❊st✐♠❛t✐✈❛s ❞❡ ♣❛râ♠❡tr♦s
❝♦s♠♦❧ó❣✐❝♦s ♣❛r❛ ♦ ❉❛r❦ ❊♥❡r❣②
❙✉r✈❡②
❚❊❙❊
❛♣r❡s❡♥t❛❞❛ ❛♦ ■♥st✐t✉t♦ ❞❡ ❋ís✐❝❛ ❚❡ór✐❝❛
❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛✱ ❇r❛s✐❧
❡♠ ❝✉♠♣r✐♠❡♥t♦ ❛♦s r❡q✉✐s✐t♦s ♣❛r❛ ❛ ♦❜t❡çã♦ ❞♦ tít✉❧♦ ❞❡
❉♦✉t♦r
❙❡t❡♠❜r♦ ❞❡ ✷✵✶✶
❈♦♠✐ssã♦ ❊①❛♠✐♥❛❞♦r❛
❉r✳ ❘♦❣ér✐♦ ❘♦s❡♥❢❡❧❞ ■❋❚✲❯◆❊❙P
❉r✳ ❏ú❧✐♦ ❈❡sár ❋❛❜r✐s ❉❋■❙ ✲ ❯❋❊❙
❉r✳ ■♦❛✈ ❲❛❣❛ ■❋✲❯❋❘❏
❉r✳ ▼❛r❝♦s ❱✳ ▲✐♠❛ ❉❋▼❆✲❯❙P
❉r✳ ❏♦sé ●❡r❛❧❞♦ P❡r❡✐r❛ ■❋❚✲❯◆❊❙P
❖r✐❡♥t❛❞♦r✿ ❉r✳ ❘♦❣ér✐♦ ❘♦s❡♥❢❡❧❞
❉❡❞✐❝♦ ❡st❡ tr❛❜❛❧❤♦ ♣❛r❛ ♦ ♠❡❧❤♦r ♣r❡s❡♥t❡ q✉❡ ❣❛♥❤❡✐ ❞✉r❛♥t❡ ♦ s❡✉ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦✿
♠❡✉ ❛♠♦r ❇r✉❝✐t♦✳
❘❡s✉♠♦
◆❡st❛ t❡s❡ ❡st✉❞❛♠♦s ♣r❡✈✐sõ❡s s♦❜r❡ ♦s ❡rr♦s ♥♦s ♣❛râ♠❡tr♦s ❝♦s♠♦❧ó❣✐❝♦s ✉s❛♥❞♦ ❛ ❢♦r♠❛ ✐♥t❡❣r❛❧ ❞❛ ❢✉♥çã♦ ❞❡ ❝♦rr❡❧❛çã♦ ❛♥❣✉❧❛r ❞❡ ❞♦✐s ♣♦♥t♦s ❡♠ ❞✐❢❡r✲ ❡♥t❡s ❝❡♥ár✐♦s ♣❛r❛ ♦ ♣r♦❥❡t♦ ❉❛r❦ ❊♥❡r❣② ❙✉r✈❡②✳ ❖ ♠♦❞❡❧♦ ❛❞♦t❛❞♦ t❡♠ ✷✻ ♣❛râ♠❡tr♦s ❡ ✐♥❝❧✉✐ ❡❢❡✐t♦s ❞❡ ❞✐st♦rçã♦ ♥♦ r❡❞s❤✐❢t✱ ❡rr♦s ❣❛✉ss✐❛♥♦s ❞❡ r❡❞s❤✐❢t ❢♦t♦♠étr✐❝♦✱ ✈✐és ❞❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❣❛❧á①✐❛s ❡ ♠❛tér✐❛ ❡s❝✉r❛ ❡ ♥ã♦✲❧✐♥❡❛r✐❞❛❞❡ ♥♦ ❡s♣❡❝tr♦ ❞❡ ♣♦tê♥❝✐❛✳ ❆ ♠❛tr✐③ ❞❡ ❋✐s❤❡r ❢♦✐ ❝♦♥str✉í❞❛ ✉s❛♥❞♦ ❛ ♠❛tr✐③ ❞❡ ❝♦✈❛r✐â♥❝✐❛ ❝♦♥s✐❞❡r❛♥❞♦ ❛ ❝♦rr❡❧❛çã♦ ❡♥tr❡ ❞✐❢❡r❡♥t❡s ❢❛✐①❛s ❞❡ r❡❞s❤✐❢t✳
▼♦str❛♠♦s q✉❡✱ s♦❜r❡ ❛❧❣✉♠❛s ❤✐♣ót❡s❡s✱ ♦ ❉❛r❦ ❊♥❡r❣② ❙✉r✈❡② t❡♠ ♦ ♣♦✲ t❡♥❝✐❛❧ ❞❡ ✈✐♥❝✉❧❛r ♦ ♣❛râ♠❡tr♦ ❞❛ ❡q✉❛çã♦ ❞❡ ❡st❛❞♦ ❞❛ ❡♥❡r❣✐❛ ❡s❝✉r❛ w ❡ ♦
♣❛râ♠❡tr♦ ❞❛ ❞❡♥s✐❞❛❞❡ ❞❡ ♠❛tér✐❛ ❡s❝✉r❛ ❢r✐❛ Ωcdm ❝♦♠ ✐♥❝❡rt❡③❛ ❞❡ 21% ❡ 13% r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ◗✉❛♥❞♦ ❝♦♠❜✐♥❛♠♦s ✐♥❢♦r♠❛çõ❡s ❞❡ ♦✉tr❛s ♦❜s❡r✈❛çõ❡s ❛
♣r❡❝✐sã♦ ♥❛ ❞❡t❡r♠✐♥❛çã♦ ❞❡st❡s ♣❛râ♠❡tr♦s ❛✉♠❡♥t❛ ♣❛r❛ 11% ❡ 4% r❡s♣❡❝t✐✈❛✲
♠❡♥t❡✳
P❛❧❛✈r❛s ❝❤❛✈❡s✿ ❆❝❡❧❡r❛çã♦ ❞♦ ✉♥✐✈❡rs♦❀ ❊♥❡r❣✐❛ ❊s❝✉r❛❀ ♣❛râ♠❡tr♦s ❝♦s✲ ♠♦❧ó❣✐❝♦s❀ ❉❛r❦ ❊♥❡r❣② ❙✉r✈❡②✳
➪r❡❛s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦✿ ❈♦s♠♦❧♦❣✐❛✳
❆❜str❛❝t
■♥ t❤✐s t❤❡s✐s✱ ✇❡ st✉❞② ❢♦r❡❝❛sts ♦❢ ❝♦s♠♦❧♦❣✐❝❛❧ ♣❛r❛♠❡t❡rs ❢r♦♠ t❤❡ ✉♣❝♦♠✲ ✐♥❣ ❉❛r❦ ❊♥❡r❣② ❙✉r✈❡② ♣r♦❥❡❝t ♦❜t❛✐♥❡❞ ✉s✐♥❣ t❤❡ ❢✉❧❧ s❤❛♣❡ ♦❢ t❤❡ ✷✲♣♦✐♥t ❛♥✲ ❣✉❧❛r ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥ ✐♥ ❞✐✛❡r❡♥t s❝❡♥❛r✐♦s✳❚❤❡ ❛♥❣✉❧❛r ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥ ♠♦❞❡❧ ❛❞♦♣t❡❞ ❤❛s ✷✻ ♣❛r❛♠❡t❡rs ❛♥❞ ✐♥❝❧✉❞❡s t❤❡ ❡✛❡❝ts ♦❢ ❧✐♥❡❛r r❡❞s❤✐❢t s♣❛❝❡ ❞✐st♦rt✐♦♥✱ ♣❤♦t♦♠❡tr✐❝ r❡❞s❤✐❢t ❣❛✉ss✐❛♥ ❡rr♦rs✱ ❣❛❧❛①② ❜✐❛s ❛♥❞ ♥♦♥✲❧✐♥❡❛r✐t✐❡s ✐♥ t❤❡ ♣♦✇❡r s♣❡❝tr✉♠✳❚❤❡ ❋✐s❤❡r ✐♥❢♦r♠❛t✐♦♥ ♠❛tr✐① ✐s ❝♦♥str✉❝t❡❞ ✇✐t❤ t❤❡ ❢✉❧❧ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐①✱ ✇❤✐❝❤ t❛❦❡s ✐♥t♦ ❛❝❝♦✉♥t t❤❡ ❝♦rr❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ♥❡❛r❜② r❡❞s❤✐❢t s❤❡❧❧s ❛r✐s✐♥❣ ❢r♦♠ t❤❡ ♣❤♦t♦♠❡tr✐❝ r❡❞s❤✐❢t ❡rr♦r✳❲❡ s❤♦✇ t❤❛t ✉♥❞❡r s♦♠❡ r❡❛s♦♥❛❜❧❡ ❛ss✉♠♣t✐♦♥s t❤❡ ❉❛r❦ ❊♥❡r❣② ❙✉r✈❡② s❤♦✉❧❞ ❜❡ ❛❜❧❡ t♦ ❝♦♥str❛✐♥ t❤❡ ❞❛r❦ ❡♥❡r❣② ❡q✉❛t✐♦♥ ♦❢ st❛t❡ ♣❛r❛♠❡t❡r w ❛♥❞ t❤❡ ❝♦❧❞ ❞❛r❦ ♠❛tt❡r ❞❡♥s✐t②
Ωcdm ✇✐t❤ ❛ ♣r❡❝✐s✐♦♥ ♦❢ t❤❡ ♦r❞❡r ♦❢21%❛♥❞13%r❡s♣❡❝t✐✈❡❧② ❢r♦♠ t❤❡ ❢✉❧❧ s❤❛♣❡
♦❢ t❤❡ ❛♥❣✉❧❛r ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥ ❛❧♦♥❡✳❲❤❡♥ ❝♦♠❜✐♥❡❞ ✇✐t❤ ♣r✐♦rs ❢r♦♠ ♦t❤❡r ♦❜s❡r✈❛t✐♦♥s t❤❡ ♣r❡❝✐s✐♦♥ ✐♥ t❤❡ ❞❡t❡r♠✐♥❛t✐♦♥ ♦❢ t❤❡s❡ ♣❛r❛♠❡t❡rs ✐♥❝r❡❛s❡ t♦
11% ❛♥❞ 4% r❡s♣❡❝t✐✈❡❧②✳
❙✉♠ár✐♦
❘❡s✉♠♦ ✐①
❆❜str❛❝t ①✐
❆❣r❛❞❡❝✐♠❡♥t♦s ①✈
▲■❙❚❆ ❉❊ ❋■●❯❘❆❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ▲■❙❚❆ ❉❊ ❚❆❇❊▲❆❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺
✶ ■♥tr♦❞✉çã♦ ✼
✷ ❆ ❡①♣❛♥sã♦ ❞♦ ✉♥✐✈❡rs♦ ✶✶
✷✳✶ ❈♦s♠♦❧♦❣✐❛ ❜ás✐❝❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✶✳✶ ❉✐stâ♥❝✐❛s ❝♦s♠♦❧ó❣✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✷ ❆❝❡❧❡r❛çã♦ ❝ós♠✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✸ ▼♦❞❡❧♦s ♣❛r❛ ❛ ❛❝❡❧❡r❛çã♦ ❝ós♠✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✸✳✶ ❈♦♥st❛♥t❡ ❈♦s♠♦❧ó❣✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✸✳✷ ▼♦❞❡❧♦s ❝♦♠ ❝❛♠♣♦ ❡s❝❛❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✸✳✸ ●r❛✈✐t❛çã♦ ♠♦❞✐✜❝❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✸ ❆ ❡str✉t✉r❛ ❡♠ ❣r❛♥❞❡ ❡s❝❛❧❛ ❞♦ ✉♥✐✈❡rs♦ ✷✺ ✸✳✶ ❉✐♥â♠✐❝❛ ❞❛s ♣❡rt✉r❜❛çõ❡s ❞❡ ♠❛tér✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✸✳✷ ❋✉♥çã♦ ❞❡ ❝♦rr❡❧❛çã♦ ❡s♣❛❝✐❛❧ ❞❡ ❞♦✐s ♣♦♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✸✳✸ ❋✉♥çã♦ ❞❡ ❝♦rr❡❧❛çã♦ ❛♥❣✉❧❛r ❞❡ ❞♦✐s ♣♦♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✳✹ ❖s❝✐❧❛çõ❡s ❛❝úst✐❝❛s ❜❛r✐ô♥✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✹ ▼♦❞❡❧❛❣❡♠ ❞❛ ❋✉♥çã♦ ❞❡ ❈♦rr❡❧❛çã♦ ❞❡ ❞♦✐s ♣♦♥t♦s ✹✶
