Takeshi Nihei, Nobuchika Okada and Osamu Seto- Dark Matter Abundance in Brane Cosmology

8/3/2019 Takeshi Nihei, Nobuchika Okada and Osamu Seto- Dark Matter Abundance in Brane Cosmology

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Dark Matter Abundance in Brane Cosmology

Takeshi Nihei

Department of Physics, College of Science and Technology, Nihon University, 1-8-14, Kanda-Surugadai,Chiyoda-ku, Tokyo 101-8308, Japan

Nobuchika Okada

Theory Division, KEK, Oho 1-1, Tsukuba, Ibaraki 305-0801, Japan

Osamu Seto

Department of Physics and Astronomy, University of Sussex, Brighton BN1 9QJ, United Kingdom

We investigate the thermal relic density of a cold dark matter in the brane world cosmology. Since the expansionlaw in a high energy regime is modified from the one in the standard cosmology, if the dark matter decouples insuch a high energy regime its relic number density is affected by this modified expansion law. We derive analyticformulas for the number density of the dark matter. It is found that the resultant relic density is characterizedby the transition temperature at which the modified expansion law in the brane world cosmology is connectingwith the standard one, and can be considerably enhanced compared to that in the standard cosmology, if thetransition temperature is low enough. As an application, the thermal relic density of the neutralino dark matterin the brane world cosmology is studied. For the neutralino, if the five dimensional Planck mass M5 is lower than104 TeV, the brane world cosmological effect is significant at the decoupling time and the resultant relic densityis enhanced. We calculate the neutralino relic density in the Constrained Minimal Supersymmetric StandardModel (CMSSM) and show that the allowed region is dramatically modified from the one in the standardcosmology and eventually disappears as M5 is decreasing. Thus, we find a new lower bound on M5 >

600 TeV

based on the neutralino dark matter hypothesis, namely the lower bound in order for the allowed region of the

neutralino dark matter to exist.

1. INTRODUCTION

Recent cosmological observations, especially theWilkinson Microwave Anisotropy Probe (WMAP)satellite [1], have established the CDM cosmologicalmodel with a great accuracy and the relic abundanceof the cold dark matter is estimated as (in 2 range)

CDMh2 = 0.1126+0.0161

0.0181. (1)

On the other hand, theoretically, if the dark matter isthe thermal relic, the present number density is esti-mated as

Y|0 nCDM

s

0 xd

0for n = 0,

2x2d1

for n = 1, (2)

with a constant

= 0.26

gS

g1/2

m

G(3)

for models in which the thermal averaged productof the annihilation cross section and the relativevelocity v, v, is approximately parametrized asv = nxn with x = m/T, where g is the ef-fective total number of relativistic degrees of freedom,xd = m/Td, Td is the decoupling temperature and mis the mass of the dark matter particle and G is theNewtons gravitational constant [2].

Note that the thermal relic density of the dark mat-ter depends on the underlying cosmological model aswell as its annihilation cross section. A brane world

cosmological model which has been intensively inves-tigated [3] is a well-known example as such a non-standard cosmological model. The model is a cos-mological version of the so-called RS II model firstproposed by Randall and Sundrum [4], where our 4-dimensional universe is realized on the 3-brane lo-cated at the ultra-violet boundary of a five dimen-sional Anti de-Sitter spacetime. In this setup, theFriedmann equation for a spatially flat spacetime isfound to be

H2 = 8G3

1 + 0

, (4)

where

0 = 96GM65 , (5)

H is the Hubble parameter, is the energy densityof matters, M5 is the five dimensional Planck mass,and we have omitted the four dimensional cosmolog-ical constant and the so-called dark radiation term.The second term proportional to 2 is a new ingre-dient in the brane world cosmology and lead to anon-standard expansion law. Since at a high energy

regime this term dominates and the universe obeys anon-standard expansion law, some results previouslyobtained in the standard cosmology can be altered.In fact, some interesting consequences in the braneworld cosmology have been recently pointed out. Forexample, the novel possible solution to the gravitinoproblem by the modified expansion law is pointedout [5].

Here, we investigate the brane cosmological effectfor the relic density of the dark matter due to the non-standard expansion law. If the new term in Eq. (4)

22nd Texas Symposium on Relativistic Astrophysics at Stanford University, Dec. 13-17, 2004

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dominates over the term in the standard cosmologyat the freeze out time of the dark matter, it can causea considerable modification for the relic abundance ofthe dark matter [6].

After deriving an analytic formula for the relic den-sity of the dark matter [6], we apply this result toa concrete candidate, neutralino. The lightest super-symmetric particle (LSP) is suitable for a cold dark

matter, because they are stable owing to the conser-vation of R-parity. In the minimal supersymmetricstandard model (MSSM), the lightest neutralino istypically the LSP and the promising candidate for thecold dark matter. In the light of the WMAP data, theparameter space in the Constrained MSSM (CMSSM)which allows the neutralino relic density suitable forthe cold dark matter has been recently re-analyzed [7].It has been shown that the resultant allowed region isdramatically reduced due to the great accuracy of theWMAP data. By taking account of the modified ex-pansion law in brane world cosmology, we estimatenumerically the neutralino relic density in the brane

world cosmology and show that the allowed region forthe neutralino dark matter in the CMSSM is dramat-ically modified in the brane world cosmology [8].

