The laws of physics are covariant with respect to the Lorentz transformations. All observers use the same value c, including when the relative speeds become large [2].

With | V x | ≤ c and V x > 0 when x is moving away from the observer.

With | V y | ≤ c and V y > 0 when y is moving away from the observer.

The space direction of V x being the opposite of V y .

V = relative speed between x and y [m∙s⁻¹]

V x = relative speed between x and observer [m∙s⁻¹]

V y = relative speed between y and observer [m∙s⁻¹]

c = speed of light in vacuum, 0.3 × 10⁹ [m∙s⁻¹]

With respect to the temporal dimension at cosmological large scales, Lorentz transformations are also necessary when the laws of physics are extended with a second spacetime constant (b). All observers, at any time, use the same value b when measuring the time span during which the light emitted could have traveled freely from the cosmological horizon to the observer [3].

With | T x | ≤ b and T x > 0 when x is in the future of the observer.

With | T y | ≤ b and T y > 0 when y is in the past of the observer.

Note that the time-direction of T x being opposite of T y .

T = relative time span between x and y [billion orbital revolutions]

T x = relative time span between x and observer [billion orbital revolutions]

T y = relative time span between y and observer [billion orbital revolutions]

b = time-invariant maximum orbit count, 13.8 [billion orbital revolutions]

The temporal dimension that operates at cosmological large scales is defined by means of having a time-invariant maximum orbit (e.g. Earth’s orbit around the Sun) count, bounding the age of the universe when determining the time span during which the light emitted could have traveled freely from the cosmological horizon to the observer. The literature shows that the duration of the Earth’s anomalistic year increases on average by one second every 400 calendar years [7]-[10]; see Section 3.1.

This research shows that the value of the Hubble constant (H₀), interpreted here as the relative rate of the expansion of the universe, correlates with Equation (2).

Note that Einstein’s cosmological constant has a very small positive value, curving the universe just slightly; when compared to the 13.8-billion-year-old age of the universe, this leads to minute observed curvatures. At the current observed relative rate of expansion of the universe, it takes approximately one billion years for an unbound large-scale structure to grow by approximately 7% [5][15].

With respect to the temporal dimension at cosmological large scales, the relative time span between one billion years in the future and the observed cosmic microwave background radiation (CMBR) can be calculated by means of Equation (2).

T = relative time span between x and y [billion orbital revolutions]

T x = one billion years in the future of the observer, 1 [billion orbital revolutions]

T y = CMBR is in the past of the observer, 13.8 [billion orbital revolutions]

b = time-invariant maximum orbit count, 13.8 [billion orbital revolutions]

Thus, the relative time span between one billion orbital revolutions (Earth’s orbit around the Sun) in the future and the CMBR remains at 13.8 billion orbital revolutions. The aforementioned calculated correction factor of 1.072 aligns well with the observed relative rate of expansion of the universe in everyday experience; it takes one billion years for an unbound large-scale structure to grow by approximately 7% [5][15]. The dark energy is nothing more than the calculated correction factor (T_x × T_y /b²) in Equation (2).

With respect to the temporal dimension at cosmological large scales, distant galaxies do not disappear in the future beyond the cosmological horizon. For example, the relative time span between one billion years in the future and the observed galaxy HD1, which is approximately 13.47 billion years old [16], can be calculated by means of Equation (2).

T = relative time span between x and y [billion orbital revolutions]

T x = one billion years in the future of the observer, 1 [billion orbital revolutions]

T y = galaxy HD1 is in the past of the observer, 13.47 [billion orbital revolutions]

b = time-invariant maximum orbit count, 13.8 [billion orbital revolutions]

Utilizing new theoretical frameworks, this study examines how they correlate with Type Ia supernovae and distant quasar measurements to map the expansion of the universe and investigate dark energy. Type Ia supernovae originate from galactic white dwarf explosions with known physics. The uniform peak luminosity can be used as a calibration rod, thus determining the expansion rate of the universe by using the inverse-square law of light to accurately calculate extragalactic distances, thus building the cosmic distance ladder [17]. By combining these distance measurements with galactic redshifts, one can determine the expansion rate of the universe, a value known as the Hubble constant (H₀). Type Ia Supernovae are thus used as cosmic standard candles [4] to measure the distance and redshift of exploding stars. Their luminosity-redshift measurements were the foundational data set used to discover the accelerating expansion of the universe in 1998 [5][6], revealing the presence of dark energy. Note that distant quasars [18] are another set of cosmic beacons, energized by supermassive black holes in the early universe, which also reveal the presence of dark energy.

This paper must explain how new theoretical concepts of dark energy relate to supernovae and quasar measurements, which trace how the scale of the universe and the distances between galaxies change over cosmic time. Three primary datasets are analyzed: low- and high-redshift supernovae, alongside high-redshift quasars.

