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Physics in Many-Sheeted Space-Time

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Year 2014

Pioneer and Flyby anomalies almost decade later

There was a very interesting link at Thinking Allowed Original in this morning - a lot of thanks for Ulla. The link was to two old anomalies discovered in the solar system: Pioneer anomaly and Flyby anomalies with which I worked for years ago. I remember only the general idea that dark matter concentrations at orbits of planets or at spheres with radii equal that of orbit could cause the anomaly. So I try to reconstruct all from scratch and during reconstruction become aware of something new and elegant that I could not discover for years ago.

The article says that Pioneer anomaly is understood. I am not at all convinced about the solution of Pioneer anomaly. Several "no new physics" solutions have been tailored during years but later it has been found that they do not work.

Suppose that dark matter is at the surface of sphere so that by a well-known text book theorem it does not create gravitational force inside it. This is an overall important fact, which I did not use earlier. The model explains both anomalies and also allow to calculate the total amount of dark matter at the sphere.

  1. Consider first Pioneer anomaly.
    1. Inside the dark matter sphere with radius of Jupiter's orbit the gravitational force caused by dark matter vanishes. Outside the sphere also dark matter contributes to the gravitational attraction and Pioneer's acceleration becomes a little bit smaller since the dark matter at the sphere containing the orbit radius of Jupiter or Saturn also attracts the space-craft after the passby. A simple test for spherical model is the prediction that the mass of Jupiter effectively increases by the amount of dark matter at the sphere after passby.
    2. The magnitude of the Pioneer anomaly is about Δ a/a=1.3× 10-4 and translates to Mdark/M≈ 1.3× 10-4. What is highly non-trivial is that the anomalous acceleration is given by Hubble constant suggesting that there is a connection with cosmology fixing the value of dark mass once the area of the sphere containing it is fixed. This follows as a prediction if the surface mass density is universal and proportional to the Hubble constant. The value of acceleration is a=.8× 10-10× g, g=9.81 m/s2 whereas the MOND model () finds the optimal value for the postulated minimal gravitational acceleration to be a0=1.2× 10-10 m/s2. In TGD framework it would be assignable to the traversal through a dark matter shell. The ratio of the two accelerations is a/a0=6.54.

      Could one interpret the equality of the two accelerations as an equilibrium condition? The Hubble acceleration associated with the cosmic expansion (expansion velocity increases with distance) would be compensated by the acceleration due to the gravitational force of dark matter? Could one interpret the equality of the two accelerations as an equilibrium condition? The Hubble acceleration H associated with the cosmic expansion (expansion velocity increases with distance) would be compensated by the acceleration due to the gravitational force of dark matter. The formula for surface density of dark matter is from Newton's law GMdark/R2= H given by σdark= H/4π G. The approximate value of dark matter surface density is from Hc=6.7 × 10-10 m/s2 equal to σ =.8 kg/m2 and surprisingly large.

    3. TGD inspired quantum biology requiring that the universal cyclotron energy spectrum of dark photons heff=hgr transforming to bio-photons is in visible and UV range for charged particles gives the estimate Mdark/ME≈ 2× 10-4 and is of the same order of magnitude smaller than for Jupiter. The minimum value of the magnetic field at flux tubes has been assumed to be BE=.2 Gauss, which is the value of endogenous magnetic field explaining the effects of ELF em radiation on vertebrate brain. The two estimates are clearly consistent.
  2. In Flyby anomaly spacecraft goes past Earth to gain momentum (Earth acts as a sling) for its travel towards Jupiter. During flyby a sudden acceleration occurs but this force is on only during the flyby but not before or after that. The basic point is that the spacecraft visits near Earth, and this is enough to explain the anomaly.

    The space-craft enters from a region outside the orbit of Earth containing dark matter and thus experiences also the dark force created by the sphere. After that the space craft enters inside the dark matter region, and sees a weaker gravitational force since the dark matter sphere is outside it and does not contribute. This causes a change in its velocity. After flyby the spacecraft experiences the forces caused by both Earth and dark matter sphere and the situation is the same as before flyby. The net effect is a change in the velocity as observed. From this the total amount of dark matter can be estimated. Also biology based argument gives an estimate for the fraction of dark matter in Earth.

This model supports the option in which the dark matter is concentrated on sphere. The other option is that it is concentrated at flux tube around orbit: quantitative calculations would be required to see whether this option can work. One can consider of course also more complex distributions: say 1/r distribution giving rise to constant change in acceleration outside the sphere.

A simple TGD model for the sphere containing dark matter could be in terms of boundary defined by gigantic wormhole contact (at its space-time sheet representing "line of generalized Feynman diagram" one has deformation of CP2 type vacuum extremal with Euclidian signature of induced metric) with radius given by the radius of Bohr orbit with gravitational Planck constant equal to hgr =GMm/v0. This radius does not depend on the mass of the particle involved and is given by rn= GM/v03 where 2GM is Schwartschild radius equal to 3 km for Sun. One has v0/c≈ 2-11.

An interesting possibility is that also Earth-Moon system contains a spherical shell of dark matter at distance given by the radius of Moon's orbit (about 60 Earth's radii). If so the analogs of the two effects could be observed also in Earth Moon system and the testing of the effects would become much easier. This would also mean understanding of the formation of Moon. Also interior of Earth (and also Sun) could contain spherical shells containing dark matter as the TGD inspired model for the spherically symmetric orbit constructed for more than two decades ago suggests. One can raise interesting questions. Could also the matter in mass scale systems be accompanied by dark matter shells at radii equal to Bohr radii in the first approximation and could these effects be tested? Note that a universal surface density for dark matter predicts that the change of acceleration universally be given by Hubble constant H.

For details see the chapter TGD and Astrophysics or the article Pioneer and Flyby anomalies for almost ten years later.

Sagittarius A and many-sheeted space-time

One of the basic predictions of TGD is the variation of effective light velocity determined in terms of the time taken to travel from point A to B. This time depends on the space-time sheet along which the photon, neutrino or some other elementary particle propagates (note that neutrinos are not quite massless). This effect is one of the key signatures of many-sheeted space-time manifesting as anomalies of general relativity.