✹✳✶ ❱✐és ❡♥tr❡ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❣❛❧á①✐❛s ❡ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ♠❛tér✐❛ ❡s❝✉r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✹✳✷ ❉✐st♦rçã♦ ♥♦ r❡❞s❤✐❢t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✹✳✷✳✶ ❆♣r♦①✐♠❛çã♦ ❞❡ ♦❜s❡r✈❛❞♦r ❞✐st❛♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✹✳✸ ❈♦rr❡çõ❡s ♥ã♦✲❧✐♥❡❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽
✹✳✹ ❋✉♥çã♦ ❞❡ s❡❧❡çã♦ ❢♦t♦♠étr✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✹✳✺ ❈♦♥s❡q✉ê♥❝✐❛ ❞♦s ❡❢❡✐t♦s ❡♠ ω(θ) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸
✹✳✻ ■♥❢♦r♠❛çõ❡s ❝♦s♠♦❧ó❣✐❝❛s ❡♠ ω(θ) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸
✺ ▼♦❞❡❧❛❣❡♠ ❞❛ ♠❛tr✐③ ❞❡ ❝♦✈❛r✐â♥❝✐❛ ❞❛ ❢✉♥çã♦ ❞❡ ❝♦rr❡❧❛çã♦
❛♥❣✉❧❛r ✺✼
✺✳✶❊s♣❡❝tr♦ ❞❡ ♣♦tê♥❝✐❛ ❛♥❣✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✺✳✷ ❈♦rr❡çõ❡s ♥♦ ❡s♣❡❝tr♦ ❞❡ ♣♦tê♥❝✐❛ ❛♥❣✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✺✳✸ ❋✉♥çã♦ ❞❡ ❝♦rr❡❧❛çã♦ ❛♥❣✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✺✳✹ ▼❛tr✐③ ❞❡ ❝♦✈âr✐❛♥❝✐❛ ♣❛r❛ ✉♠❛ ❢❛✐①❛ ❞❡ r❡❞s❤✐❢t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✺✳✺ ▼❛tr✐③ ❞❡ ❝♦✈âr✐❛♥❝✐❛ ❡♥tr❡ ❢❛✐①❛s ❞❡ r❡❞s❤✐❢t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✻ ❊st✐♠❛t✐✈❛s ❞❡ ♣❛râ♠❡tr♦s ❝♦s♠♦❧ó❣✐❝♦s ✻✼ ✻✳✶▼❛tr✐③ ❞❡ ❋✐s❤❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ✻✳✶✳✶ ▼❛tr✐③ ❞❡ ❋✐s❤❡r ♣❛r❛ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❣❛❧á①✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✻✽ ✻✳✶✳✷ Pr♦♣r✐❡❞❛❞❡s ❞❛ ♠❛tr✐③ ❞❡ ❋✐s❤❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ✻✳✷ ▼❛tr✐③ ❞❡ ❋✐s❤❡r ♣❛r❛ ❛s ♠❡❞✐❞❛s ❞❡ ω(θ)♥♦ ❉❛r❦ ❊♥❡r❣② ❙✉r✈❡② ✼✸
✻✳✷✳✶Pr♦❝❡❞✐♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✻✳✷✳✷ ▲❛r❣✉r❛ ❞❛ ❢❛✐①❛ ❞❡ r❡❞s❤✐❢t ❡ s❤♦t ♥♦✐s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺ ✻✳✷✳✸ ❈♦♥str✉çã♦ ❞❛ ♠❛tr✐③ ❞❡ ❋✐s❤❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼ ✻✳✷✳✹ ■♠♣❧❡♠❡♥t❛çã♦ ♥✉♠ér✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵ ✻✳✷✳✺ ❊st✐♠❛t✐✈❛s ❞♦s ♣❛râ♠❡tr♦s ❝♦s♠♦❧ó❣✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶ ✻✳✸ ▼❛tr✐③ ❞❡ ❋✐s❤❡r ✈❡rs✉s r❡❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✽
❈♦♥❝❧✉sõ❡s ❡ P❡r♣❡❝t✐✈❛s ✾✶
❆ ❋✉♥çã♦ ❞❡ ❝♦rr❡❧❛çã♦ ❡s♣❛❝✐❛❧ ♥♦ ❡s♣❛ç♦ ❞❡ r❡❞s❤✐❢t ✾✸
❇ ▼❛tr✐③ ❞❡ ❋✐s❤❡r ♥✉♠ér✐❝❛ ✾✼
❆❣r❛❞❡❝✐♠❡♥t♦s
P❛r❛ ♠✐♠ é ✉♠ ♣r❛③❡r ❡ ✉♠❛ ♥❡❝❡ss✐❞❛❞❡ ❡①♣r❡ss❛r t♦❞♦s ♠❡✉s ♣r♦❢✉♥❞♦s s❡♥t✐♠❡♥t♦s ❞❡ ❣r❛t✐❞ã♦ ♣❛r❛ ❝❛❞❛ ✉♠ ❞❡ ♠❡✉s ❛♠✐❣♦s✱ q✉❡♠ tê♠ ❢❡✐t♦✱ s❡♠ ❞ú✈✐❞❛ ❛❧❣✉♠❛✱ ❡st❛ t❡s❡ ❞❡ ❞♦✉t♦r❛❞♦ ♣♦ssí✈❡❧✱ ❝♦♠ s✉❛ ❛♣r♦♣r✐❛❞❛ ❡ ❣❡♥t✐❧ ❛❥✉❞❛ ❞✉r❛♥t❡ ❡ss❡s q✉❛tr♦ ❛♥♦s✳ ❍♦❥❡✱ ❛ ♠❛♥❡✐r❛ ♠❛✐s s✐♥❝❡r❛ q✉❡ ❡♥❝♦♥tr♦ ❞❡ ❝♦♠♣❛rt✐❧❤❛r ❛ ✐♠❡♥s❛ ❛❧❡❣r✐❛ q✉❡ ✐♥✈❛❞❡ ♠❡✉ ❝♦r❛çã♦ ♣♦r t❡r♠✐♥❛r ❡st❛ ❡t❛♣❛ tã♦ ✐♠♣♦rt❛♥t❡ ❞♦ ♠❡✉ ♣r♦❝❡ss♦ ❞❡ ❢♦r♠❛çã♦✱ é ❢❛③ê✲❧♦s ❝♦✲❛✉t♦r❡s ❞❡st❡ tr❛❜❛❧❤♦✳ P❛r❛ s❡r ♠❛✐s ♦❜❥❡t✐✈❛✱ ❞✐r❡✐ q✉❡ ♦ ♣r✐♠❡✐r♦ ❛✉t♦r é ♦ ❙❡♥❤♦r ♠❡✉ ❉❡✉s✱ q✉❡♠ t❡♠ ♠❡ ❢♦rt❛❧❡❝✐❞♦ ❝♦♠ s❡✉ ✐♠❡♥s♦ ❛♠♦r✳ ▼❡✉s ♣❛✐s✱ q✉❡♠ sã♦ ♦ ❛❧✐❝❡r❝❡ ❞❛ ♠✐♥❤❛ ✈✐❞❛✱ ❡stã♦ ✐♥❞✐s❝✉t✐✈❡❧♠❡♥t❡ ❡♠ t♦❞❛s ❛s ❧✐♥❤❛s ❞❡st❛ t❡s❡✳ ▼❡✉s ✐r♠ã♦s ❆❞r✐❛♥♦✱ ❉é❜♦r❛✱ ❖tá✈✐♦ ✭❡♠ ♠❡♠ór✐❛✮✱ ◆❛tá❧✐❛ ❡ ▲✐❧✐❛♥❡ ♣♦r s✉❛s ❛♠✐③❛❞❡s ❡ s❡✉ ❛♠♦r✳ ❆❣r❛❞❡ç♦ t❛♠❜é♠ ❛ ❙❛♥❞r❛✱ ❋❛❜rí❝✐♦✱ ▲❡❛♥❞r♦ P❡r❡✐r❛✱ ❆❞r✐❛♥❛✱ ▲❡❛♥❞r♦ ●✉❛r♥✐❡r✐ ❡ ❛s ♠✐♥❤❛s ❞✉❛s ♣r✐♥❝❡s❛s✱ ◆✐♥❛ ❡ ❊st❡r✱ q✉❡ ✈✐❡r❛♠ ❛❧❡❣r❛r ❛✐♥❞❛ ♠❛✐s ❛ ♠✐♥❤❛ ✈✐❞❛✳ ❯♠ ❛❣r❛❞❡❝✐♠❡♥t♦ ♠✉✐t♦ ❡s♣❡❝✐❛❧ ❞❡❞✐❝♦ ❛ ❢❛♠í❧✐❛ ❙❛♥❝❤❡③✲ ❱❡❣❛✱ ♣♦r s❡✉ s✐♥❝❡r♦ ❝❛r✐♥❤♦ ❡ ❛♣♦✐♦✳ ●r❛♥❞❡s ❛♠✐❣♦s q✉❡ ❢❛③❡♠ ♣❛rt❡ ❞❛ ♠✐♥❤❛ ❢♦r♠❛çã♦ ❝✐❡♥tí✜❝❛✱ t❛✐s ❝♦♠♦ ■❧②❛ ❙❤❛♣✐r♦✱ s❡♠♣r❡ ❡st❛rã♦ ♣r❡s❡♥t❡s ❡♠ ♠✐♥❤❛ ✈✐❞❛✳
▼✉✐t♦s ♦✉tr♦s ❞❡ ✈♦❝ês q✉❡ t❛♠❜é♠ t❡♠ ♣❡r❝♦rr✐❞♦ ❝♦♠✐❣♦ ♦s ❝♦♠♣❧✐❝❛❞♦s✱ ❡ ♠✉✐t❛s ✈❡③❡s s♦♠❜r✐♦s✱ ❧❛❜✐r✐♥t♦s ❧ó❣✐❝♦s q✉❡ r❡s✉❧t❛r❛♠ ♥❡st❡ tr❛❜❛❧❤♦✱ ❞❡❞✐❝♦ t♦❞❛ ♠✐♥❤❛ ❣r❛t✐❞ã♦✳ ❊♥tr❡ ❡st❡s✱ ❞❡s❡❥♦ ❛❣r❛❞❡❝❡r ❡s♣❡❝✐❛❧♠❡♥t❡ ❛ ❘♦❣ér✐♦ ❘♦s❡♥❢❡❧❞✱ ♣♦r s✉❛ ♦r✐❡♥t❛çã♦ ❡ ❝♦♥s❡❧❤♦s❀ ❛ ❋❡r♥❛♥❞♦ ❞❡ ❙✐♠♦♥✐ ♣♦r s✉❛ ♣❛rt✐❝✲ ✐♣❛çã♦ ❞✐r❡t❛ ❡ ❛t✐✈❛ ♥❛ r❡❛❧✐③❛çã♦ ❞❡st❛ t❡s❡❀ ❛ ❇r✉❝❡ ❙❛♥❝❤❡③ ♣♦r s✉❛s ❧❡✐t✉r❛s ❝✉✐❞❛❞♦s❛s ❡♥tr❡ ❛s ❧✐♥❤❛s ❡ ❡♥tr❡❧✐♥❤❛s ❞♦ t❡①t♦✳
❚❛♠❜é♠ ❞❡✈♦ ❛❣r❛❞❡❝❡r ❛ t♦❞❛ ❡q✉✐♣❡ ❞❡ ♣r♦❢❡ss♦r❡s✱ tr❛❜❛❧❤❛❞♦r❡s ❡ ❡s✲ t✉❞❛♥t❡s ❞♦ ■❋❚✳ ❊♠ ❡s♣❡❝✐❛❧ ❛❣r❛❞❡ç♦ ❛ ❊❧❛✐♥❡ ❋♦rt❡s✱ ▼❛r✐❛ ❞❡❧ P✐❧❛r✱ ▲✉✐s ❙♦r✐❛♥♦✱ ▲✉✐s ❨♦✉♥❣✱ ❆❧❜❡rt♦ ❙❛♥♦❥❛✱ ▲❡❛♥❞r♦ ❨♠❛✐✱ ▼❡✐r❡ ❈r✐st✐♥❛✱ ▼❛r❝❡❧♦ ■❛s♥✐s✇s❦✐✱ ❘♦s❛♥❡ ❙✐♠õ❡s✱ ▼❛r✐❛ ❆♣❛r❡❝✐❞❛ ❡ ♠✐♥❤❛ q✉❡r✐❞❛ ❏ô ♣❡❧❛ ❣r❛♥❞❡ ❛♠✐③❛❞❡ q✉❡ ❝♦♠♣❛rt✐❧❤❛♠♦s✳ ❆❣r❛❞❡ç♦ à ❈❆P❊❙ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦ q✉❡ ❢❡③ ♣♦ssí✈❡❧ ❛ r❡❛❧✐③❛çã♦ ❞❡st❡ ♣r♦❥❡t♦ ❡✱ ✜♥❛❧♠❡♥t❡✱ ❛ t♦❞♦ t✐♠❡ ❝✐❡♥tí✜❝♦ ❞♦ ❉❊❙✲ ❇r❛s✐❧✳
