The Cosmological Constant Problem

1. The Cosmological Constant Problem Jérôme Martin Institutd’Astrophysique de Paris

2. Outline 1- Introduction: the Cosmological Constant (CC) in the Einstein equations & observational constraints on the CC. 2- What we do not understand about the CC. 3- Possible loopholes in our approach to the CC problem. 4- General conclusions.

3. The cosmological constant (CC): introduction • In presence of a Cosmological Constant, the Einstein field equations read geometry CC matter • Preserves covariance • Covariant derivative vanishes hence compatible with a conserved energy momentum tensor • Dimension length^ (-2) • The CC can always been seen as an extra source of matter: • The equation of state of the CC is: . The effective pressure is negative.

4. The cosmological constant: constraints Planck energy density 2011 Nobel prize Parker & Pimentel, PRD25, 3180 (1982) Wright, astro-ph/9805292 Detection??

5. The cosmological constant in cosmology • If the acceleration is indeed due to the cosmological constant, then • or • This value is so small than it will be very difficult (to say the least) to • detect the influence of the CC in other regimes (other than on • cosmological scales) • If the acceleration is not due to the cosmological constant, then we have • a new upper bound …

6. The cosmological constant: ground state contributions But the ground state of a system also participates to the CC and the nature of the discussion is then drastically modified [A. Sakharov, Sov. Phys. Dokl. 12, 1040 (1968)]. There are two extra contributions • 1- “Classical contribution”: The vacuum • state has the following stress-energy tensor • In flat spacetime, only differences • of energy are measurable so not important … In curved spacetime, the • absolute value is important. Classical contribution

7. The weigh of the vacuum An example is the Electro-Weak transition

8. The value of the cosmological constant Planck energy density “prediction?” Parker & Pimentel, PRD25, 3180 (1982) Wright, astro-ph/9805292 detection

9. The cosmological constant: the quantum side • 2- Quantum contribution: • Because of Heisenberg principle the position • and the velocity of a quantum harmonic oscillator • cannot vanish at the same time • A quantum field=infinite collections of • quantum oscillators • This should not cause any panic since we are • used to tame infinities in QFT: renormalization. • However, this particular type of infinity is usually not renormalized but • ignored on the basis that, in flat spacetime, only differences of energies • are measurable. Quantum contribution

10. The cosmological constant & QFT • When gravity is taken into account, the vacuum energy density • becomes observable and we must therefore regularize it • In QFT, this is done by renormalizing the parameter of the theory • The vacuum contribution is expressed in terms of Feynman bubble diagrams, • ie diagrams with no external leg. • These diagrams have bad convergence properties, worst than ordinary • loop diagrams: they remain infinite even in the QM limit. • In non-gravitational physics, these graphs always cancel out.

11. The cosmological constant & QFT Regularizing the bubble graphs …

12. The cosmological constant & QFT Regularizing the bubble graphs … Introducing a cut-off breaks Lorentz invariance and leads to a wrong equation of state M

13. The cosmological constant & QFT Regularizing the bubble graphs Lorentz invariant methods (i.e. dimensional regularization) leads to the correction equation of state and : renormalization scale parameter

14. Regularizing the cosmological constant • Usually, the results depends on another scale, ie the in/out coming momentum External momentum (external leg of the graph)

15. Regularizing the cosmological constant • Usually, the results depends on another scale, ie the in/out coming momentum • Usually, the dependence in is removed by expressing the final result in terms of the value of the constant at the renormalization scale at External momentum (external leg of the graph) We never try to calculate the value of e at the renormalization scale

16. Regularizing the cosmological constant • Usually, the results depends on another scale, ie the in/out coming momentum • Usually, the dependence in is removed by expressing the final result in terms of the value of the constant at the renormalization scale • The value of the CC must be fixed at the observed value at a given scale • and, in this sense, there is no need to explain its value … in QFT, we never • attempt to calculate the mass of the electron or of the Higgs boson. at

17. Regularizing the cosmological constant • Usually, the results depends on another scale, ie the in/out coming momentum • Usually, the dependence in is removed by expressing the final result in terms of the value of the constant at the renormalization scale • The value of the CC must be fixed at the observed value at a given scale • and, in this sense, there is no need to explain its value … in QFT, we never • attempt to calculate the mass of the electron or of the Higgs boson. • But there is no “external” scale in a bubble diagram … what is the interpretation of in a cosmological context?? at

18. The cosmological constant: possible loopholes • A possible loophole is that vacuum fluctuations are just an artifact of • QFT. However, we observe their influence in the Casimir effect or in the • Lamb shift effect.

19. The cosmological constant: possible loopholes • Maybe vacuum fluctuations have abnormal gravitational properties?? But • vacuum fluctuations participate for a non-negligible amount to the mass of • nuclei … and they are observed to obey the UFF (WEP). Lamb shift in the nucleus Eotvos ratio:

20. Summary: so, what is the CC problem? • QM+Relativity implies virtual particles. The CC problem is the question of the gravitational field created by those virtual particles … • QFT cannot predict the value of the CC as it cannot predict the value of • the electron mass … the 120 orders of magnitude are just a simple indication. • Understanding its value probably requires something beyond QFT.