The space-time of general relativity corresponds to effective space-time obtained by replacing the sheets of space-time with single region of Minkowski space with metric replaced with the sum of empty space Minkowski metric with the deviations of the induced metrics of the space-time sheets (effects of classical fields on space-time sheets on test particle sum up since it touches all the space-time sheets: linear superposition of fields is replaced with that for their effects).

SN1987A supernova provides first evidence for the presence of several space-time sheets. Neutrinos came as two bursts and before photons. The variation of effective light-velocity was of order Δ c/c ≈ 2× 10-9. Opera experiment claimed much larger variation of order: Δ c/c ≈ 10-5: unfortunately there was an error in the analysis of the experiments. Now Lubos has a posting about galactic blackhole Sagittarius A as neutrino factory. Chandra X-ray observatory and also Nustar and Swift Gamma-Ray Burst Mission detected some X-ray flares from Sagittarius A. 2-3 hours earlier IceCube detected high energy neutrinos by IceCube on the South Pole. As a good conservative Lubos of course denies the effect as he denies also climate warming. He did not however claim that experimenters are communists;-).

Could neutrinos arrive from galactic center? If they move with the same (actually somewhat lower) velocity than photons, this cannot be the case. The neutrinos did the same trick as SN1987A neutrinos and arrived 2-3 hours before the X-rays! What if one takes TGD seriously and estimates Δ c/c for this event? The result is Δ c/c ∼ (1.25-1.40 )×10-8 for 3 hours lapse using the estimate r= 25,900+/- 1,400 light years (see this). Δ c/c is by a factor 4 larger than for SN1987A at distance about 168,000 light years (see this). This distance is roughly 8 times longer. This would suggests that the smaller the space-time sheets the nearer the velocity of neutrinos is to its maximal value. For photons the reduction from the maximal signal velocity is larger.

For details see the chapter TGD and potential anomalies of GRT.

SN1987A and many-sheeted space-time

Lubos has written a highly rhetoric, polemic, and adrenaline rich comments posting about the mediabuzz related to supernova SN 1987 A. The target of Lubos is the explanation proposed by James Franson from the University of Maryland for the findings discussed in Physics Archive Blog. I do not have any strong attitude to Franson's explanation but the buzz is about very real thing: unfortunately Lubos tends to forget the facts in his extreme orthodoxy.

What happened was following. Two separate neutrino bursts arrived from SN1987 A. At 7.35 AM Kamionakande detected 11 antineutrons, IMB 8 antineutrinos, and Baksan 5 antineutrinos. Approximately 3 hours later Mont Blanc liquid scintillator detected 5 antineutrinos. Optical signal came 4.7 hours later.

The are several very real problems as one can get convinced by going to Wikipedia:

  1. If neutrinos and photons are emitted simultaneously and propagate with the same speed, they should arrive simultaneously. I am notspecialist enough to try to explain this difference in terms of standard astrophysics. Franson however sees this difference as something not easy to explain and tries to explain it in his own model.
  2. There are two neutrino bursts rather than one. A modification of the model of supernova explosion allowing two bursts of neutrinos would be needed but this would suggest also two photon bursts.
These problems have been put under the carpet. Those who are labelled as crackpots often are much more aware about real problems than the academic career builders.

In TGD framework the explanation would be in terms of many-sheeted space-time. In GRT limit of TGD the sheets of the many-sheeted space-time time are lumped to single sheet: Minkowski space with effective metric defined by the sum of Minkowski metric and deviations of the metrics of the various sheets from Minkowski metric. The same recipe gives effective gauge potentials in terms of induced gauge potentials.

Different arrival times for neutrinos and photons would be however a direct signature of the many-sheeted space-time since the propagation velocity along space-time sheets depends on the induced metric. The larger the deviation from the flat metric is, the slower the propagation velocity and thus longer the arrival time is. Two neutrino bursts would have explanation as arrivals along two different space-time sheets. Different velocity for photons and neutrinos could be explained if they arrive along different space-time sheets. I proposed for more than two decades ago this mechanism as an explanation for the finding of cosmologists that there are two different Hubble constants: they would correspond to different space-time sheets.

The distance of SN1987A is 168,000 light- years. This means that the difference between velocities is Δ c/c ≈ Δ T/T≈ 3 hours/168 × 103≈ 2× 10-9. The long distance is what makes the effect visible.

I proposed earlier sub-manifold gravity as an explanation for the claimed super-luminality of the neutrinos coming to Gran Sasso from CERN and mentioned in this context also SN1987A but did not compare the deviations from the light velocity. In this case the effect would have been Δ c/c≈ 2.5× 10-5 and thus four orders of magnitude larger than four supernova neutrinos. It however turned out that the effect was not real.

For details see the chapter TGD and potential anomalies of GRT.

Further progress concerning the relationship between TGD and GRT and Kähler-Dirac action

The earlier attempts to understand the relationship between TGD and GRT have been in terms of solutions of Einstein's equations imbeddable to M4× CP2 instead of introducing GRT space-time as a fictive notion naturally emerging from TGD as a simplified concept replacing many-sheeted space-time. This resolves also the worries related to Equivalence Principle. TGD can be seen as a "microscopic" theory behind TGD and the understanding of the microscopic elements becomes the main focus of theoretical and hopefully also experimental work some day.

The understanding of Kähler Dirac action has been second long term project. How can one guarantee that em charge is well-defined for the spinor modes when classical W fields are present? How to avoid large parity breaking effects due to classical Z0 fields? How to avoid the problems due to the fact that color rotations induced vielbein rotation of weak fields? The common answer to these questions is restriction of the modes of induced spinor field to 2-D string world sheets (and possibly also partonic 2-surfaces) such that the induced weak fields vanish. This makes string picture a part of TGD.

For details see the chapter TGD and GRT or the article Further progress concerning the relationship between TGD and GRT and Kähler-Dirac action.

