- Email: [email protected]
Neutrinos in cosmology
Progress in Particle and Nuclear Physics PERGAMON
Progress
in Particle
and Nuclear
Physics 40 ( 199X) 1 -15
Neutrinos in Cosmology
We review the main bounds on neutrino properties from cosmological arguments.
1
Introduct
ion
Neutrinos may play an important yet unknown parameters, are complementary
role in cosmology depending on the values of their fundamental
such as masses, mixing and lifetimes. Conversely, cosmological arguments
to laboratory experiments and observations of astrophysical solar and atmospheric
neutrinos, in providing strong bounds on neutrino properties. The Hot Big Bang (BB), the standard
model of cosmology, establishes that the Universe is
homogeneous, isotropic and expanding from a state of extremely high temperature the early universe can be described as an adiabaticlly
and density. Thus
expanding classical gas of relativistic
namely radiation, in local thermal equilibrium at a temperature
particles,
T, that changes with time, Z’(t). The
lifetime of the Universe t is counted from the moment the expansion started, taken to be t = 0. This model is based on General Relativity, the Cosmological Principle and four major empirical pieces of evidence, namely the Hubble expansion, the cosmic blackbody microwave background radiation (CMBR), the anisotropies of the CMBR, and the relative cosmic abundance of the light elements (up to ‘Li). Because the model is based on General Relativity, it is for sure not valid in the realm of Quantum
Gravity, T > &&n& = 1.22
x
1O”GeV and t < 10-43sec. The Cosmological Principle
postulates that we do not live in a special place in the Universe, by requiring that every comoving observer in the “cosmic fluid” has the same history. The “cosmic fluid” has as particles clusters of galaxies, and “comoving” in practice means at rest with the galaxies within a 100 Mpc radius (one pa.rsec,
1
pc = 3.26 light years = 3.09
x
1018cm). The Hubble parameter H provides the proportionality
between the velocity tr of recession of faraway objects, and their relative distance d,
(1)
v=Hd, 0146-6410:98/%19.00+0.00 PII: SOl46-6410(9X)00003-9
0
1998 Elsevier Science
BV. All rights reserved.
Printed
in Great
Britain
where H = h 100 km/set
Mpc and h is a parameter
The CMBR wss produced It has a blackbody
Explorer) satellite
N 3 x 105y, the recombination
at t,,
spectrum
measured
found a blackbody
and it is remarkably
the CMBR spectrum
of the CMBR photons, n., = 2C(3)Tj/a2 Anisotropies
COBE measured
p-, = 7r2c/15
inhomogeneities
a dipole anisotropy
of galaxies with respect
between
in the Universe
0.05 and 1 cm, and
of less than 0.03% [2].
the number and energy density by several orders of magnitude,
= 4.71 x 10-34(g/cm3). due to our motion with respect
that triggered
corresponding
structure
formation
to the CMBR rest frame, in the Universe.
to a velocity of 6.27&22km/sec
to the CMBR rest frame (COBE even saw the rotation
the Sun!), and measured
In fact
of our Local Group of the Earth
around
(bT/T)o for angles 0 = 7” to 90”. At 90” COBE measured
anisotropies
of (6T/T)go. N 0.5 x lob5 131. Results
anisotropy
The COBE (Cosmic Background
we know with great accuracy
in the CMBR are expected
and due to the density
isotropic.
in 1992 for wavelengths
that are the most abundant
= 411/cm3,
epoch, when atoms first became
To = 2.726 f O.OlO”K, with deviations
with temperature
Knowing so well the CMBR temperature,
quadrupole
determinations
to h IS!0.55 - 0.75 [l].
are converging
stable.
of order one, whose observational
from other experiments
in balloons
a are
angles, 0 = 0,5” to 90” and the results show 6T/T 5 10m5 after subtracting
available now at smaller the dipole. Finally,
the earliest
cosmological by several
abundance orders
available
proof of the consistency
of the Hot BB model is provided
of 4He and of the trace elements
of magnitude,
D, and ‘Li.
Their
abundances,
are well accounted
for in terms
of nuclear
tNS = 1o-2 - lo2 set, the Big Bang nucleosynthesis
(NS epoch),
when the photon
TNSN 10 - O.lMeV Cosmological radiation
(necessarily bounds
and mass content
below the binding
on neutrinos
nucleosynthesis
and structure
after reviewing
the main known cosmological
2
matter
constant).
density
N lo-‘.
of the Universe
(non-relativistic 4l = p/p,
PC = 10.5h2(keV/cm3) find
occur at
temperature
wss
from limits
from measurements
arguments.
on the total
of the CMBR,
We will present
energy
density
and
and from primordial
them in turn in Sects.
3 to 5,
data in Sect. 2.
Content, Expansion and Age of the Universe
The energy cles),
formation
that
differing
energy of the light nuclei).
come mostly
in the Universe,
reactions
by the
p can be in radiation
particles)
is the density
or vacuum
mass,
= A/&G,
in units of the density
= 1.88 10-29h2(~/n2).
The luminous
(pne
(photons
namely
and other where
relativistic
A is the cosmological
of a flat universe,
the critical
Using the value of p7 given above,
the matter
associated
parti-
with typical
stellar
density
we see that populations,
for Qum _N 6.7 IO-*. However, there is much more than the luminous
accounts
The gravitationally
dominant
emission or absorption ferent measurements
of the Universe
of any type of electromagnetic
determinations
is “dark”, i.e. it is not seen either in
radiation.
give fIo~ 2! 0.02/z-’ (from the rotation
(using the virial theorem ¢
mass component
This is called dark matter
CUTveS of Spiral gahC&s),
fI oM > 0.3 (from the peculiar
in clusters),
of the baryonic
density
msss in the Universe.
using NS arguments
velocities
(DM). Dif-
RDM
N
0.2 -
0.4
of large scale flows).
give on 2 0.8 - 2.6 h-*10-*
(see
more about this in Sect. 4). With h 2 0.6(0.4) we have fig 1 0.07(0.16), what means that all the DM in the haloes of galaxies could be baryonic. The NS estimate
of fin combined
with the measurement
of a “large” amount of gas in rich clusters
of galaxies has led to what some call “the x-ray cluster baryon crisis” [4]. In fact, the gas in the central part (until a radius of about by the gas in hydrostatic
lh-’
equilibrium
mass in gas in those regions question PB/PM
Mpc) of the cores of rich clusters and it has been recently
is large, f = (M,,/M,,i)
h > 0.16 (as all measurements open Universe R 0 = R,,,,
+ 0,d)
is spatially
from structure
light (m < keV) neutrinos
formation
of the density
in the past,
degrees
of freedom
either we live in an
at
of matter
lensing of
on A, because
is larger than for A = 0.
(see Sect.
dominated
and radiation,
5) indicate
at present. prnatt N
T > Teq, where Tq is the temperature
the present small radiation (g. = 3.36 with photons
physical size of the horizon
of gravitational
of lensings depends
that the density
in massive
Due to the different
contribution) and three
then (at the moment
evolution
T3,prd - P, the radiation of matter-radiation
= pm,,tt.(Tes), which turns out to be TeqN 5.8eVR,h2(3.36/g,). (neglecting
if
is 0, < 0.2 (this is for h = 0.5, so R,h* _< 0.05).
These data show that the Universe is matter
density
f and the NS upper
This would mean that,
flat, namely Q = R, + CIA= I, the frequency
to a quasar of a given redshift
arguments
with temperature
in the Universe
with A # 0 (if we want it fiat we need R, + Rh = 1, where
by nearer galaxies gives a bound of RA 5 0.7. The number
Finally,
large value of
in
or both.
with A > 0 the distance
dominant
of the baryon fraction
limit of R 5 (0.2 - 0.4)h-‘I*.
of the total
Since the clusters
and the bound on Rs from NS is correct,
(if A = 0) or in a Universe
If our Universe quasars
confirm)
the x-rays emitted
that the fraction
z (0.05 - 0.10)h-3/2.
Ifso, nDh4 = fiBf-' and using the measured
bound of O.O2h-* on Ra one gets an upper
prd(Tq)
estimated
are large, one can think that this ratio is representative = QB/nDM.
is seen through
equality,
R, is the present
was when
matter
and g. is the number of effective relativistic relativistic
of radiation-matter
neutrino
species).
equality)
is
The present
(2) a result we will use later, in Sect. 5.
G. Gchtittil
4
The expansion Universe 0,).
rate,
h = H/(lOOkm/secMpc),
are not independent.
For an empty
Universe
down the expansion, dominated
so that
Universe
t, and decreasing
Universe
makes t, longer.
the present
lifetime,
t,, and the content
of the
x logy) is a function
of G,,,.,,., Rd
and
Ht,
attraction
and radiation
slow
Ht,
= 1. The gravitational < 1 in a matter
(or radiation)
because gravitation
the matter
determinations
( 1998) i-15
In fact, Ht, = (h/0.75)(&,/13
Ht, > 1 instead,
a shorter
Present
Prog. PNI.I. Nrrd. Plt,vs. 40
or radiation
dominated
is repulsive.
content
of matter
Universe.
In synthesis,
or increasing
a larger H implies
the vacuum
of h include values from 0.4 to 0.8, with a confluence
the range 0.55 to 0.75 [l]. There are three main methods
In a vacuum
content
of the
of measurements
in
t,. Nuclear cosmochronology
to determine
gives t, = (10 - 20) Gyr [5], the cooling of white dwarfs gives t, = (9 - 10) Gyr [6] (what is considered an absolute
lower bound)
it is very difficult a reasonable
matter
by the dark matter
due to its simplicity
t,, > 13 Gyr requires
or open or both.
and aesthetic
This tension
between
+ RrJ,
the lower bound
to be namely
the model preferred
one has Ht, = 2/3.
appeal),
by
This means that
h 5 0.65, values lower than most present
h and t, is called sometimes
the bound
(so t, 1 13(10) Gyr can be accommodated
3
for a flat G, = 1 Universe,
larger than 0.65, then we live in a Universe with a non-zero
Qnc = 1 - R. > 0 (n, = n,,,
Gyr requires
gives t, = (13 - 15) Gyr and
with A = 0 (until recently
h 5 0.50 and t, 2 10 Gyr requires
If h is actually
open Universe,
clusters
However,
of the age of the Universe.
dominated
most cosmologists
determination.
globular
to get to t, < (11 - 12) Gyr [7] [I]. Th us, t, 2 I3 Gyr is taken at present
lower bound
a Universe
constant
and the age of the oldest
mentioned
With
0, 2 0.1 from DM measurements,
the “age crisis”.
