Subject: The back-reaction
Date: Fri, 09 Aug 2002 19:30:49 +0300
From: Dimi Chakalov <>
To: Robert Brandenberger <>

Dear Professor Brandenberger,

In your recent astro-ph/0208103 [Ref. 1], you mentioned your hypothesis that the back-reaction of cosmological fluctuations might lead to a dynamical relaxation of the cosmological constant [Ref. 2]. Please keep me in touch with your work on this issue.

I'm wondering why you didn't mention QSSC; see [Ref. 3] for a recent review. It seems to me that the challenges of vacuum energy are severe,

I'm trying to grasp the vacuum energy and the so-called curvature of spacetime along the lines suggested by C. Isham and J. Butterfield [Ref. 4]. I believe there is a fourth, completely unexplored, road to quantum gravity: "the power of the universe as a whole to organize itself" [Ref. 5]. Some details are available at

I suspect the problems with your back-reaction [Ref. 2] are relevant to the putative global mode of time pertaining to the universe 'as a whole' (please see the URLs above). Without it, you encounter the enormous, if not insurmountable, difficulties of uniquely defined energy density of the gravitational field at a given point,

and the "talk" between matter and geometry (J. Wheeler) sounds like a jabberwocky,

Am I wrong?

I will appreciate the opinion of your colleagues as well.


Dimiter G. Chakalov
Dead matter makes quantum jumps; the living-and-quantum matter is smarter.


[Ref. 1] Robert H. Brandenberger. Principles, Progress and Problems in Inflationary Cosmology. Mon, 5 Aug 2002 21:52:20 GMT,

"As is illustrated in Fig. 2, at the time the microwave radiation last scattered (which occurred when the temperature was about a factor 10^3 of the present temperature), the maximal distance which light could have communicated information starting at the Big Bang (the forward light cone) is much smaller than the distance over which the microwave photons are observed to have the same temperature (the past light cone). This is the famous "horizon problem" of SBB cosmology. Within the context of SBB cosmology it is also a mystery why the Universe today is observed to be approximately spatially flat, since a spatially flat Universe is an unstable fixed point of the FRW equations in an expanding phase. This problem is called the "flatness problem". Finally, as illustrated in Fig. 3, within standard cosmology there is no causal mechanism which can explain the nonrandom distribution of the seed inhomogeneities which develop into the present-day large-scale structure. This constitutes the "formation of structure problem".

"The inflationary Universe scenario [5] is based on the simple hypothesis that there was a time interval in the early Universe beginning at some time t_i and ending at a later time t_R (the "reheating time") during which the scale factor is exponentially expanding.

"In extrapolating the evolution of cosmological perturbations according to linear theory to these very early times, one is implicitly making the assumption that the theory remains perturbative to arbitrarily high energies, and that the classical theory of general relativity remains the appropriate framework for describing space-time. Both of these assumptions are clearly not justified.

"The Achilles heel of our current inflationary models in without doubt the "cosmological constant problem". There is some as of yet unknown mechanism which prevents the bare cosmological constant, which in theories with quantum fields is predicted to be at least 62 orders of magnitude larger than the observational limit (this number comes from assuming the cancellation of vacuum energies on scales larger than the supersymmetry breaking scale taken to be about 1TeV), from gravitating. How do we know that this unknown mechanism does not also lead the transient "cosmological constant" given by the potential energy of the scalar field to be gravitationally inert, thus eliminating the basis of scalar field-driven inflation?

"In particular, it is possible that some of the problems are consequences of neglecting the intrinsically nonlinear structure of general relativity (see e.g. [24] for some speculations along these lines)."

([24] R.H. Brandenberger, Back reaction of cosmological perturbations, hep-th/0004016.)

[Ref. 2] R.H. Brandenberger, Back Reaction of Cosmological Perturbations. Invited lecture at COSMO-99 (ICTP, Trieste, Sept. 27 - Oct. 2, 1999).

"It is well known that gravitational waves propagating in some background space-time affect the dynamics of the background. This back-reaction can be described in terms of an effective energy-momentum tensor [x] . (...) In recent work [1; 2], we demonstrated that the back reaction problem can be set up in a way which is gauge-invariant (under linear coordinate transformations).

"5. Speculations

"Since the back-reaction of cosmological fluctuations in an inflationary cosmology acts (see (14)) like a negative cosmological constant, and since the magnitude of the back-reaction effect increases in time, one may speculate (12) that back-reaction will lead to a dynamical relaxation of the cosmological constant (see Tsamis & Woodard (4) for similar considerations based on the back-reaction of long wavelength gravitational waves). (...) Furthermore, we speculate that this dynamical relaxation mechanism for  [lambda]  will be self-regulating.

"The most interesting result which emerges is that, in an inflationary background, the effective energy-momentum tensor which describes the back-reaction has the form of a negative cosmological constant. The absolute value of the induced effective energy density grows in time and, in a model with a long period of inflation, can become significant, which leads to the speculation that the effect may lead to a dynamical relaxation of the cosmological constant. However, the effective energy-momentum tensor defined in this work describes the effect of fluctuations on the homogeneous mode of the gravitational field. If the speculations in the previous section are to hold up, the analysis must be extended to give back-reaction effects on local quantities. Work on this issue is in progress."

[Ref. 3] J.V. Narlikar, R.G. Vishwakarma, G. Burbidge. Interpretations of the Accelerating Universe. To appear in the October 2002 issue of the "Publications of the Astronomical Society of the Pacific".

[Ref. 4] C.J. Isham, J. Butterfield. On the Emergence of Time in Quantum Gravity. Imperial/TP/98-99/23.

"The usual tools of mathematical physics depend so strongly on the real-number continuum, and its generalizations (from elementary calculus 'upwards' to manifolds and beyond), that it is probably even harder to guess what non-continuum structure is needed by such radical approaches, than to guess what novel structures of dimension, metric etc. are needed by the more conservative approaches that retain manifolds. Indeed, there is a more general point: space and time are such crucial categories for thinking about, and describing, the empirical world, that it is bound to be ferociously difficult to understand their emerging, or even some aspects of them emerging, from 'something else'."


[Ref. 5] L. Smolin (2001). Three Roads to Quantum Gravity. London: Weidenfeld & Nicolson, p. 206.