right: present-day constraints from oscillations (gray regions) and from cosmology in the Σ, Δ plane.
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4. Outlook towards the future: can cosmology help constraining the hierarchy? Neutrinos of different masses have different transition redshifts from relativistic to non relativistic behavior, and their individual masses and their mass splitting change the details of the transition of the Universe evolution from the radiation-dominated to matterdominated regime. Following [29] we define the neutrino mass splitting Δ, depending on wether one is considering normal (NH) or inverted (IH) hierarchy as: NH
:
IH
:
(M − m) ; Σ = 2m + M Σ (m − M) ; Σ = 2M + m . Δ= Σ Δ=
(1)
Δ is negative for IH and positive for NH. The mass splitting is |Δ| and with the sign of Δ the hierarchy is univocally determined. Fig.6 shows on the left panel, the “response” of the linear matter power spectrum to a change in Δ . On the right panel are the present-day constraints from oscillations (gray regions) and from cosmology in the Σ, Δ plane. It is clear that cosmology is chiefly sensitive to the absolute value of Δ i.e., on the mass splitting, and less sensitive to the sign of Δ which is needed to fully characterize the hierarchy. In other words for a given fiducial model (say, for example an inverted hierarchy model with Σ = 0.1eV) the forecasted likelihood as function of Δ for a combination of Planck satellite data plus a given large-scale structure data will have two maxima at the same value of |Δ| = 0.5. One will be a global maximum (Δ < 0) and one a local one (Δ > 0). At this point there are two different questions one might ask: “Is there enough information in the sky to distinguish the two?” and “Can this be done with a specific survey?” The first question is relevant because in cosmology the next generation surveys will be very close to be what I call the ultimate survey. Since in cosmology we have only one observable Universe from which we try to infer properties of the (unobservable) “ensemble”, once a sizable fraction of the observable Universe has been measured, statistical error-bars cannot be significantly reduced further even increasing drastically the observing time: one has performed the ultimate experiment. It turns out that there is enough information in the sky to measure both |Δ| and its sign: one would need something close to the ultimate largescale structure experiment. (Luckily this may become available with the next generation of space-based galaxy surveys). It should also be noted that the standard approach to address this type of questions, i.e., the Fisher matrix approach to forecasts, has important limitations
Figure 7: Figure from [17]. Effect of the hierarchy on the matter power spectrum. We show the % difference between IH and NH. For a Σ = 0.1eV. Lines correspond to linear theory predictions and symbols to the N-body simulations results at different redshifts. Again nonlinearities enhance the effect on the scales of interest.
in this specific context: the likelihood is multi-peaked while the Fisher approach assumes a Gaussian likelihood. A Fisher-type approach is still possible, but extra care must be taken given the unusual form of the likelihood surface e.g., [29]. What about non-linearities? In [17] we have address this issue by running 1 billion cold dark matter particles and 2 billion neutrino particles N-body simulation of a volume of (600Mpc/h)3 . The effect of the hierarchy parameter Δ on the matter power spectrum is shown in fig.7. We show the % change in matter power spectrum between IH and NH for a Σ = 0.1eV. Lines correspond to linear theory predictions and symbols to the N-body simulations results at different redshifts. Again non-linearities enhance the effect on the scale of interest, making linear theory based forecasts conservative in this respect. Of course here we have quantified the effect on a relative quantity which is much more robust agains systematic/numerical errors. The magnitude of the effect is small (smaller that 1%), and one should deep in mind that even without massive neutrinos, it is challenging to compute the non-linear power spectrum to sub-percent precision. Even if the effect is small, if all cosmological parameters are fixed, a survey covering a not too large cosmological volume (few (Gpc/h)3 ) is enough to measure the hierarchy. Investigations on the effect of cosmological degeneracies is ongoing, but we can anticipate that with forthcoming surveys it will soon be possible to determine |Δ|. This offers a powerful consistency check: if a survey reveals a detection of a non-zero neutrino mass, the confidence on this measurement not being affected by systematic errors could
