As a summary of the last decade of neutrino experiments, two hierarchical neutrino mass splittings and three mixing angles have been measured. Furthermore, the standard model has three neutrinos: the motivation for considering deviations from the standard model in the form of extra sterile neutrinos has disappeared [655, 13]. Of course, deviations from the standard effective numbers of neutrino species could still indicate exotic physics which we will discuss below (Section 2.8.4).
New and future neutrino experiments aim to determine the remaining parameters of the neutrino mass
matrix and the nature of the neutrino mass. Within three families of neutrinos, and given all neutrino
oscillation data, there are three possible mass spectra: a) degenerate, with mass splitting smaller than the
neutrino masses, and two non-degenerate cases, b) normal hierarchy (NH), with the larger
mass splitting between the two more massive neutrinos and c) inverted hierarchy (IH), with
the smaller spitting between the two higher mass neutrinos. Figure 36 [480
] illustrates the
currently allowed regions in the plane of total neutrino mass,
, vs. mass of the lightest
neutrino,
. Note that a determination of
would indicate normal hierarchy and
that there is an expected minimum mass
. The cosmological constraint is from
[762
].
Cosmological constraints on neutrino properties are highly complementary to particle physics experiments for several reasons:
The hot big bang model predicts a background of relic neutrinos in the universe with an average number
density of , where
is the number of neutrino species. These neutrinos decouple
from the CMB at redshift
when the temperature was
, but remain
relativistic down to much lower redshifts depending on their mass. A detection of such a neutrino
background would be an important confirmation of our understanding of the physics of the early
universe.
Massive neutrinos affect cosmological observations in different ways. Primary CMB data alone can
constrain the total neutrino mass , if it is above
[526
, finds
at 95% confidence]
because these neutrinos become non-relativistic before recombination leaving an imprint in the
CMB. Neutrinos with masses
become non-relativistic after recombination altering
matter-radiation equality for fixed
; this effect is degenerate with other cosmological
parameters from primary CMB data alone. After neutrinos become non-relativistic, their free
streaming damps the small-scale power and modifies the shape of the matter power spectrum below
the free-streaming length. The free-streaming length of each neutrino family depends on its
mass.
Current cosmological observations do not detect any small-scale power suppression and break many of
the degeneracies of the primary CMB, yielding constraints of [762] if we assume the neutrino
mass to be a constant. A detection of such an effect, however, would provide a detection, although indirect,
of the cosmic neutrino background. As shown in the next section, the fact that oscillations predict a
minimum total mass
implies that Euclid has the statistical power to detect the cosmic
neutrino background. We finally remark that the neutrino mass may also very well vary in time [957
]; this
might be tested by comparing (and not combining) measurements from CMB at decoupling with low-
measurements. An inconsistency would point out a direct measurement of a time varying neutrino mass
[959].
Particle physics experiments are sensitive to neutrino flavours making a determination of the neutrino absolute-mass scales very model dependent. On the other hand, cosmology is not sensitive to neutrino flavour, but is sensitive to the total neutrino mass.
The small-scale power-suppression caused by neutrinos leaves imprints on CMB lensing: forecasts
indicate that Planck should be able to constrain the sum of neutrino masses , with a
error of
0.13 eV [491, 557, 289].
Euclid’s measurement of the galaxy power spectrum, combined with Planck (primary CMB only) priors
should yield an error on of 0.04 eV [for details see 211
] which is in qualitative agreement with previous
work [e.g. 779
]), assuming a minimal value for
and constant neutrino mass. Euclid’s weak lensing
should also yield an error on
of 0.05 eV [507
]. While these two determinations are not fully
independent (the cosmic variance part of the error is in common given that the lensing survey and the
galaxy survey cover the same volume of the universe) the size of the error-bars implies more than
detection of even the minimum
allowed by oscillations. Moreover, the two independent techniques will
offer cross-checks and robustness to systematics. The error on
depends on the fiducial model
assumed, decreasing for fiducial models with larger
. Euclid will enable us not only to detect
the effect of massive neutrinos on clustering but also to determine the absolute neutrino mass
scale.