Still about TGD and inflation

Quantum criticality is the TGD counterpart of the inflation and the flatness of 3-space follows from the condition that no local dimensional quantities are present in 3-geometry. Also the imbeddability fo M4 is an important piece of story and restricts the set the parameters of imbeddable cosmologies dramatically.

One can try to understand the situation microscopically in terms of the cosmic strings which gradually develop higher than 2-D M4 projection during cosmic evolution and become magnetic flux tubes carrying magnetic monopole fluxes explaining the presence of magnetic fields in cosmology.

At microscopic level magnetic flux tubes are the key structural elements. The phase transitions increasing Planck constant for the matter associated with flux tubes and thus also the lengths of magnetic flux tubes should be important as also the phase transitions increasing p-adic prime and reducing Planck constant originally emerged in the modelling of TGD inspired quantum biology are highly suggestive. First transitions would mean adiabatic expansion with no heat generation and latter transitions would liberate magnetic field energy since flux conservation forces field strength to be reduced and leads to liberation of magnetic energy producing ordinary matter and dark matter. Dark energy in turn is identifiable as magnetic energy.

The key question concerns the mechanism causing the isotropy and homogeny of the cosmology. There are two possible identifications.

  1. According to two decades old TGD proposal primordial cosmology before the emergence of space-time sheets could be regarded as string gas in M4+× CP2 at Hagedorn temperature determined by CP2 radius: TH∼ hbar/R(CP2). This phase could be present also after the transition to radiation dominated cosmology and consist of strings, whose thickness is gradually increasing and which contain carry dark energy and dark matter. The horizon radius is infinite for this cosmology thus providing at least partial explanation for the homogeny and isotropy and visible matter would represent deviations from it.
  2. The accelerating expansion period towards the end of the critical period could smooth out inhomogenities and thus provide an additional mechanism leading to homogenous and isotropic Big Bang. This for given space-time sheet representing R-W cosmology: in many-sheeted cosmology one can imagine distribution of parameters for the cosmology. The rapid expansion period could however also develop large fluctuations! Indeed, the time aF<a1 (density would be infinite for a1) for its end - and therefore local mass density - must have a distribution after the rapid expansion ends. This expansion would generate separate smoothed out radiation dominated space-time sheets with slightly different mass densities and cosmic temperatures. A splitting to smooth radiation dominated sub-cosmologies would take place.
Therefore TGD scenario could be very different from inflationary scenario. The problem is to decide which option is the most feasible one.

The formulas used to make back of the envelope (see this) calculations in inflation theory discussed in a guest posting in Lubos's blog given some idea about TGD counterpart for the generation of gravitons. Inflationary period is replaced with essentially unique critical cosmology containing only its duration as a free parameter. The fluctuations in the duration of this parameter explain scalar temperature fluctuations assoiated with CMB.

How the local polarization of CMB is generated?

There is a nice discussion about the mechanism leading to the generation of CMB polarization (see this). The polarization is generated after the decoupling of CMB photons from thermal equilibrium and is due to the scattering of photons on free electrons during decoupling. This scattering is known as Thomson scattering. The page in question contains schematic illustrations for how the polarization is generated. The scattering from electrons polarizes the photons in direction orthogonal to the scattering plane. In thermal equilibrium the net polarization of scattered radiation vanishes. If however the scattered photons from two perpendicular directions have different intensities a net polarization develops.

Polarized photons could be produced only during a short period during recombination scattering from free electrons was still possible and photons could diffuse between regions with different temperature. Polarized photons were generated when electrons from hot and cold regions where scattering on same electrons. CMB polarization indeed varies over sky but not in long length scales since photons could not diffuse for long lengths.

So called quadrupole anisotropy of CMB temperature contains information about the polarization. There are three contributions: scalar, vector, and tensor.

  1. Scalar contributions is due to density fluctuations reflecting themselves as temperature fluctuations and does not distinguish between polarizations: this is what has been studied mostly hitherto. A natural TGD mechanism for their generation would be different time for the end of the critical period leading to splitting of critical cosmology to radiation dominated cosmologies with slightly different temperatures.
  2. There is also so called vorticity distribution due to the flow which has vorticity and would due to defects/string like objects present also in TGD. The simplified situation corresponds to are region in which one has two flows in opposite direction locally. Depending on whether the scattering photons are upstream or down stream they are blue-shifted or red-shifter so that the temperatures are slightly different in up-stream and down.The flows in opposite direction give rise to a situation in which photons with different temperatures scatter and produce polarization. The effects of vorticity are expected to disappear during the fast expansion period. Probably because the gradients of velocity giving rise to vorticity are smoothed out.
  3. The third contribution is tensor contribution and due to gravitons generating stretching and squeezing of space in two orthogonal directions defining polarization tensor. Stretching increases wavelengths and decreases temperature. Squeezing does the opposite. Therefore temperature differences distinguishing between the two directions are generated and the outcome is polarization of the CMB background much later. This corresponds to the so called E and B modes. One can decompose polarization as vector field to two parts: the first one - the E-mode - is gradient and thus irrotational and second is curl and thus rotational and with vanishing divergence (incompressible liquid flow is a good concrete example).

How the polarization anisotropies could be generated in TGD Universe?

One can try to understand microscopically how the polarization anisotropies are generated in TGD framework using poor man's arguments.

  1. One can introduce a vision vision about fractal 3-D network of cosmic strings forming a kinds of grids with nodes in various scales. These grids would be associated with different levels of the hierarchy of space-time sheets associated with many-sheeted space-time. Coordinate grid is of course an idealization since three coordinate lines would meet in single node. A weaker form of grid would involve meeting of two coordinate lines at given node. There is data about our own galactic nucleus understood if it correspond to the node at which two magnetic flux tubes meet. Ordinary visible matter would be generated in nodes.

    One might say that galaxies are due to traffic accidents in which dark matter arriving along two cosmic strings collides in the crossing of the roads. Flux tubes would be attracted together by gravitational attraction so from the crossing.

