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The formation of cosmic structure with a 17 keV neutrino
Physics Letters B 265 ( 1991 ) 245-250 North-Holland
PHYSICS LETTERS B
The formation of cosmic structure with a 17 keV neutrino J.R. B o n d
a
a n d G. E f s t a t h i o u b
CIAR Cosmology Program, Canadian Institutefor TheoreticalAstrophysics, Toronto, Canada b Departmentof Physics, Oxford OX1 3NP, UK
Received 20 May 1991
Adding a 17 keV neutrino with a lifetime of ~ 1 yr can explain the puzzling large-scale structure observed in the galaxy distribution which is difficult to understand in the "standard" cold dark matter model with the adiabatic scale invariant density fluctuations predicted by inflation. Decay times > 10 yr are incompatible with microwave background anisotropy limits and galaxy clustering observations.
There have been several experiments recently [ 14 ] apparently confirming Simpson's report [ 5 ] of a 17 keV neutrino mass eigenstate. The significance of these experiments can hardly be overstated, for a 17 keV neutrino would have important implications for particle physics and cosmology. Within the context o f the standard hot big bang model, a 17 keV neutrino must decay on a timescale zd< 10 6 yr if the decay products are to contribute less than the critical density at the present epoch. Much stronger constraints on the lifetime, of order ~ 106 s, can be derived from limits on the IR and optical background radiations and microwave background distortions if the decays have a significant branching ratio to photons (see refs. [6,7] ). Laboratory bounds from double beta decay suggest that the 17 keV mass must be almost entirely Dirac (e.g., ref. [ 8] ). This requires the existence of righthanded chirality states leading to astrophysical problems associated with the length o f the neutrino bursts from SN 1987A if they are very weakly coupled [ 9 ], or with nucleosynthesis bounds if they participate in new interactions (i.e., beyond those of standard electroweak theory) that keeps them equilibrated with the left handed states below the q u a r k / h a d r o n transition [ 10 ]. The situation could be even worse if there are light neutrino singlet states in addition to the three SU (2) doublets allowed by LEP. Despite these problems, the cosmological implications of a 17 keV neutrino deserve careful investiga-
tion. In this letter, we show that a 17 keV neutrino could have played an important role in the formation o f cosmic structure and could explain recent observations of large-scale structure in the galaxy distribution [ 1 1,12 ] if it has a lifetime o f ~ 1 yr. Until recently, the C D M theory was thought to provide an excellent description of cosmic structure [ 13 ]. In the standard version of the C D M theory, fluctuations in the early universe are assumed to be gaussian with a scale-invariant spectrum, as seems most natural if the perturbations were generated during an inflationary phase [ 14,15 ]. The universe is assumed to have a mean density £2= 1 made up almost entirely of C D M with a small contribution from baryons, photons and three light ( < < e V ) neutrino types (where £2 is the mean density divided by the critical density of the Einstein-de Sitter model). The evolving fluctuation spectrum is characterized by a single physical length scale associated with the Hubble radius when the universe passes from relativistic (er) matter domination to nonrelativistic (nr) matter domination, which we express in terms o f a comoving wavenumber scale keq: k~qt..~5(£2nrh2)-~O~/2Mpc,
0 - / T J 1.68/~y,
(1)
where h is the Hubble constant Ho in units of 100 km s - ~ M p c - ~. With the standard choice of parameters (£2nr~ 1, h ~ 0 . 5 , 0 = 1 ), this length scale is relatively small and so it proves difficult to explain observations of structure on scales > 1Oh - ~ Mpc. Ways
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out of this difficulty include non-scale invariant initial spectra [ 15 ], or the introduction of a cosmological constant to allow £2.r ~ 0.1-0.2 in a spatially flat universe [ 16 ]. However, neither o f these proposals is particularly attractive since each requires fine tuning which is not understood at present. A decaying neutrino with a mass in the keV range can also lead to an increase in kyq ~ via an increase in 0 as first pointed out in ref. [ 17 ]. In this letter, we investigate these models further by fixing the mass of the decaying neutrino to be 17 keV (denoted by m ~7 ) and treating the lifetime 3o as a free parameter. For simplicity, we assume that the decaying neutrinos are non-relativistic (and hence behave like cold dark matter) with mean density/~,ra and decay into relativistic particles with mean d e n s i t y / ~ . The rest of the universe is assumed to consist of photons (T), primordial relativistic neutrinos (v) and stable cold particles ( X ) . We also assume that the baryon abundance is small (E2B<< £2x) SO that their dynamical role can be neglected. The mean densities vary as
finrd =~]ffX e x p ( - - t / r d ) , e = 168mlT(f2xh 2) - l ,
(2a)
~x =ap~, fi~=afi~
(2b)
1 d (a4fierd) dt
~/4
--
/gnrd , rd
(2C)
where a = ao/( 1 + z) is the cosmological scale factor, z is the redshift and a o = 4 × 104f2xh 2 is the present value of a. For two massless neutrino types ¢x= 0.45 and for three neutrino types ¢x= 0.68, If the decay time exceeds 2 × 106m i-72 S, the universe will be matter dominated when the neutrinos decay. In this case, the universe will go through two periods of matter domination, the first occuring when the energy density in 17 keV neutrinos dominates over relativistic matter and the second when the C D M density dominates over the er decay products. The fluctuation spectrum is therefore characterized by two length scales, associated with the Hubble radii at these two epochs, kyqll ~ 28m i-7j k p c ,
(3a)
k;-q~~25mZ/3(zd/1 yr)'/3h -t Mpc.
