## [astro-ph/0410281] What does inflation really predict?

 Authors: Max Tegmark Abstract: If the inflaton potential has multiple minima, as may be expected in, e.g., the string theory "landscape", inflation predicts a probability distribution for the cosmological parameters describing spatial curvature (Omega_tot), dark energy (rho_Lambda, w, etc.), the primordial density fluctuations (Q, n_s, dn_s/dlnk, etc.) and primordial gravitational waves (r, n_t, etc.). We compute this multivariate probability distribution for various classes of models, exploring its dependence on the characteristic inflationary energy scales, the shape of the potential V and and the choice of measure underlying the calculation. We find that unless the characteristic scale Delta-phi on which V varies happens to be near the Planck scale, the only aspect of V that matters observationally is the statistical distribution of its peaks and troughs. For all energy scales and plausible measures, we obtain the predictions Omega_tot ~ 1+-0.00001, w=-1 and rho_Lambda in the observed ballpark but uncomfortably high. The high energy limit predicts n_s ~ 0.96, dn_s/dlnk ~ -0.0006, r ~ 0.15 and n_t ~ -0.02, consistent with observational data and indistinguishable from single-field eternal phi^2-inflation. The low-energy limit predicts 5 parameters but prefers larger Q and redder n_s than observed. We discuss the coolness problem, the smoothness problem and the pothole paradox, which severely limit the viable class of models and measures. Our findings bode well for detecting an inflationary gravitational wave signature with future CMB polarization experiments, with the arguably best-motivated single-field models favoring the detectable level r ~ 0.03. [PDF]  [PS]  [BibTex]  [Bookmark]

Discussion related to specific recent arXiv papers
Antony Lewis
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### [astro-ph/0410281] What does inflation really predict?

This seems like an interesting paper just asking for discussion! I have two comments

* Presumably there could easily be a physical prior (e.g. from string theory) that changes the conclusion about it boding "well for detecting an inflationary gravitational wave signature". For example if the underlying theory gives a prior distribution for the energy scale of inflation sharply peaked at TeV scales, then the quest may be hopeless. The actual physical prior over functions (as opposed to the Gaussian process assumed) could presumably change the conclusions about $n_s$ and Q.

* I'm a bit worried about using observations to determine the correct measure and ordering - e.g. using the "coolness problem" to rule out t-ordering. Surely the correct statement is if t-ordering is correct, inflation is highly disfavoured compared to almost any other model. Otherwise however wrong inflation is you can probably always keep changing your ordering/measure until you find one which gives acceptable predictions - inflation becomes even more untestable! I would think that only if you assume inflation is right can you use observation to determine the ordering: using observations to argue about the correct logic for calculation seems odd to me. (However I completely agree that t-ordering is almost certainly wrong.)

I would be interested to know how well people think the Gaussian process prior over functions encapsulates the predictions of the string landscape.

Anze Slosar
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Affiliation: Brookhaven National Laboratory
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It's a long paper, I must admit I didn't quite manage to read it all!
• Where is this omega_lambda constraint coming from, phyiscally? I always thought that the fact that omega_lambda is so small is one of the biggest problems in cosmolgoy, and yet in the Figure 1 he finds that log wlambda is between −125 and −110, in natural units, I presume, so it sort of solves this long-standing problem simply by assuming a gaussian process?
• Gaussian random process: Firstly it is not clear to me, why should this be 1-dimensional (i.e. why not d-dimensional in any random direction), or phrasing it differently: why should it be exactly white noise. Also, surely in the fourier expansion of the potential landscape, setting n=10e100 would give different results as the power in high f would just blow up (?!)
• Similar as before: not sure that putting some sort of prior on distribution of V(phi) is qualitativelly any better than putting say a gaussian prior on slow roll parameters and V(phi_0). In other words, this paper just calcualtes the implied prior on derived parameters from assuming a prior on some basic parameter. Interesting enough, but not sure whether it tells us something about physics, *unless one assumes that inflation works in a statistical-mechanical manner*, which is not at all stressed in the abstract.
• Conditioning on reference objects: one need to be careful here as reference objects are a-posteriori, i.e. you are putting in data in some form. In extreme case, if you condition on universes with omega_m=0.3 omega_lambda=0.7, etc. you are going to recover exactly this parameters. Of course, conditioning on halos or galaxy formation is much less strong but still all it gives you are the implied priors.
• This ordering ambiguity: surely this again depends on what is your basic theory, that selects how universes are picked. Unless, you know this basic theory, I don't see how this could be resolved observationally, based on one point only (our universe). (But we played this game before: if I draw a screw with a serial number 5 from a box of screws, what is the pdf for the number of screws in the box?)