This document discusses analyzing imprints of the semiclassical phase of the universe's evolution on modern quantum gravity measurements. It suggests that any quantum modifications to general relativity can be detected by precisely measuring lengths in quasi-Minkowski spacetime and distinguished from other quantum effects. Length operators and their expectation values are discussed, as are descriptions of the universe near the Planck scale and implications for cosmology from quantum gravity models.

1. Imprints of the semiclassical phase of the
evolution of the universe on modern
quantum-gravity measurements
O.M. Lecian
Department for Astronautics Engineering, Electrical and
Energetics- DIAEE, Sapienza University of Rome, Rome, Italy
Theoretical Physics Workshop and Summer School
Models of Modern Physics
10 - 17 September 2017, Svit, Slovakia
6 August 2017
September 27, 2017O.M. Lecian Department for Astronautics Engineering, Electrical and Energetics- DIAEE, Sapienza University of Rome, Rome, ItaImprints of the semiclassical phase of the evolution of the unive

2. Abstract
The features of the semiclassical wavefunction of the universe,
established in the course of classicalization, are analyzed from the point
of view of their consistency with Einstein field equations, independently
of the quantum theory of gravity chosen for the description of the
cosmological quantum epoch. Any quantum modification of general
relativity can be detected by precision measurement of lengths in
quasi-Minkowski spacetime. Such modifications can be distinguished, by
analyzing their expansion in the Planck constant, from quantum effects
in the emerging Minkowski spacetime, and also from experimental
indeterminacy and quantum gravitational effects. They can also be
compared with the features of the evolution of the universe appearing
due to deformations and modifications of general relativity.
Based on Semiclassical length measure from a quantum-gravity wave function
submitted to Technologies Special Issue ”Quantum Gravity Phenomenology
and Experimental Implications” 15 Jul 2017
O.M. Lecian Department for Astronautics Engineering, Electrical and Energetics- DIAEE, Sapienza University of Rome, Rome, ItaImprints of the semiclassical phase of the evolution of the unive

3. Starting from the Planck age, the classicalization of perturbations
| ¯δg
must be consistent with the age and length scale at which QG
effects are due. This is achieved by selecting the relevant
contributions from the Hamiltonian. The corresponding quantum
states encode information about the solution to the Einstein field
equations and about the initial conditions.
O.M. Lecian Department for Astronautics Engineering, Electrical and Energetics- DIAEE, Sapienza University of Rome, Rome, ItaImprints of the semiclassical phase of the evolution of the unive

4. The same information items must be encoded not only in
expectation values of operators, but also in its expression when
evaluated on quantum (matter) settings, such as a length operator
for non-geodesic curves
L ∼
1
0 qab(c(t)) ˙ca(t) ˙cb(t)
1/2
Results of measure operations must be independent of the
definition of operators but consistent with the matter content of
the theory.
O.M. Lecian Department for Astronautics Engineering, Electrical and Energetics- DIAEE, Sapienza University of Rome, Rome, ItaImprints of the semiclassical phase of the evolution of the unive

5. Introduction
• Perturbed FRW at the Planck scale
• Minimal Lengths at the Plank scale
• The ’Length’ operator
• Observed anisotropies from Quantum Gravity: Cosmology
• Observed anisotropies from Quantum Gravity: Astrophysical
data
• Observed anisotropies from Quantum Gravity: emerging
Minkowski gravity
O.M. Lecian Department for Astronautics Engineering, Electrical and Energetics- DIAEE, Sapienza University of Rome, Rome, ItaImprints of the semiclassical phase of the evolution of the unive

6. Length operators
0806.4710
ψΓ,c cylindrical function of a given group symmetry of the
connections
The expectation value of the elementary length operator L for a
wedge of ω on the state ψΓ,c scales as
ψΓ,c, ˆLψΓ,c =
k,h
c∗
k (j0)(L)h
ck(j0) ∼ j0
where the first-order expansion is obtained from numerical study
O.M. Lecian Department for Astronautics Engineering, Electrical and Energetics- DIAEE, Sapienza University of Rome, Rome, ItaImprints of the semiclassical phase of the evolution of the unive

7. given two curves γ, γ1
the two operators measuring their lengths do not commute if the
two curves intersect or almost intersect ˆLγ, ˆLγ1 = 0, tγ ∩ tγ1 = 0
they commute if the curves do not intersect tγ ∩ tγ1 = 0
O.M. Lecian Department for Astronautics Engineering, Electrical and Energetics- DIAEE, Sapienza University of Rome, Rome, ItaImprints of the semiclassical phase of the evolution of the unive