(3b)
The first of these scales, kZq~, follows from eq. ( 1 ) with 0 = 1.45/1.68 and I 2 , ~ 168h-2 (the contribu246
15 August 1991
tion of the 17 keV neutrinos if t << rd). It corresponds to a mass scale of ~ 108m i-73~2xMo, i.e., about the mass of small galaxy. Objects on this and smaller mass-scales would collapse at high redshifts (z ~ 100), in contrast to the standard C D M model where little structure forms above a redshift of ~ 20 [ 18 ]. The second scale, keq2, - follows from eq. ( 1 ) with Onr ~ 1 and
O~ 1.45/1.68+ 5.5(mlTzd/yr) 2/3 , the latter deriving from the energy density in relativistic decay products, ~ceao/( 1+ Zd). The decay redshift (when the Hubble time equals the decay time) is 1 +Zd -~ 105"9m 1-71/3( r J y r ) - 2 / 3 . The scale k~q~ is larger than the scale kyq ~ of the standard C D M model (eq. ( 1 ) ) and can be comparable to the scales on which large-scale structure has been observed ifrd ~ 1 yr. Indeed, if the initial fluctuations were scale-invariant then a 17 keV neutrino must decay on a timescale of < 1 yr to avoid producing too much larger scale structure and excess fluctuations in the microwave background radiation. We now elaborate on these points using detailed calculations of the evolution of fluctuations with 17 keV neutrinos. To evolve the fluctuations in the different components of matter present, we utilize linear perturbation theory and the synchronous gauge [ 18 ]. We define a characteristic timescale r * = [ (8nG/3)pva 4] -J/2 = 1.22× 103(.Qxh 2) -2 y r . Dots denote derivatives with respect to q = r / r * , where the conformal time is r=fdt/a. The trace h of the metric perturbation hij then obeys the equation
d (adh~
3(2A~o+20~Avo+2P-~rdAerdO
+ a g x + ct/~nro_ ~nrd], / PX
(4)
where the Ao'S and ~'s are the fractional energy density perturbations in the various components. The C D M and decaying neutrinos are treated as cold eollisionless particles and we use the perfect fluid model
Volume265, number 3,4
PHYSICS LETTERS B
(valid in the tight coupling limit) to evolve the photons
i000
,~
fix=½/~,
J.rd=Jx,
........
i
........
,
........
,
........
i
........
,, , > ~ . . ~ , .
LSSV
I00
//..~,]I~
(5a) -~
Z],o = ]/~-ikr%lr,,
15 August 1991
z~r, = - ]ikr*Aro,
lo ~
w,,(o>
..~./..'" "._M.S---~
(5b)
where k is the comoving wave number of the plane wave perturbation and/~rAv ~ is the photon flux. We solve the collisionless Boltzmann equations for the relativistic decay products and the massless neutrinos. The relative perturbation Aerato the energy density for these components is defined by
~
o l -- COBE - .
0.0011
'
" /I klllV
~ Z"
0.:-0001 0.001
HIll
0.01
.....
~
.......
]
0.1 1 k hMpc -1
.......
]
10
......
li
]
100
Perd( k, ~', q ) =Perd(~') [1 "JI-Aerd(k, r, 0 ) ] ,
where ~ is the direction of the decay products. The perturbation A~d evolves according to ~l~d +ikr*/tA~rd- 2h/tz + 2 (h-/t33)' ½(3/~ 2 - 1 ) =afi.~, r* (6nrd __Z~erd)
,
fferd "[d
(6)
where/z=#-/~, and the three-axis is chosen to be in the direction of k. The Boltzmann equation for the massless neutrinos is identical to (6) except that the right-hand side is zero. These equations are solved by expanding A in Legendre polynomials, a = Z (2~+ 1 )a~P~(u),
retaining as many terms as are necessary for an accurate solution (~ >> kz). The h-/~33 term required for eq. (6) is related to the ~ = 1 moments of the A's through the constraint equation /~_//33=
i 6 ( fferd ) kr--~ 5 Av~+azl~j+ _ Ae~d, . Pv
(7)
We use initial conditions appropriate to a growing adiabatic mode in the radiation era and we assume a spatially fiat background universe with f2= 1 with two massless neutrino types ( a = 0.45 ). Fig. 1 shows the power spectra of the linear CDM density fluctuations, dcr2/d In k = k 3 p x ( k ) /21r 2 , where P× (k) = I~x (k, ~o) 12, for models with h = ~ and for lifetimes rd ranging from 104 yr to zero (i.e., the standard CDM model). The spectra are plotted in this way to provide a measure of the contribution of a band around the given wavenumber to the overall
Fig. 1. Power spectra for CDM models with decaying 17 keV neutrinos are plotted against comoving wavenumber k (referred to current length units). There is progressively more large scale power as the v lifetime increases through the values rd = 0, 1, 10, 102, 103, 104 yr shown in the figure. The lines under the labels indicate approximate regions in k space that various probes of structure are sensitive to, such as: microwave background anisotropy experiments of large angle (COBE, [19] ) and of intermediate angle (e.g., SPole is the 1 o experiment of ref. [20] ); clustering o b s e r v a t i o n s for galaxies in the APM Galaxy Survey (w~ [ 11 ] ) and the QDOT redshift survey [ 11 ] and for clusters ((~c [21 ]); and large scale streaming velocities (LSSV [21]). The normalization is a s = 1. The high amplitude at small scales implies that early reionization of the universe is more likely in these models than in CDM. However our calculations are not valid for wavenumbers above the collisionless damping scale, corresponding to the gaussian filter R~ ~~ 133 (h - ~Mpc) - ~ shown by the heavy vertical line.