With
< 0.7, implies Ht, < 0.96
above R,
with h < 0.72(0.94)).
cosmological
R = R, < 1, the case of an
gives Ht, 5 0.90 (so t, 2 13(10)
h < 0.68(0.88)).
Bounds from Cl
Maybe
the most
neutrinos
important
cosmological
(m, < 1 MeV) with standard
tons and photons
in the cosmic plasma
constraint
weak interactions
the rate of weak processes
verse and neutrinos
“decouple”
due to the Hubble
the number
of photons
be formed),
due to entropy
nui = (3/4)(Ty/T)3n, light neutrino
species.
or “freeze out”. expansion)
is increased
when
conservation,
= (3/11)n,
neutrinos
becomes
e+e-
smaller
Their number
becomes
constant
annihilate
the present
than the expansion (per comoving
T N 1 MeV. rate of the Uni-
volume,
Taking
both neutrinos
i.e. a volume
into account
(at T’ 5 me, when e+etemperature
Light
with charged ‘lep
until a temperature
afterwards.
neutrino
= 112 cmm3 (including
is the mass bound.
are kept in equilibrium
(due to weak interactions)
At this temperature
increasing
on stable
that
can no longer
is T, = (4/ll)‘lsT
and anti-neutrinos)
and
for each
Knowing radiation
Ty, we can compute
energy
density
is the effective (Ty/TJ4
number
p,d,
the contribution
that
of relativistic
= 2 x 7/8 x (4/11)4/3
species
are still relativistic,
nldh2
= 4 x 10-5(g./3.36).
is usually
parametrized
degrees
of freedom.
and neutrinos
2.3 x lob4 eV). In this case, the contribution the Universe
Ref.
[8]).
for g.(Z)
Assuming
where g.(T) = 2 x 7/8 x
all three
neutrino
= 2 + 3 x 0.454 = 3.36, and
are non-relativistic
of the non-relativistic
to the present
g.(T,)
For every v species
account
that one or more neutrinos
still relativistic = (n2/30)g0(T)T”,
as p,d
= 0.454 (see for example
photons
However, it is possible
of each v-species
(if m, > To =
at present
light neutrinos
to the density
now is pu = Cf=, mvinq = Q,p,, which means 5
m,
= 92 eVR,h2
.
(3)
i=l
Only left-handed contribution
neutrinos
are considered
of right-handed
states,
from the lifetime of the Universe. in a matter Universe Cim,
of
dominated
Universe,
is excluded
by structure
here (for Dirac masses < 1 keV this is correct,
even if they exist, is negligible).
For example,
the
The best bound on R,h2 comes
t, 1 13 Gyr with h > 0.5 gives R,h2 5 0.25, for A = 0 into Ci m, _< 23 eV. However, a neutrino
what translates formation
because
arguments,
dominated
which give 0, 5 0.2 (with h = 0.5), namely
5 5 eV (see Sect. 5). Up to now we dealt
would decouple
only with neutrinos
lighter
while they are non-relativistic.
R, N exp[-my/T].
In this case the bound
Their density
m, > Mzf2
neutrino
non-standard
apply to neutrinos
only because
range mentioned
Unstable
neutrinos
of the Universe
neutrino,
they have a non-zero
mass).
with m < 1 MeV whose relativistic until the present
The bound a continuous
the energy density
in Eq. contour
1 MeV
by a Boltzmann
factor
for m, 2 few GeV [8]. Only
its existence
unless
R, -K 10e2.
(i.e. to neutrinos
that are
masses could be in the forbidden beyond the standard
or decays. decay products
dominate
the energy
must have a lifetime
of the decay products,
(4) and the corresponding line. If the Universe
2
excludes
interactions
Neutrino
that allow for faster annihilations
m,
they are higher for the latter
would only have a small density
with only standard
with
Q,h2 is satisfied
heavy, but the LEP bound
r 5 (92eV/m,)2(R,h2)2t, to ensure that
is thus reduced
above, namely 30eV 5 m, 5 few GeV, if they have interactions
model of weak interactions
density
could be that
N 45 GeV. These heavy neutrinos
These constraints
1 MeV. Neutrinos
in the energy density
[9]. The mass bounds differ for a Dirac or a Majorana a fourth-generation
than
bound
is assumed
,
(4)
R nr, is not too large, i.e. 00~ 5 R, [lo].
for heavier
masses are shown in Fig.
to become dominated
by matter
1.a with
at some time
after the neutrino
decay, the bound on the lifetime is more restrictive
Fig. 1.a). In fact, the massive
neutrinos
respect
to a matter
structure
formation
of radiation ordinary
density
domination
matter
component
arguments.
that decreases
of the decay products,
replaces
contour
of effective
weakly interacting
extra
particles
SN1987A,
include
limits (distortions
photons
neutrino
fluctuations should
t,
is suppressed
with
required
finish at most when structure
more stringent
by
in the period
5 10d5t,, when atoms become
a bound
contour
Fig.
species
or e+e-
by a factor
1.a shows the nucleosynthesis
6N, < 1, see Sect.
the cosmological
lifetimes
by bounds
pairs.
4).
in
stable. of lob5
bounds
area is excluded
bound
Only decays into
on stable
neutrinos,
coming from the CMBR and on unstable
The grey area in Fig.
of the CMBR and other photon
flux from the supernova
epoch
This bound is shown for m < 1 MeV in Fig. 1.a
for the relevant
figure 5.6 of Ref [S], while the hatched with the observed
neutrino
the present
as can be seen in Fig. 1.b. This figure shows bounds
whose main decay products cosmological
[ll]).
could help evading
all visible modes have been rejected the supernova
bound
line (while the dot-dashed
on the allowed number neutral
this period
in (4), thus yielding
t, by t,
before
more slowly, as T3. This is actually
i.e. at recombination
(usually called the “galaxy formation” with a dashed
subdominant
Because the growth of density
could start forming,
This argument
have to decay earlier for the energy of their decay products
as T4, and become
to have a longer time to decrease,
than Eq. (4) (and the bound in
backgrounds),
by the non-observation
neutrinos
l.b., excluded
is reproduced of photons
by
from the
in coincidence
SN1987A (see Ref [12] for details).
24
24
21
21
18
18
15
15
b
Get
g 12 u1 E $ 9 6 3
bnd I
-4
-2
0
2
4
6
8
1012
-4-2
0
109 IWWI
Figure 1: Bounds
on unstable
modes, i.e. including
photons
neutrinos or e+e-
2
4
6
810
109 tmWl1
that apply to any decay mode (l.a, left) and to visible decay pairs as decay products
(I.b, right).
See the text for details.
4
Bounds from Nucleosynt hesis
When the temperature
of the Universe
MeV, nuclei first became than protons, mately
nn/np
stable.
became
Because
N exp[-(m,
neutrons
- m,)/T]
25% of the mass of the Universe)
protons
stayed
as H (approximately
and 7Li were produced The predictions
predicted
(with nD,qe/nH
of the reactions
that
of 4He increases
over protons
of light (i.e. relativistic
estimates.
values (which require relatively
4He assumed then, which required
either
of D-abundance
obtained
with data
with new data
important
recent developement
lead in 1996 to abandon than bounds but it cannot
by Tytler
started
a tension
between
N, = 3 + JN,
to point to values of the these new relatively
low
low upper bound on
at LEP, based on a low estimate
al.
[14] obtained
that Hata et in quasars
and a chemical pointed
al.
indicator
lines.
in the processing of the primordial
had been that D is easily destroyed
composition
to, has been largely
Ly o absorption
of uncertainties
on D+3He as a better
on D alone (the argument
depends
lower values of r] with N, = 3, or N, < 3. In fact, in 1995
has been the recognition
the bound
end up in 4He
species,
large values of 17)and the relatively
on the solar wind and meteorites
et
to a larger value
all neutrons
during NS) neutrino
This created
code. Since then, the low range of D-abundance
confirmed
The reason
p, during NS, which is expressed
[13] claimed a bound of N, < 2.6, less than the 3 observed
al.
species.
of the 4He-abundance
with NY). In 1995, measurements
D-abundance
evolution
of neutrino
the prediction
number
lower than previous
While the
with r~ (as the rate
and, consequently,
in terms of the equivalent
D-abundance
that there is a range
sharply
of 4He (since as a first approximation
of 4He increases
during
leads to an earlier freeze out of the weak
to the total energy density of the Universe,
Hata et
ratio q E na/nT
for all the light elements.
on any extra contribution
(so, the abundance
3He
N O(lO-lo).
with the number
to fi,
of D,
to 77).
rate of the Universe
rate is proportional
The remaining
and trace amounts
with 9, those of D and 3He decrease
the ratio of neutrinos
Since the expansion
of protons.
are vastly different,
are obtained
of 4He also increases
in the expansion determine
there were less neutrons
ended up into 3He (approxi-
with an equal number
that the abundances abundances
of this ratio and to overproduction nuclei).
together
N 0(10m4) and (n,r,i/nu)
considering
abundance
an increase
interactions
< 1. Most of the neutrons
that burn them into 4He are proportional
The predicted is that
are heavier than protons
on the value of the baryon-to-photon
for which realistic
abundance
energy of nuclei, at T N 0.1
75% of the mass of the Universe),
of NS depend
NS, and it is remarkable, of q N O(lO-lo)
smaller than the binding
Another
of 3He which D-abundance
in stars, mostly into 3He,
be produced).
A range in r] is translated With T = 2.73”K one obtains due to observational
uncertainties
into a range of R&,
since
71 =
(nn/nT)
= (nnp,/mp)/(2~(3)T3/~*).
