Graviton contributions to the graviton self-energy at one loop order during inflation

Graviton contributions to the graviton self-energy at one loop order during inflation PEDRO J. MORA DEPARTMENT OF PHYSICS UNIVERSITY OF FLORIDA PASI2012

1. Description of my thesis problem. i. Graviton self-energy at one loop order. ii. Dynamical gravitons and the force of gravity. 2. Importance of my thesis problem. i. Inflation and de Sitter ii. Quantum effects during inflation iii. What I expect from my calculation. 3. Strategic plan for solving my thesis problem. i. Feynman rules ii. Calculations in de Sitter space iii. Example: 3-pt vertex.

! Using dimensional regularization, I will calculate and BPHZ renormalize the one loop contribution to the graviton self-energy from gravitons on a de Sitter background.! I will study the effects that quantum corrections have on dynamical gravitons and on the force of gravity.! The context of this work is quantum gravitational perturbation theory about a de Sitter background. This means that we express the full metric as:!! µ" is the flat space metric with spacelike signature.! h µ" is a small perturbation to this background.! k 2 = 16#G = loop counting parameter of QG.! a = a(!) = -1/H! is the scale factor.

! This part of my calculation consists of finding the first order correction to the graviton mode functions u(!,k). To study the effects of quantum corrections to dynamical gravitons and the force law I need to solve the quantum corrected version of the classical field equations:! The gauge fixed kinetic operator in a de Sitter background is, in D=4 : where D A is the type-a de Sitter covariant derivative operator:! I will only calculate the graviton self-energy to one-loop order:! Hence I can only solve for the graviton field perturbatively at order k 2 :

! To this order, the equation I need to solve is:! Energy-momentum tensor: " dynamical gravitons.! Zeroth order perturbation: " transverse, traceless, purely spatial.! To the same order the mode functions are:

! I m also interested in finding the quantum corrections to the force of gravity due to graviton contributions.! Calculate the force as the response of a point mass in the field.! I will solve the same equation as before:! Energy-momentum tensor: " point particle.! Zeroth order perturbation:

! First, this problem has not been done before.! Calculating the graviton contributions to the graviton self-energy allows us to determine the quantum corrections to the mode functions and the force of gravity.! Knowledge about the mode functions during inflation allows us to know what happens to dynamical gravitons.! The results are in principle measurable and can be compared to actual cosmological data (WMAP, Planck).

! Consider our universe to be homogeneous and isotropic on the largest scales.! Based on these two assumptions, the invariant line element appropriate for cosmological models with zero spatial curvature (k=0) is given by the FRW metric: comoving coordinates conformal coordinates! Cosmological parameters:! Can also define:

! The tensor-to-scalar ratio is then:! Data collected with the WMAP satellite give : r < 0.22 (95% CL) (at k=0.002 Mpc -1 ) ( upper bound )! Hence,! This means that primordial inflation was very close to the de Sitter limit $=0 which allows us to work in exact de Sitter without large deviations from the data.! With these values, we can also find the value of H during primordial inflation: ( G = 6.71 x10-39 GeV -2 )! We will henceforth use de Sitter as a paradigm for inflation.

! It is important to note that quantum effects are enhanced during inflation.! Quantum loop effects are produced by the classical response to virtual particles. This means that in order to enhance these effects we need a mechanism that increases the number density for these particles.! In this section I will show how de Sitter expansion provides this mechanism and leads to virtual particles being produced out of the vacuum for gravitons and massless, minimally coupled scalars.! Consider the Lagrangian density of a massless minimally coupled scalar:! When expanded, the Lagrangian is:

! None of the different modes are coupled " we can treat them independently and write the Lagrangian for one of them: No stationary states! Following the harmonic oscillator analogy, the minimum energy eigenstate at time t has:! Here because of inflation, we have defined our vacuum to be a Bunch-Davies vacuum which is the state with minimum energy in the distant past.! The equation of motion for q(t) that follows from the previous Lagrangian is:! Grishchuk found that dynamical gravitons obey the same equation as massless minimally coupled scalars.

! The mode functions for dynamical gravitons are obtained from the linearized Einstein field equations: (transverse, traceless and purely spatial)! Expanding h TT ij in terms of its mode sum this equation reduces to the equation of motion for the mode functions: Hard to solve for general a(t)! A solution for de Sitter exists of the form: Solution is known for de Sitter (and in general for $ = const.)! Can write the general solution as: where the Bunch-Davies vacuum is annihilated by the operator % :! The expectation value of the Hamiltonian is:! The particle number can be read off in analogy with the quantum mechanical SHO:

Calculation of graviton self-energy Quantum-corrects the linearized gravitational field equations! This result will allow me to find what happens to dynamical gravitons propagating through the de Sitter background. In other words, we are interested in finding the mode functions of the graviton field expansion.! We perturbatively expand the graviton field mode functions to order k 2.! This way we can make contact with the power spectrum:! The graviton field has the mode function expansion:

! Substituting the mode function expansion in & h2 (p)in the case of a transverse traceless graviton field gives the following result: first order correction! In this work I will compute the one loop correction to the mode functions u(t,p) and this will contribute to the measured result above.! Although these are very hard to observe in practice, they are still measurable in principle.

