Fixed Functionals in Quantum Einstein Gravityusers.uoa.gr/~papost/TALKS_FILES/Demmel.pdf · introduction 3/20 setup tool: kdΓ k dk = 1 2 Tr Γ(2) k + R k −1 kdR k dk truncation:

1 / 20

Fixed Functionals in Quantum Einstein Gravity

Maximilian Demmel, Frank Saueressig, Omar Zanusso

THEPUni Mainz

22.09.2014

Maximilian Demmel Fixed Functionals in QEG

2 / 20

1 introduction

2 pole crossing - analytic example

3 fixed functions in d = 3

4 fixed functions in d = 4

5 summary


introduction 3 / 20

setup

tool: k dΓkdk = 1

2 Tr[(

Γ(2)k +Rk

)−1k dRk

dk

]

truncation: Γgravk [gµν ] =

∫ddx√g fk(R)

[0705.1769, 0712.0445, 1204.3541, 1211.0955, ...]

background field method gµν = gµν + hµν

conformal reduction: hµν = 1d gµν φ

[0801.3287, ...]

maximally symmetric background: Sd (R > 0)


introduction 4 / 20

type II regulator

operator: � := −D2 + E (potential term E containing R)

define regulator Rk(�)

Γ(2)k (�) +Rk(�) def.= Γ(2)

k (� + Rk(�)),

where Rk is the profile function (Litim’s cutoff).

flow equation ∫ddx√g ∂tfk(R) = 1

2 Tr W (�)


introduction 5 / 20

operator trace

spectral sum:

Tr W (�) =∑

iDiW (λi)

eigenvalues λi and multiplicities Di of �

integrating out eigenmodes:

optimised cutoff =⇒ W (λi) ∝ θ(k2 − λi)=⇒ W (λi) 6= 0 ⇐⇒ k2 ≥ λi

every time k crosses λi the eigenmode is integrated out

spectral sum is a finite sum (ACHTUNG: λi ∝ R)


introduction 6 / 20

definition: fixed function

definition:

R =: k2r , E =: k2e, fk(R) =: kdϕk(R/k2)

flow equation: partial differential equation for ϕk(r)

fixed functions: (global) k stationary solutions ⇐⇒ ∂tϕk(r) = 0

third order equation: three parameter family of solutions


introduction 7 / 20

pole structure of flow equation

the coefficient of ϕ′′′(r)

1 2 3 4 5 6

r

1000

2000

3000

4000

5000

6000

three poles r0, r1 and r2 (in d = 3, 4)

global solutions should cross the poles


pole crossing - analytic example 8 / 20

regular singular points and pole crossing

generic ODE: y(n)(x) = f (y(n−1), . . . , y′, y, x)

r.h.s. f can have singular points

f (y(n−1), . . . , y′, y, x) = e(y(n−1)(x0), . . . , y′(x0), y(x0), x0)x − x0

+O((x − x0)0)

pole crossing (global) solution ⇐⇒ e|x=x0= 0 (regularity

condition)

additional boundary condition reduces number of free parameters



analytic example

initial value problem:

y′′(x) = − y(x)x − 1 , y(0) = y0, y′(0) = y1

solutions are modified Bessel functions In. For most y0, y1: no globalsolution.

add BC y(1)=0: pole crossing solution

y(x; y0) = y0

√1− x I1

(2√

1− x)

I1(2)

self similarity in initial values y(x; y0) = λ−1y(x;λy0)



analytic example

initial value problem:

y′′(x) = − y(x)x − 1 , y(0) = y0, y′(0) = y1

polynomial expansion: y(x) =∑k

n=0 anxn

fixing all coefficients at x = 0 =⇒ ai = 0

improved strategy: fix an at x = 0 for n ≥ 2 and fix a1 at x = 1=⇒ series expansion of analytic solution recovered

self similarity y(x) = y0∑k

n=0 anxn


fixed functions in d = 3 11 / 20

numerical shooting I

expand around r = 0

ϕ(r ; a0, a1) = a0 + a1r +k∑

n=2an(a0, a1)rn

1 fix an, n ≥ 2 by using regularity condition at r = 0

2 initial conditions for numerical integration (ε > 0)

ϕ(n)init(ε) = ϕ(n)(ε; a0, a1), n = 0, 1, 2

3 regularity condition at r1

e : (a0, a1) 7→ R



pole crossing solutions

0.2 0.4 0.6 0.8 1.0

a0

-2.0

-1.5

-1.0

-0.5

a1

green: e(a0, a1) > 0red: e(a0, a1) < 0black line: e(a0, a1) ≈ 0



polynomial expansion

polynomial expansion

ϕ(r) =n∑

k=0akrk

boundary condition atr = 0 (a2, a3, . . . )

boundaty condition atr = r1 (a1)

a1 as function of a0(dashed blue line)

0.1 0.2 0.3 0.4

a0

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

a1

n=17



numerical shooting II

algorithm for second shooting

1 parametrize regular line by a0

2 initial conditions for second numerical integration

ϕ(n)init(r1 + ε) =ϕ(n)

num(r1 − ε; a0), n = 0, 1, 2

3 numerically integrate up to r2 and check regularity condition

e3 : a0 7→ R



regularity at r2

0.24 0.32 0.4

a0

-0.04

-0.02

0.02

0.04

0.06

0.08

e2

-6 -4 -2 2 4 6

r

-1

1

2

3

4

5

6

jHrL

There are two distinct zeros =⇒ two fixed functionsimproved stability: at most three relevant directions



preliminary results in d = 4

0.02 0.03 0.04 0.05

a0

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

a1

n=17

0.015 0.020 0.025 0.030 0.035 0.040

a0

0.00

0.02

0.04

0.06

0.08

e3

numerical shooting and polynomial expansion yield regular line

one (nontrivial) fixed function identified



fixed function in d = 4

1 2 3 4 5

r

0.0

0.1

0.2

0.3

jHrL


summary 18 / 20

summary

flow of conformally reduced f (R)-gravity

interpretation of integrating out eigenmodes

numerical and analytical techniques for pole crossing

two fixed functions in d = 3

one fixed function in d = 4


summary 19 / 20

Porto Katsiki - 21. Sept. 2014


summary 20 / 20

Thank You!


Documents

Fixed Functionals in Quantum Einstein Gravityusers.uoa.gr/~papost/TALKS_FILES/Demmel.pdf · introduction 3/20 setup tool: kdΓ k dk = 1 2 Tr Γ(2) k + R k −1 kdR k dk truncation: