Results from Planck and AMS-02, but also the way they are often presented, have created a very exciting situation. Its exceptional character has led me to quickly write an article, *Planck data, spinorial space-time and asymptotic Universe*, and to make it public for open discussion.

Are dark matter and dark energy really at the origin of the present value of the cosmic velocity/distance ratio ? Is dark energy really necessary ? Unfortunately, present analyses are based on the ΛCDM model right from the beginning.

I present here an alternative approach. My suggestion is based on a pre-existing (spinorial) space-time geometry that would already have been there when standard matter appeared in the Universe.

Then, the limit *H t* → 1 as *t* → ∞ (*t* = age of the Universe, *H *= speed/distance ratio at cosmic scale) would be a natural geometric property of the original space-time on which conventional matter has been generated.

More on the subject will follow soon.

A downloadable (7 pages) PDF copy of the paper is here: Planck data, spinorial space-time and asymptotic Universe

**Planck data, spinorial space-time and asymptotic Universe**

**Luis Gonzalez-Mestres ^{*}**

(Posted on April 11, 2013, to **mp_arc**, **archive.org** and **HAL**)

**Abstract - The Planck collaboration reports an age of the Universe t close to 13.82 Gyr and a present ratio H
between relative speeds and distances at cosmic scale around 67.8
km/s/Mpc. The product of these two measured quantities is then slightly
below 1 (about 0.96), while it can be exactly 1 in the absence of matter
and cosmological constant in suitable patterns. An example is the
cosmology based on a spinorial space-time we have considered in previous
papers, where such an expansion law is of purely geometric origin and
can reflect an equilibrium between the dynamics of the ultimate
constituents of matter and the geometry of space and time. Taking also
into account the observed cosmic acceleration, the present situation
suggests that the value of 1 can be a natural asymptotic limit for the
product H t in the long-term evolution of our Universe up to possible small corrections. No ad hoc combination of dark matter and dark energy would then be needed to get an acceptable value of the cosmic speed/distance ratio H. We briefly comment on possible realistic versions of cosmologies exhibiting this property.**

**1. Introduction**

Obviously, WMAP [1] and Planck [2] data require a new close study after the 2013 Planck results have been made available (see [3,4] and other recent papers by the Planck Collaboration). In particular, even if WMAP and Planck have systematically used the ΛCDM model as the basic tool for data analysis, nothing prevents from exploring other possible cosmologies [5] potentially related to new physics at ultra-high energy [6] and beyond Planck scale.

The question of the space curvature is a major one, but measuring it at large cosmological scales may be extremely difficult if the actual Universe is much larger than the observable one. If the effective global curvature is very small for this reason, it can even be masked by other (more local) phenomena. But it may also happen that such a small space curvature generates the leading contribution to the expansion of the Universe [5].

And do we really understand the meaning of concepts such as dark matter and dark energy that, according to Planck analysis, would account for 95% of the energy in our Universe? The recent AMS results [7,8] are not yet conclusive and, even assuming that a dark matter signature would have been detected, alternatives to fashionable theories already exist [9].

The situation seems even more unclear concerning what is usually called inflation, in spite of the remarkable effort in model building for more than thirty years. Suitable alternatives to inflation can naturally be provided by pre-Big Bang cosmologies [5,10].

Similarly, in the standard ΛCDM cosmology, the
ratio between relative velocities and distances at cosmic scales given
by the Lundmark–Lemaître-Hubble (LLH) constant (see [5,11] and
references [12] to [16]) depends crucially on a set of poorly identified
phenomenological parameters. However, if *H* is the LLH constant and *t* the age of the Universe, the product *H t* is identically equal to 1 in some specific cosmological geometries before introducing matter, energy and gravitation.

A potential example can be built using the well-known Friedmann equations [17,18] as a guide. We start considering the Friedmann-like relation :

*H*^{2} = 8π*G*ρ/3 - *k R*^{-2}* c*^{2} + Λ *c*^{2}/3 (1)

where *H* =* a _{s}*

^{-1}

*da*/

_{s}*dt*is the LLH constant,

*a*the scale factor,

_{s}*G*the gravitational constant, ρ the energy density,

*c*the speed of light,

*k R*

^{-2}the curvature parameter,

*R*the present curvature distance scale of the Universe (the curvature radius, and possibly the radius of the Universe, for

*k*= 1) and Λ the cosmological constant. Taking ρ = 0, Λ = 0 and

*a*

_{s}^{-1}

*da*/

_{s}*dt*=

*R*

^{-1}

*dR/dt*, one would get for

*k*= -1 (negative curvature, hyperbolic space) the simple relation

*dR/dt = c*. But such a relation does not appear to be compatible with cosmological data, as

*c t R*

^{-1}would be too small for a realistic fit. A possible modification of equation (1) will be discussed below.

A more clear situation concerning the geometric origin of the relation *H* = *t*^{-1} is obtained with the spinorial space-time we introduced in 1996 [5,19], where the cosmic time* t* is given by the modulus of a cosmic spinor and the cosmic space at time *t*
is described the associated hypersphere. In this case, the natural
space curvature is positive and no critical speed is introduced, but the
law *H* = *t*^{-1}holds automatically in the absence of matter and cosmological constant.

In the spinorial space-time under consideration, the inverse
square of the age of the Universe plays the role of the curvature term
replacing the term - *k* *R*^{-2}* c*^{2} in the equation equivalent to (1). The expression *c R*^{-1}
is directly replaced by an inverse time scale, and there is no - sign
associated to the space curvature. At this stage, no space units or
critical speed(s) have been introduced. Similarly, matter and
cosmological constant are not required to get a sensible value of *H*.

Even assuming the existence of a realistic concept similar to
the cosmological constant, in alternative approaches to quantum field
theory [5] one can expect it to be generated only in the presence of
standard matter. In this case, its contribution to equations like (1)
decreases like the matter density as the Universe expands and the
relation *H = t*^{-1} will be preserved as a limit at large *t* except if new (small) corrections to the effective geometry must be taken into account.

In the present note, we discuss the possibility that the geometric relation *H = t*^{-1} remains, up to small extra terms, an asymptotic limit of cosmic evolution at large* t*
in suitable cosmologies. We defer to later work a more precise
insertion of general relativity within the spinorial space-time
framework we have suggested.

**2. The spinorial space-time**

As explained in [5,19], for a SU(2) spinor ξ describing space-time coordinates, and taking the positive SU(2) scalar |ξ|^{2} = ξ^{†} ξ where the dagger stands for hermitic conjugate, a definition of the cosmic time can be *t* = |ξ| with an associated space given by the S^{3} hypersphere |ξ| =* t*. Other definitions of *t* in terms of |ξ| (f.i. *t* = |ξ|^{2}) lead to similar cosmological results as long as a single-valued function is used.

Then, using the definition *t* = |ξ|, if ξ_{0} is the observer position on the |ξ| =* t*_{0}
hypersphere, space translations inside this hypersphere correspond to
SU(2) transformations acting on the spinor space, i.e. ξ = *U *ξ_{0} where:

*U* = exp (i/2 *t*_{0}^{-1}** σ.x**) ≡* U* (**x**) (2)

**σ** being the vector formed by the usual Pauli matrices. The vector **x** is the spatial position of ξ with respect to ξ_{0} at constant time * t*_{0}. The antipodal point - ξ_{0} corresponds to *U* (2 π) = -1. The spinorial position ξ - ξ_{0} violates causality but can be relevant at very small scales [5,21].

Space rotations with respect to a fixed point ξ_{0} are SU(2) transformations of the spatial position vector **x**.

A standard spatial rotation around ξ_{0 }corresponds now to *U* (**y**) turning any *U* (**x**) into *U* (**y**) *U* (**x**) *U* (**y**)^{†}, where the vector **y** provides the rotation axis and angle.

The origin of our time can then be associated to the point ξ = 0. One
thus gets a naturally expanding Universe where cosmological comoving
frames would correspond to straight lines crossing the origin ξ = 0.
In the absence of matter and of a cosmological constant, such a
geometry can be applied to relative velocities and distances at cosmic
scale for comoving frames and automatically yields the LLH law *H = t*^{-1}.

More precisely, for two cosmological comoving frames separated by a constant angular distance θ, the spatial distance *D* between the two corresponding points on the |ξ| = *t* hypersphere will be *D* = θ *t*, with a relative velocity *v* = θ. The ratio between relative velocities and distances is therefore given by *t*^{-1} [5,21].* t* is actually the only physical scale available, and no critical speed has been introduced at this stage.

The situation does not basically change if the distance on the S^{3} hypersphere is taken to be equal to *t* times an arbitrary function of the angle θ. It is even possible, for each observer at a given ξ_{0}, to send to infinity the antipodal point θ = 2π
turning the hypersphere into a hyperboloid. Contrary to standard
cosmology, such a transformation would not change the sign of the
spinorial curvature term* t*^{-2} [5] in the equation describing the Universe expansion through *H*^{2}.

Together with conventional matter, standard relativity can be introduced as a local low-energy limit of space-time as seen by such matter [5,6], just as low-momentum phonons in a solid can exhibit a Lorentz-like symmetry [20]. Friedmann-like equations emerging at this stage must take into account the pre-existing global spinorial space-time on which matter has been generated and the LLH law inherent to this space-time.

**3. Geometry in our Universe**

The ΛCDM-based data analyses presented by Planck and WMAP do not report any significant space curvature. But this is not a surprise if the actual Universe is much larger than the observed one, in which case the effective curvature can be very small even if the space geometry is spherical or hyperbolic. Furthermore, as just explained, it may turn out that the standard Friedmann equations ignore fundamental pre-matter information and fail for this reason to correctly describe the cosmological role of space curvature.

**3.1 A modified Friedmann equation**

Even within the ΛCDM model framework, if the
radius of the actual Universe is more than ∼ 100 times larger than that
of the observable one, a curved space does not appear to be excluded by
the bounds Planck has recently presented. If most of this space is
empty, and in the absence of dark matter and dark energy, a hyperbolic
curvature term can become the dominant contribution to *H* in a Friedmann-like equation of the type (1) describing this global Universe.

Furthermore, as the matter density is expected to decrease with
the Universe expansion faster than the curvature term, the relation *H = t*^{-1} clearly appears as the natural asymptotic limit at large t in the absence of a cosmological constant.

It remains to fit the measured value of *H* without using dark matter and dark energy. A possibility would be to modify the value of the constant *c*^{2} multiplying the term - *k R*^{-2} in (1). Instead of the square of the speed of light, a larger effective constant* c*'^{2}
allowing to account for superluminal relative speeds in the global
Universe can in principle compensate the smallness of the curvature
parameter itself. More precisely, the ratio *c*' *R*^{-1} should have a value close to the observed value of *H* (and therefore, to that of *t*^{-1}).

The possible physical and cosmological meaning of this modification of the Friedmann equations will be further discussed in a forthcoming paper. But the basic idea is to give the global cosmic curvature a weight accounting for its role at cosmic scale, including reminiscent effects from an inflationary or pre-Big Bang era. The spinorial space-time provides an illustration of such a new approach to space curvature in cosmology. In this case, the sign of the leading cosmic curvature term does not depend on that of the standard curvature parameter in equations like (1).

**3.2 Cosmology and spinorial space-time**

As explained in Section 2, the spinorial space-time considered
in [5,19] and in [21,22] (see also [23,24]) presents a direct geometric
description relating the expansion rate of the Universe to its age, and
automatically reproducing the relation *H* =* t*^{-1} in the absence of matter and of a conventional cosmological constant.

As no critical speed or space units have been introduced to obtain the *H* =* t*^{-1}
law, the spinorial space-time can naturally be much larger than our
observable Universe and allow for critical speeds much larger than that
of light. The speed of light would be just the critical speed of
standard matter. Pre-Big Bang scenarios allowing for superluminal motion
would thus provide a natural alternative to inflation [5,10].