Using low-redshift Type Ia supernovae to map the local cosmic distance ladder typically limits accurate measurements to approximately 1.5 billion light-years. For the Hubble Constant (H₀), calibrating these low-redshift supernovae is helpful to minimize cosmic bias [19]. Note, however, that these limited light-travel times are insufficient for evaluating this paper’s new theoretical concepts related to dark energy; the hypothesized additional temporal dimension (see Section 1) becomes apparent only at larger scales.

With respect to high-redshift datasets, the Type Ia supernova SN UDS10Wil (nicknamed SN Wilson), whose light took approximately 10.4 billion years to reach Earth, is a heavily studied event [20]. Although high-redshift Type Ia supernovae are not used as the primary anchor of the cosmic distance ladder, they act as distant indicators. Recent studies by researchers at Charles University [21] and Yonsei University [22] claim that observations of cosmic time dilation in distant Type Ia supernova light curves imply that the passage of cosmic time varies throughout the evolution of the universe. Consequently, the rate of proper time is not constant, but rather is time-dependent. More studies are required to confirm or falsify these controversial findings. If confirmed, then this would align with the methodological framework of this paper.

Similarly, a recent study published in the Monthly Notices of the Royal Astronomical Society [23] detected cosmological time dilation in the variability timescales of high-redshift quasars. If validated, i.e., by establishing quasars as reliable cosmic clocks, then these findings align with the theoretical concepts of this paper and highlight the value of further research.

Studies of Type Ia supernovae and distant quasars are important observational pillars of modern cosmology, mapping the expansion history of the universe and probing the nature of dark energy. To evaluate the validity of extending the postulates of relativity with a second invariant parameter (as described in Section 1), additional observations are required. The validation of double relativity, which has two universal spacetime constants, aims to provide a theoretical explanation for dark energy [24].

The anomalistic year is the time taken for the Earth to complete one revolution through its elliptical orbit with respect to the extreme points of that elliptical orbit, i.e., to complete one full revolution around the Sun with respect to its own elliptical orbit’s orientation instead of against the background of fixed stars. With respect to these extreme points, the perihelion is where the Earth is closest to the Sun, and the aphelion is where the Earth is farthest from the Sun. The anomalistic year is usually defined as the time between the perihelion passages.

The duration of the anomalistic year of epoch J2011.0 has been determined to be approximately 3.1558433 × 10⁷ seconds [25]. Note that the orbital period of Earth for an anomalistic year is slightly larger than a Julian year Jy of 365.25 days, approximately 1.0000264 Jy.

The literature shows that over the period 4000 BC to AD 6000, the duration of the Earth’s anomalistic year increases on average by one second every 400 calendar years [7]-[10]¹. This constant change in the duration of the anomalistic year corresponds well to having a time-invariant count of Earth’s orbital revolutions when measuring the time span during which the light emitted could have traveled freely from a specific cosmological large distance (close to the CMBR) to the observer. This time-invariant orbit count can be calculated using Equation (3).

b l o c a l = local time-invariant orbit count, 12.623 [billion orbital revolutions]

J c a l = calendar year numbering [−]

a a n o m = duration of Earth’s anomalistic year for epoch J2011.0, 3.1558433 × 10⁷ [s]

Δ a a n o m = average delta of Earth’s orbital period compared to epoch J2000.0 [s]

Compared to the present duration of Earth’s anomalistic year, the value of the orbital period (a) changes on average by one second every 400 orbital revolutions of Earth around the Sun [7]-[10]. Synchronizing Δ a a n o m with the calibration of a so-called anomalistic year clock leads to a clock that gradually ticks just slightly slower over time in everyday experience, but is rather time-invariant when counting the clock’s cycles at cosmological large scales, as calibrated in the present epoch of the observer. At any time between 4000 BC and AD 6000, the time span during which the light emitted could have traveled freely between a specific cosmological large distance (a ×b×c = number of seconds of Earth’s anomalistic year in the present epoch of the observer× 12.623 × 10⁹ × 0.3 × 10⁹) and Earth is aligned with having a time-invariant orbit count. Using this anomalistic year clock, continuously calibrated with Earth’s orbital period in the present epoch of the observer, leads to this time-invariant count of 12.623 billion orbital revolutions. Thus, Equation (3) shows that the number of revolutions of the hand on this clock dial plate is rather time-invariant when measuring at cosmological large time scales, i.e., close to the age of the universe of 13.8 billion orbital revolutions [12].

The time-invariant value of b l o c a l over a period of 1 × 10⁴ years is calculated by means of Equation (3) to be 12.623 billion orbital revolutions. This research extrapolates this beyond 1 × 10⁴ years, making the value of b l o c a l also time-invariant over a much longer period. All observers on Earth measure changes in the duration of the anomalistic year to correspond to having a time-invariant orbit count b l o c a l [3].

This section introduces the Hubble tension as a correction factor between Earth’s time-invariant count b l o c a l of 12.623 billion orbital revolutions and the age of the universe, measured to be approximately 13.8 billion Earth’s orbital revolutions [12].