RMS fluctuations; the ordinate roughly gives 1 WZnl(k), where z,~(k) is the redshift at which the RMS fluctuations in the band become nonlinear. The power spectra of fig. 1 show the two features corresponding to the scales k~l and kzq~ ofeq. (3) and illustrate that decay times of > 1 yr can produce more power at large scales in comparison to the standard CDM model. We normalize the spectra by requiring that the RMS fluctuations in the mass distribution within spheres of radius 8 h - l Mpc, as, are given by a8 = b 8 ~, where b8 is the "biasing'factor. If bs= 1 galaxies are clustered like the mass distribution while if b8 > 1 galaxies are more strongly clustered than the mass; bs is thought to lie in the range 1-2 for CDM models [ 21 ]. To place thrther constraints on the range of acceptable lifetimes, we have used linear theory and the power spectra of fig. 1 to compute the angular corre247
V o l u m e 265, n u m b e r 3,4
PHYSICS LETTERS B
lation function of galaxies (see refs. [ l 1,22 ] for further details). These predictions are compared with the observational results from the APM Galaxy Survey [11] in fig. 2. This shows that lifetimes in the range ~ 0.5-10 yr can explain the excess large-scale power, and that longer lifetimes produce unacceptably large fluctuations. Table 1 gives further details of these models. The first column lists the lifetime in years and the second column lists the mean-square mass fluctuations in spheres of radius 0.5 h - l Mpc at the present epoch if one extrapolates their growth by linear theory, (AM/M)o.5. This gives a rough estimate of the redshift zgf at which large galaxies would form, ( 1 + zgf) -'g
.
'
.
.
.
.
.
.
.
i
.
.
.
.
.
.
.
.
i
0.1
105
0.01
10 - 3
0.1
1
10
0 (degrees) Fig. 2. The m o d e l s of fig. I are c o m p a r e d with the angular correlation functions d e t e r m i n e d from the A P M G a l a x y Survey [ l 1 ] scaled to the depth of the k i c k catalogue at which 1 ° corresponds to a physical scale of ~ 5 h - ~Mpc. No nonlinear corrections were a p p l i e d to the theoretical power spectra, but for angular scales above ~ 1 ° and for biasing factors b s > 1, the linear a p p r o x i m a tion is accurate.
Table 1 G a l a x y f o r m a t i o n and b u l k flows. ra(yr)
0 1 5 10
248
<(~I/M)g.5)
6.2b~ 4.5b~ 4.0b~ 3.8b8
~ t ~ ~
(l+z~f)
8.7bff h 6.3b21 5.6b~ ~ 5.3bg ~
a, 25h-lMpc
40h-IMpc
437b~ 754b8 978b~ 1106b2
331bff 634b~ 857bff 986b2
~ ~ I L
~ ~ t L
15 August 1991
(AM/M)o.5. The third column provides another estimate of the redshift of galaxy formation using the "peaks" method of Bardeen et al. [ 17 ] which gives the average redshift at which peaks with a number density equal to that of bright galaxies would collapse when the matter density is filtered with a gaussian of width 0.35 h - i Mpc. A more careful analysis of star formation history would be required to improve upon these numbers, but they illustrate that there are no grave difficulties with the redshift of galaxy formation in these models provided b8 is not too large. Fig. 1 shows that small scale structure can form early in decaying particle models. In our computations, we have neglected collisionless damping of fluctuations caused by the free-streaming of the 17 keV neutrinos when they become semi-relativistic (which occurs at redshifts ~ 1 0 7 5 ) . From the collisionless damping calculations of Bond and Szalay [ 23 ], the damping can be approximated by multiplying the power spectra of fig. 1 with a gaussian, exp(-k2R~s) with Rfs~ 15 kpc. Thus, in these models the first structures collapse at redshifts z ~ 100 and have masses equal to those of dwarf galaxies. Unlike the standard cold dark matter model [ 18 ], this early structure formation could reionize the universe shortly after the normal epoch of photon-baryon decoupling at z ~ 1000, thereby reducing small angle anisotropies in the microwave background radiation. The fourth and fifth columns in table 1 list values for the three-dimensional peculiar velocities in km saveraged over spheres of radius 25 h ~Mpc and 40 h - I Mpc, respectively, This illustrates how the extra large-scale power in the decaying neutrino models increase bulk flow speeds. Decay times of > l0 yr are probably incompatible with the observed velocity flows in the local universe [21 ] unless one invokes a large biasing factor. Some of the tightest constraints on these models come from the background radiations. Even with a lifetime of ~ 1 yr, we have to worry about distortions in the microwave background since the radiation will not thermalize. The temperature of the background radiation at the decay redshift is Tya~ 200(rd/yr) -2/3 eV, if little decay energy goes into photons, lfthe neutrinos decay into photons, the decay radiation will be injected at energies of about 8.5 keV, so the most important mechanisms relevant to the reprocessing of the photon spectrum are Comp-
Volume 265, number 3,4
PHYSICS LETTERS B
ton scattering, double Compton scattering and bremsstrahlung. With current constraints from COBE [24], we conclude that no more than ~ 1% of the CMB energy can be liberated as photons if the neutrinos have a decay time of ~ 1-10 yr, using the methods in ref. [ 7 ]. Thus the branching ratio to photons needs to be strongly suppressed. Since small scale structure forms early in these models, predictions o f small scale anisotropies in the microwave background are uncertain because of the possibility that the intergalactic medium is reionized. However, anisotropies at angles > 5 ° from fluctuations in the gravitational potential (the SachsWolfe effect [25,18 ] ) are unaffected by reionization. If we assume that the universe is matter dominated from photon decoupling to the present, the meansquare temperature fluctuations for the lth multipole of the radiation field can be expressed in terms of the power spectrum for gravitational potential fluctuations, d a 2 / d In k, by AT 2 (2l+1) --T--z ~ 9