8. for the measure of non-intersecting curves, above the Planck
length, the state ψΓ,c can be compared to the ’heuristic’ state-
grqc/9809038-
for which a discrete approximation of smooth flat geometry leads
to the expectation value
qab = δab + O
lPl
Λ
(1)
with Λ > lPl a characteristic length scale
O.M. Lecian Department for Astronautics Engineering, Electrical and Energetics- DIAEE, Sapienza University of Rome, Rome, ItaImprints of the semiclassical phase of the evolution of the unive

9. Plank-age anisotropies
1009.3935
The linearizationof homogeneous quasi-isotropic cosmological
models is consistent with perturbations of an FRW model, for
which the contributions to the potential are (exactly)
second-order for the field equations, such that the potential
term R stays unchanged
O.M. Lecian Department for Astronautics Engineering, Electrical and Energetics- DIAEE, Sapienza University of Rome, Rome, ItaImprints of the semiclassical phase of the evolution of the unive

10. The description of the appearance of spikes
arXiv:0710.0628
within the oscillatory regime can be consistent, at each time, with
a change (not always a rescaling) of the structure constants for the
symmetry group characterizing the space part of the metric tensor.
This description is not only consistent with the linearization
technique, but also suited for the investigation of the behavior of
QG length operators.
O.M. Lecian Department for Astronautics Engineering, Electrical and Energetics- DIAEE, Sapienza University of Rome, Rome, ItaImprints of the semiclassical phase of the evolution of the unive

11. The gravitational potential for the heat kernel
the heat kernel k, k = k(x, x ; s)
∂
∂s − ∆x k = 0 (2)
for the diffusion process of a particle from x to x (time t) within
the gravitational field and back (time s)
is dimension-dependent only as
k(x, x ; s) ∼ (4πs)−ds/2
(3)
0909.3329
1009.1136
O.M. Lecian Department for Astronautics Engineering, Electrical and Energetics- DIAEE, Sapienza University of Rome, Rome, ItaImprints of the semiclassical phase of the evolution of the unive

12. Hamidew expansion
k ∼
1
4πs2
a0 + a1s + a2s2
+ ... (4)
the a1 term, proportional to the scalar curvature, vanishes only for
exact vacuum solutions
in presence of matter,
the contributions to a1 are diverging
O.M. Lecian Department for Astronautics Engineering, Electrical and Energetics- DIAEE, Sapienza University of Rome, Rome, ItaImprints of the semiclassical phase of the evolution of the unive

13. Futamase-Berkin expansion
k ∼
1
4πs2
1 +
a
t2
s + ... (5)
for fixed t, there always exists s small enough for 1 to dominate
for fixed s, there can always exist a time t for the scalar-curvature
term to dominate
O.M. Lecian Department for Astronautics Engineering, Electrical and Energetics- DIAEE, Sapienza University of Rome, Rome, ItaImprints of the semiclassical phase of the evolution of the unive

14. Before-thermalization scenarios
The quasi-isotropization mechanisms acting on an isotropic
cosmology are model independent (i.e., independent of the
particular scalar-field gravity coupling of the resulting toymodel)
for the cold phase of any (also strongly) anisotropic Early Universe.
T. Futamase, Coleman-Weinberg symmetry breaking in an anisotropic
spacetime, Phys. Rev. D 29, 2783 (1984)
This simplifying assumption partially reconciles the difficulties of
the thermalization process
Doroshkevich, A. G., Novikov, I. D., Mixmaster Universes and the
Cosmological Problem, Soviet Astronomy, 14, p.763, (1971)
O.M. Lecian Department for Astronautics Engineering, Electrical and Energetics- DIAEE, Sapienza University of Rome, Rome, ItaImprints of the semiclassical phase of the evolution of the unive

15. ’Reduced’ models
In ’Quantum Reduced Gravity’, the symmetry group describing the
metric tensor is interpreted as three independent U(1) fields, who
then undergo suitable gauge fixing
1210.45404
The connection variables internal indices are therefore summed as
ˆAa = Aa
ˆI
The corresponding cosmological model at the Planck scale and
briefly after can offer a simplification wrt LQC for the definition of
length measures
O.M. Lecian Department for Astronautics Engineering, Electrical and Energetics- DIAEE, Sapienza University of Rome, Rome, ItaImprints of the semiclassical phase of the evolution of the unive

16. Modifications of the free dynamics term in Quantum
Cosmology
The corresponding qg problem allows one to understand its
Planck-scale modifications at most as modifying the free-particle
dynamics as from the modifications of the kinetic-energy term in
the Lagrangian
far from the singularity-removal regions of ’loopy’ cosmologies, the
picture can be at most that of a ’frozen’ anisotropic model
O.M. Lecian Department for Astronautics Engineering, Electrical and Energetics- DIAEE, Sapienza University of Rome, Rome, ItaImprints of the semiclassical phase of the evolution of the unive

