1. Varying Physical Constants from Astrometric and Cosmological Analysis
Rajendra Gupta
SUSY2019 Conference– Corpus Christie, TX May 20-24, 2019
Macronix Research Corporation

2. OBJECTIVE
Explore relaxing the constancy constraint on c, G & Λ in Einstein field equation in cosmology.
Derive: 𝑐 𝑐 =1.8𝐻; 𝐺 𝐺 =5.4𝐻; Λ Λ =−1.2𝐻; ħ /ħ=1.8𝐻; 𝐻 𝐻 =−0.6𝐻; 𝑘 𝐵 𝑘 𝐵 =3.6𝐻.
Calculate three astrometric anomalies with the above variability of c and G.
Establish that VcGΛ fits supernovae 1a redshift data as well as ΛCDM and with better predictive capability.
Seek collaboration to improve theoretical model and to explore how universe will evolve with the above variable constants – funds available. (Contact: ;
H is the Hubble parameter. VcGL with one parameter as well as LCDM with two parameters. We have funds available to support a graduate student or a post doc at any university for up to 3 years.

3. Modified Einstein field Eqns.
𝑑 𝑠 2 = 𝑐 2 𝑑 𝑡 2 −𝑎 𝑡 2 [ 𝑑 𝑟 2 1−𝜅 𝑟 2 + 𝑟 2 (𝑑 𝜃 2 + sin 2 𝜃𝑑 𝜙 2 )]
𝐺 𝜇𝜈 +Λ 𝑔 𝜇𝜈 =− 8𝜋𝐺 𝑐 4 𝑇 𝜇𝜈 ; for flat universe 𝜅=0
𝑎 𝑎 𝑎 𝑎 2 =− 4𝜋𝐺 𝑐 2 𝑝+ 1 2 Λ; 𝑎 2 𝑎 2 = 8𝜋𝐺 3 𝑐 2 𝜀+ 1 3 Λ≡ 𝐻 2
𝜀 + 3 𝑎 𝑎 𝜀+𝑝 +( 𝐾 𝐾 𝜀+ Λ 8𝜋𝐾 )=0 cont. eqn.; 𝐾≡ 𝐺 𝑐 2
𝐾 𝐾 =𝑘 𝑎 𝑎 , Λ Λ =𝑙 𝑎 𝑎 and 𝐻 𝐻 =𝑚 𝑎 𝑎 Machian scenario
𝑎 2 𝑎 2 = 𝐻 0 2 𝑎 2𝑚 = 8𝜋 3 (𝐾 0 𝑎 𝑘 )𝜀 0 𝑎 −3 1+𝑤 Λ 0 𝑎 𝑙
2𝑚=𝑘−3−3𝑤=𝑙
Start with Robertson Walker metric, plug in the Eeinstein filed equation, then for flat universe we obtain standard equations; standard continuity equation and ancillary continuity equation. a is the scale factor. w is the equation of state parameter p=w𝜀. Substitute Machian scenario in the second einstein eq. and we get the last equation. Equating the exponents of scale factor, we get relation among k,l,m &w.

4. Modified Einstein field Eqns.
𝑎 𝑡 = 𝑎 𝑡 𝑎 𝑡 0 = 𝑡 𝑡 𝑤−𝑘 ; 𝑎 𝑎 = 2 3+3𝑤−𝑘 𝑡 −1 ;
𝑎 𝑎 = 𝑎 𝑎 1− 3+3𝑤−𝑘 2 ; −𝑞≡ 𝑎 𝑎 𝑎 2 = −1−3𝑤+𝑘 2 .
Taking 𝑞 0 =−0.4* and w=0, we get 𝑘=1.8, and also 𝑙=−1.2, and 𝑚=−0.6. We thus have:
𝐾 𝐾 = 𝐺 𝐺 − 2 𝑐 𝑐 =1.8𝐻, Λ Λ =−1.2𝐻 and 𝐻 𝐻 =−0.6𝐻
Not generally Lorentz invariant
Not derived from scalar-tensor action
So treat the VcGΛ model quasi-phenomenological * R. P. Gupta, IJAA 8, 219 (2018); Universe 4, 104 (2018).
q0 is obtained analytically as -0.4 assuming tired light and expansion of the universe both contribute to the observed redshift; w=0 for matter only universe (since lambda is implicitly included). We still don’t have independently the variation of G and c.

