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Gμν = Rμν − �

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fJ ��������� ���!��� �� " ���

��%�� Tμν ) ~�� ��%�� Rμν !re.is ����<� �� .��� (�# �� &�� (−, +, +, +) ��0 �� G��&� $�>%��% ��

a -H�%� ��%�� x�� ���� �1� ) =��� ���"�x�� re.is ����<� �? �1� .��� �� �%� � a -H�%�

� "<� -H�%� <]� >&���I �� Gμν �1� , # .�� �� �� �� �� �%� �

T μν;ν =

∂T μν

∂xν= � =⇒ Gμν

;ν =∂Gμν

∂xν= � re.js

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��#��� ���C -�"3%� -���� �� ��� ��&� !���� %�[1�

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��#�� $��� �� �&���) �%��� G��&� ��M� �1�X 7�

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dl� = a(t)�gijdxidxj re.Ls

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fe ��������� ���!��� �� " ���

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�������"� .�#�� G��&� ��M� �1�X �� ,+&�� !���� $��%

f(r) =�

�− kr�re.Jms

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��� ��� �����

ds� = dt� − a(t)�[

dr�

�− kr�+ r�(dθ� + sin(θ)�dφ�)

]re.JJs

-�M� �1�X k < � &X) .�#� b[�� R�curve = �/k ��0 �� �� ��� -�<�� �� -�M� -�"3%� k

@��<� �� �� k �� ��� \��"� .�#�� = ��� �&�� =��� k > � -��� ) �[� =��� G� k = � -��� !��� G��&�

�= "� vp5 *� +� ,���

a(t) ≡ a(t)√|k| re.Jes

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ds� = dt� − a(t)�[

dr�

�− Kr�+ r�(dθ� + sin(θ)�dφ�)

]re.Jfs

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ds� = dt� − a(t)�[

dχ�

�− Kr�+ sinn(χ)�(dθ� + sin(θ)�dφ�)

]re.Jis

K = +� -�� � sin(χ) ) rΩtot < �s K = −� -�� � sinh(χ) � � � ��� � sinn(χ) ��� S � re.Jis � ��� < � ��

��� ���� ��� ��0 �� G��&� rΩtot = �s K = � -��� .��� rΩtot > �s

ds� = dt� − a(t)�[dχ� + χ�(dθ� + sin(θ)�dφ�)

]re.Jjs

���� �� $��� a �M� G ��"�� �� ��� (� �� $R �X) $"�� !=���� �S30 $��� a �M� G ��1" � �� $"� ��

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ff ��������� ���!��� �� " ���

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Tμν = (ρ + P )uμuν + Pgμν re.Jgs

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����

�πGT �� = �πGρ = Aa

a+ B

[(a

a

)�+

K

a�

]

�πGT ii = �πGP = Ca

a+ D

[(a

a

)�+

K

a�

]re.Jds

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ρ =(

A

C

)P − �

C[(C + D)ρ − (B + A)P ]

a

are.JLs

�>&� I ����<� �� �5� ��

∂ρ

∂t= −�(ρ + P )

a

are.JKs

�WI D = −B/� ) C = −�B/� !A = � �=����

�πGρ = B

[(a

a

)�+

K

a�

]

�πGP = −B

[�a

a+(

a

a+

K

a�

)]re.ems

��� ��0 �� *� +� ,��� G ��"�� -��� FRW �_��<� .�Jg��#��� B = � "� % ��� �� �� ����+� ��

��� ����

�πG

∑i

ρi =(

a

a

)�+

K

a�

�πG∑

i

Pi = −� a

a−(

a

a

)�− K

a�re.eJs

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fi ��������� ���!��� �� " ���

) ')� ����<� \ ��� �� .�#� `��� =��� (�%��� �0�"� @7&[� -��8% �� `��� re.eJs �_��<� �� i W��%�

���R� ���� ��� ��0 �� =��� v�&# ����<� re.eJs Y)�

a

a= −�πG

∑i

(ρi + �Pi) re.ees

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��0 �� >&� I ����<� re.Jgs ����<� �� (��O&�� �� ) rre.Jis ����<�s FRW G��&� �� ��� ��� $��� �� �

���R� ��� �� ���

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a(ρ + P ) = � re.efs

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��#� ,��S� ��� ��0 �� �:Z� ����<� �� �� re.efs -H�%�

ρr + �a

a(ρr + Pr) = �

ρm + �a

a(ρm + Pm) = �

ρλ + �a

a(ρλ + Pλ) = �

re.eis

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fj ��������� ���!��� �� " ���

���� ���� �� ��� �

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��� ��� ����� ����� -)� % ����� �C� �� .�""�

F = �pzvzAnr re.ejs

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� WI !�""�� ���; z �Su� ��5 �� �� ���

f =�

�F =

P

A= pzvznr re.egs

��P ��

pz = py = px =p

�re.eds

�=���� !�"#��� c ���� -���� �� ��%�� �"%�� & S�% ���| -��� ������"�

P =�

�pcnr =

�mc�nr =

�ρr re.eLs

.wr = �/� �� ��� ����� 9��� -��� ���; �&����I ������"�

����� ���� ���� �� ��� �

��� G���� (��� $ � Y�5 �� ����+� ,��X $�� � -��� �� � �)� $�� � $��)� �� (�| ��� -��� %�� � (��# ����

$R ���� �� $��� !�� &O5�) �� �<� $��)� -��� .�#��� P ∼ ρ/� ��� ����� �VS��+� & S�% ��0 �� �#���

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fg ��������� ���!��� �� " ���

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����� (��� ��� ��0

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Rμν − �

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) $���� &� � <]) �� � � � X� ������� ��� .�#�� = ��� = ��� -�� � � &� �� v�5 G� Λ > � -�� � $ "��

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− K

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a

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=�

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∝ η�

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a(t) = a�eH(t−t�), , K = �, H =

a

a= const

= a� sinh(H(t − t�)

), , K = −� re.iis

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H�

= Ωr + Ωm + Ωλ re.jJs

critical density��

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eH(tf−ti) − �

Hre.gfs

��� 'P ���� $R �S�% $��� Y�� $��)� �� $R '3� �� �5� �� ��� >� �B% �� l(t) 'P �� ��&��� G� ��� '�;