2. DARK MATTER RELIC DENSITY INBRANE WORLD COSMOLOGY

In this section, we derive the general analytic for-mula for the relic density of the dark matter in thebrane world cosmology with a low five dimensionalPlanck mass [6].

In the context of the brane world cosmology, the

thermal relic density of a dark matter particle is eval-uated by solving the Boltzmann equation

dn

dt+ 3Hn = v(n2 n2EQ), (6)

with the modified Friedmann equation Eq. (4), wheren is the actual number density of the dark matterparticle, nEQ is the equilibrium number density. It isuseful to rewrite Eq. (6) into the form,

dY

dx=

x2

1 + xtx

4v(Y2 Y2EQ), (7)

where xt is defined as

x4t

0

T=m

. (8)

At the era x xt the 2 term dominates in Eq. (4),while the 2 term becomes negligible after x xt andthe expansion law in the standard cosmology is real-ized. Hereafter we call the temperature defined asTt = mx

1t (or xt itself) transition temperature at

which the expansion law of the early universe changesfrom the non-standard one to the standard one. Sincewe are interested in the effect of the 2 term for thedark matter relic density, we consider the case thatthe decoupling temperature of the dark matter Tdis higher than the transition temperature Tt, namelyxt > xd = m/Td.

At the early time, the dark matter particles are in

the thermal equilibrium and Y = YEQ + tracksYEQ closely. After the temperature decreases, the de-coupling occurs at xd roughly evaluated as (xd) Y(xd) YEQ(xd). The solutions of the Boltzmannequation during the xt > x > xd epoch are given as

1

(x) 1

(xd)=

0x2t

(x xd) for n = 0,1x2t

ln

x

xd

for n = 1. (9)

Note that (x)1 is continuously growing withoutsaturation for n 1. This is a very characteris-tic behavior of the brane world cosmology, comparingthe case in the standard cosmology where (x) satu-rates after decoupling and the resultant relic density isroughly given by Y() Y(xd). For a large x xdin Eq. (9), (xd) and xd can be neglected. When xbecomes large further and reaches xt, the expansionlaw changes into the standard one, and then Y obeysthe Boltzmann equation with the standard expansionlaw for x xt. Since the transition temperature issmaller than the decoupling temperature in the stan-dard cosmology (which case we are interested in), thenumber density freezes out as soon as the expansionlaw changes into the standard one and can be roughlyevaluated as Y(

)

(xt) in Eq. (9).

By adopting the approximate parametrization forv, we can obtain simple analytic formulas for therelic number density of the dark matter as

Y(x ) 0.54 xt0

for n = 0,

x2t1 ln xt

for n = 1, (10)

in the limit xd xt, where xd is the decoupling tem-perature. Note that the results are characterized bythe transition temperature rather than the decouplingtemperature. It is interesting to compare these resultsto that in the standard cosmology. Using the well-

known approximate formulas in the standard cosmol-ogy, Eq. (2), we obtain the ratio of the relic energydensity of the dark matter in the brane world cosmol-ogy ((b)) to the one in the standard cosmology ((s))such that

(b)(s)

0.54

xtxd(s)

for n = 0,

1

2 ln xt

xt

xd(s)

2for n = 1, (11)


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0 500 1000 1500 2000

0

500

1000

1500

2000

0 500 1000 1500 2000

0

500

1000

1500

2000

0 500 1000 1500 20000

500

1000

1500

2000

Figure 1: Contours of the neutralino relic density h2 in the (m1/2, m0) plane for M5 = (upper left window), 4000

TeV (upper right window), and 2000 TeV (lower window) in the case of tan = 50, A = 0 and > 0. The dotted,dashed, solid and short-dashlong-dash lines correspond to h

2 = 1.0, 0.3, 0.1 and 0.05, respectively. The shadedregions (in red) are allowed by the WMAP constraint. The region outside the bold line, including the two axes, areexcluded by experimental constraints or the condition for successful electroweak symmetry breaking.

where xd(s) denotes the decoupling temperature in thestandard cosmology.

Thus, the consequence for the dark matter in braneworld cosmology is that the relic energy density canbe enhanced from the one in the standard cosmologyif the transition temperature is low enough.

If the above discussion is applied to detailed analysisof the relic abundance of the neutralino dark matter,we can expect a dramatic modification of the allowedregion in the CMSSM. In the next section, we presentthe results.

3. THE APPLICATION TO THENEUTRALINO DARK MATTER

In this section, we study the enhancement effecton the parameter space in a neutralino dark mattermodel [8]. We calculate the relic density of the neu-tralino, h

2, in the CMSSM with the modified Fried-mann equation Eq. (4). For this purpose, we havemodified the code DarkSUSY [9] so that the modi-fied Friedmann equation is implemented.

