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Dark energy is argued by the accelerating expansion of the Universe but has not been directly measured. In this article, some uncertainties are pointed out, first one being the determination of the Hubble constant. And the main parameters (magnitude, distance, redshift, velocity) are checked. Distinguishing the instantaneous velocity from the average velocity, it is then concluded from the same data that the expansion would not be accelerating, and that the Gravity would slightly slow down the motion of explosion. Moreover, at the end of the paper, it is proposed a neo-Newtonian approach to get the computed values in a closer agreement with the observed values; this Neo-Newtonian Mechanics is in coherence with the Quantum Mechanics.

In 2011, the Nobel Prize in Physics was awarded “for the discovery of the accelerating expansion of the Universe through observations of distant supernovae”. It rewards both difficult observations and mainly a clever explanation. The trouble comes from that a Nobel Prize is understood not only as a justified reward but also as a proof or an irremovable truth. Then the accelerating expansion is today a dogma which should not be criticized or even seriously discussed with the partisans of the theory of Relativity, despite their own inconsistencies: if the theory of Relativity can explain gravitational effect (by distortion of space-time), it cannot explain the “Standard Model” of Quantum Mechanics with the (other) three forces. And although a large majority (69%) of the Universe would content it, in fact the dark energy has never been measured. It looks like a mysterious form of energy, and a few people begin to wonder about its reality [

A supernova is a sudden appearance of star (due to its explosion) before slowly fading from sight over several weeks or months. Supernovae of type I are considered to supply the same brightness and so are considered as candles whose distances can be computed. Cepheids too are used to estimate distances due to their relation between pulsation period and luminosity.

The redshift is the increase of the wavelength of light emitted from most of the stars. It is linked to the radial velocity.

Then it can be built a diagram to show the link between the magnitude (or the brightness) and the redshift. This link was initially established by Lemaître and Hubble, and recently completed by Pr. Perlmutter [

“The high-redshift SNeIa from both teams are 0.28 mag dimmer or 14% farther than expected in a Universe with this much matter and no cosmological constant. The statistical uncertainty of these values is 0.08 and 0.06 mag (or 4% and 3% in distance)” [

That means that the distance would be farther from 18% to only 10%. Where could these uncertainties come from?

Edwin Hubble [

V = H o ⋅ D (1)

with Ho being a Constant which carries his name and which value he found was close to

H o ≈ 500 km / s / Mpc (2)

Today, the value of this constant is considered to be between 50 and 80, close to 67 or 72 (see

How could Pr Hubble do such a mistake in his measurements? The measure of the redshift is very precise, and the relation between the redshift and the velocity has not changed. The difference comes from the measure of the brightness and the estimation of distance which, for a poorly explained reason, were absolutely wrong. A difference of magnitude of 6 is a difference of observed brightness of about 40 times (see Appendix A).

If the values of magnitudes and distances have been now corrected, however it does no match exactly with a Constant value. But the idea of the constant link between velocity and distance remains in the scholar community, and it is the meaning of the blue dashed lines in

^{1}Terms in square brackets [

Due to the Hubble “constant”, it was then considered that the expansion of the Universe should be “constant” (i.e. along the dashed lines of ^{1} [

Let us illustrate this argument by a graph in

May we discuss the fuzzy assertion about the “expected” redshift in the past with the consequence that expansion would have been slower in the past? Can another analysis be proposed?

Some diagrams are given in “observed” magnitude [

Of course we are confident that such a correction is already done in observed magnitude (see Appendix B). But because the value of absorption on such a distance cannot be precisely measured and could be so huge (see Appendix C) and because the hypotheses taken on the absorption coefficient are not given in the articles, this trivial explanation is given “for memory”.

It is difficult to determine its value. For Pr. Edwin Hubble, it was about 500. With the measures of Hubble telescope on Cepheids, it is 72 km/s/Mpc but with the measures of Plank satellite on far supernovae, it is 67 km/s/Mpc. We can note that Cepheids are closer to Earth than Supernovae; maybe is there a link between the computed Hubble value and the distance. Then it would be conclude that the Hubble constant decreases with the distance, which would mean that the Hubble constant is not a constant!