17. Sampling quantum-gravity-effects errors from Minkowski
spacetime
The error in length measurements due to both quantum
mechanical effects and relativistic ones is calculated as
δgµν > (lP/l)2/3
> (tP/t)2/3
(6)
Ng, J.N., van Dam, H. Limitation to quantum measurements of space-time distances. Annals N. Y. Acad. Sci.
755 1995 , 579 [hep-th/9406110].
A lower bound on the quantum mechanical contribution to the
total error can be estimated to be local, and, in particular, to
depend on the inverse of the initial position xi of the measurement
system,
δx ≥ δxi +
h
c
2L
mδxi
. (7)
Salecker, H., Wigner, E.P., Quantum limitations of the measurement of space-time distances. Phys. Rev. 109,
571 1958
Amelino-Camelia, G. Limits on the measurability of space-time distances in the semiclassical approximation ofO.M. Lecian Department for Astronautics Engineering, Electrical and Energetics- DIAEE, Sapienza University of Rome, Rome, ItaImprints of the semiclassical phase of the evolution of the unive

18. Quantum-gravitational models can allow an intuition of the
Planck-scale description of the spacetime as a most general
expansion of the white noise as as powers of the Planck length
, whose peculiarities can account for those a particular
quantum-gravity model. Such a power spectrum of the frequency
ϕ of the strain noise, according to its most general features for the
detection of ’foamy’ features of the spacetime, needs be
ρ
n≥0
an
LP
c
ϕn
(8)
with an numerical coefficients; inverse powers need not be included,
as they are inconsistent with the theoretical classical limit LP → 0.
Amelino-Camelia, G. Phenomenological description of space-time foam; gr-qc/0104005.
The total uncertainty of an experimental measure having to
depend on the sum of the uncertainty of the initial and final
position, which cannot be smaller than the eigenvalue spacing (i.e.
the Planck length).
Calmet, X. On the Precision of a Length Measurement. Eur. Phys. J. C 54, 501 2008 , [Subnucl. Ser. 44, 625
2008] [hep-th/0701073].O.M. Lecian Department for Astronautics Engineering, Electrical and Energetics- DIAEE, Sapienza University of Rome, Rome, ItaImprints of the semiclassical phase of the evolution of the unive

19. Length Observations from Quantum Cosmology
Any (observed) modification of the isotropy of the Universe must
be therefore ascribed to non-second-order measured length ratios
ˆL(γxa (t2 − t1))
ˆL(γxb
(t2 − t1))
∼ 1 + (fa − fb) + O
lPl
Λ
(9)
for segments of length γxa < Λ
evaluated along two of the three space directions xa, xb = x, y, z
in the time interval t2 − t1
with ± ∼ 1
O.M. Lecian Department for Astronautics Engineering, Electrical and Energetics- DIAEE, Sapienza University of Rome, Rome, ItaImprints of the semiclassical phase of the evolution of the unive

20. (because the expectation values are non-vanishing) a semiclassical
state ψ contains information on the metric tensor;
because of the non-trivial ratios observed, a semiclassical state is
characterized non only by the space three-geometry, but also by
the initial conditions for the Einstein field equations
O.M. Lecian Department for Astronautics Engineering, Electrical and Energetics- DIAEE, Sapienza University of Rome, Rome, ItaImprints of the semiclassical phase of the evolution of the unive

21. Comparison with emerging gravity
the investigation of quantum-gravitational (spinfoamy) effects
exhibited by (quasi) Minkowsky flat spacetime ˆX(t)/ ˆY (t) is
performed by
ˆX(t)
ˆY (t)
1 −
ˆx1(t − 2L/c) − ˆx2(t − 2L/c)
ˆY (t)
, (10)
i.e. where the synchronization of measurements is performed
according to Special Relativity, for which the difference in definition
is of gravitational origin, i.e. according to the (non-Minkowski)
metric tensor used for (reconstructed after following) the
description of cosmologically-originated metric anisotropies
O.M. Lecian Department for Astronautics Engineering, Electrical and Energetics- DIAEE, Sapienza University of Rome, Rome, ItaImprints of the semiclassical phase of the evolution of the unive

22. Hamiltonian limit
In particular, any dependence on quantum-gravity effects fi (from
the ratio of different length observations from quantum
cosmology) of semiclassical states must be such that the quantities
fi (x; β) become
lim
β→0
fi (x; β) = fi (| x |; xi ), (11)
where the normalization | x | is performed with respect to the
metric tensor.
O.M. Lecian Department for Astronautics Engineering, Electrical and Energetics- DIAEE, Sapienza University of Rome, Rome, ItaImprints of the semiclassical phase of the evolution of the unive