Since cosmology is insensitive to flavour, one might expect that cosmology may not help in determining the
neutrino mass hierarchy. However, for , only normal hierarchy is allowed, thus a mass
determination can help disentangle the hierarchy. There is however another effect: neutrinos of different
masses become non-relativistic at slightly different epochs; the free streaming length is sightly different for
the different species and thus the detailed shape of the small scale power suppression depends on the
individual neutrino masses and not just on their sum. As discussed in [480
], in cosmology one
can safely neglect the impact of the solar mass splitting. Thus, two masses characterize the
neutrino mass spectrum: the lightest
, and the heaviest
. The mass splitting can be
parameterized by
for normal hierarchy and
for inverted
hierarchy. The absolute value of
determines the mass splitting, whilst the sign of
gives the
hierarchy. Cosmological data are very sensitive to
; the direction of the splitting – i.e.,
the sign of
– introduces a sub-dominant correction to the main effect. Nonetheless, [480
]
show that weak gravitational lensing from Euclid data will be able to determine the hierarchy
(i.e., the mass splitting and its sign) if far enough away from the degenerate hierarchy (i.e., if
).
A detection of neutrino-less double- decay from the next generation experiments would indicate that
neutrinos are Majorana particles. A null result of such double-
decay experiments would lead to a
definitive result pointing to the Dirac nature of the neutrino only for degenerate or inverted mass spectrum.
This information can be obtained from large-scale structure cosmological data, improved data on the
tritium beta decay, or the long-baseline neutrino oscillation experiments. If the small mixing in the
neutrino mixing matrix is negligible, cosmology might be the most promising arena to help in this
puzzle.
Neutrinos decouple early in cosmic history and contribute to a relativistic energy density with an effective
number of species . Cosmology is sensitive to the physical energy density in
relativistic particles in the early universe, which in the standard cosmological model includes only
photons and neutrinos:
, where
denotes the energy density in photons
and is exquisitely constrained from the CMB, and
is the energy density in one neutrino.
Deviations from the standard value for
would signal non-standard neutrino features or
additional relativistic species.
impacts the big bang nucleosynthesis epoch through its effect
on the expansion rate; measurements of primordial light element abundances can constrain
and rely on physics at
[158]. In several non-standard models – e.g., decay
of dark matter particles, axions, quintessence – the energy density in relativistic species can
change at some later time. The energy density of free-streaming relativistic particles alters the
epoch of matter-radiation equality and leaves therefore a signature in the CMB and in the
matter-transfer function. However, there is a degeneracy between
and
from
CMB data alone (given by the combination of these two parameters that leave matter-radiation
equality unchanged) and between
and
and/or
. Large-scale structure surveys
measuring the shape of the power spectrum at large scale can constrain independently the
combination
and
, thus breaking the CMB degeneracy. Furthermore, anisotropies
in the neutrino background affect the CMB anisotropy angular power spectrum at a level of
through the gravitational feedback of their free streaming damping and anisotropic stress
contributions. Detection of this effect is now possible by combining CMB and large-scale structure
observations. This yields an indication at more than
level that there exists a neutrino background
with characteristics compatible with what is expected under the cosmological standard model
[901, 285].
The forecasted errors on for Euclid (with a Planck prior) are
at
level [507
], which is
a factor
better than current constraints from CMB and LSS and about a factor
better than
constraints from light element abundance and nucleosynthesis.
A recurring question is how much model dependent will the neutrino constraints be. It is important to recall
that usually parameter-fitting is done within the context of a CDM model and that the neutrino effects
are seen indirectly in the clustering. Considering more general cosmological models, might degrade neutrino
constraints, and vice versa, including neutrinos in the model might degrade dark-energy constraints. Here
below we discuss the two cases of varying the total neutrino mass
and the number of relativistic species
, separately.
In [211] it is shown that, for a general model which allows for a non-flat universe, and a redshift dependent
dark-energy equation of state, the
spectroscopic errors on the neutrino mass
are in the
range 0.036 – 0.056 eV, depending on the fiducial total neutrino mass
, for the combination
Euclid+Planck.
On the other hand, looking at the effect that massive neutrinos have on the dark-energy parameter
constraints, it is shown that the total CMB+LSS dark-energy FoM decreases only by 15% – 25% with
respect to the value obtained if neutrinos are supposed to be massless, when the forecasts are computed
using the so-called “
-method marginalized over growth-information” (see Methodology
section), which therefore results to be quite robust in constraining the dark-energy equation of
state.