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be boosted if the measured value for |Δ| is in agreement with that predicted from oscillations for the same total mass. For example a detection of Σ ∼ 0.1eV should be accompanied by a measurement of |Δ| consistent with 0.5. Should |Δ| = 0.5 be excluded then the “detection” of Σ should be considered an systematic effect. 5. Conclusions Precision cosmology means that we can start (or prepare for) constraining interesting physical quantities. I have presented here the case for neutrino properties from cosmology: absolute mass scale, number of families, maybe hierarchy. To date, firm constraints from cosmology on neutrino properties are: • the sum of the masses is Σ < 0.3eV at 95% confidence for a baseline standard ΛCDM model where the Universe is spatially flat, the primordial power spectrum is a power law, and dark energy is a cosmological constant. • The cosmic neutrino background has been (indirectly) detected at more than 4σ. • The effective number of species Neff is consistent with Nν = 3 at 2 σ, but is also consistent with 4. Large future surveys means that sub % effects become detectable, which brings in a whole new set of challenges and opportunities (e.g., mass hierarchy). Acknowledgments I acknowledge support from ERC grant FP7-IDEASPhys.LSS 240117. References [1] Keisler, R., Reichardt, C. L., Aird, K. A., et al. 2011, ApJ, 743, 28 [2] Das, S., Marriage, T. A., Ade, P. A. R., et al. 2011, ApJ, 729, 62 [3] Anderson, L., Aubourg, E., Bailey, S., et al. 2012, arXiv:1203.6594 [4] Blake, C., Brough, S., Colless, M., et al. 2012, MNRAS, 425, 405 [5] Riess, A. G., Macri, L., Casertano, S., et al. 2011, ApJ, 730, 119 [6] Moresco, M., Cimatti, A., Jimenez, R., et al. 2012, JCAP, 8, 6 [7] Laureijs, R., Amiaux, J., Arduini, S., et al. 2011, arXiv:1110.3193 [8] Komatsu, E., Smith, K. M., Dunkley, J., et al. 2011, ApJS, 192, 18 [9] Reichardt, C. L., Shaw, L., Zahn, O., et al. 2012, ApJ, 755, 70 [10] Reid, B. A., Verde, L., Jimenez, R., & Mena, O. 2010, JCAP, 1, 3
[11] Thomas, S. A., Abdalla, F. B., & Lahav, O. 2010, Physical Review Letters, 105, 031301 [12] Gonzalez-Garcia, M. C., Maltoni, M., & Salvado, J. 2010, Journal of High Energy Physics, 8, 117 [13] Riemer-Sørensen, S., Blake, C., Parkinson, D., et al. 2012, PRD, 85, 081101 [14] de Putter, R., Mena, O., Giusarma, E., et al. 2012, arXiv:1201.1909 [15] LSST Science Collaboration, Abell, P. A., Allison, J., et al. 2009, arXiv:0912.0201 [16] Carbone, C., Verde, L., Wang, Y., & Cimatti, A. 2011, JCAP, 3, 30 [17] Wagner, C., Verde, L., & Jimenez, R. 2012, ApJL, 752, L31 [18] Bird, S., Viel, M., & Haehnelt, M. G. 2012, MNRAS, 420, 2551 [19] Saito, S., Takada, M., & Taruya, A. 2008, Physical Review Letters, 100, 191301 [20] Lesgourgues, J., Matarrese, S., Pietroni, M., & Riotto, A. 2009, JCAP, 6, 17 [21] Brandbyge, J., & Hannestad, S. 2010, JCAP, 1, 21 [22] Brandbyge, J., & Hannestad, S. 2009, JCAP, 5, 2 [23] Viel, M., Haehnelt, M. G., & Springel, V. 2010, JCAP, 6, 15 [24] Agarwal, S., & Feldman, H. A. 2011, MNRAS, 410, 1647 [25] Abazajian, K. N., Acero, M. A., Agarwalla, S. K., et al. 2012, arXiv:1204.5379 [26] Moresco, M., Verde, L., Pozzetti, L., Jimenez, R., & Cimatti, A. 2012, JCAP, 7, 53 [27] Gonzalez-Moreles et al. in prep. [28] Giusarma, E., Corsi, M., Archidiacono, M., et al. 2011, PRD, 83, 115023 [29] Jimenez, R., Kitching, T., Pe˜na-Garay, C., & Verde, L. 2010, JCAP, 5, 35
Nuclear Physics B (Proc. Suppl.) 235–236 (2013) 321–328 www.elsevier.com/locate/npbps
Neutrino properties and Cosmology L. Verdea a ICREA
& ICC-UB/IEEC, Marti i Franques 1, 08028 Barcelona, Spain
Abstract I will present some recent work on constraints on neutrino properties from cosmology, touching upon present and possibly future constraints on neutrino masses, number of families and mass hierarchy. Keywords: cosmology, large-scale structure