Another way to express the Hubble law is:

D = V ⋅ T o (3)

with

T o = 1 / H o (4)

For example, for Ho = 70 km/s/Mpc, To = (3.085 × 10^{19}/70 = 4.4 × 10^{17} s/3.15 × 10^{7} =) 14 billions of years. If the Hubble constant were not the same for distant and for close galaxies, that would mean the time since the Big Bang would be different either we are far or close from the Earth, which would be strange.

Let us check another hypothesis. The photons of the far galaxies were emitted a long time ago. Let us call it this moment T', with T' < To. Then by definition of H' (see

With the Hubble law and the redshift, quite all the galaxies seem to move away from each other. It is usually given the picture of a pancake in an oven where all the grapes move away from each other. In the theory of Relativity, there is no centre. We can check in Appendix D that we can explain it within the Newton Mechanics, with the single difference that there would be a centre for the Big Bang.

Then the Relativity theory is not the single explanation for the Big Bang and expansion of the universe can be seen from another point of view, with a centre.

Let us now say that all the observations of redshift z and magnitude M are correct, that the computing of velocity V and distance D are correct too, and that the Earth is not so far from the centre.

And let us suppose that at the Big Bang origin, all the galaxies were ejected at different velocities (like in a bomb). The speeder one will go farther. In a sphere,

Symbol | Property | Definition |
---|---|---|

Ho | Hubble constant(an average of calculated values) | |

To | 1/Ho | Present time or duration from the origin |

T' | T' < To | Time of the observed event |

H' | 1/T' | H value calculated from time T' at a specific point |

D' | D (T') | Distance at the time T' |

v' | v (T') | Observed velocity at the time T' |

v ″ a v | D'/T' | Average velocity from the origin to the time T' |

H (obs) | v'/D' | Observed H value, or H value calculated from the velocity and the distance at a specific point. |

the value of gravity is in correlation with the radius of the sphere, with the volume more precisely. Then the influence of gravity will be higher for distant galaxies than for close galaxies (from the centre).

In fact, the more the star or galaxy is fast, the more the distance is important, and so the older the event is.

Let us have the definitions hereafter (see

The path the photon used from a distant event is:

D ( T ′ ) = c ( T o – T ′ ) (5)

D ( T ′ ) = v ″ T ′ (6)

So

H ′ = ( 1 + v ″ / c ) H o (7)

Because velocity is linked to z, It confirms that H' (see the dashline in

Note: it could give the impression that the observed universe is more increasing than previously

Because observed points are above the dashed curve, we can conclude that

H ( o b s ) < H ′ (8)

By definition of H ( o b s ) :

D ′ ⋅ H ( o b s ) = v ′ (9)

And by definition of the time T' when occurred the event:

D ′ ⋅ H ′ = v ″ ( a v . ) (10)

Then

D ′ = v ″ ( a v . ) / H ′ = v ′ / H ( o b s ) (11)

Because H ( o b s ) is lower than H ′ (cf. Equation (8)), then

v ′ < v ″ ( a v . ) (12)

Velocity at the event time T' is lower than the average velocity, so the initial velocity was greater. Then we can conclude that the expansion is not accelerating, and it is even decreasing from our point of view.

Then the velocity will slightly decrease with the time (see

It would mean the gravity slow down the expansion, and that the formula would look like:

D ′ = ∫ T initial T ′ ( V initial − G ⋅ M ⋅ t D ( t ) 2 ) d t (13)

To sum up the purpose, let us check

The other axis of

v c = ( 1 + z ) 2 − 1 ( 1 + z ) 2 + 1 (14)

According to the Newtonian mechanics, it is:

1 + z = 1 + v source / c 1 + v o b s / c (15)