dlnk
j~(kro)
(8)
0
where jz is a spherical Bessel function. The gravitational potential spectrum is related to that for the density by dtr~/d In k = ( 3 H ~ k - 2 ) 2 dtrp2/d In k , where the comoving wavenumber k is referred to current length units. Table 2 gives computations for the microwave background anisotropies for various decay times (listed in the first column). Column 2 gives results for the quadrupole component (the R E L I C T 1 limit [26] for scale invariant spectra is 1 . 5 × 1 0 -5 at the 95% confidence level). The third column gives the RMS anisotropies expected within the COBE beam, found by summing over multipole contribuTable 2 Microwave background anisotropies. rd(yr)
bs(AT/T)~=2
bs(AT/T) 7o
0 1 5 I0 100 1000
4.31 >( 10 - 6 1 . 4 6 X 10 -5 2 . 6 2 X 10 - s 3 . 4 7 X 10 -5 9 . 4 3 X 10 - 5 2 . 4 9 X 10 - 4
1.01X 3.13X 5.49X 7.19X 1.85X 4.58X
10 -5 10 - s 10 - s 10 -5 10 - 4 10 - 4
15 August 1991
tions given by eq. (7), weighted with e x p [ - ( 1 2 / ldmr)2], where ldnar = 19.3 corresponds to the 7 ° (full width half m a x i m u m ) beam appropriate for the D M R experiment on COBE [ 19 ]. The current D M R limit is 4X 10 -5 at the 95% confidence level on such fluctuations. These limits rule out models with decay times > 10 yr (ifbs < 2 ) and are not too far above our predictions for models with rd ~ 1 yr. Particle physics models with decay times of ~ 1 yr and with a small radiative branching ratio are not easy to construct by are certainly not impossible. Holdom [27] constructed models in an extended technicolour theory in which a 17 keV neutrino can decay by techniphoton emission on a timescale of ~ 10 yr. Glashow [28] considered a scheme in which a massive neutrino decays into light neutrinos and majorons while Fukugita [29] considered familon models. In either case a judicious choice of parameters can give lifetimes of order o f years. We have shown that lifetimes of order o f years result in attractive models of structure formation which can explain recent observations of large-scale structure in the universe. However, it is also important to stress that lifetimes > 10 yr conflict with observations if the initial fluctuations are scale invariant. This lifetime limit is even stronger if other kinds of dark matter in addition to, or instead of, C D M is invoked, or if there is a nonzero cosmological constant. To accommodate a 17 keV neutrino with a longer lifetime, we require carefully tuned fluctuation spectra in the early universe to avoid over-producing perturbations on large scales. We thank Andrew Hime, Rocky Kolb, G r a h a m Ross and Subir Sarkar for interesting discussions.
References [ 1 ] J.J. Simpson and A. Hime, Phys. Rev. D 39 (1989) 1825. [2] A. Hime and J.J. Simpson, Phys. Rev. D 39 (1989) 1837. [3] A. Hime and N.A. Jelley, Phys. Lett. B 257 ( 1991 ) 441. [4] B. Sur et al., LBL preprint (1991). [ 5 ] J.J. Simpson, Phys. Rev. Lett. 54 ( 1985) 1891. [6]J.R. Bond, B.J. Carr and C.J. Hogan, Astrphys. J. 306 (1986) 428. [ 7 ] J.R. B o n d , in: The early universe, ed. W.G. Unruh (Reidel, Dordrecht 1988), [8 ] A. Hime, R.J.N. Phillips, G.G. Ross and S. Sarkar, preprint ( 1991 ). [ 9 ] R. Gandhi and A. Burrows, Phys. Lett. B 246 (1990) 149.
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[ 10] E.W. Kolb and M.S. Turner, preprint ( 1991 ); K.A. Olive, D.N. Schramm, G. Steigman and T.P. Walker, Phys. Lett. B 236 (1990) 454. [ 11 ] S.J. Maddox, G. Efstathiou, W.J. Sutherland and J. Loveday, Mon. Not. R. Astron. Soc. 242 (1990) 43P. [ 12] G. Efstathiou et al., Mort. Not. R. Astron. Soc. 247 (1990) 10P; W. Saunders et al., Nature 349 ( 1991 ) 32. [13] M. Davis, G. Efstathiou, C.S. Frenk and S.D.M. White, Astrophys. J. 292 ( 1985 ) 371. [ 14] J.M. Bardeen, P.J. Steinhardt and M.S. Turner, Phys. Rev. D 2 8 (1983) 679. [ 15 ] D.S. Salopek, J.R. Bond and J.M. Bardeen, Phys. Rev. D 40 (1990) 1753. [ 16] G. Efstathiou, W.J. Sutherland and S.J. Maddox, Nature 348 (1990) 705. [ 17] J.M. Bardeen, J.R. Bond and G. Efstathiou, Astrophys. J. 321 (1987) 28. [ 18 ] J.R. Bond and G. Efstathiou, Astrophys. J. Lett. 285 ( 1984 ) 45; J.R. Bond and G. Efstathiou, Mon. Not. R. Astron. Soc. 226 (1987) 655.
250
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[ 19] G. Smoot et al., Astrophys. J. Lett. 371 ( 1991 ) 1. [20] P. Meinhold and P. Lubin, Astrophys. J. Lett. 370 (1991) II. [21 ] See papers in: V.C. Rubin and G.V. Coyne, eds., Large-scale motions in the universe (Princeton U.P. Princeton, NJ, 1988). [22] H.M.P. Couchman and J.R. Bond, in: Large scale structure and motions in the universe, eds. M. Mezzetti et al., (Kluwer, Dordrecht, 1989) p. 335. [23] J.R. Bond and A. Szalay, Astrophys. J. 274 (1983) 433. [24] J. Mather et al., Astrophys. J. Lett. 354 (1990) 37. [ 25 ] R.K. Sachs and A.M. Wolfe, Astrophys. J. 147 ( 1967 ) 73. [26] A.A. Klypin, M.V. Sazhin, I.A. Strukov and D.P. Skulachev, Sov. Astron. Lett. 13 (1987) 104. [27] B. Holdom, Phys. Lett. B 160 (1985) 403. [28] S.L. Glashow, Phys. Lett. B 187 (1987) 367; B 256 ( 1991 ) 255. [ 29 ] M. Fukugita and T. Yanagida, Phys. Lett. B 157 ( 1985 ) 403; see also M. Fukugita and T. Yanagida, Yukawa Institute preprint YITP/K-907 ( 1991 ).