1. Introduction In the past few years there have been new developments in the effort of constraining neutrino properties with cosmology, both concerning new data sets and new theoretical results. On the data side, the Cosmic Microwave Background (CMB) angular power spectrum damping tail has been accurately measured by ground-based experiments like The South pole Telescope (SPT,[1]) and the Atacama Cosmology Telescope (ACT, [2]). Large scale structure surveys like the Sloan III BOSS and WiggleZ [3, 4] have measured the baryon acoustic oscillations (BAO) at redshift z > 0. There have new been direct measurements of the Universe expansion rate and history [5, 6]. Finally, future, massive large-scale strutter survey have been presented and approved (e.g., the Euclid mission [7]). On the theory side, better modeling of non-linearities via N-body simulations (and perturbation theory) have become available. Also, a new possibility has emerged of constraining the mass hierarchy with cosmology. In what follows we will use Σ to denote the total neutrino mass (i.e., the sum of the masses of the different families), Nν the number of neutrino species (or families), m the lightest mass state and M the heaviest mass state. Since oscillations indicate that one mass splitting is much smaller than the other one we will make the approximation that the small mass splitting can be neglected. For cosmological applications this is an extremely good approximation. Before we start, one remark is in order. In cosmol0920-5632/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nuclphysbps.2013.04.028
ogy many (if not all) parameters constraints are modeldependent, and the constraints discussed here are no exception. In cosmology we do observations (although we call them experiments), we only have one observable Universe and we can’t perturb it slightly and observe what happens. From these observations we want to infer the properties of the (fictitious) “ensemble” of all possible observable Universes as this is a good proxy for the properties of the whole Universe (beyond the observable one). So, interpretations of observations are inevitably made in the framework of the standard big bang model. In most cases a reference model is chosen, typically a minimal ΛCDM, where the Universe is spatially flat, dark energy is a cosmological constant and the primordial power spectrum of fluctuations is a power law. In this “reference” or “baseline” model, neutrinos are massless and the model has 6 cosmological (interesting) parameters which are to be constrained by data. Then typically one parameter at the time is being added and constrained (e.g., Σ, Nν etc.).
2. Neutrino masses Neutrinos with total mass Σ 1eV become nonrelativistic before recombination, leaving an imprint (and thus being constrained by) the primary CMB. If the total mass is instead Σ 1eV they become nonrelativistic after recombination. They still alter the matter radiation equality (compared to a “reference” model
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L. Verde / Nuclear Physics B (Proc. Suppl.) 235–236 (2013) 321–328
Figure 1: Left: Figure from [15]; total neutrino mass constraint forecast from the ground-based Large Synoptic Survey telescope (LSST) survey. The curves correspond to the normal and inverted hierarchies as constrained by oscillations experiments. The horizontal bands are the 1 and 2 σ errors on the sum of the masses (mtot is Σ). The forecast uses the weak gravitational lensing signal and assumes a prior on cosmological parameters corresponding to the statistical power of Planck mission’s data. Right: total neutrino mass forecast for the space-based Euclid mission. The signal used is galaxy clustering and Planck prior is also assumed. The calculations were presented in [16]. Stars indicated the fiducial models considered, shaded (blue) region and (blue) vertical lines indicate the 1 and 2 σ errors around two fiducial models (Σ = 0.05, 0.2eV). Note how in both cases, the error depend on the fiducial model chosen. For low Σ a detection is achieved with Euclid if the background cosmology is assumed to be a minimal ΛCDM. For higher Σ more flexible “baseline” models can be assumed where extra parameters such as curvature, equation of state parameter for dark energy and Nν can also be simultaneously constrained.