17= 272.2 x 10-r’ flBh*. Bounds on 77have changed and different treatment
of errors.
over time, mainly
The ranges of the “Chicago group”
have changed from 0.015 5 Rnh* < 0.026 (and NV < 3.4) in 1980’s, to 0.010 _< f&h* 5 0.015 (and NV < 3.3) in 1991, 0.009 5 S2Bh2_< 0.022 (and NV < 3.4) in 1995 and 0.018 s Reh* 5.0.026 (and N, c 3.7) in 1997 (using now mostly the lower D-abundance
of Tytler et al. and a relatively large
‘He upper bound, YH, < 0.250; see (151 and references therein).
The “Ohio Group” gave the bounds
0.016 5 RBh* 5 0.022 (with N, < 2.6) in 1995, in the paper mentioned above and made an exhaustive analysis of the bounds in 1996 claiming a bound on NV < 3, if the upper bound on 4He is relatively low (YHc< 0.243) [13]. During 1996 the situation WBSmade even less clear by the existence of another using the same technique of Tytler et al. claiming much
measurement of the primordial D-abundance larger values. Several groups reanalyzed
then the NS bounds pointing to uncertainties
At present, the systematic errors on the upper bound of ‘He abundance consequently the bound on aN, is uncertain.
in the data.
are being m-evaluated and,
On the contrary, the prospects for the determination
of
Ra through NS have never been better, since the quasar Ly a absorption lines technique could soon determine Ga with a 20% precision (15). The possibility of a bound NV < 3 cannot be excluded at present. This would require at least one of the known neutrinos to contribute to p less than a relativistic neutrino during NS. For example, a heavy neutrino could decay before NS, i.e. with a lifetime < 10 set (161. One should also keep in mind that non-standard
NS scenarios, such as inhomogeneous NS or NS in the presence of very large
lepton asymmetries, are still viable at present and would lead to different results. For example in both of these scenarios Rn could be up to ten times larger [17].
5
Bounds from Structure Formation
The Universe looks lumpy at scales X N 100 Mpc, we see galaxies, clusters, superclusters,
voids,
walls. But it was very smooth at the surface of last scattering of the CMBR (i.e. at electromagnetic decoupling, when ions and electrons first formed atoms) and later. Inhomogeneities as anisotropies
in the CMBR,
so the density
contrast
6p/p E (p(r) - p)/p
density) cannot be much larger than &T/T c* 10m5. So inhomogeneities grow through the Jeans (or gravitational)
instability:
gravitation
regions and to further increase the density of overdense regions.
have been seen
(where p is the average
in density start small and
tends to further empty underdense One can follow analytically
the
evolution due to gravity of the density contrast in the linear regime, where 6p/p < 1. In a static fluid the rate of growth of Q/p
is exponential,
but in the Universe (an expanding fluid) it slows down into
either a power law, ~5p/p - a(t), in a matter dominated
Universe, or it stops, 6p/p N constant,
in a
radiation or a curvature dominated
Universe (a matter dominated open Universe becomes curvature
dominated for a(t) > 0,/(1
Here a(t) is the scale factor of the Universe, which accounts for
- 0,)).
the Hubble expansion linear dimensions
of the Universe.
Perturbations
where A,,,, are linear dimensions
X = a(t)&,,
(those that expand the Universe).
of the linear dimensions
with the Hubble flow and, therefore,
have different
measured
in comoving
physical
coordinates
do not change in time due to the expansion A,,,, are the present
With the usual choice of a = 1 at present,
of
actual linear dimensions.
Since the horizon ct grows linearly with t, while in a matter
(or radiation)
dominated
with a < 1, the horizon increases
coordinates,
encompassing
more material
of size X “enters” into the horizon,
we could better
with time even in comoving
ss time goes. When X = ct we say the perturbation say the perturbation and it happens
is first encompassed
at different
Independently spectrum
times for different
of the origin of these
at horizon-crossing,
(~~lP)hLX = constant, spectrum
are formed
interactions
which determines
of how fluctuations
particles
t,d each of the particles a distance expand
Thus,
At the moment
negligible.
Thus the smallest
Light neutrinos
(HDM) (such as light neutrinos
10”Mo
which structures The following
for the first time a
the dark halo), where M.3 turns out to be
and the volume occupied
become
by the horizon
T N lkec’.
In this case, at
by the particles
is erased.
structure
that
of m 2 30eV) superclusters
the fragmentation
of the larger structure.
there is not enough
time to form galaxies.
of galaxies Simulations
Thus HDM is rejected
of mass m N keV, that are just becoming
non-relativistic
becomes
of relativistic
particles
contains
to a supercluster.
by free
their motion
are becoming
can survive
will still
This is damping
non-relativistic,
when the particles
which corresponds
DM galaxies form first, but barely.
(including
at that moment
a “pro-
of these three types.
that can survive the “free streaming”
form later, through
even of the size of the smallest
the differences
later the fluctuation
are hot DM and the smallest
MJ-. = 3 x 1015Mo/(m,/30eV)2,
This feature,
types of DM: hot, warm and cold.
when the DM particles
and grow, is that one encompassed
with
of the size of the galaxy has moved from its original position
a moment
structure
namely
have shown the
at &,I, i.e. their mass is m < 1keV.
ct equal to the size of the fluctuation
streaming.
Particles
are formed first.
of the Universe
are still relativistic
their
and generate
at which the growing horizon encompasses
in a fluctuation
with the horizon.
COBE observations
to specify
or very close to it.
of the size of a typical galaxy that contains,
Hot DM particles
at horizon-crossing,
act upon the inhomogeneities
of three
is the mass of the Sun. The temperature
“horizon-crossing”
it is convenient
spectrum
evolve helps understanding
Let us define as t,d the moment perturbation
fluctuations,
spectrum.
which structures
first, leads to the distinction
naive picture
primordial
is in fact scale invariant physical
is called
a N tQ
linear scales X, larger scales cross later.
is called a Harrison-Zel’dovich
After horizon-crossing,
This moment
A scale invariant
(6p/p)k,.
at horizon-crossing
cessed” spectrum,
by the horizon.
Universe
Therefore,
non-relativistic. the Jeans
with hot DM
form first and galaxies have shown, however,
as the dominant
mass,
must that
DM component.
at t,d, are warm DM. With warm
Cold DM (CDM) is such that perturbations
smaller than a galaxy,
dwarf galaxy (106Mo) are not erased and can grow.
Simulations uprocessed”
have shown
spectrum
the highest
perturbations
spectrum
(2)). Perturbations dominated.
crossing,
distribution
entered
Perturbations
when the Universe
is matter
dominated,
at larger scales entered (the location
an overall normalization
-
by which only
There is only one feature
of matter-radiation at t < t,,
is radiation
so they started
equality,
now provided
perturbations
dominated, today,
if they all started entered
growing with the
into the horizon at Consequently, is smaller
at
free parameter
is
by the CMBR anisotropy
at these large scales entered of (Q/p)
(see Eq.
is radiation
so they all start
growing immediately.
a measurement
&
when the Universe
with X > A,, instead,
(so they did not grow much), thus providing
(6p/p)hor (for more details,
“biasing”
of the change slope) is fixed, the only remaining
COBE at large scales, B > 20”. Density very recently
N R = 1, A =
later, had less time to grow and their amplitude
of the CDM spectrum,
the
(ss can be seen in Fig.2), a change of slope at the
have the same amplitude
crossing.
because
the observations
namely Rcn~+Rs
and a scale independent
into the horizon
at horizon
Once X,
assumptions,
end up forming galaxies.
grow while the Universe
and they roughly
form of matter,
CDM models reproduces
to the horizon at the moment
with X < &
at t = f
perturbations present.
at horizon
of CDM perturbations
They cannot
same amplitude
in standard
CDM models, make the simplest
present scale that corresponds
t > t,,
generated
peaks in the CDM density
in the processed
together
CDM must be the most abundant
of perturbations
within 10%. Standard 0, scale invariant
that
measured
by
into the horizon
at horizon crossing,
see e.g. Ref [8]).
CDY
n=l
----- HDMn=l --CDU n=0.8 ............. YDY nr 1
Figure 2: Comparison
of the measured
power spectrum
of several DM models
(notice that k- X-l).
of density
perturbations
and the predictions
While both the shape
and normalizationof
the spectrum
not fit the observations
(191. In Fig. 2 (201 the power spectrum
standard
(solid line) normalized
CDM models
(the points
with vertical
of standard clusters
CDM models
and smaller.
short-dashed
solutions
consists
primordial
with n=0.8)
limits.
models
given in Eq.
&,
to that
older Universe),
of the standard
This is called
R, < 0.2.
at horizon
crossing,
one that slightly
CDM (TCDM)
require
works well.
r, would provide
favors larger
spectrum)
and some “tilt” arguments
In fact another
larger scales of the only feature
since it effectively (X > X,)
of the standard
changes amounts
defines
[26]), for example
also does
give the bound
family of solutions
(see Fig. 2), is enough to increasing
the power
to those smaller
or 2): increase g. (namely increase the radiation
CDM models
[24]
(with the
1): lower h [25] (what
content
that provides
= 0.18, Rs = 0.02, R,
than
a larger
In order to lower the value of I with
with data.
CDM model one needs to either,
with h = 1, Rcn~
line
the “shape parameter”
r N 0.25 f 0.05, while standard
constant
within
in the CDM spectrum,
with respect
X, G (lOh_‘&,,)I’-’
agreement
that
[23]).
(2), the scale where the slope in the spectrum
The data
models
[22]. See the long-dashed
formation
may also provide a solution.
with observations,
crossing,
Harrison-Zel’dovich
these structure
(see for example
A = 0 or in a Universe with a cosmological
above.
in what are called mixed
CDM models work as well, for example
at horizon
“tilted”
Hence,
(2) the relation
the
at small scales, one of
line in Fig. 2. In particular,
lower R, (i.e. take 51, < l), so that we either live in an open Universe if
(see the
to change
choices of h = 0.5, R, = 1, g. = 3.36) has I = 0.05. In fact, as we have explained,
thus a smaller
respect
neutrinos,
of fluctuations
at scales larger than the break point
I E R&7./3.36)-‘I*. standard
spectrum
that a shift towards
Using Eq.
for the data
_ N 5 eV, and the rest of R, in CDM plus some baryons,
of the flat, scale invariant,
neutrinos
good agreement
it (X < X,).
the scales of galaxy
possibilities
HDM tends to erase structure
of fluctuations
massive neutrinos
by realizing
of the spectrum
X-l),
in Fig. 2. A mixed model with both some neutrinos
Heavy unstable
to provide
variations
scales (instead
R, 5 0.2 for light massive
i.e. &,
to Cim,
spectrum
the COBE observational
is obtained
Because
h = 0 and a scale invariant
work, and in these
2 (solid line) the spectrum
are several
DM (HCDM) models [21]. See the dotted
However many other possible
(labelled
is lixed, there
to the CDM a bit of HDM, namely
with R, = 0.2, what corresponds
scales over smaller
As can be seen in Fig.
given by COBE
to agree with observations.
