New Perspective on the Cosmological Constant ...

New Perspective on the Cosmological Constant Problem Vesselin G. Gueorguiev, Ph.D. [email protected]

Sat. November 14, 2009 3:12PM - 3:24PM

S4-7-Gueorguiev, CA Section of APS meeting, Naval Postgraduate School, Monterey, CA

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Outline of the Relevant Ideas • How big is the CC problem? • Solution: actually, there isn’t any problem! – the QFT estimate is OK for a small universe. – the CC for Big Universe scales as 1/a2.

• The bedrock of the solution: – Scale related solutions of the EFE:

• Technicalities:

˜ , g˜ }, {",g} # {" g

%$˜ " = a g˜$% , " = 2 a 2

– Partition function for a multiverse ensemble, – Characteristic scale variable ‘a’, – Planck ensemble of solutions, ! – Consistency check ! Sat. November 14, 2009 3:12PM - 3:24PM


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Vacuum Energy Density (Quantum Filed Theory Estimate) • Dimensional argument (Planck Energy Density):

" Planck

c7 114 3 = = 1 Planck Units (# 4.6 $10 erg/cm ) 2 hG

• Zero point energy of a quantum field (c=1,ℏ=1):

!

E cut >>m

" ZPE =

$ 0

4 4 #k 2 dk 1 2 E cut 2 -3 k + m = % 6 &10 ; 2 2 (2# ) 2 16#

(E cut = E Planck =

!


hc 5 = 1) G

S. Weinberg RMP (1989)


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Vacuum Energy Density (Cosmological Constant) • Einstein Field Equation (EFE):

1 8&G R"# $ g"# (R $ 2%) = 4 T"# 2 c • Cosmological Observations:

c4 " #!= # = 4.6 %10&114 erg /cm 3 ' 10&123 [Planck Units] 8$G D. N. Spergel, astro-ph/0603449

!



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HUGE PROBLEM

"# " Planck


!

%123

$ 10


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Characteristic Scale • Empty Space Einstein Field Equation: 1 R"# $ g"# (R $ 2%') = 0 2

• Labels for the Ensemble of Solutions: !{"',{boundary conditions},g} # $

Characteristic Scale as Auxiliary Label

!



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WHAT IF? What if QFT or dimensional estimate of the CC is OK only for small Planck scale Universe? What should be the CC for a Big Universe?



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Scale Related Solutions 1 R"# $ g"# (R $ 2%) = 0 & {%,g} solution, 2 1 ˜ ˜ ) = 0 & {% ˜ , g˜} solution, R"# $ g˜"# ( R˜ $ 2 % 2 ˜ % 2 ˜ {%,g} ' {%, g˜ }, g"# = a g˜"# , % = 2 ; a SCALE independent : ) + + + ("# * g)+ ," g#+ , R"#) * ("(#) , R"# * ("(#+ ;

Inverse SCALING a -2 : R = R"# g"# .


!


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Change in Variables • Characteristic scale variable ‘a’. ˜ , g˜ ,#˜,a}, {",g,# } $ {" g%& = a g˜%& ' # = a#˜ = 2

( ds

g%&

dx % dx & , ds ds

˜ " 1 ˜ " = 2 ) g%& " = g˜%& " ) R%& * g%& (R * 2") = 0. a 2

!



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Planck Ensemble {ğ} Solutions of the EFE with Planck scale {",g,# = 1} characteristic scale: ˜ , g˜ ,#˜ = 1,a}, {",g,# } $ {" ! ˜ " 2 g%& = a g˜%& , # = a#˜ = a, " = 2 a

!



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a

!

Cosmological Constant for LARGE UNIVERSES • Planck ensemble:

a

˜ = 8# , g˜ ,$˜ = 1} % & " = 1 {" ˜ 8& % 2 g"# = a g˜"# , $ = a, % = 2 = 2 a a

• Our universe: !

˜ 8& % a " 8 #10 $ % = 2 = 2 " 4 #10'121 a a ! 8& # 6 #10'3 '123 %= " 2.4 #10 (QFT estimate +Scaling) 2 a 60



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Conclusions and Discussion Resolving the Cosmological Constant Problem: – The QFT estimate is correct for small " # 1, characteristic scale universe: $ ZPE # 6 %10 -3 [Planck Units]

– CC for large size universe should be scale related to its Planck size partner: a " 8 #10 60 , ! ˜$QFT at Planck Scale -123 $= " 2.4 #10 [Planck Units] 2 a


!