  2. Amusingly, the notion grid emerged also TGD inspired quantum biology as a proposal for how living system codes morphogenetic position information. Flux tubes carry dark matter and ordinary matter is associated with the nodes at which coordinate lines meet each other. This web can give rise to a generalization of topological quantum computation using 2-braids. Coordinate lines define strings which can be knotted in 3-dimensions and define braids making possible topological quantum computation using macroscopic quantum phases defined by the dark matter. The time evolutions of coordinate lines defines string world sheets and in 4-D space-time the string world sheets can be knotted and braided so that also higher level TQC becomes possible with string reconnection and going above or below the other define two bits in each node.
  3. The presence of grid could also explain the honeycomb like structure of Universe with the recent typical size of honeycomb about 108 ly.
  4. In this framework the illustrations for how the gravitational waves induce the polarization of CMB. The radiation beams entering from opposite directions can be assigned with two magnetic flux tubes meeting at the node and in slightly different temperatures due to the interaction with gravitons much earlier. The gravitons can be regarded as larger space-time sheets at which the two flux tubes had contacts so that space associated with the flux tubes was forced to stretch or squeeze. This in turn increased of reduced photon wavelength so that photon temperature at flux tubes was different and the difference were preserved during subsequent evolution.

Back on the envelope calculations in TGD framework

One can modify the back on the envelope calculations of John Preskill (see this) in Lubos's blog to see what could happen in TGD framework. Now one however starts from the critical cosmology fixed apart from its duration and looks what it gives rather than starting from Higgs potential for inflaton field. The obvious counterpart for inflaton scalar field would be magnetic field intensity having same dimension but one should avoid too concrete correspondences.

The key question is whether the critical period generates the rapid expansion smoothing out inhomogenities or whether it generates them. The original guess that it smooths them out turns out be wrong in closer examination.

  1. The basic equation in inflationary model is given by

    (da/dt)2= V/mp2

    If V is small this has as solution a(t)= a(0)exp(Ht) if H= V1/2/mp is constant. De Sitter cosmology allows partial imbedding in TGD but the imbedding is naturally static and has interpretation as black-hole interior with constant mass density. One can find coordinates in which the solution looks like expanding cosmology without Big Bang but these coordinates are not natural from the view of imbedding space.

  2. In TGD the expression for da/dt for critical cosmology is

    da/dt= [a02-a2]1/2/[a02-R2-a2]1/2 .

    a0 is roughly the duration of cosmology and R is CP2 radius of order 103.5 Planck lengths. The almost uniqueness follows from the condition that the imbedding is such that the induced metric at the 3-surfaces defined by intersections with hyperboloids of M4+ is flat rather than hyperbolic. This cosmology differs from de-Sitter cosmology.

  3. For a→ 0 one has

    da/dt ≈ a02/[a02-R2]≈ 1 .

    so that one has da/dt ≈ 1 and a≈ t for small values of a in accordance with the replacement of Big Bang with a "silent whisper amplified to a Big Bang" (density of matter goes as 1/a2) Hubble constant goes like H∝ 1/a so that Hubble radius divergence. This does not guarantee that horizon radius becomes infinite. Rather, the horizon is finite and given in good accuracy by the duration a1=[a02 R2]1/2 of the period. One can however explain the isotropy and homogenity of the string gas in light-cone M4+ carrying flux tubes carrying dark matter and energy in terms of the infinite horizon of M4.

    There is no exponential time evolution at this period since one has a≈ t in good approximation for a/a0<<1. The TGD counterpart of V would behave like 1/a2, which conforms with the idea that V corresponds to energy density.

  4. As the limit a→ a1=[ta02-R2]1/2 is approached, the expansion rate approaches infinite and for a>a1 at the latest one expects radiation dominated cosmology: otherwise a region of Euclidian signature of the induced metric results. The expectation is that a transition to radiation dominated cosmology takes place before a=a1 at which also energy density would diverge. The question is whether this period means smoothing out of inhomogenities or generation of them or both.
Consider now what could happen near the end of the Minkowskian period of critical cosmology.
  1. Although it is not clear whether rapidly accelerating expansion is needed to to smooth out homogenities, one can just find what conditions this would give on the parameters. For ai= kR at which phase transition began the condition that a was increased at least by factor e50∼ 5× 1021 (50 e-folds) this would give a1≈ a0>e50kR. For k∼ 1 this gives something like 10-18 seconds, which happens to correspond atomic length scale. Below it will be found that this period more naturally corresponds to the period during which large fluctuations in density distribution and metric are generated.
  2. The earlier estimate for the emergence of radiation dominated cosmology assumed that the transition to radiation dominated cosmology takes place at CP2 temperature defining Hagedorn temperature at which temperature of the string gas cannot be raised anymore since all the energy goes to the generation of string excitations rather than to kinetic energy, gives aF∼ 10-10 seconds, which is by factor 108 larger. If this were true, the fast expansion period aF would increase the scale factor to about 68 e-folds equivalent to 98 2-folds. p-Adic prime p≈ 2196 would correspond to p-adic length scale about L(196) ∼ .1 meters. The crucial assumption would be that the the time aF at which the expansion ends is same everywhere. There is no reason to assume this and this would mean that the period in question generates inhomogenities and isotropies of mass distribution and temperature distribution.

    Note that if the distribution of the time aF<a1 at which the critical period ends is responsible for the CMB fluctuations then the number of foldings characterizes the smoothness of given local radiation dominated cosmology and could be rather large.

  3. The rapid accelerating expansion occurs as gaa approaches zero. Indeed, for

    a→ a1= [a02-R2]1/2

    a very rapid expansion occurs and da/dt approaches infinite value. Near to a1 one can write a/a1=1-δ and solve δ approximately as function of t as

    δ =[3R2/4a12]2/3 [t-t1/a1]2/3 , t1= ∫0a1 [1-a2/a121/2/[1-a2/a12]1/2 .

    Hubble constant behaves as

    H= (da/dt)/a = (R2/2a13-1/2 .

  4. What is interesting is that applying the naive dimensional estimate for the amplitude of gravitational fluctuations to be δ hT2∼ H2/mP4. This would mean that at the limit a→ aF< a1 gravitational fluctuations become very strong and generate the strong graviton background. Same applies to fluctuations in mass density.


The possibility of very rapid expansion near a=aF<a1 leading to radiation dominated cosmology should have some deep meaning. The following tries to catch this meaning.

  1. The explosive period could lead to a radiation dominated cosmologies from string dominated cosmology with Hagedorn temperature. It could involve heff increasing phase transitions for string gas during the initial period and liberation of magnetic energy during the end period as massless particles: this would explain why the mass density of the space-time sheet increases dramatically. The critical cosmology could correspond to a phase transition from a phase with Hagedorn temperature identified as TH∝ hbar/RH to radiation dominated cosmology.
  2. The cooling of string gas would lead to the generation of hierarchy of Planck constants and liberation of the magnetic energy of strings as massless particles during the end of critical period topologically condensing to space-time sheets such as massless extremals. This process could correspond to the rapid increase of energy density towards the end of the critical period.
  3. Isotropy and homogenity appear both at the level of imbedding space and space-time sheets. The infinite horizon of M4+ would explain the isotropy and homogenity of string gas in H both before and after the emergence of space-time sheets at Hagedorn temperature around a∼ R(CP2). In particular, the smoothness of the cosmology of dark matter and dark energy would find explanation. The rapid expansion would in turn smooth out inhomogenities of individual space-time sheets.
  4. The Hubble scale 1/H approaches to zero as a=aF<a1 is approached. The rapid expansion destroys anisotropies and inhomogenities of radiation dominated space-time sheet corresponding to particular value of aF. The distribution for values of aF in turn explains CMB scalar fluctuations since the energy density in final state is highly sensitive to the precise value of aF. This distribution would be Gaussian in the first approximation. One can say that the fluctuation spectrum for inflaton field is replaced with that for aF.
  5. Also the generation of gravitational radiation and its decoupling from matter could take place during the same end period. After this gravitational fields would be essentially classical and assignable to space-time sheets. Essentially formation of gravitationally bound states would be in question analogous to what happens photons decouple from matter much later. The reduction of the temperature of string gas below Hagedorn temperature could generate also the massless graviton phase decoupling from matter and inducing the temperature fluctuations and polarization during decoupling.

    Gravitons and also other particles would topological condense at "massless extremals" (MEs,topological light rays) and particles - in particular photons - would interact with gravitons by generating wormhole contacts to gravitonic MEs. The interaction between MEs assignable to gravitational radiation and photons would have caused the fluctuations of CMB temperature.

To sum up, if the TGD inspired picture is correct then Penrose would have been correct in the identification of string theory as fashion and inflationary cosmology as fantasy ( Lubos has reacted strongly to this). Also the fact that inflationary cosmology is at the verge of internal contradiction due the fact that the assumption of field theoretic description is in conflict with the large graviton background suggests that inflationary cosmology is not for long with us anymore.

For details see the chapter TGD and Cosmology or the article BICEP2 might have detected gravitational waves.

Quantum critical cosmology of TGD predicts also very fast expansion

TGD inspired critical cosmology (see this) relies on the identification of 3-space as a= constant section, where a is Lorentz invariant cosmological time defined by the light-cone proper time a=(m0)2-rM2)1/2, and from the assumption that (quantum) criticality corresponds to a vanishing 3-curvature meaning that 3-space is Euclidian.

The condition that the induced metric of the a= constant section is Euclidian, fixes the critical cosmology apart from its duration a0 from the existence of its vacuum extremal imbedding to M4× S2, where S2 homologically trivial geodesic sphere:

ds2 = gaada2 -a2 (dr2 +r22) ,

gaa= (dt/da)2=1- ε2 /(1-u2) , u=a/a0 , ε=R/a0 .

sin(Θ)= +/- u , Φ= f(r) ,

1/(1+r2) -ε2(df/dr))2=1 .

From the expression for dt/da one learns that for the small values of a it is essentially constant equal to dt/da=(1 ε2)1/2. When a/a0 approaches to (1-ε2)1/2, dt/da approaches to zero so that the rate of expansion becomes infinite. Therefore critical cosmology is analogous to inflationary cosmology with exponential expansion rate. Note that the solution is defined only inside future or past light-cone of M4 in accordance with zero energy ontology.

After this a transition to Euclidian signature of metric happens (also a transition to radiation dominated cosmology is possible): this is something completely new as compared to the general relativistic model. The expansion begins to slow down now since dt/da approaches infinity at a/a0=1. In TGD framework the regions with Euclidian signature of the induced metric are good candidates for blackhole like objects. This kind of space-time sheets could however accompany all physical systems in all scales as analogs for the lines of generalized Feynman diagrams. For sin(Θ)=1 at a/a0=1 the imbedding ceases to exist. One could consider gluing together of two copies of this cosmology together with sin(Θ)= sin(π-Θ)= a/a0 to get a closed space-time surface. The first guess is that the energy momentum tensor for the particles defined by wormhole contacts connecting the two space-time sheets satisfies Einstein's equations with cosmological constant.

Quantum criticality would be associated with the phase transitions leading to the increase of the length and thickness of magnetic flux tubes carrying Kähler magnetic monopole fluxes and explaining the presence of magnetic fields in all length scales. Kähler magnetic energy density would be reduced in this process, which is analogous to the reduction of vacuum expectation value of the inflation field transforming inflaton vacuum energy to ordinary and dark matter.

At the microscopic level one can consider two phase transitions. These phase transitions are related to the hierarchy of Planck constants and to the hierarchy of p-adic length scales corresponding to p-adic primes near powers of 2.