DM (MDM) or hot-cold
by
above.
spectrum
in adding
predicted
(the box on the 1.h.s.) and other data
Also in Fig. 2 one can see the failure of HDM to account
Once the normalization
of a “tilted”
perturbations
has too much power on small scales (large k-
curve) mentioned
with R = 1,
of density
by COBE data
error bars) are shown.
are almost right, they do
so obtained
of the Universe
implies
at teq), or 3):
(open CDM models, 0,
an
OCDM)
= 1 - 0, (ACDM models
= 0.80, or 4): a combination
of all three
A way of obtaining
the large amount of radiation needed for the second possibility is through a
heavy neutrino decaying into relativistic particles, i.e. radiation, with the right combination
of mass
and lifetime, in so-called rCDM models (27, 281. A massive neutrino matter dominates the energy density of the Universe as soon ss it becomes non-relativistic, and pv = n,m,,
i.e. as soon as m, 2 T (since n, N n_,
prd cz n,T), thus their decay products radiation-dominate
the Universe at decay. For
m, < 1 MeV the right mass-lifetime combination lie on a narrow strip around the previously mentioned “galaxy formation” bound [ll], shown in Fig. 1-a. Near this bound, at the boundary between being irrelevant and harmful, unstable neutrinos could help in the formation of structure in the Universe [27]. A heavier neutrino,
of m, N 1 - 10 MeV, necessarily v+, decaying at or just before nucleosynthesis,
r = 0.1 - 160 set, would also provide a solution [28]. The r+ decay modes involved here should all be into neutral particles, vr + 3 v’s or v, -+ v+, with 4 a Majoron (a zero mass Goldstone boson) for example. All visible modes, i.e. producing electrons or photons, are forbidden in the necessary range, and shown in Fig. 1.b.
8
6
200
Figure 3: CMBR anisotropy from top to bottom: KDM
uncertainty
600
800
1000
power spectra predicted by four models (lines) of structure
formation,
CDM with low h (h= 0.35), TCDM (with IV,= 12), ACDM (with h=0.65),
with D,=O.2 and ‘tilted”CDM
anticipated
400
(other parameters as in standard CDM). The band shows the
from MAP (from Ref. [23])
All these
modified
very different
CDM models
patterns
ture autocorrelation temperature
of acoustic function,
fluctuations
multiplying
ple moments
with
that the Universe
is smooth
pick up additional
satellite
observations
dark matter
10%. The detection
experiments,
models.
the Fig. 3 [23] together on the density
because
last scattering
surface,
different
regions
0 1 1” it indicates to scales within the
Thompson
scattering
of multiple
moments
N 200(R)-‘,
the position
of the first peak
so far obtained
with the COBE
experiment
models.
to
and where the photons
(mainly
models,
in Europe.
angle 0 is
at angles B 2 1”
but the next generation with sub-degree
of
resolution,
They will give results as shown in The height of the peaks depends
and relative height of the peaks between models of galaxy formation.
larger values of P correspond so the information
matter
between
the multi-
corresponds
at the scale of the horizon e+
of five different
fin and the location
several 1000 will allow to discriminate
perturbations
peaks in the spectrum
new satellite
(ex COBRAS/SAMBA)
with the predictions
in baryons,
ct,,
of the anisotropies
Thus, in the future,
does not discriminate
defines
to causally disconnected
of the CMBR anisotropies
will do it. There are two approved
MAP in the U.S. and PLANCK
are relevant,
(or acoustic)
of the first peak should happen
R within
and balloon
from moving
by
by two angles),
at a certain
and at larger angles,
can move due to density
due to the scattering
as shown in Fig. 3 for different will determine
polynomials
size at recombination,
at recombination,
The tempera-
is computed
harmonics
so that the anisotropy
Thus the smallness
predict
from the first one by an angle 0
sky. Larger angles correspond
This gives origin to Doppler
P(P + 1)Cl. The position
i= = 2.726”K,
on very large scales. We see that P 1 200 correspond
where matter
energy
temperature
The horizon
they
where dT/T = (T - p)/!f’ are
+ 8)/T)),,
of C(6) in Legendre
C,(2P + l)C&(cosB),
was smooth
however
power spectrum.
(a + 0) separated
of the CMBR photons.
at decoupling,
on electrons).
position
in the present
the Universe
data,
Q in the sky (Q is given in spherical
I,with e N (200”/8).
at the time of emission that
to the average
The expansion
Ct, C(0) = (4z)-’
an angle 0~ N 1°(R)-1/2
horizon
respect
over all positions.
@T/T),N dm
anisotropy
C(B) = ((6T(cr)/T)(6T(a
6T/T at another
it with
and averaging
indicates
peaks in the CMBR
&“/T at some position
measuring
seem to be able to fit present
e N 200 and
Only the C, for l? < 0( 103)
to scales 8 < 8” that are inside the thickness
at those small scales is smeared
of the
out.
Acknowledgments I thank the organizers U.S. Department
of this workshop
of Energy
for their invitation.
under Grant DEFG03-91ER
This work was supported
in part by the
40662 TaskC.
References [l] W. L. Freedman, Astrophysics,
ASTROPH-9706072,
Chicago,
Invited
IL, 15-20 Dee 1996.
talk at 18th Texas Symposium
on Relativistic
[2] (COBE Coil.) J.C. Mather et al. Ap. J. 420, 439 (1994). [3] (COBE Coll.) G.F. Smoot et al. Ap. J. Mt.
396, Ll (1992); (COBE Coil.) C. L. Bennet et al.
Ap. J. 436, 423 (1994). [4] M. White, J. Navarro, A. Evrard and C. Frenk, Nature 366, 429 (1993); G. Steigman and J. Felten, OSU-TA-24-94 sstro_ph/9502029. [5] J. Cowan, F. Thielemann and J. Truran, Ann. Rev. As&on. Astroph. 29, 447(1991). [6] D. Winget et al. Ap. J. Leti. 315, L77 (1987); I. Iben and G. Laughlin, Ap. J. 341,430 [7] A. Renzini Proc. 16th Tezas Symp. on Relat Astroph. and 3rd Symp. on Particles,
(1989).
Stings
and
Cosmol. eds. C. Akerlof and M. Srednicki, N.Y. Act. of Sci., N.Y. (1992); X. Shi Ap. J. (1995). [8] E. Kolb and M. Turner, The Early Universe, Addison-Wesley, 1990, and references therein. [9] B. W. Lee and S. Weinberg, Phys. Rev. Mt.
39, 165 (1977) (and others, see e.g. [8]).
[lo] 0. Dicus, E. Kolb and V. Teplitz, Phys. Rev. L&t. 39, 168 (1977). [ll] P. Hut and S. White, ZVature310,637 (1984); G. Steigman and M. mrner, Nucl. Phys. B 253, 375 (1985). [12] G. Gelmini and E. Roulet, Rept. Prog. Phys. [13] N. Hata et al., Phys.Reu.Lett. Phys.Reu. D55,
58, 1207-1266 (1995)
75, 3977 (1995); N. Hata, G. Steigman, S. Bludman, P. Langacker
540 (1997).
[14] D. Tytler, X.M. Fan and S. Burles, Nature 381, 207 (1996). [15] D. Schramm and M. Turner, ASTRO-PH/9706069. [16] S. Dodelson, G. Gyuk and M. Turner, Phys. Rev.
D49, 5068 (1994); M. Kawasaki et al., Nucl.
Phys. B419, 105 (1994). [17] H. Kang and G. Steigman, Nucl. Phys. B372 (1992) 494. [18] J.P. Ostriker, Ann. Rev. Astron. Repts.
31, 689 (1993); A.D. Liddle and D. Lyth, Phys.
Astrophys.
31, 689 (1993); A.D. Liddle and D. Lyth, Phys.
231, 1 (1993).
[19] J.P. Ostriker, Ann. Rev. Astron. Repts.
Astrophys.
231, 1 (1993).
[20] W. Kolb, P rot. &nd
Scottish
School if Physics,
Univ. of St. Andrews, Scottland (1993) and
Fermilab-Conf-94/05&A. [21] Q. Shafi and F. Stecker, Phys.. Rev. Mt.
53, 1292 (1984); C.P. Ma and E. Bert&ringer, Ap. J..
429, 22 (1994) and Ap. J. 434, L5 (1994), J. Primak et al., Phys.. Rev. Mt.
74, 2160 (1995).
[22] R. Cen et al. Ap. J. 399, Lll (1992), R. Davis et a!.Phys. Rev. l&t. 69, 1856 (1992); F. Lucchin, S. Mattarese and S. Mollerach, Ap. J. 401, L49 (1992). (231 S. Dodelaon, E. Gates and M Turner Science 274 69 (1996) [24] G.E. Efstathiou, Pnx. of the “36th Scottish
Univs Summer Sch. in Phys”, “Phys. of the Early
Universe”, ed. J. Peacock, A. Heavens and A. Davies Adam-Hilger, N.Y.(1990) p. 361; J. Peacock and S. Dodde.
Monthly Not. of the Royal Ast. Sot. 267, 1020 (1994).
(251 J. Bartlett et al., Science 267, 980 (1995). [26] MS. Turner, G. Steigman and L. Krause, Phys. Rev. Mt.
52, 2090 (1984); L. Krause and M.
Turner, Gen. Rel. Gmu. 27 1137 (1995). [27] J. Bardeen, J. Bond and C. Efstathiou, Ap. J. 321, 28 (1997); J. Bond and G. Efstathiou, Phys.
Mt.
B265, 245 (1991); M. White, G. Gelmini and J. Silk, Phys. Rev. D51, 2669 (1995).
(281 S. Dodelson, G. Gyuk and M.S. Turner, Phys. Rev. Mt.