with massless neutrinos) but in the CMB this effect can be canceled by degeneracies with other parameters. After recombination however, finite neutrino masses suppress the matter power spectrum on scales smaller than the free-streaming length. In fact massive neutrinos free stream out of the potential wells smoothing small-scale perturbations. The effect is in principle large: the linear theory prediction is a 40% suppression at k > 0.2h/Mpc for Σ = 1eV compared to a model with massless neutrinos, this suppression is 15% at k > 0.2h/Mpc for Σ = 0.3eV. The latest Wilkinson Microwave Anisotropy Probe (WMAP7) results for the sum of neutrino masses is Σ < 1.3 eV at 95% confidence level [8]. This constraint has not improved much since the initial data release despite longer integration time of the experiment and subsequent improvement of the data (as expected from the above considerations). Even the measurements of the CMB damping tail does not improve the constraint much as seen in the analysis e.g., of STP data [9]: the collaboration needed to add low redshift universe information (secondary anisotropies, constraints on the amplitude of present-day clustering of matter, the σ8 parameter) to improve the constraint further. The above considerations indicate that large-scale structure measurements (secondary CMB anisotropies)
are needed to further constrain neutrino masses. In the past couple of years several groups have produced updated constraints on Σ using galaxy clustering in combination with CMB e.g., [10, 11, 12, 13]. At the time of this conference the most recent result of these comes from the latest data release of the BOSS survey [14]. All these measurements converge to Σ < 0.3eV at 95% confidence. This limit is valid for a 7 parameters model (a minimal ΛCDM +Σ), and gets degraded when more freedom is given to the model in terms of extra cosmological parameters. This constraint is worst that what one would have naively expected: Σ = 0.3 eV implies a 15% suppression of the linear matter power spectrum at small scales, and the galaxy power spectrum at these scales is measured to much better that 15% precision. However a number of factors enter in interpreting the galaxy power spectrum in terms of cosmological parameters: the relation between the dark matter and galaxy clustering properties become highly uncertain on these scales, there are degeneracies with other cosmological parameter that must be marginalized over, and there are effects of non-linearities which contribute to increase the error-bars. Future surveys however might have enough statistical power to detect the effect of neutrino masses in the sky. Fig.1 shows forecasts for two surveys set-ups. On
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the left the ground-based Large Synoptic Survey telescope (LSST) survey, the forecast uses the weak gravitational lensing signal. For the background cosmology and for breaking degeneracies, a prior based on the expected performance of the Planck satellite mapping the CMB primary anisotropy is assumed. Figure from [15]. On the right the space-based Euclid mission. The signal used is galaxy clustering and Planck prior is also assumed. The calculations were presented in [16]. It is important to note that errors depend on the fiducial model and on assumptions about the background cosmology one is willing to make. For low Σ fiducial values a detection can be achieved if the background cosmology is assumed to be a minimal ΛCDM. For higher Σ more flexible “baseline” models can be assumed where extra parameters such as curvature, equation of state parameter for dark energy can also be simultaneously constrained and still a detection can be achieved. Cosmological constraints on neutrino masses are already competitive compared with forthcoming particle physics experiments and forecasts from future surveys indicate that cosmology should be able not only to place strong upper limits on Σ but also to detect a non-zero neutrino mass by “seeing” the effect of the neutrino mass on the large-scale structure. However it is important to recall as discussed above that in cosmology, this is done as part as parameter-fitting in the context of a cosmological model. In this sense the constraints will be “model dependent”. All forecasts that I mentioned (and most of the forecasts present in the literature) are carried out under two important assumptions a) that systematic effects are under control and b) that linear theory predictions for the growth of structures apply on the scales of interest. Regarding systematic effects, as famously stated by Rumsfeld, “there are known unknowns and unknowns unknowns”. Known unknowns are for example real world effects in a survey or galaxy bias. The approach would be to model these and marginalize over them as much as possible. Different approaches also are affected by different systematics, for example weak gravitational lensing is not affected by galaxy bias as galaxy surveys are. “Unknown unknowns” are more difficult to deal with. It would be very useful to have an internal consistency check that allows one to cross check that a result of Σ > 0 is with confidence a neutrino mass detection. More on this later. What about non-linearities? On small scales, where most of the signal is, non-linearities become important. In the last few years there has been progress in modeling non-linearities of the cosmological structures in the presence of massive neutrinos both using perturba-