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✷✳✶ ❉✐stâ♥❝✐❛ ❞❡ ❧✉♠✐♥♦s✐❞❛❞❡ ♥♦ ✉♥✐✈❡rs♦ ♣❧❛♥♦ ❞♦♠✐♥❛❞♦ ♣♦r ♠❛tér✐❛ ♥ã♦ r❡❧❛t✐✈íst✐❝❛ ✭Ωm✮ ❡ ❡♥❡r❣✐❛ ❡s❝✉r❛ ✭ΩΛ✮ s❛t✐s❢❛③❡♥❞♦Ωm+ΩΛ= 1✳ ✶✽
✷✳✷ ❱í♥❝✉❧♦s ❞❡ ❙✉♣❡r♥♦✈❛ ❞♦ ❚✐♣♦ ■❛ ♥♦s ♣❛râ♠❡tr♦s Ωm ❡ ΩΛ✳ ❖
♣❛✐♥❡❧ ✭❛✮ ♠♦str❛ r❡s✉❧t❛❞♦s ❞❡ ✶✾✾✽ ❬✶✶❪ ❡ ♦ ♣❛✐♥❡❧ ✭❜✮ ❛♣r❡s❡♥t❛ r❡s✉❧t❛❞♦s ♠❛✐s r❡❝❡♥t❡s ❬✶✾❪✳ ❊♠ ❛♠❜♦s✱ ♦ ♠♦❞❡❧♦ ❝♦♠ Ωm = 1 é
❢♦rt❡♠❡♥t❡ ❡①❝❧✉í❞♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✸✳✶ ❊s♣❡❝tr♦ ❞❡ ♣♦tê♥❝✐❛ ❧✐♥❡❛r ❞❡ ♠❛tér✐❛✳ ❆ ❝✉r✈❛ ❝♦♥tí♥✉❛ r❡♣r❡✲
s❡♥t❛ P(k)✱ ❡♥q✉❛♥t♦ ❛ ❝✉r✈❛ tr❛❝❡❥❛❞❛ ♠♦str❛ P(k, z = 1)✳ ❆
❝✉r✈❛ ❧❛r❣❛ ♠♦str❛ ♦ ❡s♣❡❝tr♦ ❞❡ ♣♦tê♥❝✐❛ P(k) ❝♦♠ ❛✉sê♥❝✐❛ ❞❛s
♦s❝✐❧❛çõ❡s ❞❡ ❜ár✐♦♥s ✭❞✐s❝✉t✐❞❛ ♥❛ s❡çã♦ ✸✳✹✮ ✉s❛♥❞♦ Ωb ≈0✳ ✳ ✳ ✸✷
✸✳✷ ✭❛✮ ✲ ❋✉♥çã♦ ❞❡ s❡❧❡çã♦ t♦♣ ❤❛t ♦❜t✐❞❛ ❛ ♣❛rt✐r ❞❛ ❊q✳ ✭✸✳✹✽✮✳ ❆♦ ❞✐✈✐❞✐r ♦ ❧❡✈❛♥t❛♠❡♥t♦ ❡♠ ❢❛✐①❛s ❞❡ r❡❞s❤✐❢ts ❛ ❢✉♥çã♦ ❞❡ s❡❧❡çã♦ ♣❛r❛ ❛ ❢❛✐①❛ ❞❡ ✐♥t❡r✈❛❧♦ 0.6 ≤ z ≤ 0.8 é ❞❛❞❛ ♣❡❧❛ ❝✉r✈❛ ❡♥tr❡
❛s ❧✐♥❤❛s ✈❡rt✐❝❛✐s✳ ✭❜✮ ✲ ❋✉♥çã♦ ❞❡ ❝♦rr❡❧❛çã♦ ❛♥❣✉❧❛r ❝❛❧❝✉❧❛❞❛ ❡♠ z = 1✳ ❆ ❧✐♥❤❛ ♣♦♥t✐❧❤❛❞❛ ♠♦str❛ ❛ ♣♦s✐çã♦ ❞♦ ♣✐❝♦ ❞♦ ❇❆❖
✐♥tr♦❞✉③✐❞♦ ♥❛ s❡çã♦ ✸✳✹✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✸ ▼❡❞✐❞❛ ❞❛ ❢✉♥çã♦ ❞❡ ❝♦rr❡❧❛çã♦ ❡s♣❛❝✐❛❧ ❞❡ ❞♦✐s ♣♦♥t♦s ❞❡ ❣❛❧á①✐❛s
✈❡r♠❡❧❤❛s ❢❡✐t❛s ♣❡❧♦ ❙❉❙❙✳ ❖s q✉❛❞r❛❞♦s r❡♣r❡s❡♥t❛♠ ♦s ❞❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s✳ ❆s ❝✉r✈❛s ❝♦♥tí♥✉❛s r❡❢❡r❡♠✲s❡ ❛♦ ♠♦❞❡❧♦ Λ❈❉▼
❝♦♠ Ωmh2 = 0.12 ✭✈❡r❞❡✮✱ ✵✳✶✸ ✭✈❡r♠❡❧❤❛✮✱ ✵✳✶✹ ✭❛③✉❧✮✳ ❚♦❞❛s
❝♦♠ Ωbh2 = 0.024 ❡ ns = 0.98✳ ◆❛ ❝✉r✈❛ ♠❛❣❡♥t❛ ♥ã♦ t❡♠♦s ❛s
♦s❝✐❧❛çõ❡s ❞❡ ❜ár✐♦♥s✳ ❬✶✹❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✹✳✶ ●❛❧á①✐❛ ❧♦❝❛❧✐③❛❞❛ ♥❛ ♣♦s✐çã♦ x ❝♦♠ r❡s♣❡✐t♦ ❛♦ ♦❜s❡r✈❛❞♦r✳ ❙❡
|x| é ♠✉✐t♦ ❣r❛♥❞❡✱ ♣♦❞❡♠♦s ✉s❛r ❛ ❛♣r♦①✐♠❛çã♦ ❞❡ ♦❜s❡r✈❛❞♦r
❞✐st❛♥t❡ ♦♥❞❡ x∼χ =χzˆ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼
✹✳✷ ❊❢❡✐t♦s ❞❛ ❞✐st♦rçã♦ ❞♦ r❡❞s❤✐❢t ❝❛✉s❛❞♦s ♣❡❧❛s ✈❡❧♦❝✐❞❛❞❡s ♣❡❝✉✲ ❧✐❛r❡s ❞❛s ❣❛❧á①✐❛s✳ ❆ ❝✉r✈❛ tr❛❝❡❥❛❞❛ ♠♦str❛ θ2×ω(θ) ❝❛❧❝✉❧❛❞❛
♥♦ ❡s♣❛ç♦ r❡❛❧ ❡ ❛ ❝✉r✈❛ ❝♦♥tí♥✉❛ ♠♦str❛ θ2 ×ω(θ) ❝❛❧❝✉❧❛❞❛ ♥♦
❡s♣❛ç♦ ❞❡ r❡❞s❤✐❢t✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽
✷ ❆❣r❛❞❡❝✐♠❡♥t♦s
✹✳✸ ❆ ❝✉r✈❛ ❝♦♥tí♥✉❛ ♠♦str❛ ♦ ❡❢❡✐t♦ ♥ã♦✲❧✐♥❡❛r ♥❛ ❢✉♥çã♦ ❞❡ ❝♦rr❡❧❛çã♦ ❛♥❣✉❧❛r ❝♦♠♣❛r❛❞❛ ❛ ❝✉r✈❛ tr❛❝❡❥❛❞❛ q✉❡ ♥ã♦ ♣♦ss✉✐ ❡❢❡✐t♦s✳ ✳ ✳ ✳ ✹✾ ✹✳✹ ❋✉♥çã♦ ❞❡ s❡❧❡çã♦ ❡s♣❡❝tr♦❝ó♣✐❝❛ ✭❛③✉❧✮ ❡ ❢♦t♦♠étr✐❝❛ ♣❛r❛ ❞✐❢❡r✲
❡♥t❡s ✈❛❧♦r❡s ❞❡ ❡rr♦s ❣❛✉ss✐❛♥♦s ♥♦ ✐♥t❡r✈❛❧♦ ❞❡ r❡❞s❤✐❢t 0.6 ≤ z ≤0.8✳ ❆ ♠❡❞✐❞❛ q✉❡ δσ →0 ❛ ❢✉♥çã♦ ❞❡ s❡❧❡çã♦ ❢♦t♦♠étr✐❝❛ s❡
❛♣r♦①✐♠❛ à ❢✉♥çã♦ ❞❡ s❡❧❡çã♦ t♦♣✲❤❛t✱ φ(z)→dNg/dz. ✳ ✳ ✳ ✳ ✳ ✳ ✺✵
✹✳✺ ❋✉♥çã♦ ❞❡ ❝♦rr❡❧❛çã♦ ❛♥❣✉❧❛r ♣❛r❛ ❞✐❢❡r❡♥t❡s ❡rr♦s ❞❡ ❢♦t♦✲③ ♥♦ ❡s♣❛ç♦ ❞❡ r❡❞s❤✐❢t ♣❛r❛ 1.0 ≤ z ≤ 1.05✳ ❚❡♠♦s δσ = ✵✱ ✵✳✵✶
❡ ✵✳✵✸✱ ♦♥❞❡ ♦ ❡rr♦ ♥✉❧♦ ✭❝✉r✈❛ tr❛❝❡❥❛❞❛✮ ❝♦rr❡s♣♦♥❞❡ ❛ ❛♥á❧✐s❡ ❡s♣❡❝tr♦s❝ó♣✐❝❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✹✳✻ ■♠♣❛❝t♦ ❞❡ zbias ♥❛ ❢✉♥çã♦ ❞❡ ❝♦rr❡❧❛çã♦ ❛♥❣✉❧❛r✳ ❊♠ t♦❞♦s ♦s
❝❛s♦s ❝♦♥s✐❞❡r❛♠♦s ❛ ❞✐s♣❡rsã♦ ♥♦ r❡❞s❤✐❢t ✐❣✉❛❧ ❛ 0.03✳ ✳ ✳ ✳ ✳ ✳ ✺✷
✹✳✼ ❊❢❡✐t♦s ❞❡ ✈✐és✱ ❞✐st♦rçã♦ ❞❡ r❡❞s❤✐❢t✱ ♥ã♦✲❧✐♥❡❛r✐❞❛❞❡ ❡ ❢♦t♦✲③ ♥❛ ❢✉♥çã♦ ❞❡ ❝♦rr❡❧❛çã♦ ❛♥❣✉❧❛r✳ ◆❛ ❝✉r✈❛ ❝♦♥tí♥✉❛ ❝♦♥s✐❞❡r❛♠♦s t♦❞♦s ❡st❡s ❡❢❡✐t♦s✳ ❆ss✉♠✐♠♦s δσ = 0.03✱ β = 1.5✱ z0 = 0.5
♣❛r❛ ❝❛❧❝✉❧❛r ❛ ❢✉♥çã♦ ❞❡ s❡❧❡çã♦ ❢♦t♦♠étr✐❝❛✳ ◆❛ ❝✉r✈❛ tr❛❝❡❥❛❞❛ ♣❧♦t❛♠♦s θ2 × ω(θ) ♥❛ ❛✉sê♥❝✐❛ ❞❡ q✉❛❧q✉❡r ❡❢❡✐t♦✳ ❊♠ ❛♠❜♦s
❝❛s♦s ✉s❛♠♦s ♦s ♣❛râ♠❡tr♦s ❝♦s♠♦❧ó❣✐❝♦s ❞♦ ❲▼❆P✼ ♣❛r❛ ❝❛❧❝✉❧❛r
θ2×ω(θ) ♥♦ ✐♥t❡r✈❛❧♦ ❞❡ r❡❞s❤✐❢t ❞❛❞♦ ♣♦r 1.00≤z ≤1.05 ✳ ✳ ✳ ✺✸
✹✳✽ ❋✉♥çã♦ ❞❡ ❝♦rr❡❧❛çã♦ ❛♥❣✉❧❛r ❡♠ t❡r♠♦s ❞❡ w ✭♣❛✐♥❡❧ ❛✮ ❡ Ωcdm
✭♣❛✐♥❡❧ ❜✮ ♣❛r❛ ♦ ✐♥t❡r✈❛❧♦ ❞❡ r❡❞s❤✐❢t 1.00≤z ≤1.05❝♦♠ ❞✐s♣❡rsã♦
❞❡ ❢♦t♦✲③ ✐❣✉❛❧ ❛ δσ = 0.03✳ ❆ ❧✐♥❤❛ ✈❡rt✐❝❛❧ ✐♥❞✐❝❛ ❛ ♣♦s✐çã♦ ❞♦
♣✐❝♦ ❞♦ ❇❆❖ ♥❛ ❝✉r✈❛ ❝♦♥tí♥✉❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✺✳✶ ❊s♣❡❝tr♦ ❞❡ ♣♦tê♥❝✐❛ ❛♥❣✉❧❛r ❝❛❧❝✉❧❛❞♦ ♥♦ ✐♥t❡r✈❛❧♦ ❞❡ r❡❞s❤✐❢t
1.0 ≤ z ≤ 1.05 ♣❛r❛ ❞✐❢❡r❡♥t❡s ❡rr♦s ❞❡ ❢♦t♦✲③✳ ❆ ❧✐♥❤❛ só❧✐❞❛
❝♦rr❡s♣♦♥❞❡ ❛ ❛✉sê♥❝✐❛ ❞❡ ❡rr♦s ♥❛ ❞❡t❡r♠✐♥❛çã♦ ❞❡ z✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵
✺✳✷ ▼❛tr✐③ ❞❡ ❝♦✈❛r✐â♥❝✐❛ r❡❞✉③✐❞❛ ♣❛r❛ ❛ ❢❛✐①❛ ❞❡ r❡❞s❤✐❢t 1.00< z <
1.05✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹
✺✳✸ ❊s♣❡❝tr♦ ❞❡ ♣♦tê♥❝✐❛ ❛♥❣✉❧❛r ❡♥tr❡ ❛ ❢❛✐①❛ 1.0 ≤z ≤ 1.05 ❝♦♠ ❛s
❢❛✐①❛s 1.05 ≤ z ≤ 1.10 ✭❝♦♥tí♥✉❛✮✱ 1.10 ≤ z ≤ 1.15 ✭♣♦♥t✐❧❤❛❞❛✲
tr❛❝❡❥❛❞❛✮ ❡ 1.15 ≤ z ≤ 1.20 ✭♣♦♥t✐❧❤❛❞❛✮✳ ❆ ❝✉r✈❛ tr❛❝❡❥❛❞❛
r❡♣r❡s❡♥t❛ ♦ ❡s♣❡❝tr♦ ❝❛❧❝✉❧❛❞♦ s♦♠❡♥t❡ ❡♠ 1.0≤z ≤1.05✳ ✳ ✳ ✳ ✻✺
✻✳✶ ❊❧✐♣s❡ ❞❡ ❝♦♥✜❛♥ç❛ ♣❛r❛ ♦s ♣❛râ♠❡tr♦s x ❡ y✳ ❆ ❝♦♥st❛♥t❡ αk
❞❡t❡r♠✐♥❛ ❛ ♣r❡❝✐sã♦ ❞❡ ✶✲σ✱ ✷✲σ ♦✉ ✸✲σ✳ ❖s ❡rr♦s σx ❡ σy sã♦
❢♦r♥❡❝✐❞♦s ♣❡❧❛ ♠❛tr✐③ ❞❡ ❋✐s❤❡r ❡♠ r❡❧❛çã♦ ❛♦ ✈❛❧♦r❡s ✜❞✉❝✐❛✐s x¯
❡ y¯✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶
✻✳✷ ❋✉♥çã♦ ❞❡ ❝♦rr❡❧❛çã♦ ❛♥❣✉❧❛r ♣❛r❛ ❞✐❢❡r❡♥t❡s ❧❛r❣✉r❛s ❞❛ ❢❛✐①❛ ❞❡ r❡❞s❤✐❢t✳ ◆ós ❝♦♥s✐❞❡r❛♠♦s ❢❛✐①❛s ❞❡ r❡❞s❤✐❢t ❝❡♥tr❛❞❛s ❡♠ z = 1
❆❣r❛❞❡❝✐♠❡♥t♦s ✸
✻✳✸ ❊rr♦ ❞❡ w ❡ Ωcdm ❡♠ ❢✉♥çã♦ ❞❡ ∆z ♣❛r❛ ✷ ❢❛✐①❛s ❝❡♥tr❛❞❛s ❡♠ z = 1.1 ❡ z = 1.3✳ ◆❛ ❝✉r✈❛ ♣♦♥t✐❧❤❛❞❛ ♥ós ❞❡s♣r❡③❛♠♦s ♦s ❡rr♦
❞❡ s❤♦t ♥♦✐s❡ ♣❛r❛ ❛ ❢❛✐①❛ ❝❡♥tr❛❞❛ ❡♠ z = 1.3✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼
✻✳✹ ❊st✐♠❛t✐✈❛s ♣❛r❛ Ωcdm ❡ w✉s❛♥❞♦ ❛ ❢✉♥çã♦ ❞❡ ❝♦rr❡❧❛çã♦ ❛♥❣✉❧❛r
❡♠ t♦❞❛s ❛s ❢❛✐①❛s ❞❡ r❡❞s❤✐❢t s❡♠ ❝♦♥s✐❞❡r❛r ❛ ❝♦rr❡❧❛çã♦ ❡♥tr❡ ❡❧❛s✳ ❖♣r✐♠❡✐r♦ ❡ ♦ s❡❣✉♥❞♦ t❡r♠♦ ❞❛ ❊q✳ ✭✻✳✷✽✮ sã♦ r❡♣r❡s❡♥t❛✲ ❞♦s ♣❡❧❛s ❝✉r✈❛s ♣♦♥t✐❧❤❛❞❛ ❡ tr❛❝❡❥❛❞❛ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❆ ❡❧✐♣s❡ ❝♦♥tí♥✉❛ ♠♦str❛ ❛ s♦♠❛ ❡♥tr❡ ❡❧❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶ ✻✳✺ ❊rr♦ ❞❡ ✶✲σ♥♦ ♣❛râ♠❡tr♦w❡♠ ❢✉♥çã♦ ❞♦ r❡❞s❤✐❢t ❛♣ós ♠❛r❣✐♥❛❧✐③❛r
t♦❞♦s ♦s bi✱ σ8 ❡Ωcdm✳ ❖s ❝❡♥ár✐♦s ♦t✐♠✐st❛✱ ✜❞✉❝✐❛❧ ❡ ♣❡ss✐♠✐st❛
sã♦ r❡♣r❡s❡♥t❛❞♦s ♣❡❧❛s ❝✉r✈❛s só❧✐❞❛✱ tr❛❝❡❥❛❞❛ ❡ ♣♦♥t✐❧❤❛❞❛✲tr❛❝❡❥❛❞❛ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷ ✻✳✻ ❊❧✐♣s❡ ❞❡ ❝♦♥✜❛♥ç❛ ♣❛r❛ w ❡Ωcdm ♥❛s ❢❛✐①❛s ❝♦♠ ✐♥t❡r✈❛❧♦ 0.40≤
z ≤ 0.45 ✭♣♦♥t✐❧❤❛❞❛✲tr❛❝❡❥❛❞❛✮✱ 0.85 ≤ z ≤ 0.90 ✭tr❛❝❡❥❛❞❛✮ ❡ 1.35 ≤ z ≤ 1.40 ✭❝♦♥tí♥✉❛✮✳ ❆ ❡❧✐♣s❡ ♣r❡❡♥❝❤✐❞❛ ❝♦rr❡s♣♦♥❞❡ à
s♦♠❛ ❞❛s ✸ ❢❛✐①❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷ ✻✳✼ ❊❧✐♣s❡s ❞❡ ❝♦♥✜❛♥ç❛ ♣❛r❛ Ωcdm ❡w❛♥❛❧✐s❛♥❞♦ ✺ ❢❛✐①❛s ❞❡ r❡❞s❤✐❢t✳
❆ ❝✉r✈❛ só❧✐❞❛ é ♦ r❡s✉❧t❛❞♦ ❝♦♠ t♦❞♦s ♦s ♣❛râ♠❡tr♦s ✜①♦s ♥♦s s❡✉s ✈❛❧♦r❡s ✜❞✉❝✐❛✐s✱ ♥❛ ❝✉r✈❛ tr❛❝❡❥❛❞❛ ❛❞✐❝✐♦♥❛♠♦s ♣r✐♦r ❡♠ h
❡ Ωb ❡ ♥❛ ❝✉r✈❛ ♣♦♥t✐❧❤❛❞❛✲tr❛❝❡❥❛❞❛ ♠❛r❣✐♥❛❧✐③❛♠♦s s♦❜r❡ t♦❞♦s
♦s ♦✉tr♦s ♣❛râ♠❡tr♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸ ✻✳✽ ❊❧✐♣s❡ ❞❡ ❝♦♥✜❛♥ç❛ ♣❛r❛ ♦s ♣❛râ♠❡tr♦sw✱Ωcdm✱Ωb✱h❡ns✳ ▼❛r❣✐♥❛❧✲
✐③❛♠♦s s♦❜r❡ σ8 ❡ t♦❞♦s ♦s ♣❛râ♠❡tr♦s b✬s ❡ ❝♦♥s✐❞❡r❛♠♦s ♦s ✸
❝❡♥ár✐♦s✿ ♦t✐♠✐st❛✭❝♦♥tí♥✉❛✮✱ ✜❞✉❝✐❛❧✭tr❛❝❡❥❛❞❛✮ ❡ ♣❡ss✐♠✐st❛ ✭♣♦♥✲ t✐❧❤❛❞❛✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✺ ✻✳✾ ■♥t❡r✈❛❧♦ ❞❡ ❝♦♥✜❛♥ç❛ ❛ ✶✲σ ♣❛r❛ w ❡ Ωcdm ♠❛r❣✐♥❛❧✐③❛♥❞♦ s♦❜r❡
t♦❞♦s ♦s ♦✉tr♦s ♣❛râ♠❡tr♦s✳ ◆❛ ❋✐❣✉r❛ ✭❛✮ ♠♦str❛♠♦s ❛ ❞✐❢❡r❡♥ç❛ ❞♦s ❡rr♦s ♦❜t✐❞♦s q✉❛♥❞♦ ❝♦♥s✐❞❡r❛♠♦s ❛s ❢❛✐①❛s ❞❡ r❡❞s❤✐❢t ✐♥✲ ❞❡♣❡♥❞❡♥t❡s ✭❡❧✐♣s❡ tr❛❝❡❥❛❞❛✮ ❡ ❝♦♠ ❝♦rr❡❧❛çã♦ ❡♥tr❡ ❡❧❛s ✭❡❧✐♣s❡ ❝♦♥tí♥✉❛✮ ♥♦ ❝❡♥ár✐♦ ✜❞✉❝✐❛❧✳ ◆❛ ❋✐❣✉r❛ ✭❜✮ ❛❞✐❝✐♦♥❛♠♦s ♣r✐♦r ❞❡ ❲▼❆P✼ ♥♦ ❝❡♥ár✐♦ ✜❞✉❝✐❛❧ ✭❡❧✐♣s❡ ❝♦♥tí♥✉❛✮ ❡ ❝♦♠♣❛r❛♠♦s ❝♦♠ ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s s❡♠ ❛❞✐çã♦ ❞❡ ♣r✐♦rs✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✼ ✻✳✶✵ ❈♦♠♣❛r❛çã♦ ❡♥tr❡ ❛s ❞✐str✐❜✉✐çõ❡s ♣❛r❛ w ✭ ❋✐❣✉r❛ ✲❛ ✮ ❡ Ωcdm
▲✐st❛ ❞❡ ❚❛❜❡❧❛s
✻✳✶ ❚❛❜❡❧❛ ❞❡ ❞✐str✐❜✉✐çã♦ ❞❡ ❜✐♥s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✻✳✷ ❚❛❜❡❧❛ s✐♠♣❧✐✜❝❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽ ✻✳✸ ❊rr♦s ❞❡ ✶✲σ ♥♦s ♣❛râ♠❡tr♦s ❝♦s♠♦❧ó❣✐❝♦s✳ ❆ ❝♦❧✉♥❛ ✷ ❢♦r♥❡❝❡ ♦s
✈❛❧♦r❡s ✜❞✉❝✐❛✐s ❞♦s ♣❛râ♠❡tr♦s ❝♦s♠♦❧ó❣✐❝♦s ♥♦♠❡❛❞♦s ♥❛ ❝♦❧✉♥❛ ✶✳ ◆❛ ❝♦❧✉♥❛ ✸ t❡♠♦s ♦ ❡rr♦ ❞❡ ✶✲σ ❡♠ ❝❛❞❛ ♣❛râ♠❡tr♦ q✉❛♥❞♦
♠❛r❣✐♥❛❧✐③❛♠♦s s♦❜r❡ ♦s ♦✉tr♦s✳ ❆ ❝♦❧✉♥❛ ✹ r❡♣r❡s❡♥t❛ ❛ ♣♦r❝✲ ❡t❛❣❡♠ ❞❡st❡s ❡rr♦s ❡♠ r❡❧❛çã♦ ❛♦s s❡✉s ✈❛❧♦r❡s ✜❞✉❝✐❛✐s✳ ✳ ✳ ✳ ✳ ✳ ✽✻ ✻✳✹ ❘❡❞❡ ♣❛r❛ ♦s ♣❛râ♠❡tr♦s w✱ Ωcdm✱ b ❡ σ8✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✽
❇✳✶ ❊rr♦s ❞❡ ✶✲σ ♥♦s ♣❛râ♠❡tr♦s ❝♦s♠♦❧ó❣✐❝♦s✳ ❆ ❝♦❧✉♥❛ ✷ ❢♦r♥❡❝❡ ♦s
✈❛❧♦r❡s ✜❞✉❝✐❛✐s ❞♦s ♣❛râ♠❡tr♦s ❝♦s♠♦❧ó❣✐❝♦s ♥♦♠❡❛❞♦s ♥❛ ❝♦❧✉♥❛ ✶✳ ◆❛s ❝♦❧✉♥❛s ✸ ❡ ✹ t❡♠♦s ♦ ❡rr♦ ❞❡ ✶✲σ❡♠ ❝❛❞❛ ♣❛râ♠❡tr♦ q✉❛♥❞♦
♠❛r❣✐♥❛❧✐③❛♠♦s s♦❜r❡ ♦s ♦✉tr♦s ♥♦s ❝❡♥ár✐♦s ♦t✐♠✐st❛ ❡ ♣❡ss✐♠✐st❛ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✾
1
■♥tr♦❞✉çã♦
❆ ❞❡s❝♦❜❡rt❛ ♦❜s❡r✈❛❝✐♦♥❛❧✱ ♥❛ ❞é❝❛❞❛ ❞❡ ✾✵✱ ❞❛ ❡①♣❛♥sã♦ ❛❝❡❧❡r❛❞❛ ❞♦ ✉♥✐✲ ✈❡rs♦ tr♦✉①❡ ê♥❢❛s❡ ❛♦ ❛t✉❛❧ ♠♦❞❡❧♦ ❝♦s♠♦❧ó❣✐❝♦ ♦♥❞❡ ♦ ✉♥✐✈❡rs♦ é ❝♦♠♣♦st♦ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ♣♦r 4% ❞❡ ♠❛tér✐❛ ❜❛r✐ô♥✐❝❛✱ 20% ❞❡ ♠❛tér✐❛ ❡s❝✉r❛ ❡ 76%
❞❡ ❡♥❡r❣✐❛ ❡s❝✉r❛ ❬✶❪✳ ◆♦ ❡♥t❛♥t♦✱ ❛ ❝❛✉s❛ ❞❛ ❛❝❡❧❡r❛çã♦ ❝ós♠✐❝❛ ❛✐♥❞❛ é ❞❡s✲ ❝♦♥❤❡❝✐❞❛✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ r❡❧❛t✐✈✐❞❛❞❡ ❣❡r❛❧✱ ❛ ❡①♣❛♥sã♦ ❛❝❡❧❡r❛❞❛ ♣♦❞❡ s❡r ❡①♣❧✐❝❛❞❛ ✐♥tr♦❞✉③✐♥❞♦ ♥♦ ✉♥✐✈❡rs♦ ✉♠❛ ❢♦r♠❛ ❞❡ ❡♥❡r❣✐❛ ❡①ót✐❝❛ ❝♦♠ ♣r❡ssã♦ ♥❡❣❛t✐✈❛✱ ❞❡♥♦♠✐♥❛❞❛ ❊♥❡r❣✐❛ ❊s❝✉r❛✳ ❖ ❝❛♥❞✐❞❛t♦ ♠❛✐s s✐♠♣❧❡s ♣❛r❛ ❛ ❡♥❡r✲ ❣✐❛ ❡s❝✉r❛ é ❛ ❝♦♥st❛♥t❡ ❝♦s♠♦❧ó❣✐❝❛ ❬✷✱ ✸❪✱ q✉❡ ❢♦✐ ❛ss✐♠ ♥♦♠❡❛❞❛ ♣♦rq✉❡ s✉❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ é ❝♦♥st❛♥t❡ ♥♦ t❡♠♣♦ ❡ ♥♦ ❡s♣❛ç♦✳ ❖✉tr❛ ♣♦ss✐❜✐❧✐❞❛❞❡✱ ✐♥✲ ❝❧✉✐ ✉♠ ❝❛♠♣♦ ❡s❝❛❧❛rφ ❝♦♠ ♣♦t❡♥❝✐❛❧V(φ)✱ ♥♦♠❡❛❞♦ ❝♦♠♦ q✉✐♥t❡ssê♥❝✐❛ ❬✹✱✺❪✳
