# A cosmology of a trans-Planckian theory and dark energy.

## Summary (2 min read)

### 1 Introduction

- To the trans-Planckian theory and the motivations behind it.
- Some technical issues must be addressed to formulate the theory.
- A generic GUT scale inflation, to solve the horizon problem, generates a big hierarchy at least of 1027 in the scale factor.
- There are many different approaches to the dark energy problem (see, for example, the reviews [8–11] covering different aspects).
- The authors start from the beginning with a strong postulate, that of absolute equivalence between coordinates and momenta, and work out a possible implementation and the consequences of that.

### 2 A Trans-Planckian Theory

- The authors can implement translation invariance by introducing an extra U(1) gauge structure.
- In a gravitational background, it is instead possible to distinguish them, because the term with m interacts only with the metric gµν while the term with m̃ interact only with the dual metric g̃µν .
- The authors conclude this section with a comment about the definition of the generalized Fourier transform.

### 3 Cosmology and the FRW metric

- The authors take a space-time with an FRW metric ds2 = dt2 − a(t)2d~x2 , (3.1) where t is the time coordinate, a(t) the expansion factor of the three-dimensional space and the ~x the comoving spatial coordinate.
- The translational invariance of the spatial part of the FRW metrics, both in x and p, is implemented, thanks to the ansatz for the gauge bosons (3.5).
- The solution for (3.12) can be found exactly in case of a simple power-law behaviour a(t) = cαt α.
- This feature of the nodes has an important consequence for the generalised Fourier transform.

### 4 A Cosmological Solution

- In the previous sections, the authors defined the action of the trans-Planckian theory and the generalised Fourier map for cosmological manifolds.
- The authors will see that from the inversion of the action S̃ that the dual term can be responsible for an effective positive cosmological constant.
- 3) The magnitude of S̃ when ϕ̃ comes out of the suppression zone must be with the observed dark energy value (4.23).
- In general, to solve the horizon problem, the authors want the number of e-folds during inflation to be equal to or greater than the number of e-folds after inflation (with a small correction from the matter-domination period which count as half of the others).

### 5 Conclusion

- Every ordinary field theory can be made x ↔ p symmetric with the technique of the generalised Fourier transform.
- A different approach, which the authors have not pursued in this paper, would be to eliminate the gauge fields Qµ and Yµ and consider the translational invariance to be broken also in the space components.
- The inflationary stage, which by duality must also occur in the momentum manifold, produces a hierarchy which is big enough to suppress the effect of the dual terms in the action S̃, and thus make the theory consistent with low-energy experiments.
- The ideas the authors present have aspects in common with others that can be found in the literature, but also have some peculiar distinctions.
- The authors used some approximations to deal with this problem, the main ones are the homogeneity of the universe and the adiabaticity of the probe functions.

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##### Frequently Asked Questions (2)

###### Q2. What are the future works in this paper?

In view of the fact that the effect of S̃ is washed out by the dual inflationary stage, and thus visible only at the cosmological horizon scale or at very high energy, this different approach could still be a viable possibility. A recent attempt to extend this to non-compact directions can be found in [ 18 ]. Thus, the authors can roughly say that they still live in the centre of the relativistic harmonic oscillator ( 1. 1 ), and its size has been inflated from the original Planckian scale to the size of the universe now. There would be, instead, a small modification for the case of the definition ( 2. 30. ) f ( 0,0 ), which enters in the definition ( 2. 30 ), is in the extreme non-adiabatic regime, and it can be checked that f ( 0,0 ) ∝ 1/t3α−1 when a ( t ) ∝ tα.