Anthropic origin of the neutrino mass from cooling failure

The sum of active neutrino masses is well constrained, $58\mathrm{meV}\le m_{\nu }\lesssim 0.23\mathrm{eV}$, but the origin of this scale is not well understood. Here we investigate the possibility that it arises by environmental selection in a large landscape of vacua. Earlier work noted the detrimental effects of neutrinos on large-scale structure. However, using Boltzmann codes to compute the smoothed density contrast on Mpc scales, we find that dark matter halos form abundantly for $m_{\nu }\gtrsim 10\mathrm{eV}$. This finding rules out an anthropic origin of $m_{\nu }$, unless a different catastrophic boundary can be identified. Here we argue that galaxy formation becomes inefficient for $m_{\nu }\gtrsim 10\mathrm{eV}$. We show that in this regime, structure forms late and is dominated by cluster scales, as in a top-down scenario. This is catastrophic: baryonic gas will cool too slowly to form stars in an abundance comparable to our Universe. With this novel cooling boundary, we find that the anthropic prediction for $m_{\nu }$ agrees at better than $2\sigma$ with current observational bounds. A degenerate hierarchy is mildly preferred.

Figure 1

(a) The dimensionless power spectrum at $z=0$ for a range of neutrino masses with a normal hierarchy, computed using CAMB. From top to bottom at high wave number: $m_{\nu }=0$ (red), $m_{\nu }=5\mathrm{eV}$ (purple), $m_{\nu }=10\mathrm{eV}$ (blue), and $m_{\nu }=15\mathrm{eV}$ (black). Free-streaming of massive neutrinos causes a suppression of power at high wave numbers. Above a critical neutrino mass $m\sim 8\mathrm{eV}$, this effect is large enough so that the dimensionless power spectrum develops a peak near the free-streaming scale $k_{\text{nr}}<k_{\text{gal}}$. This implies that the first structure consists of cluster-size halos. (b) We obtain the smoothed density contrast ${\sigma }_R$ at $z=0$ numerically, essentially by integrating $k^{3}P(k)$ up to the wave number $1/R$; see Eq. (3) and surrounding discussion. We take $R$ to be the comoving scale of the Milky Way (${10}^{12}{M}_{\odot }$). The orange (upper) curve corresponds to a normal hierarchy; the green (lower) to a degenerate hierarchy. We see that neutrinos suppress halo formation only in the regime $m_{\nu }\lesssim 10\mathrm{eV}$ where the dimensionless power spectrum has no maximum and the integral is dominated by the large-$k$ cutoff. For larger neutrino masses, the formation of galactic and larger halos is actually enhanced, because the dimensionless power spectrum develops a peak that dominates the integral. At higher neutrino masses, the peak power increases, due to a decrease of the free-streaming scale and a lengthening of the matter era. (This is more pronounced for a normal hierarchy.) Hence ${\sigma }_R$ increases. If observers formed in proportion to the mass fraction in large dark matter halos, this would rule out an anthropic origin of $m_{\nu }$; see Fig. 2.

Figure 2

No cooling boundary: if one assumed that observers trace dark matter halos of mass ${10}^{12}{M}_{\odot }$ or greater, one would find a bimodal probability distribution over the neutrino mass sum $m_{\nu }$. This distribution is shown here for a normal hierarchy (orange/upper curve) and degenerate hierarchy (green curve). The range of values $m_{\nu }$ consistent with observation ($58\mathrm{meV}<m_{\nu }<0.23\mathrm{eV}$, shaded in red) is greatly disfavored, ruling out this model. By contrast, we shall assume here that observers trace galaxies. Crucially, we shall argue that for $m_{\nu }\gtrsim 10\mathrm{eV}$, galaxies do not form even though halos do. This novel catastrophic boundary excludes the mass range above 10 eV, leading to a successful anthropic explanation of the neutrino mass (see Fig. 3).