! The other part of my calculation is about the effect of dynamical gravitons on the force of gravity.! It is clear that in the corresponding limit the classical result should be obtained, '=-GM/r. Hence from dimensional analysis we expect the force law to modified as follows:! Our calculation is done in de Sitter space; this background furnishes an extra parameter H: [ H ] ~ length -1! We can form another dimensionless quantity:! Important : During inflation virtual gravitons are created from the vacuum and they will tend to cluster around the source, which ought to progressively increase its strength. In all previous computations for which secular growing dependence has appeared, it shows up as a factor of ln(a(t)) so we also expect corrections of the form

! In this section I will illustrate the method I will be using to perform the calculations with one example. 1. Feynman rules. 2. Calculations in de Sitter space. 3. 3-pt vertex contribution to self energy.

Barred quantities :

! The tensor factors are :! The scalar propagators satisfy:! i µ" & %( (x;x ) satisfies the propagator equation : # Ghosts and gauge fixing

! The expression for the A-type propagator ("=(D-1)/2 ; M 2 =0) is more interesting because it does not have a form like the previous two. Allen and Folacci* have shown that there exists no de Sitter invariant propagator for a massless minimally coupled scalar.! It has a similar infinite series but contains a de Sitter breaking term which depends upon the scale factor at the two points x and x :! The function A(y) has the expansion:! It is important to note that the infinite series terms of i& " (y(x;x')) vanish for D=4, hence we need to retain them only when multiplying a potentially divergent quantity.! This makes loop computations manageable. * B. Allen, A. Folacci.J. Math. Phys. 32, 2828 (1991)

! The gravity Lagrangian contains second derivatives of the metric, hence the generic form of an n-pt. graviton interaction is.! I will consider the 3-point interaction term (times k):! Indices here are raised and lowered with the background (flat space) metric.! The vertex factor for this interaction is:! Since each graviton field corresponds to a different leg in the Feynman diagram, I will label those as 1, 2 and 3.! The labels 2 and 3 indicate at which h µ" the derivative acts.! Note: I will label here the legs according to this diagram: and will consider the contribution of the vertex: 1 2 3

! The contribution of this vertex to has the analytic form:! Substitute in the graviton propagator:! This will produce nine terms quadratic in the tensor factors; I will concentrate only on the T C T A term.! Contracting all the repeated indices gives:! We can act the derivatives on the scalar propagators using the y-basis: When acting on B and C type propagators

! When acting on the A-type propagator we get an extra term from the de Sitter breaking term:! After acting the derivatives we obtain:

! At this point we must determine which terms in the infinite series must be kept.! Recall that what we are interested in is the D=4 limit of the integral:! To find the potentially divergent terms we note that the smallest power of &x in the basis tensors above are:! We can then consider only the following terms of the scalar propagators :

! Zooming a little bit more, consider only the first term :! To present the method of renormalization I will consider only the most divergent term in the above equation. Using the above expression of the propagator for the second derivatives and the smallest powers for the y-basis tensors given above the resulting expression is quartically divergent in the unregulated limit since the integrand is proportional to:! The contribution of this term to the self-energy is:! The contraction of the basis tensors gives:

! As mentioned above, this term is quartically divergent (in D=4).! To decrease the level of divergence we can integrate by parts using the following identities:! Now we can pull the derivatives out of the integral and the resulting integral is logarithmically divergent:! I will explain how to renormalize this term next.

! To eliminate the logarithmic divergence we can perform another partial integration using:! The integral becomes finite; however the price we have to pay for this is that now we have the divergent factor 1/(D-4) when D"4:! Now recall that the counterterms should be local. This is accomplished in the following way: we can absorb the divergences by local counterterms by using the scalar propagator equation absorbing the divergences! Hence we can add zero in the form of the propagator equation to the problematic term Dimensional regularization mass scale

! Important point: Non-local, finite term # Counterterms! So far we have not included the necessary counterterms which are needed at this point.! The present one-loop diagrams have a superficial degree of divergence of 4 which means that to cancel the appearing divergences we need to consider all possible counterterms constructed out of 2 h s and up to 4 derivatives.! The flat space limit correspondence restricts the number of possibilities. Local, divergent term! We will include the usual counterterms used in gravity: R 2, C %(µ" C %(µ".! Our choice of gauge breaks de Sitter symmetry" we expect sub-leading contributions to the counterterms.

! Finally it is important to mention the correspondence limits to which this work can compared. $ This work will generalize the results obtained by Tsamis and Woodard where they calculated the graviton self-energy to one loop order specifically for D=4 away from coincidence. $ Here I will present the results in general for D dimensions and at any point including x=x. $ Also this calculation will be useful to compare the results of the flat space limit where H=0 at fixed time t along with the conditions imposed by the Ward identities.

Thank you