RG flows of Quantum Einstein Gravity on maximally symmetric spaces

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RG flows of Quantum Einstein Gravity on
maximally symmetric spaces
Maximilian Demmel, Frank Saueressig, Omar Zanusso
THEP
Uni Mainz
02.06.2014
Maximilian Demmel
QEG on symmetric spaces
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1
introduction
2
heat kernel on symmetric spaces
3
construction of fixed functions
Maximilian Demmel
QEG on symmetric spaces
introduction
3 / 22
setup
tool:
k
k dΓ
dk
=
1
2
Tr
(2)
Γk
+ Rk
−1
k
k dR
dk
R
√
[gµν ] = d3 x g fk (R)
truncation: Γgrav
k
[0705.1769, 0712.0445, 1204.3541, 1211.0955, ...]
conformal reduction: hµν = d1 g¯µν φ
[0801.3287, ...]
maximally symmetric background: S 3 (R > 0) and H 3 (R < 0)
Maximilian Demmel
QEG on symmetric spaces
introduction
4 / 22
type II regulator
¯ 2 + E (potential term E containing R)
¯
operator: := −D
define regulator Rk ()
def.
(2)
(2)
Γk () + Rk() = Γk ( + Rk ()),
where Rk is the profile function (Litim’s cutoff).
flow equation
Z
√
¯ = 1 Tr W ()
d3 x g ∂t fk (R)
2
Maximilian Demmel
QEG on symmetric spaces
heat kernel on symmetric spaces
5 / 22
heat kernel on S 3
heat kernel of scalar Laplacian −D 2 :
Tr e−s(−D
2
)
Z
=
√
d3 x g
∞ X
6n 2 π 2
esR/6
12π 2 n 2
e− Rs
1
−
3/2
Rs
(4πs)
n=−∞
n = 0: local part of the heat kernel
Visible in small R expansions.
n 6= 0: nonlocal/topological part of the heat kernel
contains n-fold returning modes
due to compactness
“beyond all orders in R”
Maximilian Demmel
QEG on symmetric spaces
heat kernel on symmetric spaces
6 / 22
heat kernel on H 3
taking into account only normalizable eigenfunctions of −D 2
Z
esR/6
√
−s(−D2 )
Tr e
= d3 x g
,
R<0
(4πs)3/2
no returning modes due to the non-compactness
analytic continuation to negative R is not possible
local (polynomial) analysis will be insensitive of the background
topology
Maximilian Demmel
QEG on symmetric spaces
heat kernel on symmetric spaces
7 / 22
operator trace on S 3
spectral sum ⇐⇒ local + nonlocal heat kernel
X
Tr W () =
Di W (λi + E)
i
=
X
n 2 W (n 2 − 1)R/6 + E
n≥1
local approximation: use local heat kernel only
Z ∞
Z
s(R/6−E)
f (s) d3 x √g e
Tr W () =
ds W
(4πs)3/2
Z0
1
√
= d3 x g
Q3/2 [W (z − R/6 + E)]
(4π)3/2
Maximilian Demmel
QEG on symmetric spaces
heat kernel on symmetric spaces
8 / 22
operator trace on H 3
using the exact heat kernel
Z
√
Tr W () = d3 x g
1
Q3/2 [W (z − 1R/6 + E)]
(4π)3/2
now: R < 0!
formally the analytic continuation to negative R of the local
approximation on S 3 .
Maximilian Demmel
QEG on symmetric spaces
heat kernel on symmetric spaces
9 / 22
dimensionless quantities
definition:
R =: k 2 r,
E =: k 2 e,
fk (R) =: k 3 ϕk (R/k 2 )
flow equation: partial differential equation for ϕk (r)
fixed functions: k stationary solutions ⇐⇒ ∂t ϕk (r) = 0
third order equation: three parameter family of solutions
Maximilian Demmel
QEG on symmetric spaces
heat kernel on symmetric spaces
10 / 22
integrating out eigenmodes
optimized cutoff: Rk (z) = (k 2 − z)θ(k 2 − z)
Tr W () =
Nr
X
A(n, . . . ) θ 1 − (n 2 − 1 + α)r/6 ,
n≥1
finite sum for r > 0
for lowest eigenmode (n = 1) to contribute r < 6/α
α = 1: trace zero for r > 6
=⇒ all fluctuations are integrated out!
