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Axion miniclusters
Volume 205, number 2,3
PHYSICS LETTERS B
28 April 1988
AXION MINICLUSTERS C.J. H O G A N Steward Observatory, UniversityofArizona, Tucson, AZ 85721, USA and M.J. REES Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UK Received 1 February 1988
The formation of cold dark matter in the form of coherent axion fields is expected to produce large-amplitude (6p-~) isocurvature fluctuations in axion density. These are later the first fluctuations to collapse, forming tiny bound "miniclusters" of axions which cluster in a gravitational hierarchy until the standard inflationary fluctuations become non-linear. The existence of these miniclusters can have observable consequences for gravitational lensing, may have a role in seeding the collapse of the first stars, and could affect the signal observed in axion-detection experiments.
F r o m a particle physics point o f view, the axion, introduced to solve the "strong CP p r o b l e m " , is one o f the b e s t - m o t i v a t e d exotic particle candidates for cosmic cold dark m a t t e r ( C D M ) . Like all such matter axions convert scale-free curvature fluctuations dating from the inflationary era into density fluctuations which, especially with a p p r o p r i a t e "biasing", seem to produce galaxies and clusters like those observed, and radiation anisotropy consistent with current upper limits. The subject o f this note is to examine the astrophysical consequences o f additional fluctuations p r o d u c e d at the t i m e the axions acquire their mass at much lower t e m p e r a t u r e ~ 100 MeV when Q C D instanton effects turn on. These fluctuations are o f the " i s o c u r v a t u r e " variety, corresponding to spatial variations in the n u m ber o f axions per photon, and hence do not grow during the r a d i a t i o n - d o m i n a t e d era. As a result they have much smaller a m p l i t u d e than the inflationary fluctuations on large scales, but on smaller scales they are expected to dominate. These fluctuations would lead to the collapse o f the first non-linear axion clusters at z ~ 105 on a mass-scale M ~ 10- s M o , a n d subsequent hierarchical clustering on larger scales until the large scale perturbations intervene. In the ab228
sence o f any inflationary fluctuations, the current axion clustering scale would be ~ 104 M o . O u r axion lagrangian is simply [ 1 ] LP=Du0DU0+ V( O,T) , where 0 runs from 0 to 2n. At T>> 100 MeV, V(O) is flat and the axion is massless. The self-interactions represented by the first term ensure that spatial gradients in 0 are minimized. The 0 field is created initially at T ~ 10 jz G e V with uncorrelated values on scales larger than -~ H - ~ at that time. Subsequently, it relaxes to uniformity at the speed o f light, which means that 0 will have a coherence length ~ H - ~a n d gradients o f order H/2n, forming d o m a i n s o f app r o x i m a t e l y constant 0 on the current horizon scale. (We assume here that no late inflation occurs which would vastly increase the d o m a i n size. ) The d o m a i n s persist down to TQ~ 100 MeV when V(O) is suddenly no longer flat, but acquires a m i n i m u m at 0 = 0, a n d the axions acquire their mass. " A x i o n s " are produced as 0 undergoes spatially coherent oscillations about this m i n i m u m ; in particular, the local axion density is p r o p o r t i o n a l to the initial value o f V(O) at each point. But this means that the d o m a i n s in 0 are t r a n s f o r m e d into d o m a i n s o f axion density; places
Volume 205, number 2,3
PHYSICSLETTERSB
with 0= 0 have zero axion density, while those with 0= 7rhave twice the mean density. These are "isocurvature" perturbations because the total (axion +radiation) density remains constant ~l These isocurvature fluctuations are interesting because they are actually non-linear; the axion density across a domain varies from zero to 2p, where p is the mean density. Their behavior on large scales is similar to fluctuations in objects such as black holes, which have been discussed previously [ 3-6 ]. Becuase 0 has zero correlation on scales exceeding the particle horizon, initial RMS perturbations in axion density p go like
~ - (SpI~)M -~ (MIMcoh ) -1/2 corresponding to white noise. The initial coherence scale Mcoh may be estimated as the mean mass contained in a sphere with radius equal to the Hubble length at temperature T = Tloo 100MeV,
M~nh ~P~n [c/H( T) ]3 ~ 3 × l O - 6 M o t1"21~r3/4 1A. r0- 3/4'T'3 ~ Iv Jt 1 0 0
,
where we have used H2=87gG(NaT4)/3, p= poz3=.Qpm, z 3, and z 4 = ( N / N o ) ( T / T o ) 4, where N(T) is the number of effective photon degrees of freedom at temperature T and No = N(to) includes currently decoupled neutrino species. The exact linear theory for the growth of isocurvature fluctuations is given by Efstathiou and Bond [ 7 ]; in the limit of large k, we obtain from their formula (2.21a) (adding a missing factor o f a 2 ):
(t~/Si) = 3 . 6 X 104h2( 1 + z ) - I . We thus find that scale for which ( ( S p / p ) 2) ~/2 _~ 1 at redshift z,
Mcoh (z) ~-4000Moh 4 N - 3/4Nff 3/4T-31ooo( 1 + z ) -2 The first axion mini-clusters to collapse are already formed as non-linear cores with p - 2 p ; they would separate out at twice the redshift given by setting 8z= 1, or z ~ 7 × 104h 2. Thus, for typical parameters
~' Another interesting consequence of this domain structure is the formation of string-bounded domain walls [2]. These effects, and complications introduced in theories where V has multiple degenerate minima, are ignoredhere.
28 April 1988
they would have a mass M--- 10- SSMe ---6 × 10z7 g, a typical density p -~ z 3po---7 × 10- ~5 g cm - 3 , diameter d = 10 ~4 cm -2, column density pd=0.7 g cm -2, escape velocity ~ = 4 0 m/s, and a virial temperature of order 0.2 K. These mini-clusters subsequently undergo classical dissipationless hierarchical gravitational clustering, with the typical clustering scale increasing as M ~ a 2 because of the character of the initial perturbations. The behavior of such a gravitational hierarchy over many orders of magnitude is still imperfectly understood. There is, however, an established tendency for successive mergers of clusters to gradually build cores of increasing mass, density and binding energy [ 8 ]. If some axions form clusters with T~> 100 K (so that molecular cooling could operate) by z = 10, they would cause baryons to collapse, thereby possibly triggering the formation of the stars independently of any pre-existing "inflationary" fluctuations. But even in the standard cold-dark-matter scenario (where pre-existing fluctuations dominate over minicluster graininess on large scales), axion miniclusters would be incorporated into galaxy halos during galaxy formation. If they survive to the present they would cause gravitational microlensing (because their surface density > 1 g cm-2; see e.g. ref. [9] ) and large-amplitude fluctuations in direct axion-detection experiments. The following brief argument shows that at least the cores of miniclusters would indeed survive to the present. Suppose the galaxy halo is composed of "perturbers" of mass M o moving with typical velocity v. In the impulse approximation, if such a perturber passes with impact parameter d then the fractional energy change in a minicluster with mean density Pc is about
A E / E ~- GM 2/dav2pc. In the history of the galaxy the closest encounter is likely to have d-~ dmi, with drain given by ndZmin= My/vtHpc , where PG is the mean halo density and tn the age of the universe. Therefore, the energy change caused by the closest encounter (which dominates the total AE) is independent of M o,
AE/E~_ 10 GtnpG/Pc 22 ~_p2 /2pcpo , where Po is the mean cosmic density, related to tn 229
Volume 205, number 2,3
PHYSICS LETTERS B
through Friedmann's equations (12= 1 is still assumed). For typical galaxy and minicluster parameters, AE/E<< 1 and survival is expected. However, the "envelopes" thrown o f f b y successive minicluster mergers would be at much lower density and would therefore blend into a smooth halo axion component. We conclude that a smooth background o f halo axions (as usually assumed) is likely to exist and may dominate the halo mass (and hence the integrated signal in axion detection experiments) but in addition to this one can expect large-amplitude fluctuations on a wide variety of timescales due to surviving miniclusters passing through the solar system. These would range all the way up to those rare events, lasting for about a year, when a surviving core passes through and the axion density increases by a factor o f order 109. The frequency of such events, as well as the microlensing probability, depends on the quantitative details of the hierarchical merging of clusters; both are proportional to the density of surviving cores.
230
28 April 1988
We are grateful to Y. Rephaeli, J. Preskill and M. Wise for useful conversations, and to S.D.M. White for several crucial suggestions. C.J.H. acknowledges support from the Alfred P. Sloan Foundation and NASA grant NAGW-763, and the hospitality o f the Institute of Astronomy, Cambridge.
References [ 1] J. Preskill, M.B. Wise and F. Wilczek, Phys. Lett. B 120 (1983) 127. [2] F.W. Stecker and Q. Shaft, Phys. Rev. Lett. 50 (1983 ) 92B. [3] P. Meszaros, Astron. Astrophys. 38 (1975) 5. [4] C.J. Hogan, Astrophys. J. 252 (1982) 418. [ 5 ] B.J. Carr and J. Silk, Astrophys. J 268 (1983) 1. [6] K. Freese, R. Price and D. Schramm, Astrophys. J. 275 (1983) 405. [7] G. Efstathiou and J.R. Bond, Mon. Not. R. Astron. Soc. 218 (1986) 103. [8] R.T. Farouki, S.L. Shapiro and M.J. Duncan, Astrophys. J. 265 (1983) 597. [9] J.R. Gott, Astrophys. J. 243 ( 1981 ) 140.