��#% ��� ��0 �� 7E�

dcau(t)l(t)

=eH(t−ti) − �

Hl(ti)eH(t−ti)≥ �− e−H(t−ti) re.gis

By:

Movah

ed

www.smov

ahed

.ir

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id ��������� ���!��� �� " ���t

xp

t0

trec

t R

ti

a(t) = e tH

l p(t)

l f (t)

Y�� '�� G1� �� ��� �7l�� ,; �dae , #

Y�1 � � � (�# N�: � -�(��� %� � � 7E � � �� e − fold � "? �#p� �� WI �#� � l(ti) ≤ H−� � � & ��; -�� �

'P ��+? Y�� $��)� �� ��� ��� �#� z��� �7;�� ��� �� �� ��� .�� �� �� �� �� �: � -��*� +�

k��� ��+? �"� �#� Y�� $��)� -��&�� �� ����+� �� *� +� ,��� &���� �� %�: � ,X��; �V���� ) k����

�"<� �#�� '�; $��� �� 7��� 'P �� �&��:� Y�� $��)� $���I �� 7E� ��� &���� ��� �7l�� ,; -���

∫ tf

ti

dt

a(t)>

H�

re.gjs

��� ��� ����� q� ����<� �? �1� *� +� ,��� -��� ��1% '3� , # �&��� �B% ����

∫ tf

ti

dt

a(t)∼ �

Ha(ti)

[�− e−H(t−ti)

]∼ �

Ha(ti)re.ggs

��#�� = ��� ������"�

Ha(ti)>

a�H�

a�af

aiaf>

H

H�

re.gds

�=���� =��� -�"3%� �� $��� �B% D�0 ) $������ ')� ����<� �� (��O&�� ��

H�

H�

∼ ρr

ρm

[a�af

]�re.gLs

By:

Movah

ed

www.smov

ahed

.ir

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iL ��������� ���!��� �� " ���

�= "�� ��%��� �� re.gds ����<� re.gLs ����<� �� (��O&�� ��

af

ai> ��

−� a�af

re.gKs

$���I -��� ����� (�>%R !=�� >� �B% �� !Tf ∼ ����GeV !GUT -�l� *��� �� �� Y�� $��)� $���I -��� ��� ���

�=���� �V&���% ) !a�/af > ��� ��0 ��� �� �� !Tf = T�a�af

��� ��� ����� '�; $��� -��� �� Y�� $��)�

af

ai> ��

� ∼ e� re.dms

.�#� ,; ��� �7l�� ) �� ���� lf (tdec) > lp(tdec) � <]) ��� ��

?� ��� 0@ �� ���8 ��B+� ����8 <�� ��C/ 54945

�= "� ��% ��� ��� ��0 �� �� FRW ����<� � �� (��5�

H� + εT � =�G

�ρ re.dJs

I)�&%R $�1� s �Z"�� �� .�Ji� �� ���� s−�/� �� \��"&� ε � 1� .�#��� ε = K/(aT )� ����<� ��� �� ��

�>�� D�P �� .ε ∼ e−�H(t−ti) WI �#� '3&� s ∼ e�H(t−ti) ��0 �� � 1� ��� Y�� $��)� �� .�#���

$�# ��2]�) -��� $"�� .�#� ,; : % &[� �7l�� ) ρ−ρc

ρc∼ O(�) ������"� ρ−ρc

ρc∼ �

πGεT �

ρc�� = %���

�= "�� (��O&�� S�% ��>? = +&�� @��<� �� 8]�

Ωtot(a) =ρtot(a)ρc(a)

=ρtot(a)

�H�/�πG

=Ω�

r a−� + Ω�ma−� + Ωλ

Ω�r a−� + Ω�

ma−� + Ωλ − (Ω�tot − �)a−� re.des

m W��%� �� '�; $��� -��� �V "1] ) �#��� '�; $��� �� *� +� ,��� a� = �s �#�� a � a� �� &��; -���

�=���� r = "�� (��O&��

Ωtot(a) =H� − H�

�(�− Ω�

tot)a−�

H�= �+

(Ω�tot − �)a−�

Ωra−� + Ωma−�

By:

Movah

ed

www.smov

ahed

.ir

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iK ��������� ���!��� �� " ���

= �+Ω�

tot − �

Ω�r + Ω�

maa� re.dfs

�=���� (�>%R af/a� = ��−� "<� '�; $��� ) Y�� $���I �� *� +� ,��� �S�% �&��� �B% �� �� $"��

�− Ωtot(af ) = [�− Ω�tot] × ��

−�� re.dis

�� �� ��� ��| �� Y�_s �#��� ��0 �� Y�� -��&�� �� ���+� ��� \�; �� ,� ��>? Y�� $��)� $���I ��

� r�#��� 9��� ��>? �� ����� ,� ��>? \��+� �� Y�� $��)� -��&��

�− Ωtot(af ) =�− Ωtot(ai)

Ωtot(ai)

(ai

af

)�re.djs

� ������"� �#�� af/ai > ��� ∼ e� &���� &[� �7l�� ,; -��� �� =���� ,SX �1�X ��

�− Ωtot(af ) =�− Ωtot(ai)

Ωtot(ai)��

−�� re.dgs

�= "�� ��%��� '�; $��� �� ,� ��>? \�; �� �� re.djs ����<� �? �1� re.dis ����<� �� (��O&�� ��

�− Ω�tot =

�− Ωtot(ai)Ωtot(ai)

(ai

af

)���

�� <<�− Ωtot(ai)

Ωtot(ai)re.dds

�� ,� ��>? !Y�� -��&�� �� ,� ��>? -��� ���+� �� DZ��� �� != �� ���X ��−� ���+� ai/af -�5 �� ��� '�;

�� .���� $��% �� � <]) ��� 1 ��� ��0 �� rLaes , # .�� ���� G��:% G� ��� �� �� �� '�; $���

� ,; �� ��"# $�� � ����%�&�� '�� �/ �� Y�� $��)� �� ��� $��% $��� : % $�� � G ��"����� �� (��O&��

T5�� �� , 4O� �� �/ �� ��� .= #��� $R -��� : % �/ �� ���# !Y�� '�� -��� +�� �5) �� .�Ji� �"�

.��� (�# z��� �em�

%���� �� E��A� F�32 (�����;�� G�H .B'

����=�� .�#��� ��"#$�� � �� '�<E � ��+ +3� -�����# �� � $"�� *� +� N�:� -����&��� -�l�

��� �"���S� �#� �&����I ��%R �� ���# ��� �� �� �_��

By:

Movah

ed

www.smov

ahed

.ir

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jm ��������� ���!��� �� " ���

re T5��s Y�� '�� G1� �� =��� &[� �7l�� ,; �Lae , #

k�%#� , �� �%��� -�������I�% -�7 �) �� ����&��� ��R rJ

k �� ? G���� -(��� ���+� ) � ��� re

k �� ? �" �� '3� �� =��; ��"# $�� � '�� rf

k ��� (�� �? ��&��� , �� -��� � �)� U���# ri

) D�P G� �� -�� "� ���| G�: � ) 1&%�� -�� $�� � ���B% �� G��:% `�S��� �� �"��� % q� �_�l� �� ���I

*� +� �� =��� 2 ]� $��� ��"#$�� � D���� ����=�� �� � .�#��� ! �/� Y�� � ES�% �� �>�� D�P ��

������� G�: � �&�� -���# �� �� -�0� ������� ) �� 9����R ������"� !�#��� $R �� �&��:� ) ��$�� ��

.�%�(��� �� I -�l� ��"# $�� � �� >"��>"� `�S��� !�%��� ���X

�DE ���E�% 0��8 34;45

�� `��� ���# ) � �)� $�� � �� `��� -��-�l� � � �� ���� (��5� �� �� �� %�� � -���/&�� -�l�

�_/&�� -�� �"��� ��&��� , �� -��'���� .= "� ��Z�� +�"� `�S��� -�)��� *� +� N�:� -����&���

Cosmological perturbation��

By:

Movah

ed

www.smov

ahed

.ir

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jJ ��������� ���!��� �� " ���

�� �%��� -��-�����I� % \S� �� Wo� .�#� � �� � �)� =��� �� ���| G�: � ���C� U�� � E �)� ���4�

�#� ��� N�:� -��*� +� �� .�%�)R� ���I �� -�)��� -����&��� ) �&��� �#� =��� '3� $��)� 'P

) �� � h ���C� ,��+� �� .�"&� % �)� �� =�H� �� 7 � ��: � ) ��� ) �#�� � \��h �%��� ���C� ��

!δρ ��>? �� ">1��% G� �5) \S� �� !�&��� �" �� G� �� .�""�� � <� �� ���% � E<]) ��"��) �� �

��0 �� �%��� -)� %

δρ ∼ Gδρ

) r��R� ��� �� ���� �� ����� ��� �&��� $�� � �� $��I ����<� ) >&� I ����<� �&��� �B% �� ��s �#���

'�; �� �" �� G� �M; �� ��� !�"�� �#� T��� !ωk ∼ �√

πGρb �� δρk(t) ∼ eωkt T��� ��0 �� -�����I�% ���

.�f! eJ! ee! ef! ei� ��� $��% �� �" �� '3� �C� !δρ '3� ����<� �� (�""� �� � �715G� =���� ��B&%� `��S%�

��&�� ,��� (��# G� ��0 �� (�# ,&[� (��� �� = "����� ��>? ���S� '3� �� =��; ����<� � <� -���

�� �� $��)� �� �"� �� �����&��� -��� ) ��� �� �& ?� -���� +� -��� "� % ��&��� , �� '�� .�"��

�%��� , �%�&I Φ !(��# -H�%� ��>? ρ ��� �"���S� �X/� ��� -��� EF&� .�f! ej��� ���� q��0 �%��� ��

��"#��� ��� ��0 �� ��"��) �� � ���� �_��<� .I)�&%R ��>? S ) (��# ���� v !���� P !"� %

ρ + �∇ · (ρ�v) = �

�v + (�v · �∇)�v +�

ρ�∇Φ = �

�∇�Φ = �πGρ

S + (�v · �∇)S = �

P = P (ρ, S) re.dLs

�M; �� �7�)� ) � % ����<� (�"�� $��% ����<� � �)� !>&� I ����<� (�"�� $��% ����<� � �)� re.dLs ����<� ��

�� ���; ����<� !����� ����R ) I)�&%R -�+� Y���? ����� !"� % 9%��� -��� $��I ����<� ����� � �� !����

By:

Movah

ed

www.smov

ahed

.ir

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je ��������� ���!��� �� " ���

�� ��%R $��� ,&[� �� $"�� .= "�� -��p� Y�% b W��%� �� �� $�� � �" �� �� `��� -��� EF&� .���� $��%

���� ��0

ρ(�r, t) = ρb(t) + δρ(�r, t) ≡ ρb[�+ δ(�r, t)]

�v(�r, t) = �vb(�r, t) + δ�v(�r, t)

Φ(�r, t) = Φb(t) + δΦ(�r, t)

P (�r, t) = Pb(t) + δP (�r, t)

S(�r, t) = Sb(t) + δS(�r, t) re.dKs

) �r(t) = a(t)�x(t) �Z"�� �� �� !(�r, t) → (�x, t) "<� (��1� ��4&[� �� �: � ��4&[� ,��S� �� (��O&�� ��

�� $��� �B% D�0s -��� �� ) re.dLs �_��<� �� re.dKs �_��<� $��� ���X ) �#��� �v(�r, t) = H(t)�r(t)

��#�� = ��� r��_�� ) Y)� �S��� '/&��

δ + �Hδ − c�sa�

�∇�xδ − �πGρbδ =

σ

ρba�δS re.Lms

��""�� @ 0� �� ���; ����<� ��� ��0 �� �0 ���� �� σ ) �0 ���� cs !re.Lms ����<� ��

δP = c�s δρ + σδS

c�s =(

δP

δρ

)|S=fixed re.LJs

��0 �� ���� ,��S� ��

δρ(�x, t) ∼∫

δρk(t)eik·ax

��#� ,��S� ��� ��0 �� re.Lms ����<�

δk + �Hδk + c�sk�δk − �πGρbδk =σ

ρba�δSk re.Les

By:

Movah

ed

www.smov

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.ir

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jf ��������� ���!��� �� " ���

.�#�� = ��� )���� -��: � ) ��� ���� �� " � δS = � �"< � �#� � � &#�� % �5) I)� & %R : � ) ��� ���

��� ,��+� �� .δρ(�x, t = �) �= � �"< � �"#� �� �O0 � h ��>? �� � �)� -��: � ) ��� � <]) ��� ��

����%�&�� -�l� *��� �� .�#�� = ��� I)�&%R -��: � ) ��� ��0 $R �� δρ(�x, t) = δρ(�x, t = �) = �

� �f! ej��� ��� ����� re.Les ����<� WI !=�� �� �B% �� �� : � ) ��� ')� 8% �Z"�� �� ���

δk + �Hδk + c�sk�δk − �πGρbδk = �

δk + �Hδk = c�s δk

[�πGρb

c�s− k�

]re.Lfs

��&��� 'P ��� .�#� (� ��% ��:" 5 'P y� ��� �V;/�0� !kJ =√�πGρb/c�s � 1� !re.Lfs ����<� ��

���� ��&��� $�� N�:� \S� �� �%��� -)� % �� "<� ���� rλ >> λJ �� k << kJs �#�� :" 5 'P �� �&��:�

�� ,+&�� '3� ����<� ���; ��� �� !�"�1% �O�� 9S�� ��� ,��+� �� �+% ���� �� #�% ���C� ) (�# $R 9S��

��#��� ��� ��0 �� r= "�� Dp; �� k W��%� , �� � 1� ��s���� Y�1� -��� $�� � ��0 �� ) y� ���

δ + �H(a; {l})δ − �πGρmδ = � re.Lis

��� �Z"�� �� .�#� ���A H(a; {l}) �� $�� � �" �� '3� >%>? �� G���� -H�%� �C� re.Lis ����<� ��

=�� G���� -H�%� ��&��� , �� ������"� !�#��� ��: � ) ��� y� 'P �� �&��:� 7 � G���� -H�%� �0

����<� �� *� +� ,��� �� �S�% ��>? ���S� '3� .�#��� $��� �� �S�% �&�� re.Lis ����<� �� .�� ���[%

���� ��� ��0 �� -� Z%� �&�� G1� �� !�#��� -���"���

dt=

da

da

dt= δa

d�δ

dt�= a

da

da

dt+ δa = δa� + δa re.Ljs

�=���� != "� ��:>��5 re.Lis ����<� �� �� re.Ljs ����<� ���

d�δ

da�+

da

[a

a�+�H(a; {l})

a

]− �H�

�a�a�Ωmδ = � re.Lgs

Jeans��

By:

Movah

ed

www.smov

ahed

.ir

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ji ��������� ���!��� �� " ���

�� �� %R � C� �� G��� � -H� %� @7&[� -�� ��� �M; �� �� re.Lgs � ��� <� -��� ��0 �� $� �� $"��

.��� ,; �#� (��� $�� � -��" �� 9��� U�� �� � �)� U���# �� �#� ���A H(a; {l})

���E�% 0��8 �� �"� F��� 54;45

��� ��0 �� �# z��� ,SX 9[� �� �� ��>? ���S� G1� �� �0�� ���� $�� � !�� '/&�� -�l� ��

��f! ei���R� ����

v(x) = H�

f

�π

∫δ(y)

x − y

|x − y|�d�y re.Lds

��#� @��<� ��� ��0 �� �� !�#��� ���#� W��%� f � 1� re.Lds ����<� ��

f =d ln δ

d ln are.LLs

"� % ��� �� �� ��%�� �� �0�� -������ � �O� -��� �� �#� W��%� � 1� $��� ��>? ���S� �� (��O&�� ��

�� �#� W��%� G�: � �&� � =�� -��� .��� �S��3� !�)�� �� � �eg� ��&��� , �� & S�% ��� �� �� ) �ei�

Y� ) rw� �0� =��s Y)� �/15 �S�% �� � �� (��5� !��� = ��� -�0� � X �� �� $R � 1�� i ) f ,4� ��

�S�% ��� �� \��"&� �#� W��%� �� ���� = ��� �� ���� �� .= "� �5� re.Lis ����<� r$��I ����<� =��s

��� ����<� U�� �#� W��%� '3� re.LLs ) re.Lgs �_��<� G1� �� .�#��� f ∝ �Hδ/�πGρmδ "<�

��#� (���

df

d ln a= −f

[�− H�

�[�

H�

+Ωm

a�+ ΩΛ(a; {l})(�+ �w(a; {l}))]

]− f� +

�H�

�a�aΩm

re.LKs

D�P �� !�#� � <� �0�� ���� -� �(���%� �� �� ) ��%�� �� >&�S1� T��� �� = +&�� ��0 �� �#� W��%�

��"#$�� � @7&[� -��'�� -���&����I �� �#� W��%� !��� b[�� re.LKs ����<� �� �� ��%�1� �>��

Growth index��

By:

Movah

ed

www.smov

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∫ z

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��#��� ��� ��0 �� ��"���� �70�� !ΩK = ����; ��

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, #s ��� (��R ��� �� Ωλ = �.�� ) Ωm = �.�� -��� ��1% ���� �" � � !�#��� �.�� ≤ z ≤ �.�

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�� =�5 (���%� ��� F� !R⊥ = dAθ/(� + z) ���S� �� �5� �� .�#��� ����� �A�% ��� ,0�) U� -�&��� ��

��� ���� ��� ��0 �� �A�% ��� ,0�) U� �� �1� -�&���

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gL #�$� ��������� % ���

�A�% ��� ,0�) U� -�&��� �� =�5 (���%� ��� F� .�#��� z� = z� + Δz ) z = (z� + z�)/� _�� �_��<� ��

��� ���� ��� ��0 ��

ΔR‖ = a(z)Δr(z)