❆♦ ❝♦♥trár✐♦ ❞❛ ❝♦♥st❛♥t❡ ❝♦s♠♦❧ó❣✐❝❛✱ ❛ ❡q✉❛çã♦ ❞❡ ❡st❛❞♦ ❞❡ q✉✐♥t❡ssê♥❝✐❛ ♠✉❞❛ ❞✐♥❛♠✐❝❛♠❡♥t❡ ❝♦♠ ♦ t❡♠♣♦✳ ❊①✐st❡♠✱ ♣♦ré♠✱ ♦✉tr❛s ❛❧t❡r♥❛t✐✈❛s q✉❡ ♣♦✲ ❞❡♠ ❡①♣❧✐❝❛r ❛ ❛❝❡❧❡r❛çã♦ ❝ós♠✐❝❛ ❛tr❛✈és ❞❛ ♠♦❞✐✜❝❛çã♦ ❞❛ r❡❧❛t✐✈✐❞❛❞❡ ❣❡r❛❧✳ ❚❛✐s ♠♦❞❡❧♦s sã♦ ❞❡♥♦♠✐♥❛❞♦s t❡♦r✐❛ ❞❡ ❣r❛✈✐t❛çã♦ ♠♦❞✐✜❝❛❞❛ ❡ ❛❧❣✉♥s ❞❡❧❡s sã♦ ❜❛s❡❛❞♦s ♥❛ tr♦❝❛ ❞♦ ❡s❝❛❧❛r ❞❡ ❝✉r✈❛t✉r❛✱ R✱ ♣♦r ✉♠❛ ❢✉♥çã♦ ♠❛✐s ❣❡r❛❧✱ f(R)✱
♥❛ ❛çã♦ ❞❡ ❊✐♥st❡✐♥✲❍✐❧❜❡rt ❬✸✱ ✻❪✳ ❉●P ✶ é ♦✉tr♦ ♠♦❞❡❧♦ ❞❡ ❣r❛✈✐t❛çã♦ ♠♦❞✐✜✲
❝❛❞❛ ❜❛s❡❛❞♦ ❡♠ t❡♦r✐❛s ♠✉❧t✐✲❞✐♠❡♥s✐♦♥❛✐s ♦♥❞❡ ♥♦ss♦ ✉♥✐✈❡rs♦ ♣♦ss✉✐ (3 + 1)
❞✐♠❡s✠♦❡s ✐♥❝♦r♣♦r❛❞❛s ♥✉♠ ✈♦❧✉♠❡ ▼✐♥❦♦✇s❦✐❛♥♦ ❞❡ ❝✐♥❝♦ ❞✐♠❡♥sõ❡s ❬✼❪ ✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ❛ ❣r❛✈✐❞❛❞❡ é ♠♦❞✐✜❝❛❞❛ ❡♠ ❧♦♥❣❛s ❞✐stâ♥❝✐❛s ♠❛s ♠❛♥té♠✲s❡ q✉❛❞r✐❞✐♠❡♥s✐♦♥❛❧ ❡♠ t❡st❡s ❧♦❝❛✐s✳
❆ ❛♥á❧✐s❡ ❞❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❣❛❧á①✐❛s ♥♦ ✉♥✐✈❡rs♦ é ✉♠❛ ✈❛❧✐♦s❛ té❝♥✐❝❛ ♣❛r❛ t❡st❛r ♦s ❞✐❢❡r❡♥t❡s ♠♦❞❡❧♦s ❞❛ ❛❝❡❧❡r❛çã♦ ❝ós♠✐❝❛✳ ❈❛❞❛ ♠♦❞❡❧♦ t❡ór✐❝♦ é ❝❛r❛❝✲
✶▼♦❞❡❧♦ ♣r♦♣♦st♦ ♣♦r ❉✈❛❧✐✱ ●❛❜❛❞❛❞③❡ ❡ P♦rr❛t✐✳
✽ ❈❛♣ít✉❧♦ ✶✳ ■♥tr♦❞✉çã♦ t❡r✐③❛❞♦ ♣♦r ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣❛râ♠❡tr♦s ❝♦s♠♦❧ó❣✐❝♦s q✉❡ ♣♦❞❡♠ s❡r ❡st✐♠❛❞♦s ❛♥❛❧✐s❛♥❞♦ ♦s ❞❛❞♦s ♦❜s❡r✈❛❝✐♦♥❛✐s✳ ❆té r❡❝❡♥t❡♠❡♥t❡✱ ♦s ✈í♥❝✉❧♦s s♦❜r❡ ❡st❡s ♣❛râ♠❡tr♦s ❢♦r❛♠ ♦❜t✐❞♦s ✉s❛♥❞♦ ❝❛tá❧♦❣♦s ❡s♣❡❝tr♦s❝ó♣✐❝♦s ❞❡ ❣❛❧á①✐❛s ❬✽✱ ✾❪✳ ◆♦ ❡♥t❛♥t♦✱ ❢✉t✉r♦s ♣r♦❥❡t♦s ✐rã♦ ❝♦♥str✉✐r ❣r❛♥❞❡s ❝❛tá❧♦❣♦s ❢♦t♦♠étr✐❝♦s ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❛✉♠❡♥t❛r ♦ ♥ú♠❡r♦ ❞❡ ♦❜❥❡t♦s ♦❜s❡r✈❛❞♦s ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ ♠❡❧✲ ❤♦r❛r ❛ ❛♥á❧✐s❡ ❡st❛tíst✐❝❛ ❞♦s ❞❛❞♦s✳ ❚❛✐s ♣r♦❥❡t♦s ✐♥❝❧✉❡♠✱ ❡♥tr❡ ♦✉tr♦s✱ ♦ ❉❛r❦ ❊♥❡r❣② ❙✉r✈❡②✷✳
❖ ♣r♦❥❡t♦ ❉❛r❦ ❊♥❡r❣② ❙✉r✈❡② ✭❉❊❙✮ s❡rá ✉♠ ❞♦s ♠❛✐s ✐♠♣♦rt❛♥t❡s ♠❛♣❡❛✲ ♠❡♥t♦s ❢♦t♦♠étr✐❝♦s ❞♦ ✉♥✐✈❡rs♦ ❞❡st❛ ❞é❝❛❞❛✳ ❊st❡ ♣r♦❥❡t♦ ❡stá s❡♥❞♦ ❝♦♥❞✉③✐❞♦ ♣♦r ✉♠ ❝♦♥sór❝✐♦ q✉❡ ❡♥✈♦❧✈❡ ✈ár✐❛s ✐♥st✐t✉✐çõ❡s ❞♦ ❇r❛s✐❧✱ ■♥❣❧❛t❡rr❛✱ ❊s♣❛♥❤❛✱ ❊st❛❞♦s ❯♥✐❞♦s ❡ ❆❧❡♠❛♥❤❛✳ ❖ ❉❊❙ ❝♦❧❡t❛rá s❡✉s ❞❛❞♦s ❞✉r❛♥t❡ ✺ ❛♥♦s ♥♦ t❡❧❡s❝ó♣✐♦ ❇❧❛♥❝♦ ❧♦❝❛❧✐③❛❞♦ ♥♦ ❈❡rr♦ ❚♦❧♦❧♦ ■♥t❡r♥❛t✐♦♥❛❧ ❖❜s❡r✈❛t♦r② ✭❈❚■❖✱ ❈❤✐❧❡✮✳ ❊❧❡ ♠❛♣❡❛rá ✉♠❛ ❣r❛♥❞❡ ár❡❛ ❞♦ ❝é✉ ❝♦rr❡s♣♦♥❞❡♥❞♦ ❛ ✺✵✵✵ ❞❡❣2 ✭∼1/8
❞❛ ❡s❢❡r❛ ❝❡❧❡st❡✮ ❡♠ ❝✐♥❝♦ ❜❛♥❞❛s ❡s♣❡❝tr❛✐s✳ ❈♦♠ s❡✉ s✉❝❡ss♦ ♦❜s❡r✈❛❝✐♦♥❛❧ ♣r❛t✐❝❛♠❡♥t❡ ❣❛r❛♥t✐❞♦✱ ♦ ❛❝❡r✈♦ ❞❡ ❞❛❞♦s ❞♦ ❉❊❙ r❡♣r❡s❡♥t❛rá ♦ ♠❛✐♦r ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s ❡①✐st❡♥t❡s ♥❛ ❛t✉❛❧✐❞❛❞❡ ❡ ❝♦♥t❡rá ♠❛✐s ❞❡ ✸✵✵ ♠✐❧❤õ❡s ❞❡ ❣❛❧á①✐❛s✱ ✷✵ ♠✐❧ ❛❣❧♦♠❡r❛❞♦s ❞❡ ❣❛❧á①✐❛s ❡ ✷✵✵✵ ❞❡t❡❝çõ❡s ❞❡ s✉♣❡r♥♦✈❛s✱ ❛❧é♠ ❞❡ ♠✐❧❤õ❡s ❞❡ ❡str❡❧❛s ❞❛ ❱✐❛ ▲á❝t❡❛ ❡ ♦❜❥❡t♦s ❞♦ s✐st❡♠❛ s♦❧❛r✱ ✈✐❛❜✐❧✐③❛♥❞♦ ❡st✉❞♦s ♥✉♠❛ ❣r❛♥❞❡ ❞✐✈❡rs✐❞❛❞❡ ❞❡ t❡♠❛s ♥❛ ár❡❛ ❞❡ ❆str♦♥♦♠✐❛ ❡ ❈♦s♠♦❧♦❣✐❛✳
❯♠ ❞♦s ♣r✐♥❝✐♣❛✐s ♦❜❥❡t✐✈♦s ❞♦ ❉❊❙ é ❞❡t❡r♠✐♥❛r ♦ ♣❛râ♠❡tr♦ w ❞❛ ❡q✉❛çã♦
❞❡ ❡st❛❞♦ ❞❛ ❡♥❡r❣✐❛ ❡s❝✉r❛ ❡ s✉❛ ❡✈♦❧✉çã♦ ❛♦ ❧♦♥❣♦ ❞❛ ❤✐stór✐❛ t❡♠♣♦r❛❧ ❞♦ ✉♥✐✈❡rs♦✳ ❊st❡ ❡st✉❞♦ s❡rá r❡❛❧✐③❛❞♦ ✉s❛♥❞♦ q✉❛tr♦ té❝♥✐❝❛s ✐♥❞❡♣❡♥❞❡♥t❡s ❡ ❝♦♠♣❧❡♠❡♥t❛r❡s✿ ❛ ❛♥á❧✐s❡ ❡st❛tíst✐❝❛ ❞❛ ❡str✉t✉r❛ ❡♠ ❣r❛♥❞❡ ❡s❝❛❧❛ ❞♦ ✉♥✐✈❡rs♦✱ ❛❣❧♦♠❡r❛❞♦s ❞❡ ❣❛❧á①✐❛s✱ s✉♣❡r♥♦✈❛s ❡ ❡❢❡✐t♦ ❢r❛❝♦ ❞❡ ❧❡♥t❡s ❣r❛✈✐t❛❝✐♦♥❛✐s✳ ❊st❛ t❡s❡ é ❜❛s❡❛❞❛ ♥❛ ♣r✐♠❡✐r❛ té❝♥✐❝❛ ❝✐t❛❞❛ ❛❝✐♠❛✱ ♦♥❞❡ ✉s❛♠♦s ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❣❛❧á①✐❛s ♣❛r❛ ❢❛③❡r ♣r❡✈✐sõ❡s s♦❜r❡ ❧✐♠✐t❡s ♥♦s ♣❛râ♠❡tr♦s ❝♦s♠♦❧ó❣✐❝♦s✳
❖ ♣r♦❥❡t♦ ❉❊❙ ❛✐♥❞❛ ♥ã♦ ❝♦♠❡ç♦✉ ❛ ❝♦❧❡t❛r s❡✉s ❞❛❞♦s✳ ◆♦ ❡♥t❛♥t♦✱ é ♣♦ssí✈❡❧✱ s❡♠ ♥❡♥❤✉♠ t✐♣♦ ❞❡ ♦❜s❡r✈❛çã♦✱ ❡st✉❞❛r ❛ ♣r❡❝✐sã♦ ❝♦♠ q✉❡ ❡❧❡ ♣♦❞❡rá ❞❡t❡r♠✐♥❛r ♦s ♣❛râ♠❡tr♦s ❝♦s♠♦❧ó❣✐❝♦s✳ ❊st❡ ❡st✉❞♦ é ❢❡✐t♦ ❛tr❛✈és ❞❛ ♠❛tr✐③ ❞❡ ❋✐s❤❡r q✉❡ ❢♦r♥❡❝❡ ♦s ❡rr♦s ❞❡st❡s ♣❛râ♠❡tr♦s ❡♠ r❡❧❛çã♦ ❛ s❡✉s r❡s♣❡❝t✐✈♦s ✈❛❧♦r❡s ✜❞✉❝✐❛✐s✳ ❊♠❜♦r❛ ❛ ♠❛tr✐③ ❞❡ ❋✐s❤❡r ♣♦ss❛ s❡r ❡st✉❞❛❞❛ ♥♦ ❝♦♥t❡①t♦ ❞❛s ✹ té❝♥✐❝❛s ❛♣❧✐❝❛❞❛s ❛♦ ❉❊❙✱ ✈❛♠♦s ♥♦s ❝♦♥❝❡♥tr❛r ♥♦ ❡st✉❞♦ ❞❛ ❢✉♥çã♦ ❞❡ ❝♦rr❡❧❛çã♦ ❛♥❣✉❧❛r ❞❡ ❞♦✐s ♣♦♥t♦s✱ ω(θ)✱ ♦❜t✐❞❛ ❛tr❛✈és ❞❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❣❛❧á①✐❛s ♥♦ ✉♥✐✈❡rs♦✳ ❆ ❛♥á❧✐s❡
❝♦♠♣❧❡t❛ é ❜❛s❡❛❞❛ ♥❛ ♠♦❞❡❧❛❣❡♠ ❞❡st❛ ❢✉♥çã♦ ❡ ♥❛ ♠♦❞❡❧❛❣❡♠ ❞❛ ♠❛tr✐③ ❞❡ ❝♦✈❛r✐â♥❝✐❛ t❡ór✐❝❛ q✉❡ ❞❡s❝r❡✈❡ ♦s ❡rr♦s ❞❡ s✉❛s ♠❡❞✐❞❛s✳
❈♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❡♥t❡♥❞❡r ❛ ♠♦t✐✈❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✱ ✐♥✐❝✐❛♠♦s ♥♦ss♦ ❡s✲ t✉❞♦ ♥♦ ❈❛♣ít✉❧♦ ✷ ✐♥tr♦❞✉③✐♥❞♦ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❞❡ ❝♦s♠♦❧♦❣✐❛ ♣❛❞rã♦✱ ❛ ❡①♣❛♥sã♦ ❛❝❡❧❡r❛❞❛ ❞♦ ✉♥✐✈❡rs♦✳ ❊♠ s❡❣✉✐❞❛✱ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♥s ♠♦❞❡❧♦s q✉❡ ❡①♣❧✐❝❛♠ t❛❧ ❡①♣❛♥sã♦✳ ❆❜♦r❞❛♠♦s ♦s ♠♦❞❡❧♦s ❞❡ Λ❈❉▼ ❡ q✉✐♥t❡ssê♥❝✐❛ ♣❛r❛
❡♥❡r❣✐❛ ❡s❝✉r❛ ❡ ❢❛③❡♠♦s ✉♠ ❜r❡✈❡ r❡s✉♠♦ s♦❜r❡ ♠♦❞❡❧♦s ❞❡ ❣r❛✈✐t❛çã♦ ♠♦❞✐✜✲ ❝❛❞❛✳