The limit with the Newtonian mechanics is it does not work at very high speed, especially with the addition of velocities. But if we apply the Doppler effect of the Newtonian Mechanics, and assuming that the Sun (the observer) would be motionless (it is a simplification):

z ≈ v source c (16)

and so for a same z, the value of velocity is higher with neo-Newtonian mechanics than with Relativity. Then the distance computed with the Hubble law would be higher with neo-Newtonian mechanics (see Appendix E) than with Relativity theory (see

Dark energy is argued by the acceleration of the expansion of Universe but has never been directly measured. When we check the data, this explanation is based on the difference between the observed magnitude and the computed magnitude for a given redshift. But there are a lot of uncertainties; the first one is about the Hubble constant; the second one is on the “expected” redshift. Then it has been checked in this article the influence of the dust absorption, and the distinctions between the Hubble values. Establishing the distinction between the instantaneous

velocity and the average velocity, from the same data we can conclude at the opposite that the expansion is slightly slowing down due to the Gravity! In addition, using the neo-Newtonian mechanics, the calculated magnitude would be very close to the observed magnitude; let us recall that the Neo-Newtonian Mechanics is coherent with the Quantum Mechanics.

Serret, O. (2018) Gravity vs. Dark Energy, about the Expansion of the Universe. Journal of Modern Physics, 9, 84-97. https://doi.org/10.4236/jmp.2018.91006

Formula of Brightness F:

F = L 4 Π D 2 (A1)

with L the Luminosity (=6 ´ 10^{19} SI for SN)

Formula of apparent magnitude m:

D = 10 m − M − A + 5 5 (A2)

With M, absolute magnitude (=−19.3 for SN Ia)

According to the Hubble law:

D ⋅ H o = v (A3)

With Ho ≈ 70 km/s/Mpc

And according to the Relativity theory:

v c = ( 1 + z ) 2 − 1 ( 1 + z ) 2 + 1 (A4)

Then, it can be built

1) According to observations,

For z = 0.7, m = 24

2) According to Hubble law:

D ⋅ H o = V (B.1)

And according to the Relativity theory:

v c = ( 1 + z ) 2 − 1 ( 1 + z ) 2 + 1 (B.2)

And according to the formula of apparent magnitude:

D = 10 m − M − A + 5 5 ⋅ 10 − 6 (B.3)

with M = −19.3 for SN la, and D in [Mpc]

Then:

V = 10 m − M − A + 5 5 10 − 6 ⋅ H o = c ⋅ ( 1 + z ) 2 − 1 ( 1 + z ) 2 + 1 (B.4)

or

m = 5 ⋅ ( log 10 ( c H o ⋅ ( 1 + z ) 2 − 1 ( 1 + z ) 2 + 1 ) + 6 ) + M + A − 5 (B.5)

With the same example, for z = 0.7 (and A = 0), we get m = 22.3.

3) To get a coherent value with the observation, we have to add a dust absorption of A = 1.74.

Then for z = 0.7 and A = 1.74, we get m = 24.

On Earth, the air absorption is about 0.01/km at the atmospheric pressure [^{−0.08}. For air at atmospheric pressure, there are 6.023 ´ 10^{23} molecules per 22.4 Liters, or 3 ´ 10^{25} molecule/m^{3}. In vacuum, the intergalactic space is expected to get 10 atoms/m^{3} [^{8} less atoms, then absorption would be of about 10^{−10}/km. On a distance of 200 Mps, or 10^{20} km, it could be an absorption of 10^{10} or a transmission of 10^−(10^10), which means nothing!

Air:

Molecules: 6 ´ 10^{23}/22.4 L = 3 ´ 10^{25}/m^{3}

k(air) = 0.01/km

A = k ⋅ D = 0.01 × 8 = 0.08

Transmission = T = 10^{−0.08}

With A ≈ 0, then T ≈ 1

There is a good transmission

Vacuum:

10 atoms/m^{3} = 2 atom/direction

30 ´ 10^{24} molecule/m^{3} = 3 ´ 10^{8} molecule/direction

10^{8} less in vacuum than in space

K'(vacuum) = 10^{−10}/km

200 Mpc = 200 ´ 3 ´ 10^{19} km = 6 ´ 10^{21} km

Absorption A' = 10^{−10} ´ 6 ´ 10^{21} = 6 ´ 10^{11} = infinit

Transmission T' = 10^{−infinite}

With A ≈ infinite, then T ≈ 0

On such a huge value, it could have a poor transmission.