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tion theory e.g., [19, 20] and using N-body simulations [21, 22, 23] and also using hybrid approaches where the neutrinos are evolved analytically using perturbation theory and the dark matter is evolved numerically [24]. On the N-body side, when simulating the effect of neutrinos on non-linear growth of structure there are different options. The initial investigations simulated just the effect of a total mass, i.e. assuming degenerate hierarchy. Several different approaches have been used in the literature, using particles, grids or hybrid methods. Very recently also the hierarchy has been simulated[17]. All approaches converge and agree in their predictions in the interesting regime. In all cases non-linearities enhance the suppression of power effect predicted by linear theory. Figure 2 (from [17]) shows this. The three panels correspond to Σ = 0.3eV degenerate hierarchy, and Σ = 0.1eV normal and inverted hierarchy. Solid lines are linear theory predictions and points are the simulation results. Different colors (shades) correspond to different redshifts. At scales 0.1 < k < 1 nonlinearities enhance the effect of Σ on the power spectrum, making the predictions based on liner theory conservative in this respect. However we should bear in mind that these figures show the effect for the dark matter in real space. Only gravitational lensing probes directly the dark matter; galaxy surveys probe the galaxy field in redshift space. 3. Number of effective species The expansion rate in the early Universe is given by H(t) 8πG/3(ργ + ρν where ργ ) is the density of photons and the density of neutrinos is taken to be ρν ∝ T 4 Neff . The standard value (corresponding to Nν = 3) is Neff = 3.04 but any thermal background of light articles or anything affecting the expansion rate will be captured in Neff , not just neutrinos. This goes under the name of “dark radiation”, being ργ the radiation that is not dark. Of course, one obvious route is to look at Big bang nucleosynthesis, as the synthesis of light elements is extremely sensitive to the early Universe expansion rate. The difficult lies in inferring the primordial abundance from observations in the “late” Universe. The other route is to look at the CMB. Any change from Neff = 3.04 affects the matter-radiation equality and so the sound horizon at decoupling. A degeneracy with Ωm h2 and H0 follows. There are however other subtle effects due to anisotropic stress and the effect of matter-radiation equality on diffusion damping, which imprint small changes in the CMB damping tail. In the past couple of years a lot of attention has been devoted to whether cosmology indicates that Neff > 3.04
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Figure 2: Figure from [17]. Effect of non-linearities in the matter power spectrum in the presence of massive neutrinos. The three panels show the % effect of a non-zero neutrino mass on the matter power spectrum. Solid thin lines are the liner theory predictions (at z = 2, 1, 0 from top to bottom) and symbols are the (non-linear) N-body simulations results. Note how non-linearities enhance the effect compared to liner theory. Dotted lines are semi-analytic fits using ”halofit” by [18].
Figure 3: Posterior contours (1 and 2 σ joint confidence level) in the H0 -Neff plane. Left: WMAP7 data alone. Note the degeneracy that is artificially cut by the (top hat) prior imposed. Right: in red the result from combining WMAP7 data with the Baryon Acoustic Oscillations measurements; light (blue) shaded region shows the H0 constraint from [5]. Should the H0 measurement be overestimated by about one sigma the best fit for Neff from the combination CMB+H0 would be at 3 rather than around 4.
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and maybe indicates extra (sterile) neutrinos. A thorough review of the literature can be found in [25] and in particular their table 3. The numbers in the Neff column are indeed somewhat skewed towards Neff > 3.04 indicating that cosmological analyses consistently give best fit values > 3.04 consistent therefore with dark radiation. One however should keep in mind that a) these determinations are not independent as the WMAP data and the H0 determination are in common and b) it is always barely 2 σ, maybe except when the ACT+BAO data are included. What may be going on? Before jumping to the conclusion that cosmology indicates the presence of dark radiation and/or sterile neutrinos it is worth considering in detail how these constraints are obtained. First, it is important to recall that these determinations are obtained by first exploring a multi-dimensional parameter space and then marginalizing the corresponding posterior to derive a 1-dimensional constraint. In many cases, to shrink error-bars, many different data-sets are combined; should any two data sets be slightly in tension, a shift of the recovered parameter(s) would follow. Recall that statistically one out of three times the best fit value is more than one sigma away from the true value; thus combining three or more data-sets one should expect some “tension”. Also, most of these analyses are done in the Bayesian framework, where the posterior is used to do parameter inference. The posterior is the likelihood (which is given by the data) multiplied by the prior (which is chosen by the human). In a high-dimensional space where the data might poorly constrain one (or more) degeneracy directions, the prior can have a big effect. Figure 3 illustrates the first point. The solid (black) dark lines show the 1 and 2 σ confidence region of the WMAP posterior. The (blue) shaded band shows the H0 measurement by [5]. Note the degeneracy present when considering only WMAP data, degeneracy that is artificially cut by the (top hat) prior imposed. On the right panel, the light (blue) shaded region shows the H0 constraint from [5]. Should the H0 measurement be overestimated by about one sigma, Neff from the combination CMB+H0 would be around 3 rather than around 4. An independent determination of the low redshift expansion rate, H(z) for z < 1 has recently been presented in [6, 26]. This determination, in combination with WMAP CMB data , yields lower values for the best fit Neff : Neff = 3.38 ± 0.5 for WMAP+H(z) data and Neff = 3.37 ± 0.34 for WMAP+H(z)+SPT data. Regarding the effect of the prior on parameter inference, while a discussion is still on-going in the literature, it is instructive to look at a quantity that by con-
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Figure 4: Constraints on Neff derived from the profile likelihood ratio. The error bars are likely underestimated, but the results are, by construction, prior independent. Here ΔNeff = Neff − 3.04. Solid bars are approximately 1σ and dotted 2 σ. As we can see all dataset combinations are consistent with the standard Neff value at the 2σ level.
