Solving the Problems of Dark Energy and the Hubble Constant Using Corrected Hubble’s Law

1. Introduction

At the beginning of the last century, astronomers discovered a redshift in the spectra of distant galaxies, i.e. a shift of spectral lines towards the red end of the spectrum. In other words, was detected a frequency difference Δv = v₁ – v₂, where v₁ is the frequency of light emitted by a cosmic source, v₂ is the frequency of light received on the Earth’s surface.

In 1929, the US astronomer E. Hubble on the basis of observations discovered a linear relationship (later dubbed Hubble’s law) between the relative redshift z in the spectra of the galaxies and distance r to them [1]:

cz = rH,

(1)

where c is the speed of light in a vacuum, H is the coefficient of proportionality called the Hubble constant.

It should be said that the distance that appears in Hubble’s law is not directly measured. In reality we determine a object brightness, which provides information about its distance, and the redshift z.

Since astronomers determine the radius R of the observable universe (Metagalaxy) by the formula R = c/H, it turns out from equation (1) that the shift

z = rH/c = r/R.

(2)

Thus, from the Hubble’s law implies that the visible light sources that have redshift z >1 should be located outside the Metagalaxy, r > R.

Obviously, such an assumption is absurd, therefore, the distances can be directly determined from formula (1) only for relatively close objects with z < 0.1. Different methods are used to calculate distances at higher redshifts.

Recall that for determining distances to galaxies in astronomy is being used the method of "standard candles": the measurement of object (that has a known luminosity) which located within galaxy. The best type of standard candle for cosmological observations are Type Ia supernovae (SNe Ia).

In work [2] it is noted: "Deeper understanding of low-redshift supernovae greatly improved their cosmological utility. … A more extensive database of carefully and uniformly observed SNe Ia was needed to refine the understanding of SN Ia light curves." Further: "A strong empirical understanding of SN Ia light curves has been garnered from intensive monitoring of SNe Ia at z ≤ 0.1 through B and V passbands… We use this understanding to compare the light curves of the high-redshift and low-redshift samples at the same rest wavelength."

In 1998, two independent groups of researchers found that in distant galaxies, the distance to which it was determined from the redshift (z = 0.16÷0.83), supernovae Ia have brightness lower by 10÷15% than that they should be [3,4].

In other words, it was found an apparent contradiction in the distance estimates by different methods: the distance to these galaxies, computed using the standard candles (supernovae Ia), was greater than the distance calculated by Hubble’s law. The scientists concluded that the universe expands with acceleration under the influence of a mysterious "dark energy".

The nature of dark energy is completely incomprehensible and is the subject of speculation. Researchers discuss this problem from a variety of perspectives, trying to determine the properties of dark energy or finding alternative ways to explain the observational data [5,6,7].

Another problem is related to the Hubble constant.

To date there are two leading approaches to determining H. One of them is based on measurements of the cosmic microwave background (CMB or cmb), the other is based on measuring the redshift (rs) of distant galaxies and then determining the distances to them by luminosity of variables (Cepheids) and exploding (supernovae) stars (such stars are crucial rungs on the "cosmic distance ladder").

Measurements by these methods give slightly different values of the Hubble constant. In particular, measurements of the microwave background (for example, made by the Planck Space Observatory) show a value of H_cmb = 67÷68 (km/s)/Mpc [8,9,10]. Measurements using calibrated distance ladder methods indicate on a value of H_rs = 73÷74 (km/s)/Mpc [11,12,13]. The difference is 7÷8%:

H_rs /H_cmb = 1.07÷1.08 = 1/(0.92÷0.935).

(3)

Since astronomers determine the radius (R) of the observable universe (Metagalaxy) by the formula R = c/H, accordingly, the calculated radii of the observable universe (Metagalaxy) differ at 7÷8%:

R_cmb = c/H_cmb , R_rs = c/H_rs, R_cmb /R_rs = 1.07÷1.08.

(4)

The reason for this discrepancy is unclear and is the subject of heated debate in the astronomical community.

Here we will give a solution to both mentioned problems.

2. Corrected Hubble’s law

First of all, it should be noted that there are two relative magnitudes of redshift,

Δv/v₁ = (v₁ – v₂)/v₁ = 1 – v₂/v₁

(5)

and

z = Δv/v₂ = (v₁ – v₂)/v₂ = v₁/v₂ – 1,

(6)

between which there is a simple connection:

Δv/v₁ = z/(z+1).

(7)

From equation (5) follows that the unit (the number 1) is the limit of the relative redshift Δv/v₁:

lim Δv/v₁ = 1.

(8)

In addition, the distance r to the radiation source cannot exceed the radius R of the observable universe, therefore

lim r/R = 1.

(9)

Hence, we get a corrected formula (corrected Hubble’s law):

Δv/v₁ = z/(z+1) = rH/c = r/R

(10)

or

cz/(z+1) = rH.