Let us suppose Velocity V is proportional to the Distance D (as defined by the Hubble law) in the case of an explosion:

V = D ⋅ H (D1)

And we consider we are not at relativistic velocities (not close to the celerity c) and so we can add or subtract according to the usual Newtonian law.

Coordinate of Earth:

In the O frame, the Earth is at a distance of 2D, and then its velocity is 2D

In the E frame, the Earth is in the center of its own frame, so its distance is nil and its velocity too.

Coordinate of star A:

In the O frame, the star A is at a distance of 3D, and then its velocity is 3V

In the E frame, the star A is at a distance of (3D − 2D =) 1D.

Its velocity is (3V − 2V =) 1V

Coordinate of star B:

In the O frame, the star B is at a distance of 1D, and then its velocity is 1V

In the E frame, the star B is at a distance of (1D − 2D =) −1D.

Its velocity is (1V − 2V =) −1V

Coordinate of star C:

In the O frame, the star C is at a distance of − 1 D i ⇀ , and then its velocity is − 1 V i ⇀

In the E frame, the star C is at a distance of (−1D − 2D =) − 3 D i ⇀ .

Its velocity is (−1V − 2V =) −3V

Coordinate of star D:

In the O frame, the star D is at a distance of 2 D j ⇀ , and then its velocity is 2 D j ⇀

In the E frame, the star D is at a distance of ( ( 2 D ) 2 + ( 2 D ) 2 =) 2.8D.

Its velocity is on the horizontal (0 − 2V =) − 2 V i ⇀ and on the vertical (2V ? 0 =) + 2 V j ⇀ , so its velocity is ( ( 2 V ) 2 + ( 2 V ) 2 =) 2.8V.

Coordinate of star F:

In the O frame, the star F is at a distance such as 3D∙cosθ = 2D, so θ = 48˚.

Then its velocity is 3V and 3 V ⋅ cos ( 48 ∘ ) i ⇀ + 3 V ⋅ sin ( 48 ∘ ) j ⇀ = 2 V i ⇀ + 2.2 V j ⇀

In the E frame, the star F is at a distance of (3D∙sin(48˚) =) 2.2D.

Its velocity is on the horizontal (2V − 2V =) 0 V i ⇀ and on the vertical (2.2V ? 0 =) + 2.2 V j ⇀ , so its velocity is 2.2V.

Coordinate of star G:

In the O frame, the star G is at a distance ( 2 D ) 2 + ( 1 D ) 2 = 2.23 D and ( 2 D ) 2 + ( 1 D ) 2 ⋅ cos θ = 2 D , so θ = 26˚.

Then its velocity is 2.23V and

2.23 V ⋅ cos ( 26 ∘ ) i ⇀ + 2.23 V ⋅ sin ( 26 ∘ ) j ⇀ = 2 V i ⇀ + V j ⇀

In the E frame, the star G is at a distance of 1D.

Its velocity is on the horizontal (2V − 2V =) 0 V i ⇀ and on the vertical (1V ? 0 =) 1 V j ⇀ , so its velocity is 1V.

Coordinate of star H:

In the O frame, the star H is at a distance 3D at an angle of θ = 30˚.

Then its velocity is 3V and 3 V ⋅ cos ( 30 ∘ ) i ⇀ + 3 V ⋅ sin ( 30 ∘ ) j ⇀ = 2.6 V i ⇀ + 1.5 V j ⇀

In the E frame, the star H is at a distance of ( 0.6 D ) 2 + ( 1.5 D ) 2 = 1.6 D .

Its velocity is on the horizontal (2.6V − 2V =) 0.6 V i ⇀ and on the vertical (1.5V − 0 =) 1.5 V j ⇀ , so its velocity is 1.6V.