  1. The first phase transition increases Planck constant heff=nh in a step-wise manner and increases the length and width of the magnetic flux tubes accordingly but conserves the total magnetic energy so that no magnetic energy is dissipated and one has adiabaticity. This sequence of phase transitions would be analogous to slow roll inflation in which the vacuum expectation of inflation field is preserved in good approximation so that vacuum energy is not liberated. The flux tubes contain dark matter.
  2. Second phase transition increases the p-adic length scale by a power of 21/2 and increases the length and width of magnetic flux tubes so that the value of the magnetic field is reduced by flux conservation (magnetic flux tubes carry monopole fluxes made possible by CP2 homology). This phase transition reduces zero point kinetic energy and in the case of magnetic fields magnetic energy transforming to ordinary and dark matter.
  3. The latter phase transition can be accompanied by a phase transition reducing Planck constant so that the length of the flux tubes is preserved. In this transition magnetic energy is liberated and dark matter is produced and possibly transformed to ordinary matter. This kind of phase transitions could take place after the inflationary adiabatic expansion and produce ordinary matter. As a matter fact, I have originally proposed this kind of phase transition to be the basic phase transition involved with the metabolism in living matter (see this), which suggests that the creation of ordinary matter from dark magnetic energy could be seen as kind of metabolism in cosmological scales.

    In zero energy ontology one can ask whether one could assign to the Minkowskian and Euclidian periods a sequence of phase transitions increasing Planck constants but proceeding in opposite time directions.

  4. During the inflationary period the size scale of the Universe should increase by a factor of order 1026 at least. This corresponds to 287 - that is 87 2-foldings, which is a more natural notion than e-folding now. If the size of the sub-Universe is characterized by a p-adic length scale, this would correspond in the final state to p∼ 2174 at least: this p-adic length scale is about 4× 10-5 meters roughly and thus of order cell size. If the foldings correspond to increase of secondary p-adic length scale characterizing causal diamond, 89 foldings would correspond to Mersenne prime assignable to weak bosons.

  5. How the transition to radiation dominated cosmology takes place is an interesting question. The decay of the magnetic energy to ordinary matter should take place during the Euclidian period initiating therefore the radiation dominated period. For the radiation dominated cosmology the scale factor behaves as t∝ a2 so that dt/da approaches zero. Since this occurs also when the Euclidian period starts, the guess is that space-time sheets with radiation dominated sub-cosmologies assignable to sub-CDs (CD is shorthand for causal diamond) begin to be created.

Although this picture is only an artist's vision and although one can imagine many alternatives, I have the feeling that the picture might contain the basic seeds of truth.

For details see the chapter TGD and Cosmology or the article BICEP2 might have detected gravitational waves.

Could TGD allow inflationary cosmology?

A natural question is whether TGD could allow inflationary cosmology. In the lowest order this would require imbedding of the De Sitter space. De Sitter space allows two basic coordinate slicings.

  1. The first one corresponds to a stationary metric having interpretation in terms of interior of an object with constant mass density. The line element reads

    ds2 =Adt2-Bdr2-r22 ,

    A =1-(r/l)2 , B= 1/A .

    l has natural interpretation as outer boundary of the object in question. It will be found that TGD suggests 2-fold covering of this metric.

  2. Second coordinatization has interpretation as simplest possible inflationary cosmology having flat 3-space:

    ds2= dt12-e2t1/l dr2-r122 .

  3. The two coordinatizations are related to each other by the formulas deducible from the general transformation property of metric tensor:

    t =t1+ log[1+(r1/l)2e2t1/l]/2 ,

    r =et1/lr1 .

In TGD framework also the imbedding of space-time as surfaces matters besides the metric which is purely internal property. The most general ansatz for the imbedding of De Sitter metric into M4× CP2 is as a vacuum extremal for for Kähler action with the understanding that small deformation carries energy momentum tensor equal to Einstein tensor so that Einstein's equations would old true in statistical sense.

  1. The general ansatz for the stationary form of the metric is of same general form as that for Schwartchild metric. One can restrict the consideration to a homologically trivial geodesic sphere S2 of CP2 with vanishing induced Kähler form and standard spherical metric. This means that CP2 is effectively replaced with S2. This imbedding is a special one but gives a good idea about what is involved.

    Denoting by (m0,rM,θ,φ) the coordinates of M4 and by (Θ, Φ) the coordinates of S2, a rather general ansatz for the imbedding is

    m0= t+ h(r) , rM=r ,

    Rω × sin(Θ (r))= +/- r/l , Φ= ω t+ k(r) .

  2. The functions h(r), k(r), and Θ (r) can be solved from the condition that the induced metric is the stationary metric. For Schwartschild metric h(r) and k(r) are non-vanishing so that the imbedding cannot be said to be stationary at the level of imbedding space since t=constant surfaces correspond to m0 h(rM)=constant surfaces.

    De Sitter metric is however very special. In this case one can assume h(r)=k(r)=0 for Rω=1. The imbedding reduces simply to an essentially unique imbedding

    sin(Θ(r))=+/- r/l= rM/l , Φ= t/R= m0/R .

    This imbedding is certainly very natural and would describe stationary non-expanding cosmology with constant mass density. Not that the imbedding is defined only for rM<l. Unless one allows 3-space to have boundary, which for non-vacuum extremals does not seem plausible option, one must assume double covering

    sin(Θ(r))= sin(π-Θ(r))= +/- rM/l

    Stationarity implies that there is no Big Bang.

  3. The transition to the inflationary picture looks in TGD framework very much like a trick in which one replaces radial Minkowski coordinate with r1 =exp(-t1/l) rM and in these new coordinates obtains Big Bang and exponential expansion as what looks like a coordinate effect at the level of imbedding space. Also the transition to radiation dominated cosmology for which the hyperbolic character of M4+ metric ds2=da2 a2(dr2/(1+r2) +r22) is essential, is difficult to understand in this framework. The transition should correspond to a transition from a stationary cosmology at the level of imbedding space level to genuinely expanding cosmology.

The cautious conclusion is that sub-manifold cosmology neither excludes nor favors inflationary cosmology and that critical cosmology is more natural in TGD framework. In TGD Universe De Sitter metric looks like an ideal model for the interior of a stationary star characterized by its radius just like blackhole is characterized by its radius. It seems that TGD survives the new findings at qualitative and even partially quantitative level.

For details see the chapter TGD and Cosmology or the article BICEP2 might have detected gravitational waves.

BICEP2 might have detected gravitational radiation

BICEP2 team has announced a detection of gravitational waves via the effects of gravitational waves on the spectrum on polarization of cosmic microwave background (CMB). What happens that gravitational waves (or possibly some other mechanism) transforms so called E modes which correspond the curl free part of polarization field expressible as gradient to B modes responsible for the divergenceless part of polarization field expressible as curl of vector field.