Maximilian Demmel
QEG on symmetric spaces
e = αr/6
construction of fixed functions
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singular points and pole crossing
generic ODE: y (n) (x) = f (y (n−1) , . . . , y 0 , y, x)
r.h.s. f can have singular points
f (y (n−1) , . . . , y 0 , y, x) =
e(y (n−1) (x0 ), . . . , y 0 (x0 ), y(x0 ), x0 )
+ O (x − x0 )0
x − x0
pole crossing solution ⇐⇒ e|x=x0 = 0 (regularity condition)
additional boundary condition reduces number of free parameters
Maximilian Demmel
QEG on symmetric spaces
construction of fixed functions
12 / 22
example
initial value problem:
y 00 (x) = −
y(x)
,
x −1
y(0) = y0 , y 0 (0) = y1
solutions are modified Bessel functions In . For y0 , y1 generic: no global
solution.
add BC y(1)=0: pole crossing solution
√
√
1 − x I1 2 1 − x
y(x; y0 ) = y0
I1 (2)
self similarity in initial values y(x; y0 ) = λ−1 y(x; λy0 )
Maximilian Demmel
QEG on symmetric spaces
construction of fixed functions
13 / 22
pole structure of flow equation
poles are zeros of the coefficient of ϕ000 (r).
6000
5000
4000
3000
2000
1000
r
1
2
3
4
5
6
three poles at r0 = 0, r1 ≈ 1, r2 = 6
local approximation shows only two poles!
number of fixed functions could be reduce to a finite number
Maximilian Demmel
QEG on symmetric spaces
construction of fixed functions
14 / 22
numerical shooting I
expand around r = 0
ϕ(r; a0 , a1 ) = a0 + a1 r +
k
X
an (a0 , a1 )r n
n=2
1
fix a2 by using regularity condition at r = 0
2
initial conditions for numerical integration (ε > 0)
(n)
ϕinit (ε) = ϕ(n) (ε; a0 , a1 ),
3
n = 0, 1, 2
regularity condition at r1
e : (a0 , a1 ) 7→ R
Maximilian Demmel
QEG on symmetric spaces
construction of fixed functions
15 / 22
pole crossing solutions
a1
a0
0.2
0.4
0.6
0.8
-0.5
-1.0
-1.5
-2.0
green: e(a0 , a1 ) > 0
red: e(a0 , a1 ) < 0
black line: e(a0 , a1 ) ≈ 0
Maximilian Demmel
QEG on symmetric spaces
1.0
construction of fixed functions
16 / 22
numerical shooting II
algorithm for second shooting
1
parametrize regular line by a0
2
initial conditions for second numerical integration
(n)
ϕinit (r1,sing + ε) =ϕ(n)
num (r1,sing − ε; a0 ),
3
n = 0, 1, 2
integrate up to r = 6 and check regularity condition
Maximilian Demmel
QEG on symmetric spaces
construction of fixed functions
17 / 22
regularity at r = 6
jHrL
6
e2
0.08
5
0.06
4
0.04
3
0.02
2
1
a0
0.24
0.32
0.4
r
-0.02
-6
-4
2
-2
-1
-0.04
There are two distinct zeros =⇒ two fixed functions
improved stability: at most three relevant directions
Maximilian Demmel
QEG on symmetric spaces
4
6
construction of fixed functions
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improved expansion: example
initial value problem:
y 00 (x) = −
expansion: y(x) =
Pk
n=0
y(x)
x −1
an x n
fixing all coefficients at x = 0 yields only trivial solution =⇒ ai = 0
improved strategy: fix an at x = 0 for n ≥ 2 and fix a1 at x = 1
=⇒ Analytic solution can be reproduced (even self similarity)
Maximilian Demmel
QEG on symmetric spaces
construction of fixed functions
19 / 22
improved expansion: flow equation
Algorithm: fix an at r = 0 for n ≥ 2 and fix a1 at r1
Result: there are two (a0 -dependent) solutions (dashed lines)!
jHrL
1.0
0.5
r
0.2
0.4
0.6
0.8
-0.5
-1.0
Maximilian Demmel
QEG on symmetric spaces
1.0
construction of fixed functions
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regular lines
relation between a1 and a0 can now be computed analytically (dashed
lines)
a1
a0
0.2
0.4
0.6
0.8
-0.5
-1.0
-1.5
-2.0
qualitative agreement!
Maximilian Demmel
QEG on symmetric spaces
1.0
construction of fixed functions
21 / 22
summary
influence of background topology
interpretation of integrating out eigenmodes
numerical and analytical techniques for pole crossing
two distinct fixed functions constructed
Maximilian Demmel
QEG on symmetric spaces
construction of fixed functions
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Thank You!
Maximilian Demmel
QEG on symmetric spaces

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