Figure 3

Our main result: the probability distribution over the neutrino mass sum $m_{\nu }$ for (a) a normal hierarchy and (b) a degenerate hierarchy, assuming that observers require galaxies. The plot is the same as Fig. 2, but the mass range is cut off at 7.7 eV (10.8 eV) in the normal (degenerate) case. For greater masses, the first halos form late and are of cluster size; we argue that galaxies do not form efficiently in such halos. We use the second observer model described in Sec. 2b; results look nearly identical with the first model. We assume a flat prior over $m_{\nu }$ (see Fig. 4 for other priors). The central $1\sigma$ and $2\sigma$ regions are shaded. Vertical red lines indicate the lowest possible values for the neutrino mass consistent with available data: $m_{\mathrm{obs}}\approx 58\mathrm{meV}$ for a normal hierarchy, and $m_{\mathrm{obs}}\approx 150\mathrm{meV}$ for a degenerate hierarchy. We find that these values are within $2\sigma$ of the median. The agreement would further improve with a less conservative treatment of the detrimental effects of neutrinos on gas cooling in halos, and/or the cosmological detection of a neutrino mass sum larger than the minimal value.

Figure 4

The prior distribution of cosmologically produced vacua is assumed to favor large neutrino mass and to have no special feature near the observed magnitude: $d{\mathcal{P}}_{\text{vac}}/d\log m_{\nu }\propto m_{\nu }^n$, $n>0$. One then expects $n\sim O(1)$, and the previous two figures all show the case $n=1$. This figure shows that the same conclusions obtain for a considerable range of $n$. (a) The median of the probability distribution as a function of the multiverse force $n$. (b) The standard deviation of the worst case observed value from the median, as a function of $n$. The $2\sigma$ region is shaded. Orange (the upper curve at large $n$ in each plot) is for a normal hierarchy; green is for a degenerate hierarchy.

Figure 5

The dashed black line shows the probability distribution found by Tegmark et al. [53] for a flat prior over $m_{\nu }$. This result would seem to remain compatible at about $2\sigma$ with current observational constraints (red shaded region). However, the analytic fitting function for the density contrast ${\sigma }_R$ used in [53, 54] underestimates ${\sigma }_R$ above a few eV. The solid curves show the probability distribution that results when ${\sigma }_R$ is computed numerically from Boltzmann codes: $\text{orange}/\text{upper}=\text{normal}$ hierarchy; $\text{green}/\text{lower}=\text{degenerate}$ hierarchy. They differ slightly from Fig. 2 because Ref. [53] used a different measure and observer model. Either way, a successful anthropic explanation of $m_{\nu }$ requires the identification of a catastrophic boundary at or below 10 eV.

Figure 6

The comoving volume of the causal patch for $m_{\nu }=0$ (black), $m_{\nu }=4$ (blue), and $m_{\nu }=8$ (red).

Figure 7

The Press-Schechter factor (solid lines) and its derivative (dashed lines) at the galaxy scale, ${10}^{12}{M}_{\odot }$, for a normal hierarchy with $m_{\nu }=0\mathrm{eV}$ (black), $m_{\nu }=2\mathrm{eV}$ (brown), $m_{\nu }=4\mathrm{eV}$ (blue), $m_{\nu }=6\mathrm{eV}$ (purple) and $m_{\nu }=8\mathrm{eV}$ (red). Each is used to define either of the two observer models of Sec. 2b. Note that massive neutrinos suppress structure at all times, but much more so at early times [76, 77, 78, 79, 80, 81].

Figure 8

The growth factor Eq. (b14) (solid line), which behaves like $x^{1/3}$ (dashed line) during the matter era, and asymptotes to a constant value well above $x(t_{\mathrm{\Lambda }})=1$.

Figure 9

The parameters $C_{\text{eff}}$ and $p_{\mathrm{eff}}$, for normal (orange, top) and degenerate (green, bottom) hierarchies, obtained by fitting Eq. (b22) to camb output for ${\sigma }_R$ and its derivative at $z=0$. The resulting fitting function for ${\sigma }_R(z)$ is used to compute the Press-Schechter factor at negative redshift only. Note that $p_{\text{eff}}\approx 1$ throughout. This may seem surprising, but it is consistent with our earlier finding that at large neutrino masses, the scales whose power contributes dominantly to ${\sigma }_R$ are precisely the ones on which free-streaming is not effective. This is closely related to the discrepancy we find with Ref. [53], whose estimate $p_{\text{eff}}\approx p(k_{\text{gal}})\approx 1-8f_{\nu }$ would yield a monotonically decreasing curve in (b).