Then, it seems normal to assume that our standard matter universe nucleated at a very early stage of the evolution described by the spinorial space-time geometry, so that its age does not differ from the cosmic time thus defined. But this conventional universe does not necessarily fill the whole available space and other kinds of matter or pre-matter reminiscent from the pre-Big Bang era can exist elsewhere obeying different physical laws. In all cases, new forms of matter and pre-matter can exist everywhere, so that the stable structure of the physical vacuum is not necessarily made of the standard scalar fields and zero modes from quantum field theory [23,24].

As our standard matter universe is facing an expansion of the
global Universe generated by the original spinorial space-time,
fluctuations in its observable expansion rate may occur due to the
interaction between conventional matter and the global space-time
structure. The apparent cosmic acceleration [25] can thus actually be
the expression of a temporary fluctuation without real influence on the
long-term evolution of our conventional universe [23,24]. In this case,
dark energy is not required to explain the present acceleration and the
relation *H* =* t*^{-1} will remain asymptotically valid inside our Universe. Small corrections to the relation *H* =* t*^{-1} can be generated by the standard Friedmann-like curvature terms after the nucleation of conventional matter.

**3.3 On cosmic acceleration**

In the ΛCDM model, cosmic acceleration is linked to the second Friedmann equation:

*A* = - 4/3 π*G* (ρ + 3 *p*_{U}*c*^{-2}) + Λ *c*^{2}/3 (3)

where *A* = *dH/dt* + *H*^{2} = a_{s}^{-1} *d*^{2}*a _{s}*/

*dt*

^{2}and

*p*

_{U}is the pressure parameter. However, this equation may require a substantial modification following that of (1). Then, without using dark energy, new mechanisms can be imagined to explain the observed cosmic acceleration in our region of the Universe. In particular, a new term describing the reaction of standard matter to the geometric expansion of the Universe can provide a natural way out, together with a term describing the counter-reaction of the geometry itself.

A difference in density between this part of the Universe and
the Universe as a whole may have already produced a local gravitational
reaction of matter opposing the geometric expansion and making it
locally slower. Later in such a scenario, as the matter density has
become smaller, the local expansion would have started to accelerate
getting closer to the geometric value of *H*. Even without such a
difference in density, the same kind of gravitational reaction can occur
if the expansion of the Universe is led by other phenomena than those
considered in the standard Friedmann equations. In particular, if the
role of space curvature is stronger than predicted by (1) and obeys to a
different geometric origin.

A similar mechanism would be expected for the spinorial space-time just discussed. In this case, when standard matter is introduced, new terms accounting for the gravitational reaction to the already existing geometric expansion of space and for the geometric counter-reaction should be added to the Friedmann-like equations. Dark energy is not required to generate such a process.

In both cases, and most likely also in other scenarios, the
present cosmic acceleration would correspond to the evolution of our
Universe towards the asymptotic relation *H t* = 1 (up to possible small corrections) in the large *t* limit.

A small correction to the asymptotic expansion law *H t* = 1 can be of the form:

*H* = *D ^{-1}*

*dD/dt*=

*t*(1 + α) (4)

^{-1}where *D* is the distance at time t between two comoving frames. The (positive or negative) constant α
can be the expression of an additional (small) space curvature term
possibly related to the presence of conventional matter and reminiscent
from the standard Friedmann approach as given in equation (1). One then
readily gets the relation *D/D*_{0} = (*t/t*_{0})^{1+}^{α} where *D*_{0} is the value of the cosmic distance *D* at cosmic time *t*_{0}.

**3.4 The arrow of time**

In the spinorial description of space-time considered here, the arrow of time is of purely geometric origin, and directly related to the deepest geometric space-time structure.

If the cosmic time *t* is not a single-valued function of |ξ| and the arrow of time is to be preserved, the picture will require some modifications such as replacing* t* by a function of |ξ|
in the definition of the space coordinates. Although this is not the
scenario considered here, further work on this question is required.

