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

2. & observational constraints on the CC.
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 CC can always been seen as an extra source of matter:
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. Detection?? The cosmological constant: constraints
Planck energy density
2011 Nobel prize
Parker & Pimentel, PRD25, 3180 (1982)
Wright, astro-ph/
Detection??

5. If the acceleration is indeed due to the cosmological constant, then
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. 1- “Classical contribution”: The vacuum
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. An example is the Electro-Weak transition
The weigh of the vacuum
An example is the Electro-Weak transition

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

9. 2- Quantum contribution:
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. When gravity is taken into account, the vacuum energy density
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. Regularizing the bubble graphs …
The cosmological constant & QFT
Regularizing the bubble graphs …

12. Regularizing the bubble graphs …
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. Regularizing the bubble graphs
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. We never try to calculate the value of e at the renormalization scale
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. attempt to calculate the mass of the electron or of the Higgs boson.
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. attempt to calculate the mass of the electron or of the Higgs boson.
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. Maybe vacuum fluctuations have abnormal gravitational properties?? But
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. Understanding its value probably requires something beyond QFT.
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.