For what concerns the parameter correlations, at the LSS level, the total neutrino mass is correlated
with all the cosmological parameters affecting the galaxy power spectrum shape and BAO positions. When
Planck priors are added to the Euclid constraints, all degeneracies are either resolved or reduced,
and the remaining dominant correlations among
and the other cosmological parameters
are
-
,
-
, and
-
, with the
-
degeneracy being the largest
one.
spectroscopic errors stay mostly unchanged whether growth-information are included or marginalised
over, and decrease only by 10% – 20% when adding
measurements. This result is expected, if we
consider that, unlike dark-energy parameters,
affects the shape of the power spectrum via a
redshift-dependent transfer function
, which is sampled on a very large range of scales
including the
turnover scale, therefore this effect dominates over the information extracted
from measurements of
. This quantity, in turn, generates new correlations with
via
the
-term, which actually is anti-correlated with
[641
]. On the other hand, if we
suppose that early dark-energy is negligible, the dark-energy parameters
,
and
do not enter the transfer function, and consequently growth information have relatively more
weight when added to constraints from
and
alone. Therefore, the value of the
dark-energy FoM does increase when growth-information are included, even if it decreases by a factor
50% – 60% with respect to cosmologies where neutrinos are assumed to be massless, due to the
correlation among
and the dark-energy parameters. As confirmation of this degeneracy, when
growth-information are added and if the dark-energy parameters
,
,
are held fixed to their
fiducial values, the errors
decrease from 0.056 eV to 0.028 eV, for Euclid combined with
Planck.
We expect that dark-energy parameter errors are somewhat sensitive also to the effect of incoherent peculiar velocities, the so-called “Fingers of God” (FoG). This can be understood in terms of correlation functions in the redshift-space; the stretching effect due to random peculiar velocities contrasts the flattening effect due to large-scale bulk velocities. Consequently, these two competing effects act along opposite directions on the dark-energy parameter constraints (see methodology Section 5).
On the other hand, the neutrino mass errors are found to be stable again at , also when
FoG effects are taken into account by marginalising over
; in fact, they increase only by 10% – 14%
with respect to the case where FoG are not taken into account.
Finally, in Table 18 we summarize the dependence of the -errors on the model cosmology, for Euclid combined with
Planck.15
We conclude that, if
is
0.1 eV, spectroscopy with Euclid will be able to determine the neutrino
mass scale independently of the model cosmology assumed. If
is
0.1 eV, the sum of neutrino
masses, and in particular the minimum neutrino mass required by neutrino oscillations, can be measured in
the context of a
CDM model.
Regarding the spectroscopic errors, [211
] finds
from Euclid, and
,
for Euclid+Planck. Concerning the effect of
uncertainties on the dark-energy parameter errors, the
CMB+LSS dark-energy FoM decreases only by
with respect to the value obtained holding
fixed at its fiducial value, meaning that also in this case the “
-method marginalized over
growth–information” is not too sensitive to assumptions about model cosmology when constraining the
dark-energy equation of state.
About the degeneracies between and the other cosmological parameters, it is necessary to say
that the number of relativistic species gives two opposite contributions to the observed power spectrum
(see methodology Section 5), and the total sign of the correlation depends on the dominant one, for
each single cosmological parameter. In fact, a larger
value suppresses the transfer function
on
scales
. On the other hand, a larger
value also increases the Alcock–Paczynski prefactor in
. For what concerns the dark-energy parameters
,
,
, and the dark-matter density
, the Alcock–Paczynski prefactor dominates, so that
is positively correlated to
and
, and anti-correlated to
and
. In contrast, for the other parameters, the
suppression produces the larger effect and
results to be anti-correlated to
, and positively
correlated to
and
. The degree of the correlation is very large in the
-
case,
being of the order
with and without Planck priors. For the remaining cosmological
parameters, all the correlations are reduced when CMB information are added, except for the
covariance
-
, as happens also for the
-correlations. To summarize, after the
inclusion of Planck priors, the remaining dominant degeneracies among
and the other
cosmological parameters are
-
,
-
, and
-
, and the forecasted error is
, from Euclid+Planck. Finally, if we fix to their fiducial values the dark-energy
parameters
,
and
,
decreases from 0.086 to 0.048, for the combination
Euclid+Planck.
General cosmology | ||||||
fiducial ![]() |
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EUCLID+Planck | 0.0361 | 0.0458 | 0.0322 | 0.0466 | 0.0563 | 0.0862 |
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||||||
EUCLID+Planck | 0.0176 | 0.0198 | 0.0173 | 0.0218 | 0.0217 | 0.0224 |
for degenerate spectrum:
;
for normal hierarchy:
,
for inverted hierarchy:
,
;
fiducial cosmology with massless neutrinos
In general, forecasted errors are obtained using techniques, like the Fisher-matrix approach, that are not particularly well suited to quantify systematic effects. These techniques forecast only statistical errors, which are meaningful as long as they dominate over systematic errors. Therefore, it is important to consider sources of systematics and their possible effects on the recovered parameters. Possible sources of systematic errors of major concern are the effect of nonlinearities and the effects of galaxy bias.