Interaction of photons with gravitons would induce this polarization changing transformation: this is discussed in earlier post by Lubos. The signal is unexpectedly strong constraints on possible models, in particular to the inflationary models which are currently in fashion. There is excellent popular summary of the physics behing scalar, vector, and tensor perturbations of CMB here. The map produced by BICEP describes the vorticity of the polarization field at the sky and one can clearly see it.

There has been a lot of pre-hype about the finding as proof for inflation, which it is not. Even Scientific American falls in the sin of inflationary hyping which is a pity. Inflationary theory is only the dominating theory which might be able to explain the finding.

In the following the findings are discussed in the framework of TGD based cosmology in which the flatness of 3-space is interpreted in terms of quantum criticality rather than inflation. The key role is played by gradually thickening cosmic strings carrying magnetic monopole flux, dark energy as magnetic energy and dark matter as large heff phases at cosmic strings. Very thin cosmic strings dominate the cosmology before the emergence of space-time as we know it and quantum criticality is associated with the phase transition between these two phases. Later cosmic strings serve as seeds of various cosmological structures by decaying partially to ordinary matter somewhat like inflaton fields in inflationary cosmology. Cosmic strings also explain the presence of magnetic fields in cosmos difficult to understand in standard approch. The crucial point is that - in contrast to ordinary magnetic fields - monopole fluxes do not require for their creation any currents coherent in long scales.

Liam McAllister's summary about the findings of BICEP2 team

Liam McAllister from Cornell University has written an excellent posting about the discovery and its implications in Lubos's blog. McAllister discusses the finding from several points of view. Can one trust that the finding is real? How should one interpret the result? What are its implications? A brief summary is in order before going to details.

  1. Consideration is restricted to inflationary scenarios but it is made clear that they are not the only option. It is emphasized that a huge amount of inflationary parameter space is excluded by the unexpectedly high strength of the effect. Also the general problems of inflationary models are made explicit - a great favor for those who are not inflationary enthusiasts and might have something else in mind.
  2. Also other than gravitonic mechanisms transforming E modes to B modes can be imagined. For instance, the signal might not be primordial but caused by polarized foreground sources: BICEP claims that these contributions have been eliminated.
  3. The most important conclusion is of course that a direct detection of gravitational waves - maybe even quantal ones - has been achieved. Earlier gravitational radiation has been detected only a slowing down of rotation rate of pulsars (Hulse-Taylor binary pulsar).

Comparison of inflationary models and TGD

Further conclusions depend on the cosmological model adopted and McAllister considers the situation in the framework of inflationary models and lists the basic aspects of inflationary model.

  1. The Universe on large scales should be approximately homogenous, isotropic and flat.
  2. The primordial scalar density perturbations should be correlated on super-horizon scales and be approximately Gaussian, adiabatic, and approximately scale-invariant.
In TGD framework inflationary cosmology is replaced with a cosmology fixed almost uniquely by the criticality of the mass density when combined with imbeddability to N4× CP2 as Lorentz invariant 4- surface. The only free parameter is the finite durationkenotau of the critical period. This kind of critical - it seems even quantum critical - periods are predicted to appear in various scales so that Russian doll cosmology is strongly suggested as in case of inflationary models. Scalar fields (inflaton fields) are replaced with cosmic strings, which evolve by thickening their M4 projections from string world sheets to 4-D ones. Magnetic energy replaces dark energy and has interpretation as counterpart for the energy of inflation field. Dark matter at magnetic flux tubes corresponds to large hbar phases (see this , this , and this ).
  1. In TGD framework the long range correlations would be due to quantum criticality rather than extremely rapid expansion during inflationary period. The Universe in large scales should be also now homogenous, isotropic, and flat.
  2. The primordial density perturbations reflect the presence of cosmic strings (see this) before the phase transition period. These cosmic strings have 2-D M4 projection, which is minimal surface, so that these object behave for all practical purposes like strings, and CP2 projection is e 2-D holomorphic surface in CP2. During primordial period cosmic strings dominate and the mass density behaves like 1/a2, where a is proper time coordinate of the light-cone. The mass per comoving volume goes to zero at the moment of big bang so that initial singularity is smoothed out and big bang transforms to "a silent whisper amplified to big bang". For radiation dominated cosmology mass density would behave as 1/a4 giving rise to infinite energy per comoving volume at the moment of Big Bang.
  3. Cosmic strings gradually thicken their M4 projections and the huge primordial magnetic fields carrying quantized monopole flux weaken. These fields differ crucially from the ordinary magnetic fields in that no current is needed to create them - this is due the fact that CP2 Kähler form defines a self-dual magnetic monopole (instanton). Amazingly, even the magnetic fields penetrating to super-conductors could be this kind and perhaps even those associated with ferromagnets.

    This can explain why primordial and recent Universe is full of magnetic fields in length scales, where they should not exist since the currents creating them cannot exist in long scales. The thickening of the remnants of cosmic strings would give rise to birth of galaxies organised like pearls in necklace along big cosmic strings: galaxies are indeed known to be organized into long string like structures and density perturbations would correspond to these strings.

    No vacuum expectations of Higgs like scalar fields are needed. Even in elementary particle physics Higgs expectation is replaced with string tension assignable to string like structures accompanying elementary particles.

    Cosmic strings would carry dark energy as magnetic energy and dark matter as phases with large values of Planck constant coming as integer multiple of ordinary Planck constant. Ordinary matter would be formed when cosmic strings and dark matter "burn" to ordinary matter: this would be the TGD counterpart for the decay of inflaton field to ordinary matter.

