RG Improved Cosmological Evolution and Dark Energy

RG Improved Cosmological Evolution and Dark Energy APCTP-IEU Focus Program Cosmology and Fundamental Physics Jun. 11 (Sat) 2011 Chanju Kim (IEU, Ewha) Based on arXiv:1106.1435 [astro-ph.CO] In Collaboration with Changrim Ahn (IEU) and Eric V. Linder (IEU/LBL)

Outline • Renormalization group(RG) equation • Basic idea of effective field theory • Wilsonian RG • Exact RG equation • RG improved cosmological evolution

Idea of Effective Field Theories • World ~ various physical phenomena at all scales • In principle, Theory of Everything could explain all these phenomena • In practice, Impossible and unnecessary to use TOE • Divide the parameter space of the world into different regions. • Common relevant parameter: Energy scale • Change energy scale under consideration  Different effective action Example: Weinberg-Salam vs Fermi theory

Wilsonian RG • The idea of effective theories can be naturally formulated in Wilsonian RG • Consider a quantum field theory with a characteristic energy scale : low frequency modes : high frequency modes

Physics at : • Integrate out high frequency modes with where • : “Wilsonian Effective Action” • Serves as the action describing physics at low energy • How does change as we vary  “Renormalization Group Equation”

Alternative View of • Bare action (with initial UV cutoff : propagator : interaction part of the bare action • Generating Functional Integrate out higher freq. modes

Divide the propagator and fields into two parts where Has UV cutoff IR cutoff

Dual role of • : generating functional of connected Green functions with UV cutoff & IR cutoff • : : effective action • What if both are nonzero? • gives a relation between the generating functional and effective action

with , • This integral is gaussian. The result is which is a function of the combination • Rewrite the first term as

We can finally express as where • Then • If is precisely the effective action

For general , we obtain a precise relation between effective action and the generating function in the quantum field theory (with Keller, Kopper & Salmhofer (1991) • In: IR cutoff • In , : UV cutoff • Effective action with UV cutoff • Generating functional with amputated connected Green functions with IR cutoff

RG Equation • Take derivative w.r.t. For , Wegner, Houghton(1973), Polchinski(1984)

1PI Generating Functional • Legendre transform • Define the interaction part as

Interaction part satisfies • We finally obtain “Exact RG” Wetterich (1993) Bonini, D’Attanasio & Marchesini Morris • Note 1: The choice of the cutoff function is arbitrary • Sharp cutoff: • Smooth cutoff: e.g. • Choosing the cutoff function is a kind of “renormalization scheme” (Physical quantities should independent of it.)

Note 2: The separation of the action into the free and the interaction part is arbitrary • One may start from, e.g., : full bare action : pure cutoff term added to the action Then all the previous equations are valid with and replaced by and :

Approximation • Practically, impossible to solve RG equation exactly • An obvious approximation: derivative expansion • Nonperturbative approximation • RG equation reduces to differential equations of coefficient functions and • Scheme independence lost • Successfully applied to many problems including gauge theories • Cutoff breaks gauge symmetry • Modified Ward identity

Application to Cosmology • Effective action of Einstein gravity in “Einstein-Hilbert truncation” : -dependent • Matter: scalar field(dark energy part) + barotropic fluids : -dependent absorb into

Equations of Motion • Effective action contains quantum effects • The form of the eq. of motion unchanged in our approximation except and • In FRW universe, : total energy density and pressure • In addition, we have a Klein-Gordon equation

IR cutoff and Time • Universe with age • Quantum fluctuations with momenta smaller than do not play any role  may not be integrated out • Then the IR cutoff is a function of time • may also depend on Hubble parameter , i.e.,  are time-dependent  Extra time dependence in Eq. of motions • Need to check that the truncation of RG is consistent

Bianchi Identity • Friedmann eqs come from Einstein equation • Einstein tensor automatically satisfies Bianchi identity  covariant derivative of rhs should vanish • If is constant, this is the usual continuity eq. •  equation modified • In FRW cosmology i.e.,

For the dark energy component, where are RG parameters , • On the other hand, Consistency condition:

RG improved Eqs • Introduce • Eqs of motion can be rewritten as

Consistency condition Then • and will be determined if the potential is given • Here, we will consider possible fixed points at which

Fixed Points 1. Accelerating, dark energy dominated • If asymptotically, approaches de Sitter state • Stable fixed point for

2. Scaling • dark energy and barotropic component have densities in a constant ratio • not accelerating unless one already had an accelerating barotropic component

3. Flowing, dark energy dominated • depends on specific values of RG parameters

Relation of to Hubble Scale • Partial differentiation of Friedmanneqwrt • Apply the consistency condition : t-dependent • In the truncated RG, we cannot simply assume without explicit time dependence

Newton constant • Newton constant and • CMB & primordial nucleosynthesis abundances shows const up to a precision 10% over time: • This gives a condition on • can be achieved for small or slow flow

Conclusion • We have explored the quantum modifications to cosmological evolution at late times • In the approximation scheme used, we found a consistency condition which restrict the relation between the RG parameters and the cosmological quantities • Three classes of fixed points identified • Future works • Solve RG explicitly for various specific potentials • beyond Einstein-Hilbert truncation

RG Improved Cosmological Evolution and Dark Energy