PHYSICS LETTERS B
The formation of cosmic structure with a 17 keV neutrino J.R. B o n d
a
a n d G. E f s t a t h i o u b
CIAR Cosmology Program, Canadian Institutefor TheoreticalAstrophysics, Toronto, Canada b Departmentof Physics, Oxford OX1 3NP, UK
Received 20 May 1991
Adding a 17 keV neutrino with a lifetime of ~ 1 yr can explain the puzzling large-scale structure observed in the galaxy distribution which is difficult to understand in the "standard" cold dark matter model with the adiabatic scale invariant density fluctuations predicted by inflation. Decay times > 10 yr are incompatible with microwave background anisotropy limits and galaxy clustering observations.
There have been several experiments recently [ 14 ] apparently confirming Simpson's report [ 5 ] of a 17 keV neutrino mass eigenstate. The significance of these experiments can hardly be overstated, for a 17 keV neutrino would have important implications for particle physics and cosmology. Within the context o f the standard hot big bang model, a 17 keV neutrino must decay on a timescale zd< 10 6 yr if the decay products are to contribute less than the critical density at the present epoch. Much stronger constraints on the lifetime, of order ~ 106 s, can be derived from limits on the IR and optical background radiations and microwave background distortions if the decays have a significant branching ratio to photons (see refs. [6,7] ). Laboratory bounds from double beta decay suggest that the 17 keV mass must be almost entirely Dirac (e.g., ref. [ 8] ). This requires the existence of righthanded chirality states leading to astrophysical problems associated with the length o f the neutrino bursts from SN 1987A if they are very weakly coupled [ 9 ], or with nucleosynthesis bounds if they participate in new interactions (i.e., beyond those of standard electroweak theory) that keeps them equilibrated with the left handed states below the q u a r k / h a d r o n transition [ 10 ]. The situation could be even worse if there are light neutrino singlet states in addition to the three SU (2) doublets allowed by LEP. Despite these problems, the cosmological implications of a 17 keV neutrino deserve careful investiga-
tion. In this letter, we show that a 17 keV neutrino could have played an important role in the formation o f cosmic structure and could explain recent observations of large-scale structure in the galaxy distribution [ 1 1,12 ] if it has a lifetime o f ~ 1 yr. Until recently, the C D M theory was thought to provide an excellent description of cosmic structure [ 13 ]. In the standard version of the C D M theory, fluctuations in the early universe are assumed to be gaussian with a scale-invariant spectrum, as seems most natural if the perturbations were generated during an inflationary phase [ 14,15 ]. The universe is assumed to have a mean density £2= 1 made up almost entirely of C D M with a small contribution from baryons, photons and three light ( < < e V ) neutrino types (where £2 is the mean density divided by the critical density of the Einstein-de Sitter model). The evolving fluctuation spectrum is characterized by a single physical length scale associated with the Hubble radius when the universe passes from relativistic (er) matter domination to nonrelativistic (nr) matter domination, which we express in terms o f a comoving wavenumber scale keq: k~qt..~5(£2nrh2)-~O~/2Mpc,
0 - / T J 1.68/~y,
(1)
where h is the Hubble constant Ho in units of 100 km s - ~ M p c - ~. With the standard choice of parameters (£2nr~ 1, h ~ 0 . 5 , 0 = 1 ), this length scale is relatively small and so it proves difficult to explain observations of structure on scales > 1Oh - ~ Mpc. Ways
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out of this difficulty include non-scale invariant initial spectra [ 15 ], or the introduction of a cosmological constant to allow £2.r ~ 0.1-0.2 in a spatially flat universe [ 16 ]. However, neither o f these proposals is particularly attractive since each requires fine tuning which is not understood at present. A decaying neutrino with a mass in the keV range can also lead to an increase in kyq ~ via an increase in 0 as first pointed out in ref. [ 17 ]. In this letter, we investigate these models further by fixing the mass of the decaying neutrino to be 17 keV (denoted by m ~7 ) and treating the lifetime 3o as a free parameter. For simplicity, we assume that the decaying neutrinos are non-relativistic (and hence behave like cold dark matter) with mean density/~,ra and decay into relativistic particles with mean d e n s i t y / ~ . The rest of the universe is assumed to consist of photons (T), primordial relativistic neutrinos (v) and stable cold particles ( X ) . We also assume that the baryon abundance is small (E2B<< £2x) SO that their dynamical role can be neglected. The mean densities vary as
finrd =~]ffX e x p ( - - t / r d ) , e = 168mlT(f2xh 2) - l ,
(2a)
~x =ap~, fi~=afi~
(2b)
1 d (a4fierd) dt
~/4
--
/gnrd , rd
(2C)
where a = ao/( 1 + z) is the cosmological scale factor, z is the redshift and a o = 4 × 104f2xh 2 is the present value of a. For two massless neutrino types ¢x= 0.45 and for three neutrino types ¢x= 0.68, If the decay time exceeds 2 × 106m i-72 S, the universe will be matter dominated when the neutrinos decay. In this case, the universe will go through two periods of matter domination, the first occuring when the energy density in 17 keV neutrinos dominates over relativistic matter and the second when the C D M density dominates over the er decay products. The fluctuation spectrum is therefore characterized by two length scales, associated with the Hubble radii at these two epochs, kyqll ~ 28m i-7j k p c ,
(3a)
k;-q~~25mZ/3(zd/1 yr)'/3h -t Mpc.
(3b)
The first of these scales, kZq~, follows from eq. ( 1 ) with 0 = 1.45/1.68 and I 2 , ~ 168h-2 (the contribu246
15 August 1991
tion of the 17 keV neutrinos if t << rd). It corresponds to a mass scale of ~ 108m i-73~2xMo, i.e., about the mass of small galaxy. Objects on this and smaller mass-scales would collapse at high redshifts (z ~ 100), in contrast to the standard C D M model where little structure forms above a redshift of ~ 20 [ 18 ]. The second scale, keq2, - follows from eq. ( 1 ) with Onr ~ 1 and
O~ 1.45/1.68+ 5.5(mlTzd/yr) 2/3 , the latter deriving from the energy density in relativistic decay products, ~ceao/( 1+ Zd). The decay redshift (when the Hubble time equals the decay time) is 1 +Zd -~ 105"9m 1-71/3( r J y r ) - 2 / 3 . The scale k~q~ is larger than the scale kyq ~ of the standard C D M model (eq. ( 1 ) ) and can be comparable to the scales on which large-scale structure has been observed ifrd ~ 1 yr. Indeed, if the initial fluctuations were scale-invariant then a 17 keV neutrino must decay on a timescale of < 1 yr to avoid producing too much larger scale structure and excess fluctuations in the microwave background radiation. We now elaborate on these points using detailed calculations of the evolution of fluctuations with 17 keV neutrinos. To evolve the fluctuations in the different components of matter present, we utilize linear perturbation theory and the synchronous gauge [ 18 ]. We define a characteristic timescale r * = [ (8nG/3)pva 4] -J/2 = 1.22× 103(.Qxh 2) -2 y r . Dots denote derivatives with respect to q = r / r * , where the conformal time is r=fdt/a. The trace h of the metric perturbation hij then obeys the equation
d (adh~
3(2A~o+20~Avo+2P-~rdAerdO
+ a g x + ct/~nro_ ~nrd], / PX
(4)
where the Ao'S and ~'s are the fractional energy density perturbations in the various components. The C D M and decaying neutrinos are treated as cold eollisionless particles and we use the perfect fluid model
Volume265, number 3,4
PHYSICS LETTERS B
(valid in the tight coupling limit) to evolve the photons
i000
,~
fix=½/~,
J.rd=Jx,
........
i
........
,
........
,
........
i
........
,, , > ~ . . ~ , .
LSSV
I00
//..~,]I~
(5a) -~
Z],o = ]/~-ikr%lr,,
15 August 1991
z~r, = - ]ikr*Aro,
lo ~
w,,(o>
..~./..'" "._M.S---~
(5b)
where k is the comoving wave number of the plane wave perturbation and/~rAv ~ is the photon flux. We solve the collisionless Boltzmann equations for the relativistic decay products and the massless neutrinos. The relative perturbation Aerato the energy density for these components is defined by
~
o l -- COBE - .
0.0011
'
" /I klllV
~ Z"
0.:-0001 0.001
HIll
0.01
.....
~
.......
]
0.1 1 k hMpc -1
.......
]
10
......
li
]
100
Perd( k, ~', q ) =Perd(~') [1 "JI-Aerd(k, r, 0 ) ] ,
where ~ is the direction of the decay products. The perturbation A~d evolves according to ~l~d +ikr*/tA~rd- 2h/tz + 2 (h-/t33)' ½(3/~ 2 - 1 ) =afi.~, r* (6nrd __Z~erd)
,
fferd "[d
(6)
where/z=#-/~, and the three-axis is chosen to be in the direction of k. The Boltzmann equation for the massless neutrinos is identical to (6) except that the right-hand side is zero. These equations are solved by expanding A in Legendre polynomials, a = Z (2~+ 1 )a~P~(u),
retaining as many terms as are necessary for an accurate solution (~ >> kz). The h-/~33 term required for eq. (6) is related to the ~ = 1 moments of the A's through the constraint equation /~_//33=
i 6 ( fferd ) kr--~ 5 Av~+azl~j+ _ Ae~d, . Pv
(7)
We use initial conditions appropriate to a growing adiabatic mode in the radiation era and we assume a spatially fiat background universe with f2= 1 with two massless neutrino types ( a = 0.45 ). Fig. 1 shows the power spectra of the linear CDM density fluctuations, dcr2/d In k = k 3 p x ( k ) /21r 2 , where P× (k) = I~x (k, ~o) 12, for models with h = ~ and for lifetimes rd ranging from 104 yr to zero (i.e., the standard CDM model). The spectra are plotted in this way to provide a measure of the contribution of a band around the given wavenumber to the overall
Fig. 1. Power spectra for CDM models with decaying 17 keV neutrinos are plotted against comoving wavenumber k (referred to current length units). There is progressively more large scale power as the v lifetime increases through the values rd = 0, 1, 10, 102, 103, 104 yr shown in the figure. The lines under the labels indicate approximate regions in k space that various probes of structure are sensitive to, such as: microwave background anisotropy experiments of large angle (COBE, [19] ) and of intermediate angle (e.g., SPole is the 1 o experiment of ref. [20] ); clustering o b s e r v a t i o n s for galaxies in the APM Galaxy Survey (w~ [ 11 ] ) and the QDOT redshift survey [ 11 ] and for clusters ((~c [21 ]); and large scale streaming velocities (LSSV [21]). The normalization is a s = 1. The high amplitude at small scales implies that early reionization of the universe is more likely in these models than in CDM. However our calculations are not valid for wavenumbers above the collisionless damping scale, corresponding to the gaussian filter R~ ~~ 133 (h - ~Mpc) - ~ shown by the heavy vertical line.