Recent work on the arrow of time following different approaches can be found, for instance, in references [26,27] and in [28,29].

**4. Conclusion and comments**

Instead of trying to build *ad hoc* the observed value of
the LLH constant using large amounts of dark matter and dark energy, we
present here a natural geometric approach based on what can be the
ultimate space-time structure (as seen by spin-1/2 particles) and
automatically leading to the relation *H* = *t*^{-1} in the absence of standard matter and of a cosmological constant. The definition of the age of the Universe* t* is directly linked to the geometric size of the cosmic space.

No dark energy is required, and no conventional cosmological constant is introduced, in this new cosmology where the *H* = *t*^{-1} law can result from an equilibrium between geometry and the most fundamental form of matter or pre-mater.

A really new approach to the role of space curvature in
cosmology is thus at the origin of a new structure of Friedmann-like
equations. In particular, the leading contribution to the square of the
LLH constant comes from a curvature term equal to *t*^{-2} whose sign does not depend on the space curvature felt by standard matter.

As the leading contribution to the Universe expansion comes from this S^{3}
curvature term generated in the spinorial space-time previous to the
introduction of standard matter and outside the standard
general-relativistic framework, standard matter can react to this
geometric constraint. Such an interaction between matter and geometry
would lead to a new terms in the modified Friedmann-like equations. If
the reaction of standard matter has initially slowered the Universe
expansion when the matter density was much larger, the apparent cosmic
acceleration can be just an evolution associated to the weakening of
matter density and restoring asymptotically the expansion rate from
fundamental geometry.

Thus, it is tempting to conjecture that, contrary to many
claims, the observed acceleration of the expansion of the Universe is
just the reflect of a fluctuation due to gravitation, and perhaps to
other standard interactions, in the presence of the pre-existing
spinorial space-time geometry. As the Universe expands, the product *H t*
tends to 1 (except possibly for a small correction to this value) as
the natural asymptotic limit at *t* → ∞. Data from Planck and other
experiments appear compatible with such a hypothesis that does not
appear naturally in the ΛCDM model.

Similarly, as the spinorial space-time has been previous to the formation of conventional matter, it seems reasonable to assume that its expansion is in equilibrium with a primordial vacuum possibly formed during a short pre-Big Bang era.

Then, the condensates and zero modes usually introduced in standard quantum field theory can be just a simplified way to describe the interaction between conventional particles and the physical vacuum. In such a situation, there is no compelling reason to consider an item like the standard cosmological constant, even if a related phenomenon can occur in the presence of conventional matter [5,23]. In this last case, the cosmological weight of such an effect is expected to decrease like the matter density, contrary to the standard cosmological constant.

Cosmologies naturally leading to an asymptotic value of *H t*
equal or close to 1 deserve particular attention. As they have not been
really explored, further work in this direction is clearly necessary.

**Footnote ***

gonzalez@lapp.in2p3.fr at LAPP, Université de Savoie and CNRS/IN2P3, 74941 Annecy-le-Vieux, France

luis.gonzalez-mestres@megatrend.edu.rs at the Cosmology Laboratory, Megatrend University, Novy Beograd, Serbia

lgmsci@yahoo.fr, personal e-mail

**REFERENCES**

[1] Wilkinson Microwave Anisotropy Probe, http://map.gsfc.nasa.gov/

[2] Planck mission (European State Agency), http://sci.esa.int/science-e/www/area/index.cfm?fareaid=17

[3] Planck Collaboration, *Planck 2013 results. I. Overview of products and scientific results*, arXiv:1303.5062

[4] Planck Collaboration, *Planck 2013 results. XVI. Cosmological parameters*, arXiv:1303.5076

[5] L. Gonzalez-Mestres, *Pre-Big Bang, fundamental Physics and noncyclic cosmologies*, presented at the International Conference on New Frontiers in Physics, ICFP 2012, Kolymbari, Crete, June 10-16 2012, mp_arc 13-18, and references therein.