struction is prior-independent, like the profile likelihood ratio. Profile likelihood ratios can be constructed for exactly the same data sets combinations for which the standard Bayesian analysis has been carried out. Parameter inference in the Bayesian sense with the profile likelihood ratio is not straightforward: error-bars are likely to be under-estimated. Nevertheless if a priorindependent measurement, with error-bars likely to be under-estimated, shows no evidence for deviations from the standard Neff value, one should be very careful before taking seriously the corresponding prior-dependent evidence. A summary of the profile likelihood ratioderived constraints on Neff can be found in [27] and is summarized in Fig.4. In summary: cosmology is consistent with 3 (but also with 4) neutrinos at 2σ level. To satisfy cosmological bounds these must be light neutrinos (< 0.5eV) e.g.[28]. More wiggle room can be gained by going beyond the baseline minimal ΛCDM model, the error-bars gets larger, but this is a keen to adding epicycles to one’s theory. Of course there are always radical options like e.g., avoiding thermalization. Future (planned or proposed) surveys, like e.g., the Euclid mission, should shed light on this issue as forecasted error-bars for Neff are 1 [16]. On the other hand is is safe to say that the cosmic neutrino background has been detected at 4σ as shown in Fig. 5.
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Figure 5: Cumulative PDF for Neff . Thin dark lines: WMAP7 alone. Thick lighter lines: WMAP7+SPT. Solid: CDF = (1-CDF). The horizontal dotted lines shows the 99.7% or 0.003 level.
Neff 0
P(Nˆ eff )d Nˆ eff . Dashed
Figure 6: Figure from [29]. Left: the “response” of the linear matter power spectrum to a change in Δ. Right: present-day constraints from oscillations (gray regions) and from cosmology in the Σ, Δ plane.
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4. Outlook towards the future: can cosmology help constraining the hierarchy? Neutrinos of different masses have different transition redshifts from relativistic to non relativistic behavior, and their individual masses and their mass splitting change the details of the transition of the Universe evolution from the radiation-dominated to matterdominated regime. Following [29] we define the neutrino mass splitting Δ, depending on wether one is considering normal (NH) or inverted (IH) hierarchy as: NH
:
IH
:
(M − m) ; Σ = 2m + M Σ (m − M) ; Σ = 2M + m . Δ= Σ Δ=
(1)
Δ is negative for IH and positive for NH. The mass splitting is |Δ| and with the sign of Δ the hierarchy is univocally determined. Fig.6 shows on the left panel, the “response” of the linear matter power spectrum to a change in Δ . On the right panel are the present-day constraints from oscillations (gray regions) and from cosmology in the Σ, Δ plane. It is clear that cosmology is chiefly sensitive to the absolute value of Δ i.e., on the mass splitting, and less sensitive to the sign of Δ which is needed to fully characterize the hierarchy. In other words for a given fiducial model (say, for example an inverted hierarchy model with Σ = 0.1eV) the forecasted likelihood as function of Δ for a combination of Planck satellite data plus a given large-scale structure data will have two maxima at the same value of |Δ| = 0.5. One will be a global maximum (Δ < 0) and one a local one (Δ > 0). At this point there are two different questions one might ask: “Is there enough information in the sky to distinguish the two?” and “Can this be done with a specific survey?” The first question is relevant because in cosmology the next generation surveys will be very close to be what I call the ultimate survey. Since in cosmology we have only one observable Universe from which we try to infer properties of the (unobservable) “ensemble”, once a sizable fraction of the observable Universe has been measured, statistical error-bars cannot be significantly reduced further even increasing drastically the observing time: one has performed the ultimate experiment. It turns out that there is enough information in the sky to measure both |Δ| and its sign: one would need something close to the ultimate largescale structure experiment. (Luckily this may become available with the next generation of space-based galaxy surveys). It should also be noted that the standard approach to address this type of questions, i.e., the Fisher matrix approach to forecasts, has important limitations
Figure 7: Figure from [17]. Effect of the hierarchy on the matter power spectrum. We show the % difference between IH and NH. For a Σ = 0.1eV. Lines correspond to linear theory predictions and symbols to the N-body simulations results at different redshifts. Again nonlinearities enhance the effect on the scales of interest.