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General Partition Function • Action for Gravity & Matter: c4 4 A[g," ] = (R $ 2%) $gd x + Amatter [g,",'" ]; & 16#G i i A [g, " ] A[g," ] 1 h h O = & D[g," ]O(g,'g,",'" )e ; Z = & D[g," ]e ; Z

• “Classical Trajectories” (EFE): 1 8&G R"# $ g"# (R $ 2%) = 4 T"# 2 c Sat. November 14, 2009 3:12PM - 3:24PM

!


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Integrating out the Matter (Effective Λ) • Dominant contributions - Action Functional on solutions of the EFE : 1 8&G R"# $ g"# (R $ 2%) = 4 T"# ; ' g"# 2 c 8&G R $ 2% = 2% $ 4 trT = 2 %( ) 2%; c c4 4 ( A[g] ) 2 % $gd x * 16&G Sat. November 14, 2009 3:12PM - 3:24PM

!


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Multiverse Ensemble (Characteristic Scale) • Empty Space Einstein Field Equation: 1 c4 R"# $ g"# (R $ 2%') = 0, A[g] & 2 16'G

4 ( 2 % $gd x )

• Labels for the Ensemble of Solutions:

{"',{boundary conditions},g} # $

!

Characteristic Scale as Auxiliary Label

!



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Euclidean Formulation • Partition Function in Euclidean Form & c4 ) 4 Z = " D[gE ]exp(# " %' #gE d x + ' 8$Gh *

• Hand waving justification !

c4 A[g] " 16#G

4 $ 2 % &gd x= '

)i , i exp+ A[g]. = exp(&/E ), / = & *h h Sat. November 14, 2009 3:12PM - 3:24PM

4 ( &gd x ' %

' dt, E = ' (

(3) (3) 4 dV , dtdV = &gd x %


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Change in Variables • Characteristic scale variable ‘a’. ˜ , g˜ ,#˜,a}, {",g,# } $ {" g%& = a g˜%& ' # = a#˜ = 2

( ds

g%&

dx % dx & , ds ds

˜ " 1 ˜ " = 2 ) g%& " = g˜%& " ) R%& * g%& (R * 2") = 0. a 2

!



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Planck Ensemble {ğ} • Solutions of the EFE with Planck scale characteristic scale: {",g,# = 1} a ˜ , g˜ ,#˜ = 1,a}, {",g,# } $ {" g%&

˜ " = a g˜%&!, # = a#˜ = a, " = 2 a 2

& 1 4 ) Z = " D[g]exp(# " % #gd x + , Z = ' 8$ *

! !


˜ a 2 (4 ) ) & % " da D[g˜ ]exp('# 8$ V [g˜ ]+* 2


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Average Characteristic Scale • Consistency check… – Planck ensemble: {"˜ = 8# , g˜ ,$˜ = 1} % & " = 1 (4 ) ˜4 ˜ ˜ V [ g ] = " [ g ] # – 4-Volume: – Geometric factor: " [ g˜ ] # 1, " [R 4 ] = 1, ! " [D4 ] = $ 2 /2 # 5, " [BH] = 16$ 2 /3 # 53 !

• Partition Function:

˜ a 2 (4 ) ) & $ 1 Z = " da D[g˜ ]exp( # V [ g˜ ]+ , " da 2 exp(#a 2- ) = ' 8% * ! ˜ a 2 (4 ) ) 1 & $ 1 1 % 2 ˜ . = " da D[g˜ ]a. exp(# V [ g˜ ]+ , " da 2 a.˜ exp(#a 2- ) = ,1 Z 2 ' 8% * Z 2


!


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!

Cosmological Constant for LARGE UNIVERSES • Planck ensemble:

a

˜ = 8# , g˜ ,$˜ = 1} % & " = 1 {" ˜ 8& % 2 g"# = a g˜"# , $ = a, % = 2 = 2 a a

• Our universe: !

˜ 8& % a " 8 #10 $ % = 2 = 2 " 4 #10'121 a a ! 8& # 6 #10'3 '123 %= " 2.4 #10 (QFT estimate +Scaling) 2 a 60



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