RMS fluctuations; the ordinate roughly gives 1 WZnl(k), where z,~(k) is the redshift at which the RMS fluctuations in the band become nonlinear. The power spectra of fig. 1 show the two features corresponding to the scales k~l and kzq~ ofeq. (3) and illustrate that decay times of > 1 yr can produce more power at large scales in comparison to the standard CDM model. We normalize the spectra by requiring that the RMS fluctuations in the mass distribution within spheres of radius 8 h - l Mpc, as, are given by a8 = b 8 ~, where b8 is the "biasing'factor. If bs= 1 galaxies are clustered like the mass distribution while if b8 > 1 galaxies are more strongly clustered than the mass; bs is thought to lie in the range 1-2 for CDM models [ 21 ]. To place thrther constraints on the range of acceptable lifetimes, we have used linear theory and the power spectra of fig. 1 to compute the angular corre247
V o l u m e 265, n u m b e r 3,4
PHYSICS LETTERS B
lation function of galaxies (see refs. [ l 1,22 ] for further details). These predictions are compared with the observational results from the APM Galaxy Survey [11] in fig. 2. This shows that lifetimes in the range ~ 0.5-10 yr can explain the excess large-scale power, and that longer lifetimes produce unacceptably large fluctuations. Table 1 gives further details of these models. The first column lists the lifetime in years and the second column lists the mean-square mass fluctuations in spheres of radius 0.5 h - l Mpc at the present epoch if one extrapolates their growth by linear theory, (AM/M)o.5. This gives a rough estimate of the redshift zgf at which large galaxies would form, ( 1 + zgf) -'g
.
'
.
.
.
.
.
.
.
i
.
.
.
.
.
.
.
.
i
0.1
105
0.01
10 - 3
0.1
1
10
0 (degrees) Fig. 2. The m o d e l s of fig. I are c o m p a r e d with the angular correlation functions d e t e r m i n e d from the A P M G a l a x y Survey [ l 1 ] scaled to the depth of the k i c k catalogue at which 1 ° corresponds to a physical scale of ~ 5 h - ~Mpc. No nonlinear corrections were a p p l i e d to the theoretical power spectra, but for angular scales above ~ 1 ° and for biasing factors b s > 1, the linear a p p r o x i m a tion is accurate.
Table 1 G a l a x y f o r m a t i o n and b u l k flows. ra(yr)
0 1 5 10
248
<(~I/M)g.5)
6.2b~ 4.5b~ 4.0b~ 3.8b8
~ t ~ ~
(l+z~f)
8.7bff h 6.3b21 5.6b~ ~ 5.3bg ~
a, 25h-lMpc
40h-IMpc
437b~ 754b8 978b~ 1106b2
331bff 634b~ 857bff 986b2
~ ~ I L
~ ~ t L
15 August 1991
(AM/M)o.5. The third column provides another estimate of the redshift of galaxy formation using the "peaks" method of Bardeen et al. [ 17 ] which gives the average redshift at which peaks with a number density equal to that of bright galaxies would collapse when the matter density is filtered with a gaussian of width 0.35 h - i Mpc. A more careful analysis of star formation history would be required to improve upon these numbers, but they illustrate that there are no grave difficulties with the redshift of galaxy formation in these models provided b8 is not too large. Fig. 1 shows that small scale structure can form early in decaying particle models. In our computations, we have neglected collisionless damping of fluctuations caused by the free-streaming of the 17 keV neutrinos when they become semi-relativistic (which occurs at redshifts ~ 1 0 7 5 ) . From the collisionless damping calculations of Bond and Szalay [ 23 ], the damping can be approximated by multiplying the power spectra of fig. 1 with a gaussian, exp(-k2R~s) with Rfs~ 15 kpc. Thus, in these models the first structures collapse at redshifts z ~ 100 and have masses equal to those of dwarf galaxies. Unlike the standard cold dark matter model [ 18 ], this early structure formation could reionize the universe shortly after the normal epoch of photon-baryon decoupling at z ~ 1000, thereby reducing small angle anisotropies in the microwave background radiation. The fourth and fifth columns in table 1 list values for the three-dimensional peculiar velocities in km saveraged over spheres of radius 25 h ~Mpc and 40 h - I Mpc, respectively, This illustrates how the extra large-scale power in the decaying neutrino models increase bulk flow speeds. Decay times of > l0 yr are probably incompatible with the observed velocity flows in the local universe [21 ] unless one invokes a large biasing factor. Some of the tightest constraints on these models come from the background radiations. Even with a lifetime of ~ 1 yr, we have to worry about distortions in the microwave background since the radiation will not thermalize. The temperature of the background radiation at the decay redshift is Tya~ 200(rd/yr) -2/3 eV, if little decay energy goes into photons, lfthe neutrinos decay into photons, the decay radiation will be injected at energies of about 8.5 keV, so the most important mechanisms relevant to the reprocessing of the photon spectrum are Comp-
Volume 265, number 3,4
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ton scattering, double Compton scattering and bremsstrahlung. With current constraints from COBE [24], we conclude that no more than ~ 1% of the CMB energy can be liberated as photons if the neutrinos have a decay time of ~ 1-10 yr, using the methods in ref. [ 7 ]. Thus the branching ratio to photons needs to be strongly suppressed. Since small scale structure forms early in these models, predictions o f small scale anisotropies in the microwave background are uncertain because of the possibility that the intergalactic medium is reionized. However, anisotropies at angles > 5 ° from fluctuations in the gravitational potential (the SachsWolfe effect [25,18 ] ) are unaffected by reionization. If we assume that the universe is matter dominated from photon decoupling to the present, the meansquare temperature fluctuations for the lth multipole of the radiation field can be expressed in terms of the power spectrum for gravitational potential fluctuations, d a 2 / d In k, by AT 2 (2l+1) --T--z ~ 9