PHYSICS LETTERS B
28 April 1988
AXION MINICLUSTERS C.J. H O G A N Steward Observatory, UniversityofArizona, Tucson, AZ 85721, USA and M.J. REES Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UK Received 1 February 1988
The formation of cold dark matter in the form of coherent axion fields is expected to produce large-amplitude (6p-~) isocurvature fluctuations in axion density. These are later the first fluctuations to collapse, forming tiny bound "miniclusters" of axions which cluster in a gravitational hierarchy until the standard inflationary fluctuations become non-linear. The existence of these miniclusters can have observable consequences for gravitational lensing, may have a role in seeding the collapse of the first stars, and could affect the signal observed in axion-detection experiments.
F r o m a particle physics point o f view, the axion, introduced to solve the "strong CP p r o b l e m " , is one o f the b e s t - m o t i v a t e d exotic particle candidates for cosmic cold dark m a t t e r ( C D M ) . Like all such matter axions convert scale-free curvature fluctuations dating from the inflationary era into density fluctuations which, especially with a p p r o p r i a t e "biasing", seem to produce galaxies and clusters like those observed, and radiation anisotropy consistent with current upper limits. The subject o f this note is to examine the astrophysical consequences o f additional fluctuations p r o d u c e d at the t i m e the axions acquire their mass at much lower t e m p e r a t u r e ~ 100 MeV when Q C D instanton effects turn on. These fluctuations are o f the " i s o c u r v a t u r e " variety, corresponding to spatial variations in the n u m ber o f axions per photon, and hence do not grow during the r a d i a t i o n - d o m i n a t e d era. As a result they have much smaller a m p l i t u d e than the inflationary fluctuations on large scales, but on smaller scales they are expected to dominate. These fluctuations would lead to the collapse o f the first non-linear axion clusters at z ~ 105 on a mass-scale M ~ 10- s M o , a n d subsequent hierarchical clustering on larger scales until the large scale perturbations intervene. In the ab228
sence o f any inflationary fluctuations, the current axion clustering scale would be ~ 104 M o . O u r axion lagrangian is simply [ 1 ] LP=Du0DU0+ V( O,T) , where 0 runs from 0 to 2n. At T>> 100 MeV, V(O) is flat and the axion is massless. The self-interactions represented by the first term ensure that spatial gradients in 0 are minimized. The 0 field is created initially at T ~ 10 jz G e V with uncorrelated values on scales larger than -~ H - ~ at that time. Subsequently, it relaxes to uniformity at the speed o f light, which means that 0 will have a coherence length ~ H - ~a n d gradients o f order H/2n, forming d o m a i n s o f app r o x i m a t e l y constant 0 on the current horizon scale. (We assume here that no late inflation occurs which would vastly increase the d o m a i n size. ) The d o m a i n s persist down to TQ~ 100 MeV when V(O) is suddenly no longer flat, but acquires a m i n i m u m at 0 = 0, a n d the axions acquire their mass. " A x i o n s " are produced as 0 undergoes spatially coherent oscillations about this m i n i m u m ; in particular, the local axion density is p r o p o r t i o n a l to the initial value o f V(O) at each point. But this means that the d o m a i n s in 0 are t r a n s f o r m e d into d o m a i n s o f axion density; places
Volume 205, number 2,3
PHYSICSLETTERSB
with 0= 0 have zero axion density, while those with 0= 7rhave twice the mean density. These are "isocurvature" perturbations because the total (axion +radiation) density remains constant ~l These isocurvature fluctuations are interesting because they are actually non-linear; the axion density across a domain varies from zero to 2p, where p is the mean density. Their behavior on large scales is similar to fluctuations in objects such as black holes, which have been discussed previously [ 3-6 ]. Becuase 0 has zero correlation on scales exceeding the particle horizon, initial RMS perturbations in axion density p go like
~ - (SpI~)M -~ (MIMcoh ) -1/2 corresponding to white noise. The initial coherence scale Mcoh may be estimated as the mean mass contained in a sphere with radius equal to the Hubble length at temperature T = Tloo 100MeV,
M~nh ~P~n [c/H( T) ]3 ~ 3 × l O - 6 M o t1"21~r3/4 1A. r0- 3/4'T'3 ~ Iv Jt 1 0 0
,
where we have used H2=87gG(NaT4)/3, p= poz3=.Qpm, z 3, and z 4 = ( N / N o ) ( T / T o ) 4, where N(T) is the number of effective photon degrees of freedom at temperature T and No = N(to) includes currently decoupled neutrino species. The exact linear theory for the growth of isocurvature fluctuations is given by Efstathiou and Bond [ 7 ]; in the limit of large k, we obtain from their formula (2.21a) (adding a missing factor o f a 2 ):
(t~/Si) = 3 . 6 X 104h2( 1 + z ) - I . We thus find that scale for which ( ( S p / p ) 2) ~/2 _~ 1 at redshift z,
Mcoh (z) ~-4000Moh 4 N - 3/4Nff 3/4T-31ooo( 1 + z ) -2 The first axion mini-clusters to collapse are already formed as non-linear cores with p - 2 p ; they would separate out at twice the redshift given by setting 8z= 1, or z ~ 7 × 104h 2. Thus, for typical parameters
~' Another interesting consequence of this domain structure is the formation of string-bounded domain walls [2]. These effects, and complications introduced in theories where V has multiple degenerate minima, are ignoredhere.