Coordinate of star I:

In the O frame, the star I is at a distance 4D and 4D∙cosθ = 1D, so θ = 14.5˚.

Then its velocity is 4V and 4 V ⋅ cos ( 14.5 ∘ ) i ⇀ + 4 V ⋅ sin ( 14.5 ∘ ) j ⇀ = 3.9 V i ⇀ + 1 V j ⇀

In the E frame, the star I is at a distance of ( 1.9 D ) 2 + ( 1 D ) 2 = 2.14 D .

Its velocity is on the horizontal (3.9V − 2V =) 1.9 V i ⇀ and on the vertical (1V − 0 =) 1 V j ⇀ , so its velocity is 2.14V.

Coordinate of star J:

In the O frame, the star J is at a distance 2D and the angle is θ = 135˚.

Then its velocity is 2V and 2 V ⋅ cos ( 135 ∘ ) i ⇀ + 2 V ⋅ sin ( 135 ∘ ) j ⇀ = − 1.4 V i ⇀ + 1.4 V j ⇀

In the E frame, the star J is at a distance of ( 3.4 D ) 2 + ( 1.4 D ) 2 = 3.67 D .

Its velocity is on the horizontal (−1.4V − 2V =) − 3.4 V i ⇀ and on the vertical (1.4V − 0 =) 1.4 V j ⇀ , so its velocity is 3.67V.

And so on∙∙∙ see

STAR | O frame | E frame | ||
---|---|---|---|---|

Distance | Velocity | Distance | Velocity | |

Earth | 2D | 2V | 0 | 0 |

Star A | 3D | 3V | 1D | 1V |

Star B | 1D | 1V | −1D | −1V |

Star C | −1D | −1V | −3D | −3V |

Star D | 2D | 2V | 2.8D | 2.8V |

Star F | 3D | 3V | 2.2D | 2.2V |

Star G | 2.23D | 2.23V | 1D | 1V |

Star H | 3D | 3V | 1.6D | 1.6V |

Star I | 4D | 4V | 2.14D | 2.14V |

Star J | 2D | 2V | 3.67D | 3.67V |

To go beyond this limit of Newtonian mechanics, it can be appealed to the “Neo-Newtonian Mechanics”

This new theory is based on a “variable inertial mass” instead of a variable time.

With the Neo-Newtonian Mechanics, it can be explained:

− the Lorentz transformation: “How to demonstrate the Lorentz factor: variable time vs. variable inertial mass”, Journal of Modern Physics, http://file.scirp.org/Html/7-7502022_54203.htm

− the addition of velocities with a sum always lower than the light celerity: “Velocity addition demonstrated from the conservation of linear momenta, an alternative expression”, Journal of Modern Physics, http://file.scirp.org/Html/2-7502196_56126.htm

− the radiation emitted in a synchrotron: “Net force F=γ^{3}.m.a at high velocity”, Journal of Modern Physics, http://file.scirp.org/Html/5-7502665_66042.htm

− the Mercury perihelion precession: “About the ovoid orbits in general, and perihelion precession of Mercury in particular (2)”, Millennium Relativity,, http://gsjournal.net/Science-Journals/Research%20Papers/View/6607

− the gravitational light bending: “Hipparcos did not measure directly the light bending!”, the General Science Journal, http://gsjournal.net/Science-Journals/Research%20Papers/View/6998

− the missing of gravitational waves: “Gravitational waves or particles radiation?”, Physics Essays, https://www.physicsessays.org/browse-journal-2/product/1588-12-olivier-serret-gravitational-waves-or-particle-radiation.html

− the missing of dark matter: “The flat rotation curve of our Galaxy explained within Newtonian mechanics”, Physics Essays, https://physicsessays.org/browse-journal-2/product/1240-7-olivier-serret-the-flat-rotation-curve-of-our-galaxy-explained-within-newtonian-mechanics.html

− the missing of dark energy: present article

And let us add that based on a stable time whatever the motion of the frame is, the Neo-Newtonian Mechanics is in coherence with the Standard Model of Quantum Mechanics.