The mass spectra in the CMSSM are determined bythe following input parameters

m0, m1/2, A, tan , sgn(), (12)


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Figure 2: h2vs. M5 for tan = 50, m0 = 280 GeV, m1/2 = 360 GeV (left window) and tan = 30, m0 = 2 TeV,

m1/2 = 490 GeV (right window) with A = 0 and > 0. The range of h2 between the two dashdotted lines satisfies

the WMAP constraint.

where m0 is the universal scalar mass, m1/2 is theuniversal gaugino mass, A is the universal coefficientof scalar trilinear couplings, tan is the ratio of thevacuum expectation values of the two neutral Higgsfields, and sgn() is the sign of the higgsino mass pa-rameter . With these input parameters, renormal-ization group equations for the CMSSM parametersare solved using the code Isasugra [10] to obtain themass spectra at the weak scale. In the present analy-sis, we take A = 0 and > 0.

In Fig. 1, we show the allowed region in the(m1/2, m0) plane consistent with the WMAP 2 al-

lowed range 0.094 < h2 < 0.129 for tan = 50,

A = 0 and > 0. The shaded regions (in red) areallowed by the WMAP constraint. These figures con-tains the contour plots of h

2. The dotted, dashed,solid and short-dashlong-dashed lines correspond toh

2 = 1.0, 0.3, 0.1 and 0.05, respectively. The regionamong the bold line and the two coordinate axes is ex-cluded by various experimental constraints (the light-est Higgs mass bound, b s constraint, the lightestchargino mass bound etc.) [7, 11] or the condition forthe successful electroweak symmetry breaking. Theupper left window corresponds to the usual result inthe standard cosmology (M5 = ). The upper rightwindow and the lower window are the corresponding

results for M5 = 4000 TeV and 2000 TeV, respectively.These figures clearly indicate that, as M5 decreases,the allowed regions shrink significantly. The allowedregion eventually disappears as M5 decreases further.

Next, we present sensitivity of the relic density toM5 in Fig. 2, fixing m0 and m1/2 as well as tan , Aand sgn(). The left window, where we take tan =50, m0 = 280 GeV and m1/2 = 360 GeV, corresponds

to the point giving the smallest value of h2 in the

small mass region (m0, m1/2 < 1 TeV) for tan = 50.

The range of h2 between the two dashdotted linessatisfies the WMAP constraint. For large M5 > 104TeV, the (too small) relic density (h

2 0.02) inthe standard case is reproduced independently of M5.This is because xt


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have derived the analytic formulas for the relic den-sity and found that the resultant relic density can beenhanced. The enhancement factor is characterizedby the transition temperature xt, at which the expan-sion law changes from the non-standard law to thestandard one.

In addition, we have studied the neutralino relicdensity in the CMSSM in the brane world cosmology.

If the five dimensional Planck mass is low enough,M5 600 TeV in thebrane world cosmology based on the neutralino darkmatter within the CMSSM.

Acknowledgments

The works of T.N. and N.O. are supported inpart by the Grant-in-Aid for Scientific Research(#16740150 and #15740164) from the Ministry of Ed-ucation, Culture, Sports, Science and Technology ofJapan. The work of O.S. is supported by PPARC.

References

[1] C. L. Bennett et al., Astrophys. J. Suppl. 148, 1(2003); D. N. Spergel et al., Astrophys. J. Suppl.148, 175 (2003).

[2] E. W. Kolb, and M. S. Turner, The Early Uni-verse, Addison-Wesley (1990).

[3] For a review, see, e.g., D. Langlois, Prog. Theor.

Phys. Suppl. 148, 181 (2003).[4] L. Randall and R. Sundrum, Phys. Rev. Lett. 83,

4690 (1999).[5] N. Okada and O. Seto, Phys. Rev. D 71, 023517

(2005).[6] N. Okada and O. Seto, Phys. Rev. D 70, 083531

(2004).[7] J. R. Ellis, K. A. Olive, Y. Santoso and

V. C. Spanos, Phys. Lett. B 565, 176 (2003);A. B. Lahanas and D. V. Nanopoulos, Phys. Lett.B 568, 55 (2003).

[8] T. Nihei, N. Okada and O. Seto, hep-ph/0409219.[9] P. Gondolo, J. Edsjo, L. Bergstrom, P. Ullio and

E.A. Baltz, JCAP 0407, 008 (2004).[10] F. E. Paige, S. D. Protopescu, H. Baer and

X. Tata, hep-ph/0312045.[11] For experimental constraints, see, e.g., K. Hagi-

wara et al. [Particle Data Group Collaboration],Phys. Rev. D 66, 010001 (2002).


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Takeshi Nihei, Nobuchika Okada and Osamu Seto- Dark Matter Abundance in Brane Cosmology