  4. Cosmic strings would define the density perturbations having correlations on super-horizon scales. In the first approximation they are certainly Gaussian. Whether they are adiabatic (no exchange of heat with environment) is an interesting question: if they correspond to large values of Planck constant, this is certainly what one expects. The perturbations would be approximately scale invariant: p-adic length scale hypothesis would formulate this quantitatively by replacing continuum of scales with a hierarchy of discrete p-adic length scales coming as powers of square root of 2 (half octaves).
  5. One can of course ask about spectrum of Planck constant coming as integer multiples of ordinary Planck constant: could it realize the presence of large number of length scales characterizing criticality? Could the spectrum of length scales implied by spectrum of Planck constants be the TGD counterpart for the inflationary expansion? Does the average value of Compton length or flux tube length proportional to heff=nh increase with exponential rate during quantum criticality as larger and larger Planck constants emerge?
It seems that at this qualitative level TGD survives basic tests at qualitative level but without assuming inflation fields and exponentially fast expansion since quantum criticality predicting flat 3-space (dimensional parameters such as curvature of 3-space vanish). Cosmic strings would represent the long range fluctuations. A further bonus is that cosmic strings explain dark energy and dark matter, and also the presence of long range magnetic fields in cosmos.

Fluctuations of gravitational field

McAllister gives a nice overall summary about the physics involved if given by inflationary models.

  1. It is not yet fully clear whether the fluctuations of gravitational field are quantum mechanical or classical. In TGD framework quantum classical correspondence suggests that quantal and classical identifications might be equivalent.
  2. Just as the quantum fluctuations of inflaton field would give rise to the density fluctuations visible as temperature anisotropies and large scale structures, the quantum fluctuations of gravitational field would give rise to the observed B modes in inflationary scenario. The correlation functions of gravitons in the background metric would tell everything. The problem is that we do not yet have quantum theory of gravitation allowing to really calculate everything except in QFT approximation.
  3. In TGD framework the fluctuations should physically correspond to cosmic strings and the question is whether gravitons can be identified as massless modes for the cosmic strings so that string like objects would give all. In fact, elementary particles are in TGD framework identified as string like objects! Ironically, TGD as generalization of string model realizes stringy dream in all scales and even for ordinary elementary particles!
Since gravitons couple to energy the formula for the energy density at which inflationary period begins should determine the spectrum of gravitational waves. Inflationary models predict this energy scale as the fourth root of the energy density in the beginning of inflation: the formula is given by in the article of McAllister. This formula contains single dimensionless parameter called r, and BICEP measurements give a rather large value r=.2 for it.

The natural expectation is that any theory explaining the findings in terms of gravitons produces similar prediction but with the energy density of scalar field replaced with something else. In TGD the energy density assignable to cosmic strings so that the square root of the energy density of cosmic string multiplied by some numerical factor should be the relevant parameter now.

Inflation should begin at GUT mass scale

The first implication of the findings is that if inflation explains the findings, it should have begun in GUT scale 1016 GeV, which is very high. The findings cut off a gigantic portion of the parameter space of inflationary models and leaves only inflation potentials that are approximately translationally invariant.

In TGD framework one expects that the energy scale corresponds to that in which quantum critical period begins after string dominated primordial period. This scale should be given by CP2 mass scale apart from some numerical factor. CP2 mass corresponds to m(CP2)=hbar/R(CP2), where R(CP2) is CP2 radius. p-Adic mass calculations predict the value of electron mass and assign to electron the largest Mersenne prime M127 having the property that the p-adic length scales kenosqrtpR(CP2) is not completely super-astronomical. This fixes R(CP2) and m(CP2). The outcome is m(CP2)∼ 4× 1015 GeV.

A numerical constant can be present in the estimate for the energy scale at which quantum critical period begins. In particular, the factor 1/αK1/4 should be present since Kähler action is proportional to 1/αK, which by simple argument is in excellent approximation equal to the inverse of the fine structure constant equal to 137. This would rise the estimate for the energy scale to about 1016 GeV if the same formula for it is used also in TGD (which might of course be wrong!). With a considerable dose of optimism one could say that TGD allows to understand why the measured value of r is what it is.

Difficulties of the inflationary approach

What is nice that McAllister discusses also so the difficulties of inflationary approach.

  1. So called Lyth bound gives lower bound for the distance that inflaton's vacuum expectation must move in field space in order to generate detectably large primordial waves: that is the duration of the inflationary expansion. The lower bound is given by Planck mass MP: Δ Φ >MP.
  2. There is however a problem. This distance should be not larger than the cutoff scale Λ of the quantum field theory. But if standard wisdom is taken granted, Λ should be smaller than Planck mass MP giving Δ Φ<MP!
  3. One can certainly invent all kinds of tricky mechanisms to circumvent the problem: the proposal considered by McAllister is that the couplings of Φ are suppressed to heavy degrees of freedom so that the UV theory respects the approximate shift symmetry Φ→ Φ+Δ Φ. This is true for massless scalar field but this field does not develop vacuum expectation value. McAllister mentions that for V=m2Φ2/2 the approximate shift symmetry is true. Maybe it is for small enough values of m: exact symmetry would require m=0 .
  4. The physical interpretation of masslessness implied by strict shift invariance would be in terms of conformal invariance. In TGD framework quantum criticality implies conformal invariance also in 2-D sense and quantum criticality corresponds to the absence of dimensional parameters from Higgs potential making Higgs mechanism impossible.
To my humble opinion, this difficulty means a strong blow against the idea about Higgs mechanism as source of vacuum energy density in cosmology. As already mentioned, the decay of the dark energy identifiable as magnetic energy and large heff dark matter associated with the evolving primordial cosmic strings would produce ordinary matter in TGD Universe.

Also the ordinary Higgs mechanism is plagued by the loss of naturalness and predictivity by the fact that the Higgs particle has too low mass and SUSY has not been found in low enough mass scales to stabilize Higgs mass. In TGD framework the string tension of string like objects assignable to elementary particles would give the dominating contribution to gauge boson masses and p-adic thermodynamics in its original form the dominating contribution to fermion masses (see this and this). The couplings of fermions to Higgs are gradient couplings and the coupling is same for all fermions in accordance with naturality and universality.

The overall conclusion is that TGD survives the new findings at qualitative and even partially quantitative level.

For details see the chapter TGD and Cosmology or the article BICEP2 might have detected gravitational waves.

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