Resolving the Hubble Tension Via a Scaling Factor to Account for the Evolving Gravitational Potential of the Universe

1. Introduction

The Hubble Tension: The value of the Hubble constant determined via CMB processes during the decoupling epoch $z\approx 1100$ is different from the value seen at the current epoch $z=0$. The ratio of the Planck predicted $H_0=67.4\pm 0.5$ km/s/Mpc to the locally observed $H_0=72\pm 1$ km/s/Mpc is 0.93 [1,2]. Initially this differences were considered a tension since it was a few $\sigma$ discrepancy. However, with the improvement of the experimental uncertainties now it has become a problem at more than 5$\sigma$ discrepancy [3,4]. Even more, it has be identified as 6$\sigma$ difference but is also resolved within the Scale Invariant Vacuum (SIV) paradigm by modification to the Hubble function $H\left(z\right)$ [5]. While the temperature ($T^{*}=3000K$) and the value of the Hubble expansion at decoupling are almost the same for SIV and LCDM ($H^{*}=1.38\times {10}^6$ see Figure 4 in [5]), the value of z at decoupling is different for the two models ($z^{*}=1099.51$ and $z_{\mathrm{SIV}}^{*}=1438.08$) as well as the value of ${\Omega }_m$ is now 0.2 for SIV instead of 0.31 as for LCDM.

FFAT Scaling Symmetry Idea: The idea of rescaling suggested a possible resolution by looking into its impact on the Free-fall, Amplitude, and Thomson (FFAT) scaling [6] (see (2) for the currently adopted version). Subsequently, a study of various surrogate dark processes and matter components have been studied in details by [7,8]. The problem with such constant rescaling adjustment is that if it was present during the decoupling epoch it would manifest itself in the Thomson scattering today as well. It seems that a way out would be if the scaling $\lambda$ is time dependent so that now ${\lambda }_0=1$ but during the decoupling epoch it could have been different so it could affect the relevant processes and place its fingerprint as a different $H_0$ accordingly.

The Scale Invariant Vacuum (SIV) Idea: Within the SIV paradigm [9,10], there is an overall conformal factor $\lambda \left(t\right)$ that now is ${\lambda }_0=1$; thus, there will be no effect on current measurements but in the past it could induce modifications. If one is to consider only time rescaling then one may have $H=\dot{a}/a\to \lambda H$ where $\lambda$ is a time reparametrization factor. Since this is happening during the matter dominated epoch then one can use the corresponding functional form for $\lambda \left(t\right)$ within SIV given in dimensionless time $t\in [0,1]$ with $\lambda \left(t\right)=1/t$ and $a\left(t\right)=(t^3-{\Omega }_m)/(1-{\Omega }_m)$ to determine ${\Omega }_m$. The Big-Bang within SIV is happening near $t_{BB}={\Omega }_m^{1/3}$. Which for ${\Omega }_m=$ 5% and 30% gives 0.37 and 0.66 for $t_{BB}$. By setting $t_{dc}=1/{\lambda }_{dc}=H_{0Plank}/H_{0Local}=0.93$, which is way too late than what is usually expected, and by using $a_{dc}=1/(z+1)=9\times {10}^{-4}$, one gets ${\Omega }_m=80\%$ that seems to be at odds with the conventional matter content. However, when SIV is carefully applied, as shown in [5] one can first determine the ${\Omega }_m$ for the SIV model, and then by looking at the $H\left(z\right)$ one can show that one can arrive at $H_0$ within SIV that agrees with the local observations. Here the SIV framework is used but the above argument will be refined by implementing $\lambda \left(a\right)$ and subsequent modifications to $H\left(a\right)$ that resolves the Hubble tension and shows that the CMB spectra can be properly recovered. The current framework also provides a physical interpretation of the scaling function $\lambda$ within the RISS paradigm.

Reparametrization Invariant Scaling Symmetry (RISS) resolution: The RISS idea is very similar to the SIV framework mathematically [10,11,12], but differs in the freedom of choosing a time parametrization for the description of a physical process. Thus, $\lambda$ has to be finalized given the process under consideration and the physical situation. For the discussion here, Cosmology and the CMB, $\lambda$ will be related to the lapse function $N=\sqrt{1+2\Psi }$ see Section 4. The physical situation is an evolving gravitational potential of the universe that at recombination is bigger than at the current epoch. Evolving gravitational potential is a key component in modern CMB computer codes (see Figure 1). However, one can understand such decrease towards its current value simply by looking at the gravitational potential within a homogeneous spherical mass/energy distribution. If R is the radius of the causally connected part of the Universe with total mass/energy M within R then one has $\Psi \approx GM/R$ – as the Universe expands R grows, but G and M stay the same within the usual view point. Thus, the gravitational potential decreases as the Universe expands! This effectively modifies the unit of time. The current resolution of the Hubble tension is related to such time unit adjustment between $H_{0Planck}$ and $H_{0Local}$ as discussed in Section 4 and further validated by subsequent CMB calculations as demonstrated in Figure 3.