72, 3578 (1994).
Progress
in Particle
and Nuclear
Physics 40 ( 199X) 1 -15
Neutrinos in Cosmology
We review the main bounds on neutrino properties from cosmological arguments.
1
Introduct
ion
Neutrinos may play an important yet unknown parameters, are complementary
role in cosmology depending on the values of their fundamental
such as masses, mixing and lifetimes. Conversely, cosmological arguments
to laboratory experiments and observations of astrophysical solar and atmospheric
neutrinos, in providing strong bounds on neutrino properties. The Hot Big Bang (BB), the standard
model of cosmology, establishes that the Universe is
homogeneous, isotropic and expanding from a state of extremely high temperature the early universe can be described as an adiabaticlly
and density. Thus
expanding classical gas of relativistic
namely radiation, in local thermal equilibrium at a temperature
particles,
T, that changes with time, Z’(t). The
lifetime of the Universe t is counted from the moment the expansion started, taken to be t = 0. This model is based on General Relativity, the Cosmological Principle and four major empirical pieces of evidence, namely the Hubble expansion, the cosmic blackbody microwave background radiation (CMBR), the anisotropies of the CMBR, and the relative cosmic abundance of the light elements (up to ‘Li). Because the model is based on General Relativity, it is for sure not valid in the realm of Quantum
Gravity, T > &&n& = 1.22
x
1O”GeV and t < 10-43sec. The Cosmological Principle
postulates that we do not live in a special place in the Universe, by requiring that every comoving observer in the “cosmic fluid” has the same history. The “cosmic fluid” has as particles clusters of galaxies, and “comoving” in practice means at rest with the galaxies within a 100 Mpc radius (one pa.rsec,
1
pc = 3.26 light years = 3.09
x
1018cm). The Hubble parameter H provides the proportionality
between the velocity tr of recession of faraway objects, and their relative distance d,
(1)
v=Hd, 0146-6410:98/%19.00+0.00 PII: SOl46-6410(9X)00003-9
0
1998 Elsevier Science
BV. All rights reserved.
Printed
in Great
Britain
where H = h 100 km/set
Mpc and h is a parameter
The CMBR wss produced It has a blackbody
Explorer) satellite
N 3 x 105y, the recombination
at t,,
spectrum
measured
found a blackbody
and it is remarkably
the CMBR spectrum
of the CMBR photons, n., = 2C(3)Tj/a2 Anisotropies
COBE measured
p-, = 7r2c/15
inhomogeneities
a dipole anisotropy
of galaxies with respect
between
in the Universe
0.05 and 1 cm, and
of less than 0.03% [2].
the number and energy density by several orders of magnitude,
= 4.71 x 10-34(g/cm3). due to our motion with respect
that triggered
corresponding
structure
formation
to the CMBR rest frame, in the Universe.
to a velocity of 6.27&22km/sec
to the CMBR rest frame (COBE even saw the rotation
the Sun!), and measured
In fact
of our Local Group of the Earth
around
(bT/T)o for angles 0 = 7” to 90”. At 90” COBE measured
anisotropies
of (6T/T)go. N 0.5 x lob5 131. Results
anisotropy
The COBE (Cosmic Background
we know with great accuracy
in the CMBR are expected
and due to the density
isotropic.
in 1992 for wavelengths
that are the most abundant
= 411/cm3,
epoch, when atoms first became
To = 2.726 f O.OlO”K, with deviations
with temperature
Knowing so well the CMBR temperature,
quadrupole
determinations
to h IS!0.55 - 0.75 [l].
are converging
stable.
of order one, whose observational
from other experiments
in balloons
a are
angles, 0 = 0,5” to 90” and the results show 6T/T 5 10m5 after subtracting
available now at smaller the dipole. Finally,
the earliest
cosmological by several
abundance orders
available
proof of the consistency
of the Hot BB model is provided
of 4He and of the trace elements
of magnitude,
D, and ‘Li.
Their
abundances,
are well accounted
for in terms
of nuclear
tNS = 1o-2 - lo2 set, the Big Bang nucleosynthesis
(NS epoch),
when the photon
TNSN 10 - O.lMeV Cosmological radiation
(necessarily bounds
and mass content
below the binding
on neutrinos
nucleosynthesis
and structure
after reviewing
the main known cosmological
2
matter
constant).
density
N lo-‘.
of the Universe
(non-relativistic 4l = p/p,
PC = 10.5h2(keV/cm3) find
occur at
temperature
wss
from limits
from measurements
arguments.
on the total
of the CMBR,
We will present
energy
density
and
and from primordial
them in turn in Sects.
3 to 5,
data in Sect. 2.
Content, Expansion and Age of the Universe
The energy cles),
formation
that
differing
energy of the light nuclei).
come mostly
in the Universe,
reactions
by the
p can be in radiation
particles)
is the density
or vacuum
mass,
= A/&G,
in units of the density
= 1.88 10-29h2(~/n2).
The luminous
(pne
(photons
namely
and other where
relativistic
A is the cosmological
of a flat universe,
the critical
Using the value of p7 given above,
the matter
associated
parti-
with typical
stellar
density
we see that populations,
for Qum _N 6.7 IO-*. However, there is much more than the luminous
accounts
The gravitationally
dominant
emission or absorption ferent measurements
of the Universe
of any type of electromagnetic
determinations
is “dark”, i.e. it is not seen either in
radiation.
give fIo~ 2! 0.02/z-’ (from the rotation
(using the virial theorem ¢
mass component
This is called dark matter
CUTveS of Spiral gahC&s),
fI oM > 0.3 (from the peculiar
in clusters),
of the baryonic
density
msss in the Universe.
using NS arguments
velocities
(DM). Dif-
RDM
N
0.2 -
0.4
of large scale flows).
give on 2 0.8 - 2.6 h-*10-*
(see
more about this in Sect. 4). With h 2 0.6(0.4) we have fig 1 0.07(0.16), what means that all the DM in the haloes of galaxies could be baryonic. The NS estimate
of fin combined
with the measurement
of a “large” amount of gas in rich clusters
of galaxies has led to what some call “the x-ray cluster baryon crisis” [4]. In fact, the gas in the central part (until a radius of about by the gas in hydrostatic
lh-’
equilibrium
mass in gas in those regions question PB/PM
Mpc) of the cores of rich clusters and it has been recently
is large, f = (M,,/M,,i)
h > 0.16 (as all measurements open Universe R 0 = R,,,,
+ 0,d)
is spatially
from structure
light (m < keV) neutrinos
formation
of the density
in the past,
degrees
of freedom
either we live in an
at
of matter
lensing of
on A, because
is larger than for A = 0.
(see Sect.
dominated
and radiation,
5) indicate
at present. prnatt N
T > Teq, where Tq is the temperature
the present small radiation (g. = 3.36 with photons
physical size of the horizon
of gravitational
of lensings depends
that the density
in massive
Due to the different
contribution) and three
then (at the moment
evolution
T3,prd - P, the radiation of matter-radiation
= pm,,tt.(Tes), which turns out to be TeqN 5.8eVR,h2(3.36/g,). (neglecting
if
is 0, < 0.2 (this is for h = 0.5, so R,h* _< 0.05).
These data show that the Universe is matter
density
f and the NS upper
This would mean that,
flat, namely Q = R, + CIA= I, the frequency
to a quasar of a given redshift
arguments
with temperature
in the Universe
with A # 0 (if we want it fiat we need R, + Rh = 1, where
by nearer galaxies gives a bound of RA 5 0.7. The number
Finally,
large value of
in
or both.
with A > 0 the distance
dominant
of the baryon fraction
limit of R 5 (0.2 - 0.4)h-‘I*.
of the total
Since the clusters
and the bound on Rs from NS is correct,
(if A = 0) or in a Universe
If our Universe quasars
confirm)
the x-rays emitted
that the fraction
z (0.05 - 0.10)h-3/2.
Ifso, nDh4 = fiBf-' and using the measured
bound of O.O2h-* on Ra one gets an upper
prd(Tq)
estimated
are large, one can think that this ratio is representative = QB/nDM.
is seen through
equality,
R, is the present
was when
matter
and g. is the number of effective relativistic relativistic
of radiation-matter
neutrino
species).
equality)
is
The present
(2) a result we will use later, in Sect. 5.
G. Gchtittil
4
The expansion Universe 0,).
rate,
h = H/(lOOkm/secMpc),
are not independent.
For an empty
Universe
down the expansion, dominated
so that
Universe
t, and decreasing
Universe
makes t, longer.
the present
lifetime,
t,, and the content
of the
x logy) is a function
of G,,,.,,., Rd
and
Ht,
attraction
and radiation
slow
Ht,
= 1. The gravitational < 1 in a matter
(or radiation)
because gravitation
the matter
determinations
( 1998) i-15
In fact, Ht, = (h/0.75)(&,/13
Ht, > 1 instead,
a shorter
Present
Prog. PNI.I. Nrrd. Plt,vs. 40
or radiation
dominated
is repulsive.
content
of matter
Universe.
In synthesis,
or increasing
a larger H implies
the vacuum
of h include values from 0.4 to 0.8, with a confluence
the range 0.55 to 0.75 [l]. There are three main methods
In a vacuum
content
of the
of measurements
in
t,. Nuclear cosmochronology
to determine
gives t, = (10 - 20) Gyr [5], the cooling of white dwarfs gives t, = (9 - 10) Gyr [6] (what is considered an absolute
lower bound)
it is very difficult a reasonable
matter
by the dark matter
due to its simplicity
t,, > 13 Gyr requires
or open or both.
and aesthetic
This tension
between
+ RrJ,
the lower bound
to be namely
the model preferred
one has Ht, = 2/3.
appeal),
by
This means that
h 5 0.65, values lower than most present
h and t, is called sometimes
the bound
(so t, 1 13(10) Gyr can be accommodated
3
for a flat G, = 1 Universe,
larger than 0.65, then we live in a Universe with a non-zero
Qnc = 1 - R. > 0 (n, = n,,,
Gyr requires
gives t, = (13 - 15) Gyr and
with A = 0 (until recently
h 5 0.50 and t, 2 10 Gyr requires
If h is actually
open Universe,
clusters
However,
of the age of the Universe.
dominated
most cosmologists
determination.
globular
to get to t, < (11 - 12) Gyr [7] [I]. Th us, t, 2 I3 Gyr is taken at present
lower bound
a Universe
constant
and the age of the oldest
mentioned
With
0, 2 0.1 from DM measurements,
the “age crisis”.