The description of nonlinearities in the matter power spectrum in the presence of massive neutrinos has
been addressed in several different ways: [966, 779, 778, 780] have used perturbation theory, [555] the
time-RG flow approach and [167, 166, 168, 928
] different schemes of
-body simulations. Another
nonlinear scheme that has been examined in the literature is the halo model. This has been applied to
massive neutrino cosmologies in [1, 421, 422].
On the other hand, galaxy/halo bias is known to be almost scale-independent only on large, linear
scales, but to become nonlinear and scale-dependent for small scales and/or for very massive haloes. From
the above discussion and references, it is clear that the effect of massive neutrinos on the galaxy power
spectrum in the nonlinear regime must be explored via -body simulations to encompass all the relevant
effects.
Here below we focus on the behavior of the DM-halo mass function (MF), the DM-halo bias, and the
redshift-space distortions (RSD), in the presence of a cosmological background of massive neutrinos. To this
aim, [168] and [641] have analysed a set of large -body hydrodynamical simulations, developed with an
extended version of the code gadget-3 [928], which take into account the effect of massive free-streaming
neutrinos on the evolution of cosmic structures.
The pressure produced by massive neutrino free-streaming contrasts the gravitational collapse which is
the basis of cosmic structure formation, causing a significant suppression in the average number density of
massive structures. This effect can be observed in the high mass tail of the halo MF in Figure 38, as
compared with the analytic predictions of [824
] (ST), where the variance in the density fluctuation field,
, has been computed via camb [559
], using the same cosmological parameters of the simulations. In
particular, here the MF of sub-structures is shown, identified using the subfind package [858], while the
normalization of the matter power spectrum is fixed by the dimensionless amplitude of the primordial
curvature perturbations
, evaluated at a pivot scale
[548],
which has been chosen to have the same value both in the
CDM
and in the
CDM
cosmologies.
In Figures 38 and 39
, two fiducial neutrino masses have been considered,
and
.
From the comparison of the corresponding MFs, we confirm the theoretical predictions, i.e., that the higher
the neutrino mass is, the larger the suppression in the comoving number density of DM haloes
becomes.
As is well known, massive neutrinos also strongly affect the spatial clustering of cosmic structures. A
standard statistics generally used to quantify the degree of clustering of a population of sources is the
two-point auto-correlation function. Although the free-streaming of massive neutrinos causes a suppression
of the matter power spectrum on scales larger than the neutrino free-streaming scale, the halo
bias is significantly enhanced. This effect can be physically explained thinking that, due to
neutrino structure suppression, the same halo bias would correspond, in a
CDM cosmology, to
more massive haloes (than in a
CDM
cosmology), which as known are typically more
clustered.
This effect is evident in Figure 39 which shows the two-point DM-halo correlation function measured
with the Landy and Szalay [541] estimator, compared to the matter correlation function. In particular, the
clustering difference between the
CDM and
CDM
cosmologies increases at higher redshifts, as it
can be observed from Figures 40
and 41
and the windows at redshifts
of Figure 38
. Note also the
effect of nonlinearities on the bias, which clearly starts to become scale-dependent for separations
.
As it happens for the MF and clustering, also RSD are strongly affected by massive neutrinos. Figure 42
shows the real and redshift space correlation functions of DM haloes as a function of the neutrino
mass. The effect of massive neutrinos is particularly evident when the correlation function is
measured as a function of the two directions perpendicular and parallel to the line of sight. As a
consequence, the value of the linear growth rate that can be derived by modelling galaxy clustering
anisotropies can be greatly suppressed with respect to the value expected in a
CDM cosmology.
Indeed, neglecting the cosmic relic massive neutrino background in data analysis might induce a
bias in the inferred growth rate, from which a potentially fake signature of modified gravity
might be inferred. Figure 43
demonstrates this point, showing the best-fit values of
and
, as a function of
and redshift, where
,
being the halo effective
linear bias factor,
the linear growth rate and
the pairwise velocity dispersion.
http://www.livingreviews.org/lrr-2013-6 |
Living Rev. Relativity 16, (2013), 6
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