5. Lunar Laser Ranging
𝐺 /𝐺 (7.1±7.6 × 10 −14 yr −1 ) from LLR data analysis*
Kepler’s 3rd law: 𝑃 2 = 4 𝜋 2 𝑟 3 𝐺𝑀 for period of rotation
𝐺 𝐺 = 3 𝑟 𝑟 − 2 𝑃 𝑃 − 𝑀 𝑀 ≃0 if vary only 𝐺
But 𝑟=𝑐𝑡; 𝑟 𝑟 = 1 𝑡 + 𝑐 𝑐 (∵ c is measuring tool for LLR)
∴ 𝐺 𝐺 − 3 𝑐 𝑐 = 3 𝑡 − 2 𝑃 𝑃 − 𝑀 𝑀 ≃0; 𝑡 Hubble time 1/𝐻
∵ 𝐺 𝐺 − 2 𝑐 𝑐 =1.8𝐻 from last slide, we get
𝐺 𝐺 =5.4𝐻 and 𝑐 𝑐 =1.8𝐻
* F. Hofmann and J. Müller, Class. Quant. Grav. 35, (2018).
Gdot/G is very small from LLR analysis

6. Distance modulus 𝜇, redshift 𝑧
𝜇=5log( 𝑅 0 1− 1+𝑧 − log 𝐹 𝑧
+5 log 1+𝑧 VcGΛ model
𝐹 𝑧 =4 1− 1+𝑧 −0.6 /[1− 1+𝑧 −2.4 ]
𝐹(𝑧) is distance normalization factor
𝜇=5log[ 𝑅 0 0 𝑧 𝑑𝑢/ Ω 𝑚,0 1+𝑢 3 +1− Ω 𝑚,0 ]
+5 log 1+𝑧 ΛCDM model
𝑅 0 = 𝑐 0 / 𝐻 0
Ω 𝑚,0 is relative matter energy density
Ref: R. P. Gupta, Galaxies 7, 55 (2019).
VcGA is parameterized by one parameter R0, whereas ACDM requires two parameters R0 and Omega m,0.

7. Supernovae 1a 𝜇−𝑧 data fit

8. Supernovae 1a data fit/predict.
Focus on 𝜒2 and 𝑃 (the 𝜒2 probability). Two parameter fit is better for any data set. Fitting the data beyond what was used in parameterizing a model is better with a single parameter VcGΛ model.

9. Pioneer anomaly
If c is increasing, then e-m signal transit time to and from spacecraft will be decreasing.
This will be interpreted as if spacecraft is nearer if c const.
OR THE SPACE CRAFT HAS NEGATIVE ACCELERATION
2𝑟 𝑎 = 𝑐 0 Δ𝑡= Apparent distance
2𝑟 𝑝 = 𝑐 0 0 Δ𝑡 𝑒 1.8 𝐻 0 𝑡 𝑑𝑡 = 𝑐 𝐻 0 𝑒 1.8 𝐻 0 Δ𝑡 −1
= Proper distance, and since 1.8 𝐻 0 Δ𝑡≪1,
2𝑟 𝑝 = 𝑐 𝐻 0 [ 𝐻 0 𝛥𝑡 (1.8 𝐻 0 ) 2 𝛥 𝑡 2 …. −1]
∴ 𝑟 𝑝 = 1 2 𝑐 0 Δ𝑡+ 1.8 𝐻 𝑐 0 Δ 𝑡 2 = 𝑟 𝑎 (0.9 𝐻 0 𝑐 0 )Δ 𝑡 2 , or
𝑟 𝑎 = 𝑟 𝑝 (−𝟔.𝟏𝟐𝟗× 𝟏𝟎 −𝟏𝟎 m 𝒔 −𝟐 )Δ 𝑡 2 .
Anomalous value =−7.69±1.17× 10 −10 m 𝑠 −2 ( taking 𝐻 0 =0.716× 10 −10 yr −1 )
𝐻 0 =0.716× 10 −10 yr −1

10. Secular Increase in moon eccentricity
1− 𝑒 2 = 𝑟 𝑣 2 𝐺 𝑚 𝑒 + 𝑚 𝑚 𝑚 𝑒 2
− 𝑒 𝑒 1− 𝑒 2 = 𝑟 𝑟 + 2 𝑣 𝑣 − 𝐺 𝐺 , and since 𝑒≪1,
𝑒 𝑒 = 𝐺 𝐺 − 𝑟 𝑟 − 2 𝑣 𝑣 .
∵𝑟=𝑐𝑡 ∴ 𝑟 =𝑣= 𝑐 𝑡+𝑐 𝑡 , or 𝑟 𝑟 = 𝑐 𝑐 + 1 𝑡 .
𝑡=𝑝 𝐻 0 −1 , 1−𝑝≃0 ∵ model approx. (p is not pressure.)
𝑣 = 𝑟 = 𝑐 𝑡+ 𝑐 + 𝑐 =2 𝑐 , ∵ 𝑐 =𝑐𝑜𝑛𝑠𝑡. 𝑣 𝑣 = 2 𝑐 𝑐 𝑡+𝑐 = 2 𝑐 𝑐 𝑐 𝑐 𝑡+1 ,
∴ 𝑒 𝑒 𝐻 0 =5.4− 𝑝 − 𝑝+1 = (𝑝−1)
𝒆 =𝟑𝟕× 𝟏𝟎 −𝟏𝟐 𝐲 𝐫 −𝟏 for 𝒑=𝟏 ; 𝑒 =5× 10 −12 yr −1 for 𝑝=
(Original measured estimate 16±5× 10 −12 yr −1 , within a factor of 2)
By adjusting p by less than one percent we can get significant change in 𝑒 . 𝑒 =5±2× 10 −12 yr −1 for 𝑝= ± Originally the anomaly was 16±5× 10 −12 yr −1 but was subsequently reduced by ”better tidal modeling” to 5.