The age of the universe is close to the value of Hubble time (t_H), the inverse of the Hubble constant (H₀). The value of Hubble time is the age the universe would have been if the expansion of space had been linear. However, the expansion is not linear, as observations show that Einstein’s cosmological constant has a minute positive value, curving the universe just slightly [5][15]. This curvature is so small that the effect is detectable only with large time spans, e.g., 10 billion years. The age of the universe is approximately 13.8 billion years [12], which is eminently leading to the very small observed curvature.

Multiple methods have been used to determine the Hubble time (t_H). Estimates based on late universe data (cosmic distance ladder measurements) cluster at around (13.3 ± 0.3) billion years [26][27], and estimates based on early universe data (cosmic microwave background radiation) cluster at around (14.5 ± 0.2) billion years [26][27]. The estimated uncertainties have shrunk, but the range of measured values has not, to the point that the disagreement between late universe and early universe data is statistically significant. This discrepancy is called the Hubble tension [27].

This paper introduces a theoretical framework based on multidimensional time, with one temporal dimension in everyday experience and the effects of the other temporal dimension becoming apparent only at cosmological large scales [1]. This theoretical framework naturally accommodates the Hubble tension through temporal evolution effects [11][12]. The mismatch between the more locally measured expansion rate of the universe and the one inferred from cosmic microwave background radiation is hereby defined as a ratio, introducing a factor that accounts for this statistically significant difference [3].

r h t = Hubble tension ratio, 1.09 [−]

t H , c m b r = Hubble time t_H based on early universe data, 14.5 ± 0.2 [billion years]

t H , l o c a l = Hubble time t_H based on late universe data, 13.3 ± 0.3 [billion years]

To account for the observed difference between the early- and late universe derived values of the Hubble time, r h t is introduced as a correction factor to calculate the value of the cosmic microwave background radiation (CMBR) time-invariant orbit count ( b c m b r ) out of the locally derived time-invariant orbit count ( b l o c a l ) in Equation (3).

b c m b r = CMBR time-invariant orbit count, 13.8 [billion orbital revolutions]

b l o c a l = local time-invariant orbit count, 12.623 [billion orbital revolutions]

r h t = Hubble tension ratio, 1.09 [−]

The value of r h t is estimated to be around (1.09 ± 0.04), and thus the value of b c m b r is estimated to be around (13.8 ± 0.5). This research defines the value b c m b r as a time-invariant maximum orbit count; see also Section 2.2.

Based on the best fit to the measured data from the Planck spacecraft, the age of the universe in the present epoch is measured to be approximately (13.787 ± 0.020) billion years [12]. This paper shows that the value of the observed age of the universe also derives from Equation (5), thus seemingly aligning the local time-invariant orbit count b l o c a l , corrected with the Hubble tension ratio r h t , with the measured age of the universe (CMBR).

Additional studies are required to understand the implications of the alignment between changes in Earth’s gravitationally induced spin around the Sun and the corresponding size of the observable universe as measured from Earth, i.e., further defining this manifestation of gravity itself.

The cosmic microwave background radiation has been observed to be smooth [13], even when comparing areas of the universe that should not have been in contact.

This paper introduces a theoretical framework based on multidimensional time, with one temporal dimension in everyday experience and the effects of the other temporal dimension becoming apparent only at cosmological large scales [1]. With respect to the temporal dimension at cosmological large scales, having two independent physical spacetime constants (c and b), neither of which can be derived from the other, explains that the whole universe is in contact [3][24]. Namely, the extension of postulate 2 in Section 1 states that, with respect to the temporal dimension at cosmological large scales, the relative time span between two events cannot be further apart than the time-invariant maximum orbit count b.

Note that introducing the second constant b also means that the laws of physics do not break down at the maximum time span in the past, the so-called Big Bang [28]. The value of b is time-invariant, a new theoretical concept embedded in physics [3].

One would expect that at the largest scales of the observable universe symmetry should prevail and any one direction should be similar. However, experimental observations of fluctuations in cosmic microwave background radiation show that the random motion (heat) in these large-scale modes is not symmetric; there is a preferred direction [14]. The Wilkinson Microwave Anisotropy Probe (WMAP) images show evidence of a non-symmetric boundary condition. Namely, the red tilt in the amplitudes of energy density fluctuations in the cosmic microwave background radiation temperature is a deviation from perfect scale-invariance, having slightly smaller amplitude as the wavelength decreases.

Having two independent physical spacetime constants (c and b) explains that the observable universe is finite (closed) with respect to the temporal dimension at cosmological large scales, with all inertial frames of reference tilting towards the direction of the observer [3][24].

Note that, as the theory of relativity is inherently background independent, one could also describe the observed acceleration of the expansion of the universe as being an ever faster shrinking of space itself compared to the static boundary of the universe [3][24]. The theory of relativity allows for this without jeopardizing any physical law. Thus, the above-mentioned inward tilting, always in the direction towards the observer, may indeed explain this non-symmetric aspect.