23. Sky patches: pixellization
The necessary ratio of the two length measures is also consistent
with the analysis of the correlation function for the corresponding
initial conditions at the given time to the most general
investigation of the temperature fluctuations
∆Tθ,φ =
Tθ,φ − ¯T
¯T
=
l=∞
l=0
m=l
m=−l
| al,m | eiΦl,m
Yl,m(θ, Φ)
where | al,m | are the amplitudes, Φl,m(θ, Φ) are the phases,
Yl,m(θ, Φ) the spherical harmonics coefficients, where, for Bianchi
geometries, the amplitudes are not stochastically generated, while
the distribution of the phases can be generated according to
different methods.
1004.1925
O.M. Lecian Department for Astronautics Engineering, Electrical and Energetics- DIAEE, Sapienza University of Rome, Rome, ItaImprints of the semiclassical phase of the evolution of the unive

24. Comparison with Astrophysical data
According to the different patches, i.e. by naming (the center of)
each patch according the radial coordinates θα, φα and so on, the
ratio of the expectation values of measured lengths defines, up to a
normalization in observational distance X,
Xα
Xβ
1 −
Fα − Fβ
Xβ
(12)
as functions of the F’s, i.e. the (correspondingly-normalized)
differences in the temperature pixellization, with a numerical
factor with the dimensions of [length].
O.M. Lecian Department for Astronautics Engineering, Electrical and Energetics- DIAEE, Sapienza University of Rome, Rome, ItaImprints of the semiclassical phase of the evolution of the unive

25. Geometric quantum tests
The consequences of quantum spacetime granularity can described
as violations of the Lorentz symmetry and Planck-scale modified
electrodynamics.
Urrutia, L.F. Corrections to at-space particle dynamics arising from space granularity. Lect. Notes Phys. 702, 299
2006 , [hep-ph/0506260].
The role payed by the Lorentz symmetry below the Planck scale
and its ’quantum’ effects at the Planck scale for the gravitational
field are investigated in
Akama, K., Oda, I. Topological pregauge pregeometry. Phys. Lett. B 259, 431 1991.
The phenomenological limits of the resulting macroscopic
Einsteinian gravity are analyzed in
Aguilar, P., Sudarsky, D., Bonder, Y. Experimental search for a Lorentz invariant spacetime granularity:
Possibilities and bounds. Phys. Rev. D 87, no. 6, 064007 2013 , [arxiv:1210.5223 [gr-qc]].,
while those in a uniform Newtonian field inO.M. Lecian Department for Astronautics Engineering, Electrical and Energetics- DIAEE, Sapienza University of Rome, Rome, ItaImprints of the semiclassical phase of the evolution of the unive

26. The possible violations of Lorentz symmetry are explained to affect
the propagation of photons and of neutrinos at astrophysical scale.
Sarkar, S. Possible astrophysical probes of quantum gravity. Mod. Phys. Lett. A 17, 1025 2002 , [gr-qc/0204092].
Lorentz violations at the Planck length are shown to produce
modifications in the dispersion relations, for which the interaction
of high-energy particles at astrophysical scales can be expected to
be different
Jacobson, T., Liberati, S., Mattingly, D. Threshold effects and Planck scale Lorentz violation: Combined
constraints from high-energy astrophysics. Phys. Rev. D 67, 124011 2003 , [hep-ph/0209264].
, for which effective experimental uncertainty windows can be
considered.
The (field theoretical) dimension of operators affected by Lorentz
symmetry violations are analyzed according to the particle on
which the effects are detectable at astrophysical scales
D. Mattingly, Have we tested Lorentz invariance enough?. arXiv:0802.1561 [gr-qc].
The phenomenological possibility to further distinguish between
such effects for CMB and gravitational waves
Girelli, F., Hinterleitner, F., Major, S. Loop Quantum Gravity Phenomenology: Linking Loops to Observational
Physics. SIGMA 8 2012 , 098 [arxiv:1210.1485 [gr-qc]].O.M. Lecian Department for Astronautics Engineering, Electrical and Energetics- DIAEE, Sapienza University of Rome, Rome, ItaImprints of the semiclassical phase of the evolution of the unive

27. Acknowledgments
OML is grateful to The National Scholarship Programme of the
Slovak Republic (NSP) SAIA for funding and to Comenius
University, Faculty of Mathematics, Physics and Informatics,
Department of Theoretical Physics and Didactics of Physics
(KTFDF), Bratislava, for hospitality.
OML is grateful to Prof. P. Maroscia for encouraging these
reaserch lines.
O.M. Lecian Department for Astronautics Engineering, Electrical and Energetics- DIAEE, Sapienza University of Rome, Rome, ItaImprints of the semiclassical phase of the evolution of the unive