dlnk
j~(kro)
(8)
0
where jz is a spherical Bessel function. The gravitational potential spectrum is related to that for the density by dtr~/d In k = ( 3 H ~ k - 2 ) 2 dtrp2/d In k , where the comoving wavenumber k is referred to current length units. Table 2 gives computations for the microwave background anisotropies for various decay times (listed in the first column). Column 2 gives results for the quadrupole component (the R E L I C T 1 limit [26] for scale invariant spectra is 1 . 5 × 1 0 -5 at the 95% confidence level). The third column gives the RMS anisotropies expected within the COBE beam, found by summing over multipole contribuTable 2 Microwave background anisotropies. rd(yr)
bs(AT/T)~=2
bs(AT/T) 7o
0 1 5 I0 100 1000
4.31 >( 10 - 6 1 . 4 6 X 10 -5 2 . 6 2 X 10 - s 3 . 4 7 X 10 -5 9 . 4 3 X 10 - 5 2 . 4 9 X 10 - 4
1.01X 3.13X 5.49X 7.19X 1.85X 4.58X
10 -5 10 - s 10 - s 10 -5 10 - 4 10 - 4
15 August 1991
tions given by eq. (7), weighted with e x p [ - ( 1 2 / ldmr)2], where ldnar = 19.3 corresponds to the 7 ° (full width half m a x i m u m ) beam appropriate for the D M R experiment on COBE [ 19 ]. The current D M R limit is 4X 10 -5 at the 95% confidence level on such fluctuations. These limits rule out models with decay times > 10 yr (ifbs < 2 ) and are not too far above our predictions for models with rd ~ 1 yr. Particle physics models with decay times of ~ 1 yr and with a small radiative branching ratio are not easy to construct by are certainly not impossible. Holdom [27] constructed models in an extended technicolour theory in which a 17 keV neutrino can decay by techniphoton emission on a timescale of ~ 10 yr. Glashow [28] considered a scheme in which a massive neutrino decays into light neutrinos and majorons while Fukugita [29] considered familon models. In either case a judicious choice of parameters can give lifetimes of order o f years. We have shown that lifetimes of order o f years result in attractive models of structure formation which can explain recent observations of large-scale structure in the universe. However, it is also important to stress that lifetimes > 10 yr conflict with observations if the initial fluctuations are scale invariant. This lifetime limit is even stronger if other kinds of dark matter in addition to, or instead of, C D M is invoked, or if there is a nonzero cosmological constant. To accommodate a 17 keV neutrino with a longer lifetime, we require carefully tuned fluctuation spectra in the early universe to avoid over-producing perturbations on large scales. We thank Andrew Hime, Rocky Kolb, G r a h a m Ross and Subir Sarkar for interesting discussions.
References [ 1 ] J.J. Simpson and A. Hime, Phys. Rev. D 39 (1989) 1825. [2] A. Hime and J.J. Simpson, Phys. Rev. D 39 (1989) 1837. [3] A. Hime and N.A. Jelley, Phys. Lett. B 257 ( 1991 ) 441. [4] B. Sur et al., LBL preprint (1991). [ 5 ] J.J. Simpson, Phys. Rev. Lett. 54 ( 1985) 1891. [6]J.R. Bond, B.J. Carr and C.J. Hogan, Astrphys. J. 306 (1986) 428. [ 7 ] J.R. B o n d , in: The early universe, ed. W.G. Unruh (Reidel, Dordrecht 1988), [8 ] A. Hime, R.J.N. Phillips, G.G. Ross and S. Sarkar, preprint ( 1991 ). [ 9 ] R. Gandhi and A. Burrows, Phys. Lett. B 246 (1990) 149.
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[ 10] E.W. Kolb and M.S. Turner, preprint ( 1991 ); K.A. Olive, D.N. Schramm, G. Steigman and T.P. Walker, Phys. Lett. B 236 (1990) 454. [ 11 ] S.J. Maddox, G. Efstathiou, W.J. Sutherland and J. Loveday, Mon. Not. R. Astron. Soc. 242 (1990) 43P. [ 12] G. Efstathiou et al., Mort. Not. R. Astron. Soc. 247 (1990) 10P; W. Saunders et al., Nature 349 ( 1991 ) 32. [13] M. Davis, G. Efstathiou, C.S. Frenk and S.D.M. White, Astrophys. J. 292 ( 1985 ) 371. [ 14] J.M. Bardeen, P.J. Steinhardt and M.S. Turner, Phys. Rev. D 2 8 (1983) 679. [ 15 ] D.S. Salopek, J.R. Bond and J.M. Bardeen, Phys. Rev. D 40 (1990) 1753. [ 16] G. Efstathiou, W.J. Sutherland and S.J. Maddox, Nature 348 (1990) 705. [ 17] J.M. Bardeen, J.R. Bond and G. Efstathiou, Astrophys. J. 321 (1987) 28. [ 18 ] J.R. Bond and G. Efstathiou, Astrophys. J. Lett. 285 ( 1984 ) 45; J.R. Bond and G. Efstathiou, Mon. Not. R. Astron. Soc. 226 (1987) 655.
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[ 19] G. Smoot et al., Astrophys. J. Lett. 371 ( 1991 ) 1. [20] P. Meinhold and P. Lubin, Astrophys. J. Lett. 370 (1991) II. [21 ] See papers in: V.C. Rubin and G.V. Coyne, eds., Large-scale motions in the universe (Princeton U.P. Princeton, NJ, 1988). [22] H.M.P. Couchman and J.R. Bond, in: Large scale structure and motions in the universe, eds. M. Mezzetti et al., (Kluwer, Dordrecht, 1989) p. 335. [23] J.R. Bond and A. Szalay, Astrophys. J. 274 (1983) 433. [24] J. Mather et al., Astrophys. J. Lett. 354 (1990) 37. [ 25 ] R.K. Sachs and A.M. Wolfe, Astrophys. J. 147 ( 1967 ) 73. [26] A.A. Klypin, M.V. Sazhin, I.A. Strukov and D.P. Skulachev, Sov. Astron. Lett. 13 (1987) 104. [27] B. Holdom, Phys. Lett. B 160 (1985) 403. [28] S.L. Glashow, Phys. Lett. B 187 (1987) 367; B 256 ( 1991 ) 255. [ 29 ] M. Fukugita and T. Yanagida, Phys. Lett. B 157 ( 1985 ) 403; see also M. Fukugita and T. Yanagida, Yukawa Institute preprint YITP/K-907 ( 1991 ).