✾ ◆♦ ❈❛♣ít✉❧♦ ✸✱ ♣❛r❛ ❞❡✜♥✐r ❛ ❢✉♥çã♦ ❞❡ ❝♦rr❡❧❛çã♦ ❛♥❣✉❧❛r✱ ❡st✉❞❛♠♦s ❛ ❞✐♥â♠✐❝❛ ❞❛s ♣❡rt✉❜❛çõ❡s ♥❛ ♠❛tér✐❛ ❞❡s❞❡ ❛ ✐♥✢❛çã♦ ❛té ❛ é♣♦❝❛ ❡♠ q✉❡ ♦ ✉♥✐✈❡rs♦ ❢♦✐ ❞♦♠✐♥❛❞♦ ♣♦r ♠❛tér✐❛ ♥ã♦✲r❡❧❛t✐✈íst✐❝❛✳ ❉❡✜♥✐♠♦s ♦ ❡s♣❡❝tr♦ ❞❡ ♣♦tê♥❝✐❛ ❝♦♠♦ ❛ ✈❛r✐â♥❝✐❛ ❞❡st❛s ♣❡rt✉r❜❛çõ❡s ♥♦ ❡s♣❛ç♦ ❞❡ ❋♦✉r✐❡r✳ ❊st❡ ❡s♣❡❝✲ tr♦ ❛♣r❡s❡♥t❛ ♣❡q✉❡♥❛s ♦s❝✐❧❛çõ❡s ❜❛r✐ô♥✐❝❛s q✉❡✱ ♥♦ ❡s♣❛ç♦ r❡❛❧✱ ❝♦rr❡s♣♦♥❞❡♠ ❛ ✉♠ ú♥✐❝♦ ♣✐❝♦ ♥❛ ❢✉♥çã♦ ❞❡ ❝♦rr❡❧❛çã♦ ❡s♣❛❝✐❛❧ ❞❡ ❞♦✐s ♣♦♥t♦s ξ(r)✳ ◆❡st❡ ❝❛♣í✲
t✉❧♦ ♠♦str❛♠♦s ❝♦♠♦ ♦❜t❡r ω(θ) ❛tr❛✈és ❞❡ ξ(r)✳ ❚♦❞❛ ❛♥á❧✐s❡ ❛♣r❡s❡♥t❛❞❛ ❛q✉✐
❢♦✐ ❢❡✐t❛ ♣❛r❛ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ♠❛tér✐❛ ♥♦ ✉♥✐✈❡rs♦✱ ✐st♦ é✱ ♠❛tér✐❛ ❜❛r✐ô♥✐❝❛ ❡ ♠❛tér✐❛ ❡s❝✉r❛✳
❊♠❜♦r❛ ❛ ❛♥á❧✐s❡ ❝♦♠ ❛ ♠❛tr✐③ ❞❡ ❋✐s❤❡r ♥ã♦ ♥❡❝❡ss✐t❡ ❞❡ ❞❛❞♦s ♦❜s❡r✈❛✲ ❝✐♦♥❛✐s✱ ❛ ❢✉♥çã♦ ❞❡ ❝♦rr❡❧❛çã♦ ❛♥❣✉❧❛r ❞❡✈❡ s❡r ♠♦❞❡❧❛❞❛ ❝♦♥s✐❞❡r❛♥❞♦ ❛❧❣✉♥s ❡❢❡✐t♦s q✉❡ s❡ ♠♦str❛♠ ♣r❡s❡♥t❡s ♥♦s ❝❛tá❧♦❣♦s ❞❡ ❣❛❧á①✐❛s✳ ◆♦ ❈❛♣ít✉❧♦ ✹ ❛♣✲ r❡s❡♥t❛♠♦s ❡st❡s ❡❢❡✐t♦s ✐♥tr♦❞✉③✐♥❞♦✲♦s ❡♠ ω(θ)✳ P❛r❛ ♦ ❉❊❙✱ ❡st❡s ❡❢❡✐t♦s sã♦✿
✈✐és ❡♥tr❡ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❣❛❧á①✐❛s ❡ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ♠❛tér✐❛ ❡s❝✉r❛✱ ❞✐st♦rçã♦ ♥♦ r❡❞s❤✐❢t✸ ❞❡✈✐❞♦ às ✈❡❧♦❝✐❞❛❞❡s ♣❡❝✉❧✐❛r❡s✱ ♥ã♦✲❧✐♥❡❛r✐❞❛❞❡ ❡ ❡❢❡✐t♦s ❞❡✈✐❞♦ à
❞✐str✐❜✉✐çã♦ ❢♦t♦♠étr✐❝❛ ❞❡ r❡❞s❤✐❢t✳
P❛r❛ ❡st✐♠❛r ♦s ♣❛râ♠❡tr♦s ❝♦s♠♦❧ó❣✐❝♦s ♣r❡❝✐s❛♠♦s ❝♦♥❤❡❝❡r ♦s ❡rr♦s ♥❛s ♠❡❞✐❞❛s ❞❡ω(θ)✳ ❊st❡s ❡rr♦s ♣♦❞❡♠ s❡r ♦❜t✐❞♦s ♣❡❧♦ ♣ró♣r✐♦ ❝❛tá❧♦❣♦ ❞❡ ❣❛❧á①✐❛s
♦✉ ♣♦❞❡♠ s❡r ♠♦❞❡❧❛❞♦s t❡♦r✐❝❛♠❡♥t❡✳ ◆♦ ❈❛♣ít✉❧♦ ✺ ❡st✉❞❛♠♦s ❛ ♠♦❞❡❧❛❣❡♠ ❝♦♠♣❧❡t❛ ❞❛ ♠❛tr✐③ ❞❡ ❝♦✈❛r✐â♥❝✐❛ q✉❡ ❢♦r♥❡❝❡ t♦❞❛s ❛s ✐♥❢♦r♠❛çõ❡s ❞♦s ❡rr♦s ♥❛s ♠❡❞✐❞❛s ❞❡ ω(θ)✳
❖ ❈❛♣ít✉❧♦ ✻ ❢♦✐ ❞❡❞✐❝❛❞♦ ❛♦ ❡st✉❞♦ ❞❛ ♠❛tr✐③ ❞❡ ❋✐s❤❡r ♣❛r❛ ♦ ❉❊❙ ❡ à ❛♣r❡s❡♥t❛çã♦ ❞❡ ♥♦ss♦s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s✳ ◆❡❧❡✱ ♠♦str❛♠♦s ❡ ❛♣❧✐❝❛♠♦s ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛ ♠❛tr✐③ ❞❡ ❋✐s❤❡r ✉s❛♥❞♦ ❛ ❢✉♥çã♦ ❞❡ ❝♦rr❡❧❛çã♦ ❛♥❣✉❧❛r ❡ ❛ ♠❛tr✐③ ❞❡ ❝♦✈❛r✐â♥❝✐❛ ✐♥tr♦❞✉③✐❞❛s ♥♦s ❈❛♣ít✉❧♦s ✹ ❡ ✺✳
❋✐♥❛❧♠❡♥t❡✱ ❛♣r❡s❡♥t❛♠♦s ♥♦ss❛s ❝♦♥❝❧✉sõ❡s ❡ ❢✉t✉r❛s ♣❡rs♣❡❝t✐✈❛s✳ ❊st❡ tr❛✲ ❜❛❧❤♦ ❢♦✐ ❜❛s❡❛❞♦ ♥❛ r❡❢❡rê♥❝✐❛ ❬✶✵❪✳
2
❆ ❡①♣❛♥sã♦ ❞♦ ✉♥✐✈❡rs♦
❊♠ ✶✾✾✽✱ ❞♦✐s ❣r✉♣♦s q✉❡ ❡st✉❞❛✈❛♠ s✉♣❡r♥♦✈❛s ❞♦ ❚✐♣♦ ■❛ ❛♣r❡s❡♥t❛r❛♠ ✐♥✲ ❞❡♣❡♥❞❡♥t❡s ❡✈✐❞ê♥❝✐❛s ❞❡ q✉❡ ♦ ✉♥✐✈❡rs♦ ❡stá ❡♠ ❡①♣❛♥sã♦ ❛❝❡❧❡r❛❞❛ ❬✶✶✱✶✷❪✳ ◆♦s ú❧t✐♠♦s tr❡③❡ ❛♥♦s ❞❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s ❞❡ ❞✐❢❡r❡♥t❡s ♦❜s❡r✈❛çõ❡s ❝♦♥✜r♠❛r❛♠ ❡st❡ ❢❛t♦ ❬✶✸✱ ✶✹❪✳ ❉❡s❞❡ ❡♥tã♦ ✈ár✐❛s ❡①♣❧✐❝❛çõ❡s t❡♠ s✐❞♦ ♣r♦♣♦st❛s ♣❛r❛ ❥✉st✐✲ ✜❝❛r ❡st❛s ❡✈✐❞ê♥❝✐❛s✳
◆❡st❡ ❝❛♣ít✉❧♦ ✐♥tr♦❞✉③✐♠♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❞❡ ❝♦s♠♦❧♦❣✐❛ ♣❛❞rã♦ ❝♦♠ ♦ ✐♥t✉✐t♦ ❞❡ ❡♥t❡♥❞❡r ❛s ❡✈✐❞ê♥❝✐❛s ❞❛ ❛❝❡❧❡r❛çã♦ ❝ós♠✐❝❛✳ ❆❜♦r❞❛♠♦s ❛❧❣✉♥s ♠♦❞❡❧♦s ♣❛r❛ ❛ ❡①♣❛♥sã♦ ❛❝❡❧❡r❛❞❛ ❞♦ ✉♥✐✈❡rs♦✱ t❛✐s ❝♦♠♦✿ ΛCDM✱ q✉✐♥t❡ssê♥❝✐❛
❡ ❣r❛✈✐t❛çã♦ ♠♦❞✐✜❝❛❞❛✳
✷✳✶ ❈♦s♠♦❧♦❣✐❛ ❜ás✐❝❛
❖ ♠♦❞❡❧♦ ♠❛✐s s✐♠♣❧❡s ❞♦ ✉♥✐✈❡rs♦✱ ❝❤❛♠❛❞♦ ♠♦❞❡❧♦ ❞❡ ❋r✐❡❞♠❛♥♥✲❘♦❜❡rts♦♥✲ ❲❛❧❦❡r ✭❋❘❲✮✱ é ❜❛s❡❛❞♦ ♥❛ ❤✐♣ót❡s❡ ❞❡ q✉❡ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ♠❛tér✐❛ ♥♦ ✉♥✐✈❡rs♦ é ❤♦♠♦❣ê♥❡❛ ❡ ✐s♦tró♣✐❝❛ ❡♠ ❣r❛♥❞❡s ❡s❝❛❧❛s✳ ❈♦♠♦ ❛s ♣r♦♣r✐❡❞❛❞❡s ❣❡♦♠étr✐❝❛s ❞♦ ❡s♣❛ç♦✲t❡♠♣♦ ❡stã♦ r❡❧❛❝✐♦♥❛❞❛s ❝♦♠ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ♠❛tér✐❛ ❛tr❛✈és ❞❛s ❡q✉❛çõ❡s ❞❡ ❊✐♥st❡✐♥✱ s❡❣✉❡ q✉❡ ♦ ❡s♣❛ç♦ tr✐❞✐♠❡♥s✐♦♥❛❧ t❛♠❜é♠ é ❤♦♠♦❣ê♥❡♦ ❡ ✐s♦tró♣✐❝♦✳ ◆♦ ❡♥t❛♥t♦✱ ✐st♦ é ✈❡r❞❛❞❡ s♦♠❡♥t❡ ❡♠ ❡s❝❛❧❛s ♠❛✐♦r❡s q✉❡ ✶✵✵ ▼♣❝✳
❖ ✐♥t❡r✈❛❧♦ ✐♥✈❛r✐❛♥t❡✱ ds2✱ ♥♦ ✉♥✐✈❡rs♦ ❞❡ ❋❲❘ é ❞❛❞♦ ♣♦r
ds2 =−dt2+a2(t)
dr2
1−kr2 +r
2(dθ2 + sin2θdφ2)
, ✭✷✳✶✮
♦♥❞❡a(t)é ♦ ❢❛t♦r ❞❡ ❡s❝❛❧❛ q✉❡ ❞❡s❝r❡✈❡ ❛ ❡①♣❛♥sã♦ ❞♦ ❡s♣❛ç♦ ❡ k é ❛ ❝♦♥st❛♥t❡
✶✷ ❈❛♣ít✉❧♦ ✷✳ ❆ ❡①♣❛♥sã♦ ❞♦ ✉♥✐✈❡rs♦ ❞❡ ❝✉r✈❛t✉r❛ q✉❡ ❛ss✉♠❡ ✈❛❧♦r❡s k = 0,1❡−1❝♦rr❡s♣♦♥❞❡♥❞♦ ❛♦ ✉♥✐✈❡rs♦ ♣❧❛♥♦✱
❢❡❝❤❛❞♦ ❡ ❛❜❡rt♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡ ❬✶✺❪✳ ◆❡st❛ ❡q✉❛çã♦ ❝♦♥s✐❞❡r❛♠♦s c = 1✶✳
❆ ❊q✳ ✭✷✳✶✮ ♣♦❞❡ s❡r r❡❡s❝r✐t❛ ❡♠ t❡r♠♦s ❞❛ ♠étr✐❝❛ gμν ❡ ❞❛s ❝♦♦r❞❡♥❛❞❛s
xμ≡(t, r, θ, φ) ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿
ds2 =gμνdxμdxν, ✭✷✳✷✮
♦♥❞❡ grr =a2(t)/(1−kr2)✱ gθθ =a2(t)r2✱gφφ =a2(t)r2sin2θ✱ g00 =−1 ❡ gμν = 0
♣❛r❛ μ=ν✳ ❆s ❝♦♦r❞❡♥❛❞❛s ❡s♣❛❝✐❛✐s ✐♥tr♦❞✉③✐❞❛s ❛❝✐♠❛ sã♦ ❝❤❛♠❛❞❛s ❝♦♦r❞❡✲
♥❛❞❛s ❝♦♠ó✈❡✐s✱ ✐st♦ é✱ ❡❧❛s ❛❝♦♠♣❛♥❤❛♠ ❛ ❡①♣❛♥sã♦ ❞♦ ✉♥✐✈❡rs♦✳ ❉❡ss❛ ❢♦r♠❛✱ ♦❜❥❡t♦s ❝♦♠ ✈❡❧♦❝✐❞❛❞❡s ♣❡❝✉❧✐❛r❡s✷ ♥✉❧❛s ♣♦ss✉❡♠ r, θ, φ ❝♦♥st❛♥t❡s✳
❆ ♠étr✐❝❛ é ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ ❡st✉❞❛r ❛ ❞✐♥â♠✐❝❛ ❞♦ ✉♥✐✈❡rs♦ ❣♦✈❡r♥❛❞❛ ♣❡❧❛s ❡q✉❛çõ❡s ❞❡ ❊✐♥st❡✐♥
Gμν ≡Rμν − 1
2gμνR= 8πGTμν, ✭✷✳✸✮
♦♥❞❡Gé ❛ ❝♦♥st❛♥t❡ ✉♥✐✈❡rs❛❧ ❞❡ ◆❡✇t♦♥ ❡ Gμν é ♥♦♠❡❛❞♦ t❡♥s♦r ❞❡ ❊✐♥st❡✐♥✳ ❖