28 April 1988
they would have a mass M--- 10- SSMe ---6 × 10z7 g, a typical density p -~ z 3po---7 × 10- ~5 g cm - 3 , diameter d = 10 ~4 cm -2, column density pd=0.7 g cm -2, escape velocity ~ = 4 0 m/s, and a virial temperature of order 0.2 K. These mini-clusters subsequently undergo classical dissipationless hierarchical gravitational clustering, with the typical clustering scale increasing as M ~ a 2 because of the character of the initial perturbations. The behavior of such a gravitational hierarchy over many orders of magnitude is still imperfectly understood. There is, however, an established tendency for successive mergers of clusters to gradually build cores of increasing mass, density and binding energy [ 8 ]. If some axions form clusters with T~> 100 K (so that molecular cooling could operate) by z = 10, they would cause baryons to collapse, thereby possibly triggering the formation of the stars independently of any pre-existing "inflationary" fluctuations. But even in the standard cold-dark-matter scenario (where pre-existing fluctuations dominate over minicluster graininess on large scales), axion miniclusters would be incorporated into galaxy halos during galaxy formation. If they survive to the present they would cause gravitational microlensing (because their surface density > 1 g cm-2; see e.g. ref. [9] ) and large-amplitude fluctuations in direct axion-detection experiments. The following brief argument shows that at least the cores of miniclusters would indeed survive to the present. Suppose the galaxy halo is composed of "perturbers" of mass M o moving with typical velocity v. In the impulse approximation, if such a perturber passes with impact parameter d then the fractional energy change in a minicluster with mean density Pc is about
A E / E ~- GM 2/dav2pc. In the history of the galaxy the closest encounter is likely to have d-~ dmi, with drain given by ndZmin= My/vtHpc , where PG is the mean halo density and tn the age of the universe. Therefore, the energy change caused by the closest encounter (which dominates the total AE) is independent of M o,
AE/E~_ 10 GtnpG/Pc 22 ~_p2 /2pcpo , where Po is the mean cosmic density, related to tn 229
Volume 205, number 2,3
PHYSICS LETTERS B
through Friedmann's equations (12= 1 is still assumed). For typical galaxy and minicluster parameters, AE/E<< 1 and survival is expected. However, the "envelopes" thrown o f f b y successive minicluster mergers would be at much lower density and would therefore blend into a smooth halo axion component. We conclude that a smooth background o f halo axions (as usually assumed) is likely to exist and may dominate the halo mass (and hence the integrated signal in axion detection experiments) but in addition to this one can expect large-amplitude fluctuations on a wide variety of timescales due to surviving miniclusters passing through the solar system. These would range all the way up to those rare events, lasting for about a year, when a surviving core passes through and the axion density increases by a factor o f order 109. The frequency of such events, as well as the microlensing probability, depends on the quantitative details of the hierarchical merging of clusters; both are proportional to the density of surviving cores.
230
28 April 1988
We are grateful to Y. Rephaeli, J. Preskill and M. Wise for useful conversations, and to S.D.M. White for several crucial suggestions. C.J.H. acknowledges support from the Alfred P. Sloan Foundation and NASA grant NAGW-763, and the hospitality o f the Institute of Astronomy, Cambridge.
References [ 1] J. Preskill, M.B. Wise and F. Wilczek, Phys. Lett. B 120 (1983) 127. [2] F.W. Stecker and Q. Shaft, Phys. Rev. Lett. 50 (1983 ) 92B. [3] P. Meszaros, Astron. Astrophys. 38 (1975) 5. [4] C.J. Hogan, Astrophys. J. 252 (1982) 418. [ 5 ] B.J. Carr and J. Silk, Astrophys. J 268 (1983) 1. [6] K. Freese, R. Price and D. Schramm, Astrophys. J. 275 (1983) 405. [7] G. Efstathiou and J.R. Bond, Mon. Not. R. Astron. Soc. 218 (1986) 103. [8] R.T. Farouki, S.L. Shapiro and M.J. Duncan, Astrophys. J. 265 (1983) 597. [9] J.R. Gott, Astrophys. J. 243 ( 1981 ) 140.