in this specific context: the likelihood is multi-peaked while the Fisher approach assumes a Gaussian likelihood. A Fisher-type approach is still possible, but extra care must be taken given the unusual form of the likelihood surface e.g., [29]. What about non-linearities? In [17] we have address this issue by running 1 billion cold dark matter particles and 2 billion neutrino particles N-body simulation of a volume of (600Mpc/h)3 . The effect of the hierarchy parameter Δ on the matter power spectrum is shown in fig.7. We show the % change in matter power spectrum between IH and NH for a Σ = 0.1eV. Lines correspond to linear theory predictions and symbols to the N-body simulations results at different redshifts. Again non-linearities enhance the effect on the scale of interest, making linear theory based forecasts conservative in this respect. Of course here we have quantified the effect on a relative quantity which is much more robust agains systematic/numerical errors. The magnitude of the effect is small (smaller that 1%), and one should deep in mind that even without massive neutrinos, it is challenging to compute the non-linear power spectrum to sub-percent precision. Even if the effect is small, if all cosmological parameters are fixed, a survey covering a not too large cosmological volume (few (Gpc/h)3 ) is enough to measure the hierarchy. Investigations on the effect of cosmological degeneracies is ongoing, but we can anticipate that with forthcoming surveys it will soon be possible to determine |Δ|. This offers a powerful consistency check: if a survey reveals a detection of a non-zero neutrino mass, the confidence on this measurement not being affected by systematic errors could
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be boosted if the measured value for |Δ| is in agreement with that predicted from oscillations for the same total mass. For example a detection of Σ ∼ 0.1eV should be accompanied by a measurement of |Δ| consistent with 0.5. Should |Δ| = 0.5 be excluded then the “detection” of Σ should be considered an systematic effect. 5. Conclusions Precision cosmology means that we can start (or prepare for) constraining interesting physical quantities. I have presented here the case for neutrino properties from cosmology: absolute mass scale, number of families, maybe hierarchy. To date, firm constraints from cosmology on neutrino properties are: • the sum of the masses is Σ < 0.3eV at 95% confidence for a baseline standard ΛCDM model where the Universe is spatially flat, the primordial power spectrum is a power law, and dark energy is a cosmological constant. • The cosmic neutrino background has been (indirectly) detected at more than 4σ. • The effective number of species Neff is consistent with Nν = 3 at 2 σ, but is also consistent with 4. Large future surveys means that sub % effects become detectable, which brings in a whole new set of challenges and opportunities (e.g., mass hierarchy). Acknowledgments I acknowledge support from ERC grant FP7-IDEASPhys.LSS 240117. References [1] Keisler, R., Reichardt, C. L., Aird, K. A., et al. 2011, ApJ, 743, 28 [2] Das, S., Marriage, T. A., Ade, P. A. R., et al. 2011, ApJ, 729, 62 [3] Anderson, L., Aubourg, E., Bailey, S., et al. 2012, arXiv:1203.6594 [4] Blake, C., Brough, S., Colless, M., et al. 2012, MNRAS, 425, 405 [5] Riess, A. G., Macri, L., Casertano, S., et al. 2011, ApJ, 730, 119 [6] Moresco, M., Cimatti, A., Jimenez, R., et al. 2012, JCAP, 8, 6 [7] Laureijs, R., Amiaux, J., Arduini, S., et al. 2011, arXiv:1110.3193 [8] Komatsu, E., Smith, K. M., Dunkley, J., et al. 2011, ApJS, 192, 18 [9] Reichardt, C. L., Shaw, L., Zahn, O., et al. 2012, ApJ, 755, 70 [10] Reid, B. A., Verde, L., Jimenez, R., & Mena, O. 2010, JCAP, 1, 3
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