❧❛❞♦ ❡sq✉❡r❞♦ ❞❛s ❊qs✳ ✭✷✳✸✮ ❝❛r❛❝t❡r✐③❛ ❛ ❣❡♦♠❡tr✐❛ ❞♦ ❡s♣❛ç♦✲t❡♠♣♦✱ ❡♥q✉❛♥t♦ ♦ ❧❛❞♦ ❞✐r❡✐t♦ ❞❡s❝r❡✈❡ ♦ ❝♦♥t❡ú❞♦ ❞❡ ♠❛tér✐❛ ❡ ❡♥❡r❣✐❛ ❝♦♥t✐❞❛ ♥♦ ✉♥✐✈❡rs♦ ♣♦r ✐♥t❡r♠é❞✐♦ ❞♦ t❡♥s♦r ❡♥❡r❣✐❛✲♠♦♠❡♥t♦ Tμν✳ ❆s q✉❛♥t✐❞❛❞❡s Rμν ❡ R ≡ gμνRμν
sã♦✱ r❡♣❡❝t✐✈❛♠❡♥t❡✱ ♦ t❡♥s♦r ❞❡ ❘✐❝❝✐ ❡ ❛ ❝✉r✈❛t✉r❛ ❡s❝❛❧❛r ❡ ❞❡♣❡♥❞❡♠ ❞♦ t❡♥s♦r ♠étr✐❝♦ ❡ s✉❛s ❞❡r✐✈❛❞❛s✳ ❖ t❡♥s♦r ❞❡ ❘✐❝❝✐ é ❞❛❞♦ ❡♠ t❡r♠♦s ❞♦ sí♠❜♦❧♦ ❞❡ ❈❤r✐st♦✛❡❧
Rμν =∂αΓαμν −∂νΓαμα+ ΓαβαΓβμν −ΓαβνΓβμα, ✭✷✳✹✮
q✉❡ é ❞❡✜♥✐❞♦ ♣♦r
Γμαβ = g μν
2 [∂βgαν +∂αgβν −∂νgαβ]. ✭✷✳✺✮
◆♦ ♠♦❞❡❧♦ ❞❡ ❋❘❲✱ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ♠❛tér✐❛ ♥♦ ✉♥✐✈❡rs♦ é ❞❡s❝r✐t❛ ♣♦r ✉♠ ✢✉✐❞♦ ♣❡r❢❡✐t♦✳ ❊st❡ é ✐s♦tró♣✐❝♦ ♥♦ s❡✉ r❡❢❡r❡♥❝✐❛❧ ♣ró♣r✐♦ ❡ ♣♦rt❛♥t♦ ♣♦ss✉✐ t❡♥s♦r ❡♥❡r❣✐❛✲♠♦♠❡♥t♦ ❞❛❞♦ ♣♦r
Tμν =pgμν+ (p+ρ)uμuν, ✭✷✳✻✮
♦♥❞❡ ρ é ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ ❡ p é ❛ ♣r❡ssã♦ ✐s♦tró♣✐❝❛ ♠❡❞✐❞❛ ♣♦r ✉♠ ♦❜✲
s❡r✈❛❞♦r ❝♦♠ó✈❡❧ ❛♦ ✢✉✐❞♦✳ ❖ ✹✲✈❡t♦r ✈❡❧♦❝✐❞❛❞❡ uμ t❡♠ ❝♦♠♣♦♥❡♥t❡s u0 = 1
❡ ui = 0✱ ♦♥❞❡ i = 1,2,3 r♦t✉❧❛ ❛s ❝♦♦r❞❡♥❛❞❛s ❡s♣❛❝✐❛✐s✳ ❆ ♣❛❧❛✈r❛ ♠❛tér✐❛
❡♠ r❡❧❛t✐✈✐❞❛❞❡ ❣❡r❛❧✱ r❡❢❡r❡✲s❡ ❛ q✉❛❧q✉❡r ❢♦♥t❡ ❞❡ ❡♥❡r❣✐❛ q✉❡ ❡stá ♣r❡s❡♥t❡ ♥♦ ✉♥✐✈❡rs♦✳ P♦r ❡①❡♠♣❧♦✱ ♦ t❡♥s♦r ❡♥❡r❣✐❛✲♠♦♠❡♥t♦ ❞♦ ✢✉✐❞♦ ❝♦♥té♠ ✐♥❢♦r♠❛çõ❡s
✶❉✉r❛♥t❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❛ t❡s❡ ❛❞♦t❛♠♦s ✉♥✐❞❛❞❡s ♥❛t✉r❛✐s ❝♦♠ c= 1✱ ❡①❝❡t♦ ♥♦s
❝❛s♦s ♦♥❞❡ ❡s❝r❡✈❡♠♦s ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❛ ❧✉③ ❡①♣❧✐❝✐t❛♠❡♥t❡✳
✷✳✶✳ ❈♦s♠♦❧♦❣✐❛ ❜ás✐❝❛ ✶✸ s♦❜r❡ ♣❛rtí❝✉❧❛s r❡❧❛t✐✈íst✐❝❛s✱ ♠❛tér✐❛ ❜❛r✐ô♥✐❝❛ ❡ ♠❛tér✐❛ ❡s❝✉r❛ ❝♦♥t✐❞❛s ♥♦ ✉♥✐✲ ✈❡rs♦✳ P♦❞❡♠♦s ❛❝r❡s❝❡♥t❛r q✉❛❧q✉❡r t✐♣♦ ❞❡ ❡♥❡r❣✐❛ ❡♠ Tμν✱ ✐♥❝❧✉s✐✈❡ ♠♦❞❡❧♦s
♠❛✐s ❡①ót✐❝♦s ❝♦♠♦ ❡♥❡r❣✐❛ ❞❡ ✈á❝✉♦✳ ❖ ♠♦❞❡❧♦ ♠❛✐s s✐♠♣❧❡s ❞❡ ❡♥❡r❣✐❛ ❞❡ ✈á❝✉♦ s❡ ❜❛s❡✐❛ ♥❛ ✐♥tr♦❞✉çã♦ ❞❛ ❝♦♥st❛♥t❡ ❝♦s♠♦❧ó❣✐❝❛✱ Λ✱ ♥❛s ❡q✉❛çõ❡s ❞❡ ❊✐♥st❡✐♥✳
■st♦ é ❡q✉✐✈❛❧❡♥t❡ ❛ r❡❞❡✜♥✐r ♦ t❡♥s♦r ❡♥❡r❣✐❛✲♠♦♠❡♥t♦ ❝♦♠♦
Tμν = Tμν− Λ 8πGgμν
= pg μν+ (p+ρ)uμuν, ✭✷✳✼✮
♦♥❞❡
p=p− Λ
8πG, ρ=ρ+
Λ
8πG. ✭✷✳✽✮
❖ t❡♥s♦r ❞❡ ❊✐♥st❡✐♥ ♦❜❡❞❡❝❡ ❛ ✐❞❡♥t✐❞❛❞❡ ❞❡ ❇✐❛♥❝❤✐ ∇νGμν = 0✱ q✉❡ ✐♠♣❧✐❝❛
♥❛ ❧❡✐ ❞❡ ❝♦♥s❡r✈❛çã♦
∇μTμν = 0. ✭✷✳✾✮
❖ sí♠❜♦❧♦ ∇ r❡♣r❡s❡♥t❛ ❛ ❞❡r✐✈❛❞❛ ❝♦✈❛r✐❛♥t❡ ❞❡✜♥✐❞❛ ♣♦r ∇μTμν =
∂Tμ ν
∂xμ + Γ μ
αμTαν−ΓανμTμα. ✭✷✳✶✵✮
P❛r❛ν = 0✱ ❛ ❊q✳ ✭✷✳✾✮ ❢♦r♥❡❝❡ ❛ ❡q✉❛çã♦ ❞❡ ❝♦♥s❡r✈❛çã♦ ❞❡ ❡♥❡r❣✐❛
˙
ρ+ 3a˙
a(ρ+p) = 0, ✭✷✳✶✶✮
♦♥❞❡ ♦s ♣♦♥t♦s r❡♣r❡s❡♥t❛♠ ❛ ❞❡r✐✈❛❞❛ ❝♦♠ r❡❧❛çã♦ ❛♦ t❡♠♣♦✳ ❈♦♥s✐❞❡r❛♥❞♦ ✉♠❛ ❡q✉❛çã♦ ❞❡ ❡st❛❞♦ ❞❛ ❢♦r♠❛ p = wρ✱ ♦♥❞❡ w é ✉♠ ♣❛râ♠❡tr♦ ❝♦♥st❛♥t❡✱ ❛
❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ ❛ss✉♠❡ ❛ ❢♦r♠❛ρ∝a−3(1+w)✳ ❊①❡♠♣❧♦s ❞❡ ✐♥t❡r❡ss❡ ✐♥❝❧✉❡♠
♦ ✉♥✐✈❡rs♦ ❞♦♠✐♥❛❞♦ ♣♦r
❘❛❞✐❛çã♦ w= 1
3 → ρr∝a
−4,
▼❛tér✐❛ w= 0 → ρm ∝a−3,
❊♥❡r❣✐❛ ❞❡ ✈á❝✉♦ w=−1 → ρΛ∝❝♦♥st.
P❛r❛ ❡♥t❡♥❞❡r ❛ ❡✈♦❧✉çã♦ ❞♦ ❢❛t♦r ❞❡ ❡s❝❛❧❛ ♣r❡❝✐s❛♠♦s ❝♦♥s✐❞❡r❛r ❛ ❝♦♠♣♦♥❡♥t❡ t❡♠♣♦r❛❧ ❞❛s ❊qs✳ ✭✷✳✸✮
R00− 1
2g00R = 8πGT00. ✭✷✳✶✷✮
◆♦ ✉♥✐✈❡rs♦ ❤♦♠♦❣ê♥❡♦ ❡ ✐s♦tró♣✐❝♦ ❛ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ s❡ r❡s✉♠❡ ♥❛ ❡q✉❛çã♦ ❞❡ ❋r✐❡❞♠❛♥♥
H2+ k
a2 =
8πG
✶✹ ❈❛♣ít✉❧♦ ✷✳ ❆ ❡①♣❛♥sã♦ ❞♦ ✉♥✐✈❡rs♦ ♦♥❞❡H é ♦ ♣❛râ♠❡tr♦ ❞❡ ❍✉❜❜❧❡ ❞❡✜♥✐❞♦ ♣♦r H = ˙a/a❡ ρ(a)r❡♣r❡s❡♥t❛ ❛ s♦♠❛
s♦❜r❡ t♦❞❛s ❛s ❝♦♠♣♦♥❡♥t❡s ❞❡ ❞❡♥s✐❞❛❞❡s ❞❡ ❡♥❡r❣✐❛ q✉❡ ❝♦♥st✐t✉❡♠ ♦ ✉♥✐✈❡rs♦✸
✭♠❛tér✐❛✱ r❛❞✐❛çã♦ ❡ ❡♥❡r❣✐❛ ❞❡ ✈á❝✉♦✮✳ ❊st❛ ❡q✉❛çã♦ ❢♦r♥❡❝❡ ❛ r❡❧❛çã♦ ❡♥tr❡ ♦ ❢❛t♦r ❞❡ ❡s❝❛❧❛ ❡ ♦ t❡♠♣♦ ❢ís✐❝♦ t✳ ❆♦ ❝♦♥s✐❞❡r❛r ♦ ✉♥✐✈❡rs♦ ♣❧❛♥♦ ♦❜t❡♠♦s ❛
s♦❧✉çã♦a ∝t1/3 ♣❛r❛ r❛❞✐❛çã♦✱ a∝t2/3 ♣❛r❛ ♠❛tér✐❛ ❡a∝exp(H0t)♣❛r❛ ❡♥❡r❣✐❛
❞❡ ✈á❝✉♦✱ ♦♥❞❡ H0 é ❞❡✜♥✐❞❛ ❡♠ a= 1 ❡ é ❞❡♥♦♠✐♥❛❞❛ ❝♦♥st❛♥t❡ ❞❡ ❍✉❜❜❧❡✳
❆ ❡q✉❛çã♦ ❞❡ ❋r✐❡❞♠❛♥♥ ♣♦❞❡ s❡r r❡❡s❝r✐t❛ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿
H2(a) =
˙
a a
2
=H02Ωma−3 + Ωra−4+ Ωka−2+ ΩΛa−3(1+w), ✭✷✳✶✹✮
♦♥❞❡ ♦s ♣❛râ♠❡tr♦sΩm,Ωr, Ωk❡ΩΛsã♦ ❞❡✜♥✐❞♦s ❡♠ t❡r♠♦s ❞❛ ❞❡♥s✐❞❛❞❡ ❝rít✐❝❛✱ ρcr✱ ❞❡ ❛❝♦r❞♦ ❝♦♠
Ωm ≡ ρ
(0)
m
ρcr
, Ωr ≡ ρ
(0)
r
ρcr
, Ωk ≡ ρ
(0)
k
ρcr
, ΩΛ ≡
ρΛ ρcr
. ✭✷✳✶✺✮
❊st❛s ❣r❛♥❞❡③❛s ❢❛③❡♠ ♣❛rt❡ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❝❤❛♠❛❞♦ ♣❛râ♠❡tr♦s ❝♦s♠♦❧ó❣✐❝♦s q✉❡ sã♦ ♥❡❝❡ssár✐♦s ♣❛r❛ ❞❡s❝r❡✈❡r ❛ ❝♦♠♣♦s✐çã♦ ❡ ❛ ❞✐♥â♠✐❝❛ ❞♦ ✉♥✐✈❡rs♦✳ ❆ ❞❡♥s✐❞❛❞❡ ❝rít✐❝❛ ❞❡♣❡♥❞❡ ❞♦ ❢❛t♦r ❞❡ ❡s❝❛❧❛ ♥♦ ❝❛s♦ ❣❡r❛❧✳ ❊❧❛ é ❞❡✜♥✐❞❛ ❢❛③❡♥❞♦
k = 0 ♥❛ ❊q✳ ✭✷✳✶✸✮✳ ◆❡st❡ tr❛❜❛❧❤♦✱ ❛❞♦t❛♠♦s ρcr ❝❛❧❝✉❧❛❞❛ ❡♠ a= 1✱ ❧♦❣♦
ρcr ≡ 3H2
0
8πG. ✭✷✳✶✻✮
◆♦t❡ q✉❡ ✭✷✳✶✹✮ ❝♦rr❡s♣♦♥❞❡ ❡①❛t❛♠❡♥t❡ ❛ ❊q✳ ✭✷✳✶✸✮ ❝♦♠ ρ(0)k = −(3/8πG)k ❡ w=−1✳ ❊①✐❣✐♥❞♦ ❛ ❝♦♥s✐stê♥❝✐❛ ❞❛ ❊q✳ ✭✷✳✶✹✮ q✉❛♥❞♦ a= 1✱ ♦❜t❡♠♦s
Ωm+ Ωr+ Ωk+ ΩΛ= 1, ✭✷✳✶✼✮
♦♥❞❡Ωmé ❛ s♦♠❛ ❞♦s ♣❛râ♠❡tr♦s ❞❡ ❞❡♥s✐❞❛❞❡ ❞❡ ♠❛tér✐❛ ❡s❝✉r❛ ✭Ωcdm✮ ❡ ♠❛tér✐❛
❜❛r✐ô♥✐❝❛ ✭Ωb✮✳
◆♦s ♠♦❞❡❧♦s ❝♦s♠♦❧ó❣✐❝♦s ❛t✉❛✐s ρΛ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❞❡♥s✐❞❛❞❡ ❞❡ ❊♥❡r❣✐❛