11. Secular AU increase
𝑎= 𝑝 𝑠 1− 𝑒 2 major axis, and 𝑏= 𝑝 𝑠 1− 𝑒 2 minor axis .
𝑝 𝑠 =𝑎=𝑏 when 𝑒=0
𝐴𝑈= 𝑎+𝑏 2 = 𝑝 𝑠 2 ( 1 1− 𝑒 − 𝑒 2 ) = 𝑝 𝑠 𝑒 𝑒 4 +𝑂 𝑒 𝑒 𝑒 4 +𝑂 𝑒 6 ≃ 𝑝 𝑠
𝑒 ′ =𝑒+Δ𝑒 in time Δ𝑡; ∴ 𝑒 ′2 = 𝑒 2 +2 𝑒 2 Δ𝑒
Δ𝐴𝑈= 𝑝 𝑠 𝑒 2 Δ𝑒, or 𝑑𝐴𝑈 𝑑𝑡 = 3 2 𝑒 2 𝑒 ×𝐴𝑈
𝑑𝐴𝑈/𝑑𝑡 =0.77 m c y −1 (within a factor of 2 of the measured value 1.5 m c y −1 )
Recall 𝑒 depends on 𝑝. 𝑝=1.010 yields measured value.
With improved Einstein field equations we should be able to eliminate the need for using p different from 1.

12. Planck’s constant ħ
𝛼=(1/4𝜋 𝜖 0 ) e 2 /ħ𝑐 is the fine structure constant
𝑐 2 =1/( 𝜖 0 𝜇 0 ),
𝛼=( 𝜇 0 /4𝜋) 𝑒 2 𝑐/ħ.
𝛼 𝛼 =2 e e − ħ ħ + 𝑐 𝑐 .
∵ 𝛼 𝛼 ≃0 and e e ≃0 as measured / estimated
∴ ħ /ħ= 𝑐 /𝑐 =1.8𝐻
(The current value of Hubble parameter 𝐻:
𝐻 0 ≃0.716× 10 −10 y r −1 )

13. Boltzmann constant 𝑘 𝐵
𝑛 𝛾 =𝛽 𝑇 3 number density of photons in blackbody radiation.
𝛽= 𝜋 2 ×( 𝑘 𝐵 3 ℏ 3 𝑐 3 )
𝛽 𝛽 = 3 𝑘 𝐵 𝑘 𝐵 − 3 ℏ ℏ − 3 𝑐 𝑐
- Assuming number density depends only on temperature 𝑇 𝛽 =0 .
∴ 𝑘 𝐵 𝑘 𝐵 = ℏ ℏ + 𝑐 𝑐 =3.6𝐻

14. 𝑐 𝑐 =1.8𝐻; 𝐺 𝐺 =5.4𝐻; 𝛬 𝛬 =−1.2𝐻; 𝐻 𝐻 =−0.6𝐻;
CONCLUSIONS
Physical constants do vary, and vary as follows:
𝑐 𝑐 =1.8𝐻; 𝐺 𝐺 =5.4𝐻; 𝛬 𝛬 =−1.2𝐻; 𝐻 𝐻 =−0.6𝐻;
ħ /ħ=1.8𝐻; 𝑘 𝐵 k B =3.6𝐻
Variation of ALL of the above constants in an expression must be included to get meaningful results.
Einstein field equations used are only approximate and thus the model quasi-phenomenological.
Correct Einstein field equations from scalar-tensor action*, that explicitly treats c, G, and Λ as variables, need to be developed to eliminate the use of factor 𝑝 in Hubble time and improve VcGΛ redshift data fit further.
Collaboration sought – financial support available for graduate student/post-doc (contact
* João Magueijo, Phys. Rev. D 62, (2000)
As we have seen for lunar laser ranging data analysis that if we ignore the variation of the speed of light, we get very low limit on the variation of G.

15. THANK YOU!
As we have seen for lunar laser ranging data analysis that if we ignore the variation of the speed of light, we get very low limit on the variation of G.