(11)

Obviously, this formula is applicable for any z.

Now it is clear that the original Hubble formula (1) needs to be corrected, but Hubble and other astronomers did not know this, because at the beginning of the last century only relatively close galaxies with very small redshift, namely z < 0.005, were known. For such small offsets, the value of z ≈ z/(z+1) or z ≈ Δv/v₁, i.e. the relative redshift magnitudes, z and Δv/v₁, are approximately equal to each other.

For example, if z = 0.005 then the value Δv/v₁ = z/(z+1) = 0.005/1.005 = 0.004975… ≈ 0.005.

However, even for small offsets, the distances calculated using the original Hubble formula (1) and the corrected formula (11) will differ.

For comparison we will carry out the calculations.

(1)

z = 0.005: according to formula (2) the ratio r/R = 0.005, and according to formula (10) the

ratio r/R = z/(z+1) = 0.005/1.005 = 0.004975, i.e. the distance to an emitting object is less by

0.005/0.004975 = 1.005 times (by 0.5%);

(2)

z = 0.01: from (2) r/R = 0.01, from (10) r/R = (0.01/1.01) = 0.0099, i.e. the distance is less

by 0.01/0.0099 = 1.01 times (by 1%);

(3)

z = 0.05: from (2) r/R = 0.05, from (10) r/R = (0.05/1.05) = 0.04761, i.e. the distance is less

by 0.05/0.04761 = 1.05 times (by 5%);

(4)

z = 0.1: from (2) r/R = 0.1, from (10) r/R = (0.1/1.1) = 0.0909, i.e. the distance is less by

0.01/0.0909 = 1.1 times (by 10%);

(5)

z = 0.16: from (2) r/R = 0.16, from (10) r/R = z/(z+1) = 0.16/1.16 = 0.1379, i.e. the distance

is less by 0.16/0.1379 = 1.16 times (by 16%);

(6)

z = 0.2: from (2) r/R = 0.2, from (10) r/R = z/(z+1) = 0.2/1.2 = 0.166, i.e. the distance is less

by 0.2/0.166 = 1.2 times (by 20%).

Thus, when using the original Hubble formula, the calculated distances are overestimated.

3. Determination of the Hubble constant

Equation (10) can be written as:

z/(z+1) = rH_rs/c = r/R_rs .

(12)

Here it is necessary to pay attention to the following circumstance.

As the distance r to the emitting objects increases, the redshift z increases and it becomes increasingly difficult to fix it. Currently, is known the object (probably, galaxy) HD1 that haves the largest measured value z ≈ 13 [14].

Consequently, for the most distant visible objects located on the edge of the Metagalaxy, the redshift z tends to its limit z_max = lim z. Taking into account equation (3), we come to the conclusion that the number z_max should lie in the interval 13÷14.5, because in this case

z_max/(z_max+1) = 0.92÷0.935 = 1/(1.07÷1.08).

(13)

Thus, formula (12) in the limiting case has the form:

z_max /(z_max+1) = 0.92÷0.935 = r_maxH_rs /c = r_max /R_rs ,

(14)

where r_max is the maximum distance at which the radiation source can be observed.

Now we will move on to the method based on measurements of the cosmic microwave background.

Recall that there is the following relationship between the speed of light c, the wavelength λ and the frequency ν:

c = λν = ƛω,

(15)

where ƛ = λ/2π is the reduced wavelength value, ω = 2πν is the cyclic frequency.

To eliminate confusion, it is necessary to establish that λ is the wavelength, and ƛ is the radius of the wave. When electromagnetic waves propagate, the wavelength and its radius increase, and the frequency, on the contrary, decreases. Therefore, the maximum radius of the wave ƛ_max corresponds to the minimum cyclic frequency ω₀:

с = ƛ_maxω₀ .

(16)

If we compare this formula with expression of

с = R_cmbH_cmb,

(17)

then it is obvious that the largest wave radius ƛ_max is equal to the Metagalaxy radius R_cmb , ƛ_max = R_cmb, and the Hubble constant Н_cmb is the smallest cyclic frequency of electromagnetic waves ω₀, H_cmb = ω₀ .

Therefore, we can write:

с = ƛ_maxH_cmb

(18)

or

1 = ƛ_maxH_cmb /с = ƛ_max /R_cmb .

(19)

Now we will rewrite the formula (14) in a short form:

0.92÷0.935 = r_maxH_rs /c = r_max /R_rs .

(20)

From equations (19) and (20) follows the relationship between the values of the Hubble constant calculated by two different methods:

1/( 0.92÷0.935) = H_rs /H_cmb = R_cmb /R_rs = 1.07÷1.08

(21)

Or

H_cmb /H_rs = 0.92÷0.935.

(22)