[6] L. Gonzalez-Mestres, *High-energy cosmic rays and tests of basic principles of Physics*, presented at the International Conference on New Frontiers in Physics, ICFP 2012, Kolymbari, Crete, June 10-16 2012, mp_arc 13-19, and references therein.

[7] The Alpha Magnetic Spectrometer Experiment, http://www.ams02.org/

[8] M. Aguilar et al. (AMS
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[9] L. Gonzalez-Mestres, *Superbradyons and some possible dark matter signatures*, arXiv:0905.4146

[10] L. Gonzalez-Mestres, *Cosmological Implications of a Possible Class of Particles Able to Travel Faster than Light*, Proceedings of the TAUP 1995 Conference, *Nucl. Phys. Proc. Suppl.* **48** (1996), 131, arXiv:astro-ph/9601090

[11] I. Steer [12,13] emphasizes that Lundmark [14] first established observational evidence of the expansion of the Universe with the now standard speed/distance relation, while Lemaître [15] established the theoretical evidence and Hubble [16] provided observational proof.

[12] I. Steer, *History: Who discovered Universe expansion?*, *Nature* **490** (2012), 176.

[13] I. Steer, *New Facts From the First Galaxy Distance Estimates*, *J. R. Astron. Soc. Can.* **105** (2011), 18. Available at the SAO/NASA Astrophysics Data System (ADS).

[14] K. Lundmark, *The determination of the curvature of space-time in de Sitter's world*, *MNRAS* **84** (1924), 747. Available at the SAO/NASA Astrophysics Data System (ADS).

[15] G. Lemaître, *Un Univers
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[16] E. Hubble, *A relation between distance and radial velocity among extra-galactic nebulae**, PNAS* **15** (1929), 168, http://www.pnas.org/content/15/3/168

[17] K. Enqvist, *Cosmological inflation*, CERN Yellow Report CERN-2012-001, pp. 207-215, available at *arXiv.org*, arXiv:1201.6164

[18] S. Gorbunov and V. Rubakov, *Introduction to the Theory of the Early Universe*, World Scientific 2011.

[19] L. Gonzalez-Mestres, *Physical and Cosmological Implications of a Possible Class of Particles Able to Travel Faster than Light*, contribution to the 28th International Conference on High Energy Physics, Warsaw 1996, arXiv:hep-ph/9610474, and references therein.

[20] L. Gonzalez-Mestres, *Vacuum Structure, Lorentz Symmetry and Superluminal Particles*, arXiv:physics/9704017

[21] L. Gonzalez-Mestres, *Cosmic rays and tests of fundamental principles*, CRIS 2010 Proceedings, *Nucl. Phys. B, Proc. Suppl.* **212-213** (2011), 26, and references therein. The *arXiv.org* version arXiv:1011.4889 includes a relevant Post Scriptum.

[22] L. Gonzalez-Mestres, *Space, Time and Superluminal Particles*, arXiv:physics/9702026

[23] L. Gonzalez-Mestres, *Pre-Big Bang, vacuum and noncyclic cosmologies*, 2011 Europhysics Conference on High Energy Physics, PoS EPS-HEP2011} (2011), 479, and references therein.

[24] L. Gonzalez-Mestres, *WMAP, Planck, cosmic rays and unconventional cosmologies*, contribution to the Planck 2011 Conference, Paris, January 2011, arXiv:1110.6171

[24] See, for instance, D.H. Weinberg et al., *Observational Probes of Cosmic Acceleration*, arXiv:1201.2434, and references therein.

[25] V.G.Gurzadyan, S.Sargsyan and G.Yegorian, *On the time arrows, and randomness in cosmological signals*, talk at Time Machine Factory conference, Torino 2012, arXiv:1302.5165

[26] G.F.R. Ellis, *The arrow of time and the nature of spacetime*, arXiv:1302.7291

[27] J.B. Hartle, *The Quantum Mechanical Arrows of Time*, arXiv:1301.2844

[28] L. Susskind, *Fractal-Flows and Time's Arrow*, arXiv:1203.6440

Happy reading!