= a(z)Δz

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Δz

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)

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dt

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%���� (�/���0 12� &�A��� ������# .B)

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�� s �� % ���� �#�� \��h ��"#$�� � ���C �� 1��� �� \��+� ��� ��� .�#� (��� lpeak ∼ ���/√

Ωm

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dm #�$� ��������� % ���

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θpeak =ηSH

η� − ηLSrf.ems

����<� ��� ��

ηSH =�

H�

√|ΩK |sinn

[√|ΩK |

∫ ∞

zdec

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dz

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)

ηLS =�

H�

√|ΩK |sinn

[√|ΩK |

∫ ∞

zdec

dz

H(z)/H�

]rf.ees

) cs(z)−� = � + /� × ρb(z)/ρr(z) !zdec = ��� ��� + � !ΩK = � − Ωm − Ωλ q � �_�� < � ��

�V&S�% ) 7 73� ����� G� JKKg '�� �� 9%��� 1� ) ��� ��)� �&S�� .�#��� ρb(z)/ρr(z) = �.��/(� + z)

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���� = ��� (��O&�� zdec -��� ���+� ��� ��

zdec = ����

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dJ #�$� ��������� % ���

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g� = �.��[�+ ��.�(Ωbh

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rf.efs

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Ωλ = � ) Ωm = ��"<� �# �&��� �B% �� �;�) =��� ,� ��>? �� ��>"� ) CDM '�� �� . �im��#���

��� ��� ����� θ ≡ θflatpeak ���S�

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�+ zdec−√

Ωr

⎞⎠ rf.eis

��� ��� ����� rf.ems ����<� 7� ���; �� )

θpeak =�cs

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�+zdec−√

Ωr

]Ωm√|ΩK |sinn

[√|ΩK | ∫ zdec

H�dzH(z)

] rf.ejs

��� ���� ��� ��0 �� $�� � -��" �� 9��� �&��� = 1<� '�+&%� �&����I $"��

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Ωm√|ΩK |sinn

[√|ΩK |

∫ zdec

H�dz

H(z)

√Ωr + �/(�+ zdec) −

√Ωr

�√

Ωr/Ωm + �/(�+ zdec) −√

Ωr/Ωm

=√

Ωm√|ΩK |sinn

[√|ΩK |

∫ zdec

H�dz

H(z)

]× A rf.egs

���R� ��� �� ��� ��0 �� '�+&%� �&����I )

R ≡ �

ΓA=

√Ωm√|ΩK |sinn

[√|ΩK |

∫ zdec

H�dz

H(z)

]

=√

Ωm

�+ zdecdL(zdec) rf.eds

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�S�% $�� � -��" �� 9��� $�� @ P �� &��R �7X ��Z��5 $�: � !Γ−� = θflatpeak/θpeak = lpeak/lflat

peak

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de #�$� ��������� % ���

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�=������

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=θflat

peak

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R

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lm = lA(m − φm) = lA(m − φ − δφm) rf.ffs

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= �.���(Ωmh�)−�(zdec/����) rf.fis

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< f(ξ(n)) >= lims→∞

s[f(ξ�(n)) + f(ξ�(n)) + ... + f(ξs(n))] rj.ejs

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P (T ≤ ξ ≤ T + dt, n) = p(T, n)dT

P (T, n) =∫ T

−∞p(T ′, T )dT ′ rj.egs

��� �"�� v�Z�� '�1&;� ��>? $�� ��Z"�� `�#

P (−∞ ≤ ξ ≤ +∞, n) =∫ +∞

−∞p(T, n)dT = � rj.eds

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〈f(ξ, n)〉 =∫ +∞

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×p(Tm−�, nm−�|Tm−�, nm−�; ...; T�, n�)

×... × p(T�, n�|T�, n�) rj.fms

� "�� `�# �� n� ��5 �� T� ���+� �&��� P�# '�1&;� ��>? p(T�, n�|T�, n�) � 1� rj.fms ����<� ��

)� �&�S1� ��>? T��� \�; �� P�# '�1&;� ��>? T��� ��� .���� ���� !�# �� I n� ��5 �� T� ���+�

��#� $� � ��� ��0 �� -���+%

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p(T�, n�)rj.fJs

m-joint probability density function��

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p(Tm, nm; Tm−�, nm−�; ...T�, n�) =∫

p(Tm, nm; Tm−�, nm−�; ...T�, n�)dT�

p(T�, n�) =∫

p(T�, n�|T�, n�)p(T�, n�)dT� rj.ffs

��#� @��<� ��� ��0 �� -���+% m '�1&;� ��>? T��� �� (��O&�� �� -���+% m >&�S1� T���

〈ξm(nm)ξm−�(nm−�)...ξ�(n�) =∫

T� × ... × Tmp(Tm, nm; Tm−�, nm−�; ...T�, n�)dT�...dTm

rj.fis

���0 �� Zξ(λ, n) !���4[�� T��� @��<� ��

Zξ(λ, n) = 〈exp(iλξ(n))〉 =∫ +∞

−∞exp(iλT )p(T, n)dT, rj.fjs

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p(T, n) =∫

exp(−iλT )Zξ(λ, n)dλ, rj.fgs

correlation angular scale��

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p(Tm, nm|Tm−�, nm−�; ...T�, n�) = p(Tm, nm) rj.fKs

��� ���� ��� ��0 �� rj.fms ����<� ������"�

p(Tm, nm, Tm−�, nm−�; ...T�, n�) = p(Tm, nm) rj.ims

moment��

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p(Tm, nm; Tm−�, nm−�; ...T�, n�) = p(Tm, nm|Tm−�, nm−�)p(Tm−�, nm−�|Tm−�, nm−�)

×... × p(T�, n�|T�, n�)p(T�, n�) rj.ies

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�;�) $R W%����) ) �O0 ��%R U�&� �� ���(��� -�� ������"� = "�� Dp; �� ′ ��/� ���� -��� .�#���

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p�k(Tk, nk; · · · ; T�, n�|T�, n�)dTkdTk−� · · · dT� = Prob{T (ni) ∈ [Ti, Ti + dTi]

for i = �,�, · · ·k and T (n�) = T�}. rj.ifs

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×pk−��

(Tk−�, nk−�|Tk−�, nk−�; ...; T�, n�; T�, n�)

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�= "� @��<� ��� ��0

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σ�PDF + σ��−joint

dT�dT�, rj.igs

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X�(xN)σN

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eem #���3 #�,; � � � �� B �:��

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σ(ai)� = cii re.ees

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eee #���3 #�,; � � � �� B �:��

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.�% =∫ x+�σ

x−�σ

e−x−x

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δa.[α].δa} re.eis

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.�#���

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eei #���3 #�,; � � � �� B �:��

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�=���� e � MX �� �5� �� .= "� � <�

χ� = −� ln(P (a�, a�, ..., aM )) + �

N∑i=�

ln(σi

√�π)

=(a� − a�

�)�

σ(a�)�+ const re.ejs

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χ� = χ�min +∂�χ�

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= χ�min +(a� − a�

�)�

σ(a�)�re.egs

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+ nσ(a�) ���

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δai.αij .δaj} re.eds

)

Δχ� =M∑ij

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L(a|(x, y)) =∫

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LB(a|(x, y)) = L(a|(x, y))Π(a) re.fjs

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∫L(a, λ|(x, y))Π(a)Π(λ)dλ re.fgs

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C �� �

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k=�

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s

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∑si=� i∑s

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s [∑s

i=� i]�,

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s

s∑i=�

i,

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s

s∑i=�

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)

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⟨x(i)�

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s∑i,j=�

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=σ�

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),

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(i�H + j�H

)− σ�

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�H + �−

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∫ �

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∼ σ�s�H+�

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). rf.Jes

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�H + �

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− �σ�

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i=�

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]�⟩

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i=�

iY (i)

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i=�

iY (i)s∑

i=�

Y (i)

⟩,

=A

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s�D +

��

s�C. ri.js

) C !B !A -��� 1� &���� $"�� .=��(��� Dp; �� ��� (�Z"I (��1# D�<� �� ν W��%� ���� -��� �Z"�� ��

�"<� fBm -�� G� ) (�#T15 fBm -�� G� increment �� = %��� .= "� � <� &���� ��D

x(i) = Y (i) − Y (i − �)

u(i) = x(i) − x(i − �) ri.gs

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efg �)���! # () #�1 HURST #�<! �1 DFA /� ;��3 ��1 #�<! =�1 >�?*�� D �:��

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� �ejJ��� ���� ���

〈x(i)x(j)〉 =σ�

[i�H + j�H − |i − j|�H

],

〈Y (i)Y (j)〉 =σ�

(H + �)�(ij)H+� ri.ds

������"�

〈[Y (i) − Y (k)]�〉 = σ�|i − k|�H ri.Ls

$"�� .�eed����⟨Y (i)�

⟩= σ�

(H+�)�i�(H+�) ��0 �� <1Z� fBm -�� W%����) .σ� =

⟨u(i)�

⟩�Z"�� ��

��#� ,��S� ��� ��0 �� ri.js ����<� ri.ds ) ri.es �_��<� G1� ��

⟨[F�(s; ν)

]⟩ν

= CH(s)�(H+�) ri.Ks

��#��� ��� ��0 �� CH ��

CH =σ�

(�H + �)(H + �)�− �σ�

[(H + �)(H + �)]�

− ��σ�

[(H + �)(H + �)]�+

��σ�

(H + �)�(H + �)(H + �). ri.Jms

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E �� �

�6� � DFA �� 5"�� �#"� �6� 7�� 89��

�":�; �. �, ��# �� ���� Hurst

� "� ��� .=��)R ���� Hurst -�1% ) DFA1 6)� �� (��R ���� -�1% � � ����� =���� �4X �1�X ��� ��

��� ���� fBm -�� G� Y (i, j) -�� ��4%R �� .�#�� ���I "<� fGn -�� G� x(k, l) ��

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WI .�#� �S��3� �:Z� ��0

bν =

∑s,mi,j=� Y (i, j)i − �

s×m

∑s,mi,j=� Y (i, j)

∑smi,j=� i∑s,m

i,j=� i� − �

s×m

[∑s,mi,j=� i

]� ,

�∑s,m

i,j=� Y (i, j)i − s�

∑s,mi,j=� Y (i, j)

m × s�/��

cν =

∑s,mi,j=� Y (i, j)j − �

s×m

∑s,mi,j=� Y (i, j)

∑smi,j=� j∑s,m

i,j=� j� − �

s×m

[∑s,mi,j=� j

]� ,

�∑s,m

i,j=� Y (i, j)j − m�

∑s,mi,j=� Y (i, j)

s × m�/��

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efL #�@1A � �)�� # () #�1 HURST #�<! �1 DFA /� ;��3 ��1 #�<! =�1 >�?*�� E �:��

aν =�

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s∑i=�

m∑j=�

Y (i, j) − bν

s × m

s∑i=�

m∑j=�

i − cν

s × m

s∑i=�

m∑j=�

j

� �

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s,m∑i,j=�

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�− cνm

�, rj.es

�"<� Fq(s, m) � 1� @��<� �� �5� ��

Fq(s, m) ≡{

�Nsm

�Nsm∑ν=�

[F�(s, m; ν)

]q/�}�/q

, rj.fs

��#� @��<� ��� ��0 �� F�(s, m; ν) $R �� ��

F�(s; ν) =�

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[Yν(i, j) − yν(i, j)]� , rj.is

�=���� rj.is ) rj.fs !rj.es �_��<� �� �5� ��

⟨[F�(s, m; ν)

]⟩=

⟨�

s × m

s,m∑i,j=�

[Y (i, j) − a − bi − cj]�⟩

�⟨

s × m

s,m∑i,j=�

Y (i, j)�⟩

+⟨a�⟩

+s�

⟨b�⟩

+m�

⟨c�⟩

−�⟨

a

s × m

s,m∑i,j=�

Y (i, j)

⟩− �

⟨b

s × m

sm∑i,j=�

iY (i, j)

