Before discussing the recent Planck results and my opinion on the present situation, let me present here an extract of my ICFP 2012 paper Pre-Big Bang, fundamental Physics and noncyclic cosmologies,
http://hal.archives-ouvertes.fr/hal-00795588
http://hal.archives-ouvertes.fr/docs/00/79/55/88/PDF/PreBBCreteNew.pdf
or
http://www.ma.utexas.edu/mp_arc-bin/mpa?yn=13-18
http://www.ma.utexas.edu/mp_arc/c/13/13-18.pdf
Page 3, I write :
Basic equations of conventional cosmology should then be reconsidered, such as the well-known Friedmann relation [31,32]:
H2 = as−2 (das /dt)2 = 8π G ρ/3 − k c2 as−2 + c2/3 (1)
where H is the usual LLH constant describing the ratio between relative speeds and distances at cosmological scale, as the space-like scale factor, G the gravitational constant, the energy density, k the curvature parameter and the cosmological constant. In this formula, leaving aside the cosmological constant term that can be
removed by a redefinition of [31], the explicit dependence on c disappears using
instead of as the time-like scale factor at = as c−1. The term 8G/3 involving
a standard coupling to gravitation is in all cases characteristic of the conventional
properties of ”ordinary” matter. The situation can be radically different with the
spinorial space-time considered in Sect. 4.
As explained in Sect. 4, the second Friedmann equation :
a−1
s d2as/dt2 = −4/3 G ( + 3pUc−2) + c2/3 (2)
where pU is the pressure parameter, must equally be reconsidered if the spinorial
space-time is used as the startpoint of a new cosmology.
WMAP data [33] are usually presented as confirming the validity of standard
cosmology. But the phenomenological analyses supporting this claim concern only
general features, use a priori parameterizations and lead to large amounts of unidentified
cosmic dark matter and dark energy. Even following these schemes, there is no
actual proof that such unknown matter and energy have conventional properties and
do not originate from a pre-Big Bang evolution involving totally new physics.
Similarly, there is no reason why the observable effects usually attributed to standard
inflation should have been generated after the Big Bang. A pre-Big Bang phase
can produce a similar expansion more naturally [8], and even replace the Big Bang
itself by a theory involving new matter and/or pre-matter [15,17].
(end of the extract)
and pages 7-8 :
4.1 Space, time and transformations
Given a space-time SU(2) spinor , and considering the positive SU(2) scalar | |2 =
† where the dagger stands for hermitic conjugate, a definition of the cosmic time
can be t = | | with an associated space given by the S3 hypersphere | | = t.
Then, if 0 is the observer position on the | | = t0 hypersphere, space translations
inside this hypersphere correspond to SU(2) transformations acting on the spinor
space, i.e. = U 0 where:
U = exp (i/2 t−1
0 .x) U(x) (5)
is the vector formed by the usual Pauli matrices, and the vector x thus defined is
the spatial position of with respect to 0 at constant time t0. x is clearly different
from the spinorial position that can be defined as − 0.
Space rotations with respect to a fixed point 0 are obtained as SU(2) transformations
acting on the spatial position vector x. A standard spatial rotation around
0 is now a SU(2) element U(y) turning any U(x) into U(y) U(x) U(y)†. The vector
y provides the rotation axis and angle.
The origin of our time can be associated to the point = 0. This leads in
particular to a naturally expanding Universe where cosmological comoving frames
would be described by straight lines crossing the origin = 0.
Contrary to the mathematical structure of the standard Poincar´e group, the spinorial
space-time transformations just described incorporate space translations and rotations
in a single compact group. Thus, the assumptions that led to the Coleman-
Madula theorem [45] concerning possible unifications of space-time transformations
with internal symmetries do not apply in the present case [17].
Such a description of space-time can be compared to a SO(4) approach where,
instead of being imaginary, the cosmic time would be given by the modulus of a fourvector
[17] obtained from the four real components of . The spinor space considered
is also in a sense similar to the subset of null (zero-norm) vectors in a SO(4,1) pattern
where the fifth dimension would be the cosmic time t, and the metric:
X2 = † − t2 (6)
X being the 5-vector formed by the four real components of the space-time and t.
4.2 Cosmological implications
The above geometry, when applied to relative velocities and distances at cosmic scale
for comoving frames, automatically yields the LLH law with a LLH constant (the
velocity/distance ratio) equal to the inverse of the age of the universe. If is a
constant angular distance between two cosmological comoving frames, the S3 spatial
distance d between the two corresponding points on the | | = t hypersphere will
be d = t, and the relative velocity v = . The ratio between relative velocities
and distances is then given by t−1. t is actually the only time scale available.
This value is in reasonable agreement with present observations while matter, radiation,
standard relativity, gravitation and specific space units have not yet been
introduced in our description of space-time. It would correspond to a situation essentially
different from equation (1), with no matter density and a positive term t−2
instead of − kc2a−2
s + c2/3. There is no track of the standard general-relativistic
explicit curvature term using such a spinorial space-time, where the space is positively
curved and spin-1/2 orbital wave functions are allowed. The cosmological constant term also disappears. The same result would be obtained with other power-like definitions
of cosmic time if simultaneously the associated space scale is suitably defined.
Similarly, it can be readily checked that the equivalent of equation (2) becomes:
dH/dt + H2 = 0 (7)
implying = 0 in a naive comparison between both equations.
The speed of light plays no special role in this geometric construction where no
specific velocity or distance scale is defined at the present stage and space dimensions
are described in time units. The overall spatial domain considered can naturally be
much larger than our conventional Universe where standard matter has been formed.
In such a situation, the effective global density of standard matter can be very weak at
the actual cosmological scale defined by the spinorial space-time. For similar reasons,
the effective space curvature of the Universe as measured in our observations would
be much smaller than naively expected for k = 1 in equations like (1).
The spinorial space-time would therefore be particularly well suited for a pre-Big
Bang (superbradyonic?) scenario. If the vacuum is made of superbradyonic matter
(or pre-matter), the actual size of the Universe can indeed be much larger than the
estimated size of the conventional observable Universe, and the nucleated standard
matter may occupy only a very small part of the space just defined.
In the spinorial space-time presented here, standard Lorentz symmetry can exist as
an approximate (low momentum) local space-time structure for conventional matter,
similar to the situation for phonons in a solid (see Sect. 2). Then, general relativity
can also remain valid as a low-energy limit in our standard Universe.
We are simultaneously assuming that the cosmic time considered in this chapter
is not fundamentally different from the age of the standard matter Universe where
we live. There is by now little observational difference between the measured Hubble
constant and the inverse of the estimated age of the Universe.
If the LLH expansion of the Universe is not generated by gravitation and geometry
through (1), and if instead a spinorial space-time is leading it at a much larger cosmological
distance scale, it is tempting to conjecture that the usually postulated dark
energy is not necessarily required to explain the observed acceleration of the expansion
of our Universe. Gravitational and other standard effects can possibly account
for past fluctuations of the velocity/distance ratio [17,18], and there is no obvious
reason for the expansion of our Universe to keep accelerating in the future.
A specific cosmological property of such a spinorial space-time is [17] that to each
point , a privileged space direction can be associated at cosmic scale through the
subspace where for any point 0 one has:
† 0 = | 0 | | | exp (i ) (8)
and exp (i ), with real, stands for a complex phase. This subspace is generated
using a matrix of which is an eigenstate. Then, the privileged space direction is
obtained by multiplying by an arbitrary complex phase.
(end of the extract)