❊s❝✉r❛✳ ◆❛ s❡çã♦ ✷✳✸ ❛❜♦r❞❛r❡♠♦s ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ✐♥tr♦❞✉③✐r ❡ss❛ ♠✐st❡r✐♦s❛ ❢♦r♠❛ ❞❡ ❡♥❡r❣✐❛ ♥♦ ✉♥✐✈❡rs♦✳
❆s ❝♦♠♣♦♥❡♥t❡s ❡s♣❛❝✐❛✐s ❞❛s ❡q✉❛çõ❡s ❞❡ ❊✐♥st❡✐♥ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛ ❊q✳ ✭✷✳✶✸✮ ❢♦r♥❡❝❡♠ ❛ ❡q✉❛çã♦ ❞❛ ❛❝❡❧❡r❛çã♦
¨
a a =−
4πG
3 (ρ+ 3p). ✭✷✳✶✽✮
❆ s❡❣✉♥❞❛ ❞❡r✐✈❛❞❛ ❞♦ ❢❛t♦r ❞❡ ❡s❝❛❧❛ ✈✐♥❝✉❧❛ ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ❛ ❡①♣❛♥sã♦ ❞♦ ✉♥✐✈❡rs♦✳ ❊❧❛ ❞❡t❡r♠✐♥❛ s❡ ♦ ✉♥✐✈❡rs♦ ❡stá ❛❝❡❧❡r❛❞♦✱ ❞❡s❛❝❡❧❡r❛❞♦ ♦✉ s❡ ❡①♣❛♥❞❡ ❝♦♠ ✈❡❧♦❝✐❞❛❞❡ ❝♦♥st❛♥t❡✳
✷✳✶✳ ❈♦s♠♦❧♦❣✐❛ ❜ás✐❝❛ ✶✺
✷✳✶✳✶ ❉✐stâ♥❝✐❛s ❝♦s♠♦❧ó❣✐❝❛s
▼❡❞✐r ♦✉ ❡st✐♠❛r ❞✐stâ♥❝✐❛s ❡♠ ❝♦s♠♦❧♦❣✐❛ ♣❡r♠✐t❡✲♥♦s ❡①tr❛✐r ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ✉♥✐✈❡rs♦ ❡♠ q✉❡ ✈✐✈❡♠♦s✳ ◆♦ ❡♥t❛♥t♦✱ ❡st❛ t❛r❡❢❛ é ♠✉✐t♦ ❞✐❢í❝✐❧ q✉❛♥❞♦ tr❛t❛✲ ♠♦s ❞❡ ❡s❝❛❧❛s ❛str♦♥ô♠✐❝❛s✳ ◆❛ s❡çã♦ ❛♥t❡r✐♦r ✈✐♠♦s q✉❡ ♥♦ s✐st❡♠❛ ❞❡ ❝♦♦r✲ ❞❡♥❛❞❛s ❝♦♠ó✈❡❧✱ r✱ ♦❜❥❡t♦s ❝♦♠♦ ❣❛❧á①✐❛s ♣❡r♠❛♥❡❝❡♠ ✜①♦s ❡♠ s✉❛s ♣♦s✐çõ❡s
q✉❛♥❞♦ ♦ ✉♥✐✈❡rs♦ s❡ ❡①♣❛♥❞❡✳ P♦ré♠✱ ❛ ❞✐stâ♥❝✐❛ ✈❡r❞❛❞❡✐r❛ ❡♥tr❡ ❞♦✐s ♦❜❥❡t♦s ❞❡✈❡ s❡r ❝❛❧❝✉❧❛❞❛ ✉s❛♥❞♦ ❛s ❝♦♦r❞❡♥❛❞❛s ❢ís✐❝❛s✱ x✱ ❞❡✜♥✐❞❛s ♣♦r
x=a(t)r. ✭✷✳✶✾✮
❚✇♦ ②❡❛rs ❧❛t❡r ❊❞✇✐♥ ❍✉❜❜❧❡ ❝♦♥✜r♠❡❞ t❤❡ ❡①✐st❡♥❝❡
◆❛ ❛✉sê♥❝✐❛ ❞❡ ✈❡❧♦❝✐❞❛❞❡s ♣❡❝✉❧✐❛r❡s r é ❝♦♥st❛♥t❡✳ ▲♦❣♦✱ ❛♦ ❞❡r✐✈❛r x ❡♠
r❡❧❛çã♦ ❛♦ t❡♠♣♦✱ ♦❜t❡♠♦s ❛ ❢❛♠♦s❛ ❧❡✐ ❞❡ ❍✉❜❜❧❡
v=Hx. ✭✷✳✷✵✮
❊st❛ r❡❧❛çã♦ ♠♦str❛ q✉❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❡ r❡❝❡ssã♦ ❞❡ ✉♠❛ ❣❛❧á①✐❛ ❡♠ r❡❧❛çã♦ ❛ ♦✉tr❛ é ♣r♦♣♦r❝✐♦♥❛❧ à ❞✐stâ♥❝✐❛ ❡♥tr❡ ❡❧❛s✳ ❆ ❧❡✐ ❞❡ ❍✉❜❜❧❡ ❢♦✐ ♣r✐♠❡✐r♦ ❞❡r✐✈❛❞❛ ❡♠ ❘❡❧❛t✐✈✐❞❛❞❡ ●❡r❛❧ ❡♠ ✶✾✷✼ ♣♦r ●❡♦r❣❡s ▲❡♠❛îtr❡✳ ❉♦✐s ❛♥♦s ❞❡♣♦✐s ❊❞✇✐♥ ❍✉❜❜❧❡ ❛ ❝♦♥✜r♠♦✉ ❝♦♠ ✉♠❛ ❡✈✐❞ê♥❝✐❛ ❡①♣❡r✐♠❡♥t❛❧ ❬✶✻❪ ❛♦ ♦❜s❡r✈❛r ♦ ❞❡s✈✐♦ ♥♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ♦♥❞❛ ❞❛ ❧✉③ ❡♠✐t✐❞❛ ✭λem✮ ❡♠ r❡❧❛çã♦ ❛♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ♦♥❞❛
♦❜s❡r✈❛❞♦ ✭λobs✮ ❞❡ ❣❛❧á①✐❛s ❞✐st❛♥t❡s✳ ❊st❡ ❞❡s✈✐♦ é ❞❡s❝r✐t♦ ♣❡❧♦ r❡❞s❤✐❢t z ✭❡♠
♣♦rt✉❣✉ês é ❝❤❛♠❛❞♦ ❞❡ ❞❡s✈✐♦ ♣❛r❛ ♦ ✈❡r♠❡❧❤♦✮
z ≡ λobs
λem −
1 = a0
a(tem)−1, ✭✷✳✷✶✮
♦♥❞❡ a0 é ✉♠❛ ❝♦♥st❛♥t❡✳ ❆ ❊q✳ ✭✷✳✷✶✮✱ ❛❧é♠ ❞❡ ❞❡✜♥✐r ♦ r❡❞s❤✐❢t✱ ✐♥tr♦❞✉③ ❛
r❡❧❛çã♦ ❡♥tr❡ ❡❧❡ ❡ ♦ ❢❛t♦r ❞❡ ❡s❝❛❧❛ a= 1/(1 +z)✳
❊①✐st❡♠ ✈ár✐❛s ♠❛♥❡✐r❛s ❞❡ ❝♦♠♣✉t❛r ❞✐stâ♥❝✐❛s ❡♠ ❝♦s♠♦❧♦❣✐❛✳ ❱❛♠♦s ♣r✐♠❡✐r♦ ❞❡✜♥✐r ❛ ❞✐stâ♥❝✐❛ ❝♦♠ó✈❡❧ ❞❡ ✉♠ ♦❜❥❡t♦ ♥♦ ❝é✉ ❡♠ r❡❧❛çã♦ ❛ ♥ós ❝♦♥s✐❞❡r❛♥❞♦ ❛ tr❛❥❡tór✐❛ ♣❡r❝♦rr✐❞❛ ♣♦r ✉♠ r❛✐♦ ❞❡ ❧✉③ ❡♠✐t✐❞♦ ♣♦r ❡❧❡✳ ❖ ✐♥t❡r✈❛❧♦ ♥♦ ❡s♣❛ç♦✲ t❡♠♣♦✱ ❛♦ ❧♦♥❣♦ ❞❛ tr❛❥❡tór✐❛ r❛❞✐❛❧ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❝♦♠ ✈❡❧♦❝✐❞❛❞❡ ❞❛ ❧✉③✱ é ③❡r♦ ✭ds2 = 0✮✳ ❆ss✐♠✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❊q✳ ✭✷✳✶✮ t❡♠♦s
dt a(t) =
dr
√
1−kr2.
❆ ❞✐stâ♥❝✐❛ q✉❡ ✉♠ r❛✐♦ ❞❡ ❧✉③ ♣❡r❝♦rr❡ ❡♥tr❡ re ❡ r0 é ❞❡♥♦♠✐♥❛❞❛ ❞✐stâ♥❝✐❛
❝♦♠ó✈❡❧✳ ❊❧❛ é ❞❡✜♥✐❞❛ ♣♦r
χ(t) = t
te
dt′
a(t′) =
r0(t)
re
dr
√
✶✻ ❈❛♣ít✉❧♦ ✷✳ ❆ ❡①♣❛♥sã♦ ❞♦ ✉♥✐✈❡rs♦ ♦♥❞❡ te ❡ re ❝❛r❛❝t❡r✐③❛♠ ♦ t❡♠♣♦ ❡ ❛ ❞✐stâ♥❝✐❛ ❡♠ q✉❡ ♦ r❛✐♦ ❢♦✐ ❡♠✐t✐❞♦✳ ❆
tr❛♥s❢♦r♠❛çã♦ ❡♥tr❡ ♦ t❡♠♣♦ ❢ís✐❝♦ ❡ ♦ r❡❞s❤✐❢t é ❢❡✐t❛ ❛tr❛✈és ❞❛ r❡❧❛çã♦ a = 1/(1 +z)❡ ❞♦ ♣❛râ♠❡tr♦ ❞❡ ❍✉❜❜❧❡✱
dt=− dz
H(z)(1 +z). ✭✷✳✷✸✮
❊♥tã♦ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❛ ❞✐stâ♥❝✐❛ ❝♦♠ó✈❡❧ ❞❡ ✉♠ ♦❜❥❡t♦ ❡♠ r❡❧❛çã♦ ❛ ♥ós ❝♦♠♦
χ(z) = z
0 dz′
H(z′), ✭✷✳✷✹✮
♦♥❞❡ z é ♦ r❡❞s❤✐❢t ❞♦ ♦❜❥❡t♦✳ P❛r❛ z ♠✉✐t♦ ♠❡♥♦r q✉❡ ❛ ✉♥✐❞❛❞❡ ❡ss❛ r❡❧❛çã♦ é
❛♣r♦①✐♠❛❞❛♠❡♥t❡ χ(z)≃z/H(z)✳
❖ ✐♥t❡r✈❛❧♦ ✐♥✈❛r✐❛♥t❡ ♣♦❞❡ s❡r ❡s❝r✐t♦ ❡♠ t❡r♠♦s ❞❛ ❞✐stâ♥❝✐❛ ❝♦♠ó✈❡❧ ♣♦r
ds2 =−dt2+a2(t)dχ2+fk2(χ)(dθ2+sin2θdφ2), ✭✷✳✷✺✮
♦♥❞❡ ❛ ❢✉♥çã♦ fk(χ)❞❡♣❡♥❞❡ ❞❛ ❣❡♦♠❡tr✐❛ ❡s♣❛❝✐❛❧ ❞♦ ✉♥✐✈❡rs♦✱
fk(χ) = ⎧ ⎨ ⎩
sinχ (k = +1), χ (k = 0),
sinhχ (k =−1).
✭✷✳✷✻✮
❊st❛ ❢✉♥çã♦ ♣♦❞❡ s❡r r❡❡s❝r✐t❛ ❞❡ ♠❛♥❡✐r❛ ♠❛✐s ❝♦♠♣❛❝t❛ ❝♦♠♦
fk(χ) = √1
−ksinh(
√
−kχ). ✭✷✳✷✼✮
P❛r❛ ♦ ✉♥✐✈❡rs♦ ♣❧❛♥♦✱ r❡t♦r♥❛♠♦s à r❡❧❛çã♦ ✭✷✳✷✻✮ ❢❛③❡♥❞♦ ♦ ❧✐♠✐t❡ ❝♦♠ k →0✳
P♦❞❡♠♦s t❛♠❜é♠ ❡st✐♠❛r ❞✐stâ♥❝✐❛s ❝♦♠♣❛r❛♥❞♦ t❛♠❛♥❤♦s ❛♥❣✉❧❛r❡s ❝♦♠ ❞✐♠❡♥sõ❡s ❢ís✐❝❛s✳ ❈♦♥s✐❞❡r❡♠♦s ✉♠ ♦❜❥❡t♦ ❞❡ t❛♠❛♥❤♦ ♣ró♣r✐♦ l ♦rt♦❣♦♥❛❧ ❛
❧✐♥❤❛ ❞❡ ✈✐s❛❞❛✳ ❆ ❞✐stâ♥❝✐❛ ❞❡ ❞✐â♠❡tr♦ ❛♥❣✉❧❛r✱ dA✱ é ❞❡✜♥✐❞❛ ❛tr❛✈és ❞♦
â♥❣✉❧♦θ q✉❡ s✉❜t❡♥❞❡ ❡ss❡ ♦❜❥❡t♦
dA≃
l
θ, ✭✷✳✷✽✮
♦♥❞❡ ✉s❛♠♦s ❛ ❛♣r♦①✐♠❛çã♦ ❞❡ â♥❣✉❧♦ ♣❡q✉❡♥♦ q✉❡ é ✈á❧✐❞❛ ❡♠ q✉❛s❡ t♦❞♦s ♦s ❝♦♥t❡①t♦s ❛str♦♥ô♠✐❝♦s✳ ❖ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ♦❜❥❡t♦ ♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞♦ ❛tr❛✈és ❞❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ s❡✉s ❡①tr❡♠♦s✳ ❯s❛♥❞♦ ❛ ❝♦♥❞✐çã♦ ❞❡ ✐s♦tr♦♣✐❛✱ ❡st❡s ❡①tr❡♠♦s ♣♦ss✉❡♠ ❝♦♦r❞❡♥❛❞❛s (t1, r1, α, φ) ❡ (t1, r1, α+θ, φ) ❡♠ r❡❧❛çã♦ ❛ ♥ós✳ ▲♦❣♦✱ ❞❡
❛❝♦r❞♦ ❝♦♠ ❛ ❊q✳ ✭✷✳✷✺✮✱ ♦ t❛♠❛♥❤♦ ❞♦ ♦❜❥❡t♦ é ❞❛❞♦ ♣♦r l =afkθ✳ ❙✉❜st✐t✉✐♥❞♦
❡st❛ r❡❧❛çã♦ ❡♠ ✭✷✳✷✽✮✱ ♦❜t❡♠♦s
dA(z) =a(z)fk(χ) = fk(χ)