With
< 0.7, implies Ht, < 0.96
above R,
with h < 0.72(0.94)).
cosmological
R = R, < 1, the case of an
gives Ht, 5 0.90 (so t, 2 13(10)
h < 0.68(0.88)).
Bounds from Cl
Maybe
the most
neutrinos
important
cosmological
(m, < 1 MeV) with standard
tons and photons
in the cosmic plasma
constraint
weak interactions
the rate of weak processes
verse and neutrinos
“decouple”
due to the Hubble
the number
of photons
be formed),
due to entropy
nui = (3/4)(Ty/T)3n, light neutrino
species.
or “freeze out”. expansion)
is increased
when
conservation,
= (3/11)n,
neutrinos
becomes
e+e-
smaller
Their number
becomes
constant
annihilate
the present
than the expansion (per comoving
T N 1 MeV. rate of the Uni-
volume,
Taking
both neutrinos
i.e. a volume
into account
(at T’ 5 me, when e+etemperature
Light
with charged ‘lep
until a temperature
afterwards.
neutrino
= 112 cmm3 (including
is the mass bound.
are kept in equilibrium
(due to weak interactions)
At this temperature
increasing
on stable
that
can no longer
is T, = (4/ll)‘lsT
and anti-neutrinos)
and
for each
Knowing radiation
Ty, we can compute
energy
density
is the effective (Ty/TJ4
number
p,d,
the contribution
that
of relativistic
= 2 x 7/8 x (4/11)4/3
species
are still relativistic,
nldh2
= 4 x 10-5(g./3.36).
is usually
parametrized
degrees
of freedom.
and neutrinos
2.3 x lob4 eV). In this case, the contribution the Universe
Ref.
[8]).
for g.(Z)
Assuming
where g.(T) = 2 x 7/8 x
all three
neutrino
= 2 + 3 x 0.454 = 3.36, and
are non-relativistic
of the non-relativistic
to the present
g.(T,)
For every v species
account
that one or more neutrinos
still relativistic = (n2/30)g0(T)T”,
as p,d
= 0.454 (see for example
photons
However, it is possible
of each v-species
(if m, > To =
at present
light neutrinos
to the density
now is pu = Cf=, mvinq = Q,p,, which means 5
m,
= 92 eVR,h2
.
(3)
i=l
Only left-handed contribution
neutrinos
are considered
of right-handed
states,
from the lifetime of the Universe. in a matter Universe Cim,
of
dominated
Universe,
is excluded
by structure
here (for Dirac masses < 1 keV this is correct,
even if they exist, is negligible).
For example,
the
The best bound on R,h2 comes
t, 1 13 Gyr with h > 0.5 gives R,h2 5 0.25, for A = 0 into Ci m, _< 23 eV. However, a neutrino
what translates formation
because
arguments,
dominated
which give 0, 5 0.2 (with h = 0.5), namely
5 5 eV (see Sect. 5). Up to now we dealt
would decouple
only with neutrinos
lighter
while they are non-relativistic.
R, N exp[-my/T].
In this case the bound
Their density
m, > Mzf2
neutrino
non-standard
apply to neutrinos
only because
range mentioned
Unstable
neutrinos
of the Universe
neutrino,
they have a non-zero
mass).
with m < 1 MeV whose relativistic until the present
The bound a continuous
the energy density
in Eq. contour
1 MeV
by a Boltzmann
factor
for m, 2 few GeV [8]. Only
its existence
unless
R, -K 10e2.
(i.e. to neutrinos
that are
masses could be in the forbidden beyond the standard
or decays. decay products
dominate
the energy
must have a lifetime
of the decay products,
(4) and the corresponding line. If the Universe
2
excludes
interactions
Neutrino
that allow for faster annihilations
m,
they are higher for the latter
would only have a small density
with only standard
with
Q,h2 is satisfied
heavy, but the LEP bound
r 5 (92eV/m,)2(R,h2)2t, to ensure that
is thus reduced
above, namely 30eV 5 m, 5 few GeV, if they have interactions
model of weak interactions
density
could be that
N 45 GeV. These heavy neutrinos
These constraints
1 MeV. Neutrinos
in the energy density
[9]. The mass bounds differ for a Dirac or a Majorana a fourth-generation
than
bound
is assumed
,
(4)
R nr, is not too large, i.e. 00~ 5 R, [lo].
for heavier
masses are shown in Fig.
to become dominated
by matter
1.a with
at some time
after the neutrino
decay, the bound on the lifetime is more restrictive
Fig. 1.a). In fact, the massive
neutrinos
respect
to a matter
structure
formation
of radiation ordinary
density
domination
matter
component
arguments.
that decreases
of the decay products,
replaces
contour
of effective
weakly interacting
extra
particles
SN1987A,
include
limits (distortions
photons
neutrino
fluctuations should
t,
is suppressed
with
required
finish at most when structure
more stringent
by
in the period
5 10d5t,, when atoms become
a bound
contour
Fig.
species
or e+e-
by a factor
1.a shows the nucleosynthesis
6N, < 1, see Sect.
the cosmological
lifetimes
by bounds
pairs.
4).
in
stable. of lob5
bounds
area is excluded
bound
Only decays into
on stable
neutrinos,
coming from the CMBR and on unstable
The grey area in Fig.
of the CMBR and other photon
flux from the supernova
epoch
This bound is shown for m < 1 MeV in Fig. 1.a
for the relevant
figure 5.6 of Ref [S], while the hatched with the observed
neutrino
the present
as can be seen in Fig. 1.b. This figure shows bounds
whose main decay products cosmological
[ll]).
could help evading
all visible modes have been rejected the supernova
bound
line (while the dot-dashed
on the allowed number neutral
this period
in (4), thus yielding
t, by t,
before
more slowly, as T3. This is actually
i.e. at recombination
(usually called the “galaxy formation” with a dashed
subdominant
Because the growth of density
could start forming,
This argument
have to decay earlier for the energy of their decay products
as T4, and become
to have a longer time to decrease,
than Eq. (4) (and the bound in
backgrounds),
by the non-observation
neutrinos
l.b., excluded
is reproduced of photons
by
from the
in coincidence
SN1987A (see Ref [12] for details).
24
24
21
21
18
18
15
15
b
Get
g 12 u1 E $ 9 6 3
bnd I
-4
-2
0
2
4
6
8
1012
-4-2
0
109 IWWI
Figure 1: Bounds
on unstable
modes, i.e. including
photons
neutrinos or e+e-
2
4
6
810
109 tmWl1
that apply to any decay mode (l.a, left) and to visible decay pairs as decay products
(I.b, right).
See the text for details.
4
Bounds from Nucleosynt hesis
When the temperature
of the Universe
MeV, nuclei first became than protons, mately
nn/np
stable.
became
Because
N exp[-(m,
neutrons
- m,)/T]
25% of the mass of the Universe)
protons
stayed
as H (approximately
and 7Li were produced The predictions
predicted
(with nD,qe/nH
of the reactions
that
of 4He increases
over protons
of light (i.e. relativistic
estimates.
values (which require relatively
4He assumed then, which required
either
of D-abundance
obtained
with data
with new data
important
recent developement
lead in 1996 to abandon than bounds but it cannot
by Tytler
started
a tension
between
N, = 3 + JN,
to point to values of the these new relatively
low
low upper bound on
at LEP, based on a low estimate
al.
[14] obtained
that Hata et in quasars
and a chemical pointed
al.
indicator
lines.
in the processing of the primordial
had been that D is easily destroyed
composition
to, has been largely
Ly o absorption
of uncertainties
on D+3He as a better
on D alone (the argument
depends
lower values of r] with N, = 3, or N, < 3. In fact, in 1995
has been the recognition
the bound
end up in 4He
species,
large values of 17)and the relatively
on the solar wind and meteorites
et
to a larger value
all neutrons
during NS) neutrino
This created
code. Since then, the low range of D-abundance
confirmed
The reason
p, during NS, which is expressed
[13] claimed a bound of N, < 2.6, less than the 3 observed
al.
species.
of the 4He-abundance
with NY). In 1995, measurements
D-abundance
evolution
of neutrino
the prediction
number
lower than previous
While the
with r~ (as the rate
and, consequently,
in terms of the equivalent
D-abundance
that there is a range
sharply
of 4He (since as a first approximation
of 4He increases
during
leads to an earlier freeze out of the weak
to the total energy density of the Universe,
Hata et
ratio q E na/nT
for all the light elements.
on any extra contribution
(so, the abundance
3He
N O(lO-lo).
with the number
to fi,
of D,
to 77).
rate of the Universe
rate is proportional
The remaining
and trace amounts
with 9, those of D and 3He decrease
the ratio of neutrinos
Since the expansion
of protons.
are vastly different,
are obtained
of 4He also increases
in the expansion determine
there were less neutrons
ended up into 3He (approxi-
with an equal number
that the abundances abundances
of this ratio and to overproduction nuclei).
together
N 0(10m4) and (n,r,i/nu)
considering
abundance
an increase
interactions
< 1. Most of the neutrons
that burn them into 4He are proportional
The predicted is that
are heavier than protons
on the value of the baryon-to-photon
for which realistic
abundance
energy of nuclei, at T N 0.1
75% of the mass of the Universe),
of NS depend
NS, and it is remarkable, of q N O(lO-lo)
smaller than the binding
Another
of 3He which D-abundance
in stars, mostly into 3He,
be produced).
A range in r] is translated With T = 2.73”K one obtains due to observational
uncertainties
into a range of R&,
since
71 =
(nn/nT)
= (nnp,/mp)/(2~(3)T3/~*).
17= 272.2 x 10-r’ flBh*. Bounds on 77have changed and different treatment
of errors.
over time, mainly
The ranges of the “Chicago group”
have changed from 0.015 5 Rnh* < 0.026 (and NV < 3.4) in 1980’s, to 0.010 _< f&h* 5 0.015 (and NV < 3.3) in 1991, 0.009 5 S2Bh2_< 0.022 (and NV < 3.4) in 1995 and 0.018 s Reh* 5.0.026 (and N, c 3.7) in 1997 (using now mostly the lower D-abundance
of Tytler et al. and a relatively large
‘He upper bound, YH, < 0.250; see (151 and references therein).