−�⟨

c

s × m

sm∑i,j=�

jY (i, j)

⟩+ s 〈ab〉 + m 〈ac〉 +

s × m

�〈bc〉 .

rj.js

-�� G� �� fBm -�� G� = %��� .=��(��� Dp; �� ��� (�Z"I (��1# D�<� �� ν W��%� ���� -��� �Z"�� ��

��#� ���� ��� ��0 �� fGn

Y (i, j) = (ij)Hx rj.gs

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efK #�@1A � �)�� # () #�1 HURST #�<! �1 DFA /� ;��3 ��1 #�<! =�1 >�?*�� E �:��

)

Y (i, j) − Y (k, l) = Y (i, l) + Y (k, j) + |i − k|H |j − l|Hx

=[(il)H + (kj)H + |i − k|H |j − l|H]x, rj.ds

������"�

〈[Y (i, j) − Y (k, l)]�〉 = σ�[(il)H + (kj)H + |i − k|H |j − l|H]� , rj.Ls

$��� U�� �� .��� �eed�⟨Y (i, j)�

⟩= σ�(ij)�H ��0 �� fBm -�� W%����) .σ� =

⟨x(i, j)�

⟩�Z"�� ��

��� ���� ��� ��0 �� Y (i, j) -�� >&�S1� T��� �� ��� b[�� ���� �� rj.Ls ����<� �? �1�

〈Y (i, j)Y (k, l)〉 =σ�

�[(ij)�H + (kl)�H − (ik)�H − (jl)�H

−�|i − k|H |j − l|H [(il)H + (kj)H]

−|i − k|�H |j − l|�H − �(ijkl)H ], rj.Ks

��#� ,��S� ��� ��0 �� rj.js ����<� rj.Ks ) rj.es �_��<� G1� �� $"��

⟨[F�(s, m; ν)

]⟩ν

= CH(s × m)�H , rj.Jms

��#� @��<� ��� ��0 �� CH � 1� )

CH =�

�σ�[

��+ H (��+ H [���+ H(�+ H)(�+ �H(�+ H))])(�+ H)�(�+ H)�(�+ �H)�

+{− ��(�Γ[� + H ]� + Γ[�+ �H ])(�+ H)�(�+ H)Γ[�+ �H ]Γ[�+ �H ]�

×{�Γ[�+ �H ](Γ[�+ H ]

{�H�Γ[H ] + (�+ H)Γ[�+ H ] + Γ[�+ H ]

}+ (�+ H)Γ[�+ �H ]

)

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.ir

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eim #�@1A � �)�� # () #�1 HURST #�<! �1 DFA /� ;��3 ��1 #�<! =�1 >�?*�� E �:��

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Γ[�+ �H ]

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]+

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)Γ[+ �H ]

))}].rj.JJs

��#��� ��� ��0 �� Γ(x) T��� !rj.JJs ����<� ��

Γ(x) ≡ (x − �)! =∫ ∞

tx−�e−tdt rj.Jes

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F �� �

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) ��� �� ��R �� �5� -�� ���)�3� \S� � � �# �3� -�� �R -����� � S��3� �1�X �� �� �� %�1�

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T��� 7� �P �� �� -���+%�� >&�S1� T��� �S��3� !$R ��� -��� 6)� ����(��� .��� ������� -�(��<��q�

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�� 1�� 9+% �� Cl-���)�� $�� @ P � 1� � "~1� .���� ���%�" � TX�) $�� �� ��� �� �� -� �= 14�

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eie ��������� #�,��<� #�5;/��!� � �!���� #�89 F �:��

\S� �� �� �>�� -��� 8% G� !���4� -����� ) -� �(���%� -����>&�� �� #�% -����� �"%�� ����

�� ��%R �X� �� &���� �� ���� �5) %�� � -��� Y�% �� != "�� ��5 ����� (�1� �&�� )� �� �� $R 9& 1��

.=�����o� $R @ 0� �� ) (��� �S��3� �� ��� 8% ��� ��Cl -��� =���� �4X ���+� ��� �� .��)R v��;

�= �� U�� -)�� ���<&� T��� \�; �� �� ΔT/T ��� : � ) ��� �� ��� >� �B% �� �>�� ���

ΔT

T(θ, φ) =

∑l=�

l∑m=−l

almYlm(θ, φ) rg.Js

)

alm =∫

ΔT

T( ˆθ, φ)Ylm(θ, φ)dΩ rg.es

�= �� ���X �B% ��� �� $R >&�S1� ���

〈alma∗l′m′〉ansmble =

∫dΩdΩ′Ylm(θ, φ)Y ∗

l′m′(θ′, φ′)⟨

ΔT

T(θ, φ)

ΔT

T(θ′, φ′)

⟩rg.fs

��#�� = ��� (�>%R C(n, n′) = C(n.n′) = "<� -���R -��>%��1� ��� ��

C(n, n′) =⟨

ΔT

T(n)

ΔT

T(n′)⟩

= C(n.n′) =�

�π

∑l

(�l + �)ClPl(cos(γ)) rg.is

��� ��� ����� rg.fs ����<� \ ��� ��� �� .��� cos(γ) = n.n′ !rg.is ����<� ��

〈alma∗l′m′〉ansmble = δll′δmm′Cl rg.js

��� ��� ���S� ��Cl -��� � 1[� ���&�� .�"�R� ���� 7S���%R -� �U�&� �� ��Cl �� ��� 2]�) ������"�

Cl =�

�l + �

+l∑m=−l

|alm|� rg.gs

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eif ��������� #�,��<� #�5;/��!� � �!���� #�89 F �:��

-���R -��>%��1� �M; �� ������"� !���� �5) �< SP �� $�� � -��" �� 9��� ��+% G� U+� �� �Z%R ��

!l �� -��� .��� Y�Z%� �� -� �U�&� ) ���� ��<&� -��,S���%R !�7�)� -���)� �3� !��+% $��)� �� $���

=���� ��B&%� ) (�� =� ���%1% ���<� !rl = π/γs G?� -��l -��� ������"� ! �#�� = ��� �%1% �l + � ���<�

��� ��� ����� Cl �� <� -��� .����� 9��:�� -���)�� $�� @ P � >%� � �� <� -��� ��

σ�l = 〈C�l 〉 − 〈Cl〉� rg.ds

��� ��� ����� rg.ds ����<� ���� �1� ')� �715 !Wick � MX �� (��O&�� �� !�#�� �� alm ���

〈C�l 〉 =

(�l + �)�∑mm′

〈alma∗lm〉 〈alm′a∗

lm′〉

= �C�l rg.Ls

��� ��� ����� rg.ds ����<� ���� �1� Y)� �715 � "~1�

〈Cl〉 =�

�l + �

∑m

〈alma∗lm〉 = 〈alma∗

lm〉 rg.Ks

�������"�

〈Cl〉� = C�l rg.Jms

��� ��� ����� ��Cl �� <� -��� )

σ�l = �C�l rg.JJs

�=���� !=���� � >%� � �� <� -��� ) �� <� -��� `�S��� ��� �� A �� I �� �� �~%R *��� ��

σ�m =σ�

N=

�l + �C�

l rg.Jes

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eii ��������� #�,��<� #�5;/��!� � �!���� #�89 F �:��

(�l + �)−�/� S�% � <�X Y�� = �� 9��� ��Cl -� �(���%� �� �� �>�� -����� (���%� �� �� ��� 2]�)

(� ��% %�� � -��� �V;/�0� � <�X Y�� ��� .�#� \S� ��Cl � X� ���+� $��)R ���� �� -�5 ���)�3�

�� ��� ��� =�� &���� : % ��"���I 2�� ����R -)� �� ��� : � ) ��� $�� � $�� �� ���� -��� .�#�

.��� (��� ���� �� $R ,4O� �egj� T5�� �� �� =��)R v��; �� -���+% �� >&�S1� T��� -���

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���<��

[1] G. Lazarides, lectures given at the Corfu Summer Institute on Elementary Particle Physics

(South European School on Elementary Particle Physics), 6-26 September 1998, Kerkyra,

Greece (to appear in the proceedings) Report-no: UT-STPD-4/99, arXive:hep-ph/9904502

[2] A. R. Liddle and D. H. Lyth, Cosmological Inflation and Large Scale Structure, Cambridge

University Press, 1998.

[3] T. Padmanabhan, Structure Formation in the Universe, Cambridge Univ. Press 1993.

[4] D. Raine and T. Thomas, An Introduction to the Science of Cosmology, IoP Press, 2002.

[5] S. Dodelson, Modern Cosmology, Academic Press, (2003).

[6] N. Breton, J.L. Cervantes-Cota, M. Salgado , The early universe and observational cos-

mology, Springer (2004).

[7] S. Perlmutter, M. S. Turner and M. White, Phys. Rev. Lett. 83, 670, (1999).

eij

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eig .��!B���

[8] B. P. Schmidt et al., Astrophys. J. 507, 46 (1998).

[9] J. L. Tonry et al., Astrophys. J. 594, 1 (2003).

[10] B. J. Barris et al., Astrophys. J. 602, 571 (2004).

[11] D. N. Spergel et al., [WMAP Collaboration], ApJ Suppl. 148, 175 (2003).

[12] A. G. Riess et al., Astrophys. J. 607, 665 (2004).

[13] Fixsen DJ et al., ApJ 473:576 (1996)

[14] A. Guth, Phys. Rev. D, 23, 2 (1981)

[15] R. Brandenberger, arXiv:astro-ph/9411049.

[16] J. Rich, Fundamental of Cosmology, Springer Press, 2001.

[17] M. Trodden and S. M. Carroll, arXiv:astro-ph/0401547.

[18] S. D. Landy, S. A. Shectman, H. Lin, R. P. Kirshner, A. A. Oemler D. Tucker, 456, L1

(LCRS), 1996.

[19] V. de Lapparent, M. Geller and J. Huchra, Ap. J. (Lett) 302, L1 (1986).

[20] R. Brandenberger, Invited lectures at the International School on Cosmology, Kish Island,

Iran, Jan. 22 - Feb. 4 1999, to be publ. in the proceedings (Kluwer, Dordrecht, 2000),

arXiv:hep-ph/9910410.

By:

Movah

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eid .��!B���

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Abstract

Standard model of cosmology including inflation is a successful model to describe the evolution

of our universe. Recent observation of cosmic microwave background radiation and supernovae type

Ia show that the expansion rate of universe is positive and almost �/� of amount of total density

of matter in universe is formed by an exotic matter with negative pressure with the equation of

state of w� < −�/�. In CDM, cosmological constant can describe this equation of state. It is well-

known that this constant has two main problems: Cosmic Coincidence and Fine tuning. The first

one states that since the energy densities of dark energy and dark matter scale so differently during

the expansion of the Universe, why are they nearly equal today?. The second one demonstrates that

the theoretical cosmological constant with ��� orders of magnitude larger than the observed value

of ��−�� GeV�. There are some efforts to solve two above problems, one of them is variable dark

energy model or quintessence model. In this thesis I investigate two phenomenological quintessence

models. We compare these model with large structure formation CMB shift parameter, Baryonic

acoustic oscillation , distance modulus of supernovea type Ia and growth index of large scale structure

observations and put constraints on the free parameters of model. In the second part, we analysis

statistically the CMB data of WMAP experiment. We show that temperature fluctuations at the last

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scattering surface is markov chain with markov angular scale, ΘMarkov = �.��+�.��−�.��

. The Langevin

equation to regenerate temperature field is given. To investigate the Gaussianity and Statistical

Isotropy of temperature fluctuations, some fractal analysis methods are used.

Keywords: Standard model, Structure formation, Drak energy, CMB, Statistical isotropy, Gaussian-

ity.

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�Sharif University of Technology

Department of Physics

Submitted in Partial Fulfillment of the Requirements for the Degree of PhD

Field: Complex Systems and Cosmology

Cosmology of Variable Dark Energy and

Statistical Analysis of Cosmic Microwave

Background Radiation

by

M. Sadegh Movahed

Under supervision of

Dr. Sohrab Rahvar & Dr. M.R. Rahimi Tabar

November 2006

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