The “Ohio Group” gave the bounds
0.016 5 RBh* 5 0.022 (with N, < 2.6) in 1995, in the paper mentioned above and made an exhaustive analysis of the bounds in 1996 claiming a bound on NV < 3, if the upper bound on 4He is relatively low (YHc< 0.243) [13]. During 1996 the situation WBSmade even less clear by the existence of another using the same technique of Tytler et al. claiming much
measurement of the primordial D-abundance larger values. Several groups reanalyzed
then the NS bounds pointing to uncertainties
At present, the systematic errors on the upper bound of ‘He abundance consequently the bound on aN, is uncertain.
in the data.
are being m-evaluated and,
On the contrary, the prospects for the determination
of
Ra through NS have never been better, since the quasar Ly a absorption lines technique could soon determine Ga with a 20% precision (15). The possibility of a bound NV < 3 cannot be excluded at present. This would require at least one of the known neutrinos to contribute to p less than a relativistic neutrino during NS. For example, a heavy neutrino could decay before NS, i.e. with a lifetime < 10 set (161. One should also keep in mind that non-standard
NS scenarios, such as inhomogeneous NS or NS in the presence of very large
lepton asymmetries, are still viable at present and would lead to different results. For example in both of these scenarios Rn could be up to ten times larger [17].
5
Bounds from Structure Formation
The Universe looks lumpy at scales X N 100 Mpc, we see galaxies, clusters, superclusters,
voids,
walls. But it was very smooth at the surface of last scattering of the CMBR (i.e. at electromagnetic decoupling, when ions and electrons first formed atoms) and later. Inhomogeneities as anisotropies
in the CMBR,
so the density
contrast
6p/p E (p(r) - p)/p
density) cannot be much larger than &T/T c* 10m5. So inhomogeneities grow through the Jeans (or gravitational)
instability:
gravitation
regions and to further increase the density of overdense regions.
have been seen
(where p is the average
in density start small and
tends to further empty underdense One can follow analytically
the
evolution due to gravity of the density contrast in the linear regime, where 6p/p < 1. In a static fluid the rate of growth of Q/p
is exponential,
but in the Universe (an expanding fluid) it slows down into
either a power law, ~5p/p - a(t), in a matter dominated
Universe, or it stops, 6p/p N constant,
in a
radiation or a curvature dominated
Universe (a matter dominated open Universe becomes curvature
dominated for a(t) > 0,/(1
Here a(t) is the scale factor of the Universe, which accounts for
- 0,)).
the Hubble expansion linear dimensions
of the Universe.
Perturbations
where A,,,, are linear dimensions
X = a(t)&,,
(those that expand the Universe).
of the linear dimensions
with the Hubble flow and, therefore,
have different
measured
in comoving
physical
coordinates
do not change in time due to the expansion A,,,, are the present
With the usual choice of a = 1 at present,
of
actual linear dimensions.
Since the horizon ct grows linearly with t, while in a matter
(or radiation)
dominated
with a < 1, the horizon increases
coordinates,
encompassing
more material
of size X “enters” into the horizon,
we could better
with time even in comoving
ss time goes. When X = ct we say the perturbation say the perturbation and it happens
is first encompassed
at different
Independently spectrum
times for different
of the origin of these
at horizon-crossing,
(~~lP)hLX = constant, spectrum
are formed
interactions
which determines
of how fluctuations
particles
t,d each of the particles a distance expand
Thus,
At the moment
negligible.
Thus the smallest
Light neutrinos
(HDM) (such as light neutrinos
10”Mo
which structures The following
for the first time a
the dark halo), where M.3 turns out to be
and the volume occupied
become
by the horizon
T N lkec’.
In this case, at
by the particles
is erased.
structure
that
of m 2 30eV) superclusters
the fragmentation
of the larger structure.
there is not enough
time to form galaxies.
of galaxies Simulations
Thus HDM is rejected
of mass m N keV, that are just becoming
non-relativistic
becomes
of relativistic
particles
contains
to a supercluster.
by free
their motion
are becoming
can survive
will still
This is damping
non-relativistic,
when the particles
which corresponds
DM galaxies form first, but barely.
(including
at that moment
a “pro-
of these three types.
that can survive the “free streaming”
form later, through
even of the size of the smallest
the differences
later the fluctuation
are hot DM and the smallest
MJ-. = 3 x 1015Mo/(m,/30eV)2,
This feature,
types of DM: hot, warm and cold.
when the DM particles
and grow, is that one encompassed
with
of the size of the galaxy has moved from its original position
a moment
structure
namely
have shown the
at &,I, i.e. their mass is m < 1keV.
ct equal to the size of the fluctuation
streaming.
Particles
are formed first.
of the Universe
are still relativistic
their
and generate
at which the growing horizon encompasses
in a fluctuation
with the horizon.
COBE observations
to specify
or very close to it.
of the size of a typical galaxy that contains,
Hot DM particles
at horizon-crossing,
act upon the inhomogeneities
of three
is the mass of the Sun. The temperature
“horizon-crossing”
it is convenient
spectrum
evolve helps understanding
Let us define as t,d the moment perturbation
fluctuations,
spectrum.
which structures
first, leads to the distinction
naive picture
primordial
is in fact scale invariant physical
is called
a N tQ
linear scales X, larger scales cross later.
is called a Harrison-Zel’dovich
After horizon-crossing,
This moment
A scale invariant
(6p/p)k,.
at horizon-crossing
cessed” spectrum,
by the horizon.
Universe
Therefore,
non-relativistic. the Jeans
with hot DM
form first and galaxies have shown, however,
as the dominant
mass,
must that
DM component.
at t,d, are warm DM. With warm
Cold DM (CDM) is such that perturbations
smaller than a galaxy,
dwarf galaxy (106Mo) are not erased and can grow.
Simulations uprocessed”
have shown
spectrum
the highest
perturbations
spectrum
(2)). Perturbations dominated.
crossing,
distribution
entered
Perturbations
when the Universe
is matter
dominated,
at larger scales entered (the location
an overall normalization
-
by which only
There is only one feature
of matter-radiation at t < t,,
is radiation
so they started
equality,
now provided
perturbations
dominated, today,
if they all started entered
growing with the
into the horizon at Consequently, is smaller
at
free parameter
is
by the CMBR anisotropy
at these large scales entered of (Q/p)
(see Eq.
is radiation
so they all start
growing immediately.
a measurement
&
when the Universe
with X > A,, instead,
(so they did not grow much), thus providing
(6p/p)hor (for more details,
“biasing”
of the change slope) is fixed, the only remaining
COBE at large scales, B > 20”. Density very recently
N R = 1, A =
later, had less time to grow and their amplitude
of the CDM spectrum,
the
(ss can be seen in Fig.2), a change of slope at the
have the same amplitude
crossing.
because
the observations
namely Rcn~+Rs
and a scale independent
into the horizon
at horizon
Once X,
assumptions,
end up forming galaxies.
grow while the Universe
and they roughly
form of matter,
CDM models reproduces
to the horizon at the moment
with X < &
at t = f
perturbations present.
at horizon
of CDM perturbations
They cannot
same amplitude
in standard
CDM models, make the simplest
present scale that corresponds
t > t,,
generated
peaks in the CDM density
in the processed
together
CDM must be the most abundant
of perturbations
within 10%. Standard 0, scale invariant
that
measured
by
into the horizon
at horizon crossing,
see e.g. Ref [8]).
CDY
n=l
----- HDMn=l --CDU n=0.8 ............. YDY nr 1
Figure 2: Comparison
of the measured
power spectrum
of several DM models
(notice that k- X-l).
of density
perturbations
and the predictions
While both the shape
and normalizationof
the spectrum
not fit the observations
(191. In Fig. 2 (201 the power spectrum
standard
(solid line) normalized
CDM models
(the points
with vertical
of standard clusters
CDM models
and smaller.
short-dashed
solutions
consists
primordial
with n=0.8)
limits.
models
given in Eq.
&,
to that
older Universe),
of the standard
This is called
R, < 0.2.
at horizon
crossing,
one that slightly
CDM (TCDM)
require
works well.
r, would provide
favors larger
spectrum)
and some “tilt” arguments
In fact another
larger scales of the only feature
since it effectively (X > X,)
of the standard
changes amounts
defines
[26]), for example
also does
give the bound
family of solutions
(see Fig. 2), is enough to increasing
the power
to those smaller
or 2): increase g. (namely increase the radiation
CDM models
[24]
(with the
1): lower h [25] (what
content
that provides
= 0.18, Rs = 0.02, R,
than
a larger
In order to lower the value of I with
with data.
CDM model one needs to either,
with h = 1, Rcn~
line
the “shape parameter”
r N 0.25 f 0.05, while standard
constant
within
in the CDM spectrum,
with respect
X, G (lOh_‘&,,)I’-’
agreement
that
[23]).
(2), the scale where the slope in the spectrum
The data
models
[22]. See the long-dashed
formation
may also provide a solution.
with observations,
crossing,
Harrison-Zel’dovich
these structure
(see for example
A = 0 or in a Universe with a cosmological
above.
in what are called mixed
CDM models work as well, for example
at horizon
“tilted”
Hence,
(2) the relation
the
at small scales, one of
line in Fig. 2. In particular,
lower R, (i.e. take 51, < l), so that we either live in an open Universe if
(see the
to change
choices of h = 0.5, R, = 1, g. = 3.36) has I = 0.05. In fact, as we have explained,
thus a smaller
respect
neutrinos,
of fluctuations
at scales larger than the break point
I E R&7./3.36)-‘I*. standard
spectrum
that a shift towards
Using Eq.
for the data
_ N 5 eV, and the rest of R, in CDM plus some baryons,
of the flat, scale invariant,
neutrinos
good agreement
it (X < X,).
the scales of galaxy
possibilities
HDM tends to erase structure
of fluctuations
massive neutrinos
by realizing
of the spectrum
X-l),
in Fig. 2. A mixed model with both some neutrinos
Heavy unstable
to provide
variations
scales (instead
R, 5 0.2 for light massive
i.e. &,
to Cim,
spectrum
the COBE observational
is obtained
Because
h = 0 and a scale invariant
work, and in these
2 (solid line) the spectrum
are several
DM (HCDM) models [21]. See the dotted
However many other possible
(labelled
is lixed, there
to the CDM a bit of HDM, namely
with R, = 0.2, what corresponds
scales over smaller
As can be seen in Fig.
given by COBE
to agree with observations.
DM (MDM) or hot-cold
by
above.
spectrum
in adding
predicted
(the box on the 1.h.s.) and other data
Also in Fig. 2 one can see the failure of HDM to account
Once the normalization
of a “tilted”
perturbations
has too much power on small scales (large k-
curve) mentioned
with R = 1,
of density
by COBE data
error bars) are shown.
are almost right, they do
so obtained
of the Universe
implies
at teq), or 3):
(open CDM models, 0,
an
OCDM)
= 1 - 0, (ACDM models
= 0.80, or 4): a combination
of all three
A way of obtaining
the large amount of radiation needed for the second possibility is through a
heavy neutrino decaying into relativistic particles, i.e. radiation, with the right combination
of mass
and lifetime, in so-called rCDM models (27, 281. A massive neutrino matter dominates the energy density of the Universe as soon ss it becomes non-relativistic, and pv = n,m,,
i.e. as soon as m, 2 T (since n, N n_,
prd cz n,T), thus their decay products radiation-dominate
the Universe at decay. For
m, < 1 MeV the right mass-lifetime combination lie on a narrow strip around the previously mentioned “galaxy formation” bound [ll], shown in Fig. 1-a. Near this bound, at the boundary between being irrelevant and harmful, unstable neutrinos could help in the formation of structure in the Universe [27]. A heavier neutrino,
of m, N 1 - 10 MeV, necessarily v+, decaying at or just before nucleosynthesis,
r = 0.1 - 160 set, would also provide a solution [28]. The r+ decay modes involved here should all be into neutral particles, vr + 3 v’s or v, -+ v+, with 4 a Majoron (a zero mass Goldstone boson) for example. All visible modes, i.e. producing electrons or photons, are forbidden in the necessary range, and shown in Fig. 1.b.
8
6
200
Figure 3: CMBR anisotropy from top to bottom: KDM
uncertainty
600
800
1000
power spectra predicted by four models (lines) of structure
formation,
CDM with low h (h= 0.35), TCDM (with IV,= 12), ACDM (with h=0.65),
with D,=O.2 and ‘tilted”CDM
anticipated
400
(other parameters as in standard CDM). The band shows the
from MAP (from Ref. [23])
All these
modified
very different
CDM models
patterns
ture autocorrelation temperature
of acoustic function,
fluctuations
multiplying
ple moments
with
that the Universe
is smooth
pick up additional
satellite
observations
dark matter
10%. The detection
experiments,
models.
the Fig. 3 [23] together on the density
because
last scattering
surface,
different
regions
0 1 1” it indicates to scales within the
Thompson
scattering
of multiple
moments
N 200(R)-‘,
the position
of the first peak
so far obtained
with the COBE
experiment
models.
to
and where the photons
(mainly
models,
in Europe.
angle 0 is
at angles B 2 1”
but the next generation with sub-degree
of
resolution,
They will give results as shown in The height of the peaks depends
and relative height of the peaks between models of galaxy formation.
larger values of P correspond so the information
matter
between
the multi-
corresponds
at the scale of the horizon e+
of five different
fin and the location
several 1000 will allow to discriminate
perturbations
peaks in the spectrum
new satellite
(ex COBRAS/SAMBA)
with the predictions
in baryons,
ct,,
of the anisotropies
Thus, in the future,
does not discriminate
defines
to causally disconnected
of the CMBR anisotropies
will do it. There are two approved
MAP in the U.S. and PLANCK
are relevant,
(or acoustic)
of the first peak should happen
R within
and balloon
from moving
by
by two angles),
at a certain
and at larger angles,
can move due to density
due to the scattering
as shown in Fig. 3 for different will determine
polynomials
size at recombination,
at recombination,
The tempera-
is computed
harmonics
so that the anisotropy
Thus the smallness
predict
from the first one by an angle 0
sky. Larger angles correspond
This gives origin to Doppler
P(P + 1)Cl. The position
i= = 2.726”K,
on very large scales. We see that P 1 200 correspond
where matter
energy
temperature
The horizon
they
where dT/T = (T - p)/!f’ are
+ 8)/T)),,
of C(6) in Legendre
C,(2P + l)C&(cosB),
was smooth
however
power spectrum.
(a + 0) separated
of the CMBR photons.
at decoupling,
on electrons).
position
in the present
the Universe
data,
Q in the sky (Q is given in spherical
I,with e N (200”/8).
at the time of emission that
to the average
The expansion
Ct, C(0) = (4z)-’
an angle 0~ N 1°(R)-1/2
horizon
respect
over all positions.
@T/T),N dm
anisotropy
C(B) = ((6T(cr)/T)(6T(a
6T/T at another
it with
and averaging
indicates
peaks in the CMBR
&“/T at some position
measuring
seem to be able to fit present
e N 200 and
Only the C, for l? < 0( 103)
to scales 8 < 8” that are inside the thickness
at those small scales is smeared
of the
out.
Acknowledgments I thank the organizers U.S. Department
of this workshop
of Energy
for their invitation.
under Grant DEFG03-91ER
This work was supported
in part by the
40662 TaskC.
References [l] W. L. Freedman, Astrophysics,
ASTROPH-9706072,
Chicago,
Invited
IL, 15-20 Dee 1996.
talk at 18th Texas Symposium
on Relativistic
[2] (COBE Coil.) J.C. Mather et al. Ap. J. 420, 439 (1994). [3] (COBE Coll.) G.F. Smoot et al. Ap. J. Mt.
396, Ll (1992); (COBE Coil.) C. L. Bennet et al.
Ap. J. 436, 423 (1994). [4] M. White, J. Navarro, A. Evrard and C. Frenk, Nature 366, 429 (1993); G. Steigman and J. Felten, OSU-TA-24-94 sstro_ph/9502029. [5] J. Cowan, F. Thielemann and J. Truran, Ann. Rev. As&on. Astroph. 29, 447(1991). [6] D. Winget et al. Ap. J. Leti. 315, L77 (1987); I. Iben and G. Laughlin, Ap. J. 341,430 [7] A. Renzini Proc. 16th Tezas Symp. on Relat Astroph. and 3rd Symp. on Particles,
(1989).
Stings
and
Cosmol. eds. C. Akerlof and M. Srednicki, N.Y. Act. of Sci., N.Y. (1992); X. Shi Ap. J. (1995). [8] E. Kolb and M. Turner, The Early Universe, Addison-Wesley, 1990, and references therein. [9] B. W. Lee and S. Weinberg, Phys. Rev. Mt.
39, 165 (1977) (and others, see e.g. [8]).
[lo] 0. Dicus, E. Kolb and V. Teplitz, Phys. Rev. L&t. 39, 168 (1977). [ll] P. Hut and S. White, ZVature310,637 (1984); G. Steigman and M. mrner, Nucl. Phys. B 253, 375 (1985). [12] G. Gelmini and E. Roulet, Rept. Prog. Phys. [13] N. Hata et al., Phys.Reu.Lett. Phys.Reu. D55,
58, 1207-1266 (1995)
75, 3977 (1995); N. Hata, G. Steigman, S. Bludman, P. Langacker
540 (1997).
[14] D. Tytler, X.M. Fan and S. Burles, Nature 381, 207 (1996). [15] D. Schramm and M. Turner, ASTRO-PH/9706069. [16] S. Dodelson, G. Gyuk and M. Turner, Phys. Rev.
D49, 5068 (1994); M. Kawasaki et al., Nucl.
Phys. B419, 105 (1994). [17] H. Kang and G. Steigman, Nucl. Phys. B372 (1992) 494. [18] J.P. Ostriker, Ann. Rev. Astron. Repts.
31, 689 (1993); A.D. Liddle and D. Lyth, Phys.
Astrophys.
31, 689 (1993); A.D. Liddle and D. Lyth, Phys.
231, 1 (1993).
[19] J.P. Ostriker, Ann. Rev. Astron. Repts.
Astrophys.
231, 1 (1993).
[20] W. Kolb, P rot. &nd
Scottish
School if Physics,
Univ. of St. Andrews, Scottland (1993) and
Fermilab-Conf-94/05&A. [21] Q. Shafi and F. Stecker, Phys.. Rev. Mt.
53, 1292 (1984); C.P. Ma and E. Bert&ringer, Ap. J..
429, 22 (1994) and Ap. J. 434, L5 (1994), J. Primak et al., Phys.. Rev. Mt.
74, 2160 (1995).
[22] R. Cen et al. Ap. J. 399, Lll (1992), R. Davis et a!.Phys. Rev. l&t. 69, 1856 (1992); F. Lucchin, S. Mattarese and S. Mollerach, Ap. J. 401, L49 (1992). (231 S. Dodelaon, E. Gates and M Turner Science 274 69 (1996) [24] G.E. Efstathiou, Pnx. of the “36th Scottish
Univs Summer Sch. in Phys”, “Phys. of the Early
Universe”, ed. J. Peacock, A. Heavens and A. Davies Adam-Hilger, N.Y.(1990) p. 361; J. Peacock and S. Dodde.
Monthly Not. of the Royal Ast. Sot. 267, 1020 (1994).
(251 J. Bartlett et al., Science 267, 980 (1995). [26] MS. Turner, G. Steigman and L. Krause, Phys. Rev. Mt.
52, 2090 (1984); L. Krause and M.
Turner, Gen. Rel. Gmu. 27 1137 (1995). [27] J. Bardeen, J. Bond and C. Efstathiou, Ap. J. 321, 28 (1997); J. Bond and G. Efstathiou, Phys.
Mt.
B265, 245 (1991); M. White, G. Gelmini and J. Silk, Phys. Rev. D51, 2669 (1995).
(281 S. Dodelson, G. Gyuk and M.S. Turner, Phys. Rev. Mt.
72, 3578 (1994).