2. Key Expressions in Cosmology

Consider the Hubble function related to the expansion factor a, and the conformal time $\eta$ given by:

$$H={\displaystyle \frac{1}{a}}{\displaystyle \frac{da}{dt}}={\displaystyle \frac{1}{a^2}}{\displaystyle \frac{da}{d\eta }},d\eta ={\displaystyle \frac{1}{a}}dt,{\displaystyle \frac{d}{d\eta }}=a^{2}H{\displaystyle \frac{d}{da}}=a{\displaystyle \frac{d}{dt}}.$$

(1)

In the above expressions, t is considered to be the cosmic time, as such the proper time for the evolution of the Universe, when utilizing the FLRW cosmology. However, later t will be a time coordinate used by an observer, while the proper time for the evolution of the Universe will be denoted by $\tau$ with $d\tau =N\left(t\right)dt$.

Dimensional considerations show that time-dependent re-scaling of units by $\lambda \left(t\right)$ satisfies re-parametrization invariance! Such rescaling can be generated by the change of the time unit, which induces change in the distance unit to maintain constant speed of light.

$$(G\rho ,Gp,H,\dot{\kappa },k)\to ({\lambda }^{2}G\rho ,{\lambda }^{2}Gp,\lambda H,\lambda \dot{\kappa },\lambda k).$$

(2)

The rescaling of the Hubble function is related to the FLRW equations that define the background cosmology:

$$H^2={\displaystyle \frac{8\pi }{3}}G\sum {\rho }_i,3H^2+2\dot{H}=-8\pi G\sum p_i.$$

(3)

By using $\dot{H}=\ddot{a}/a-H^2$, these equations can be written in the standard form containing $\ddot{a}/a$ term. If time is introduced by integrating the relationship $\dot{a}=a\sqrt{8\pi G\rho /3}$ then $\lambda$ scaling is naturally induced upon $G\rho \to {\lambda }^{2}G\rho$ and $\lambda$ can be used to induce changes in the units or it can be related to introducing a lapse function $N\left(t\right)=1/\lambda$ then $H={\displaystyle \frac{1}{Na}}{\displaystyle \frac{da}{dt}}$ for $d{s}^2=a^2{\overrightarrow{dx}}^2-N^{2}d{t}^2.$

To retain the speed of light unaltered then one has to either rescale the scale-factor $a\to Na$ or the spacial units $dr\to Ndr$ to obtain a conformally equivalent metric $d{s}^2\to N^2(a^2{\overrightarrow{dx}}^2-d{t}^2)$. Thus, by re-scaling the unit of time ($dt\to Ndt$) and keeping the speed of light fixed then one also should rescale the distance unit resulting in the rescaling of k, which has inverses distance units – usually Mpc⁻¹ for cosmology considerations. Alternatively, one can consider a Weyl transformation only for N and a$(N,a)\to (N,a){\lambda }^{-1}$ for a fixed coordinate system with fixed units. The first of these symmetries, related to units change, is obvious in the CMB equations [6,7,8] except for the term that is related to $\dot{H}$. While the second transformation does not affect the conformal time $\eta =Ndt/a$ and many of the related terms. However, a special attention and discussion is needed for $aH$ and $\dot{\kappa }$.

Note that the time reparametrization may be broken in the second equation (3) related to the pressures $p_i$ since $\dot{H}\to {\lambda }^2\dot{H}+{\lambda }^{2}H\left({\displaystyle \frac{\dot{\lambda }}{\lambda }}\right)={\lambda }^2\dot{H}+{\lambda }^2{H}^2\left(a{\lambda }^{\prime }/\lambda \right)$. This suggests considering $2H\left({\displaystyle \frac{\dot{\lambda }}{\lambda }}\right)$ to be related to a dark substance, which can be identified as $w=-2/3$ pressure component $p_w$. As such it can be related to early dark energy. Such terms appear upon generalization of the FLRW equations to reparametrization invariant form as part of the scale-invariant cosmology equations [10,11,12].

Let’s now consider the Weyl transformation $a\to \lambda a$ along with $N\left(t\right)\to \lambda N$; one can think of it as $dt\to \lambda dt$, without change in the spacial units, as proxy to changing the lapse function N. Note that this $\lambda$ is actually $1/\lambda$ when related to the units rescaling discussed earlier (2). Such Weyl transformation does not affect the conformal time $\eta$ and k and keeps the equations almost invariant as well, but one has to look closely at the $aH$ and $\dot{\kappa }$ terms. These terms are invariant upon constant rescaling, since $aH\left(\tau \right)=H\left(\eta \right)$, same for $a^{2}G\rho$ and $a^{2}Gp$ due to the [T⁻²] units in G that offset the change $a\to \lambda a$, but for time-dependent $\lambda \left(t\right)$ one must revisit the FLRW equations. For time-dependent change, one has for $aH$:

$$\begin{array}{ccc}\hfill aH=\dot{a}/N\to \dot{a}& =& d\left(\lambda a\right)/dt/\left(N\lambda \right)=\dot{a}/N+a\dot{\lambda }/\left(N\lambda \right)\hfill \\ & =& aH+a^{2}H{\lambda }^{\prime }/\lambda =aH(1+a{\lambda }^{\prime }/\lambda ),\hfill \end{array}$$

thus, no change for the usual constant rescaling. However, if one defines $H=\dot{a}/\left(Na\right)$ then $aH=da/d\tau$ should be in proper time parametrization where $\lambda$ must be constant. So, one should consider $aH$ to be $da/d\tau$ where $\tau$ is the proper time parametrization!

Let’s look at $\dot{\kappa }$ and other observables. Note that the Thomson opacity in conformal time is given by $\dot{\kappa }=a{n}_e{\sigma }_T$ but it is $n_e{\sigma }_T$ in proper time, otherwise it is $n_e{\sigma }_{T}N\left(t\right)$. Since the photon visibility function is $g\left(\eta \right)=-dexp(-\kappa (\eta \left)\right)/d\eta$ with optical depth:

$$\kappa ={\int }_a^{1}da{\displaystyle \frac{n_e{\sigma }_T}{aH}}={\int }_{\tau }^{{\tau }_0}{n}_e{\sigma }_{T}d\tau ={\int }_{\eta }^{{\eta }_0}{n}_e{\sigma }_{T}ad\eta ,$$

therefore, $\dot{\kappa }=aH{\kappa }^{\prime }=n_e{\sigma }_T$ in proper time but $a{n}_e{\sigma }_T$ in conformal time, as mentioned already. However, it should be in proper time $\tau$ for the description of the relevant processes. Therefore, here the view is to consider $\dot{\kappa }=d\kappa /d\tau =n_e{\sigma }_T$. This shows that $\kappa \left(\tau \right)$ is invariant even for time-dependent rescaling.

The photon source function $S(k,\eta )$ [7,8,13], is in conformal time, thus invariant upon constant rescaling when integrated with $d\eta$. The co-moving distance $\chi \left(a\right)={\int }_a^1{\displaystyle \frac{da}{a^{2}H}}={\int }_a^1{\displaystyle \frac{Ndt}{a}}={\eta }_0-\eta$ is related to the conformal time that is used in the photon transfer function calculations.

$${\Theta }_l(k,{\eta }_0)={\int }_0^{{\eta }_0}d\eta S(k,\eta )j_l\left(k(\eta -{\eta }_0)\right).$$

Assuming negligible effects due to time-dependent rescaling or Weyl transformation, one can move onto the CMB spectrum:

$$C_l^{TT}=4\pi {\int }_0^{\infty }{\displaystyle \frac{dk}{k}}P\left(k\right){\left|{\Theta }_l\left(k\right)\right|}^2,$$

where the primordial power spectrum is given by $P\left(k\right)=A_s{(k/k_{*})}^{n_s-1}$. Induced scaling of $A_s$ has been argued based on the scaling of k by [6,7,8] as well.

3. Symmetry Extended Cosmology

The idea of Scale Invariant Cosmology has been discussed by Eddington, Dirac, Canuto et al, and further pushed by Buvie and Maeder [14,15,16,17] within the context of cosmology and astrophysics. The theory is often and easily mistaken for some type of Jordan-Brain-Dike conformal theory due to the absence of clear delimitation of what is the physical meaning of the conformal factor $\lambda$, which within SIV is taken to be dependent only on time due to homogeneity of space [9,10]. The corresponding FLRW equations for flat ${\Omega }_k=0$ (zero ${\rho }_k\sim 1/a^2$ with $w_k=-1/3$) have the form:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{8\pi }{3}}G\rho & =& {\left({\displaystyle \frac{\dot{a}}{a}}\right)}^2+2\left({\displaystyle \frac{\dot{a}}{a}}\right)\left({\displaystyle \frac{\dot{\lambda }}{\lambda }}\right)+{\left({\displaystyle \frac{\dot{\lambda }}{\lambda }}\right)}^2-{\displaystyle \frac{1}{3}}{\Lambda }_E{\lambda }^2,\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill -8\pi Gp& =& {\left({\displaystyle \frac{\dot{a}}{a}}\right)}^2+4\left({\displaystyle \frac{\dot{a}}{a}}\right)\left({\displaystyle \frac{\dot{\lambda }}{\lambda }}\right)+2\left({\displaystyle \frac{\ddot{a}}{a}}\right)-{\left({\displaystyle \frac{\dot{\lambda }}{\lambda }}\right)}^2-{\Lambda }_E{\lambda }^2+2\left({\displaystyle \frac{\ddot{\lambda }}{\lambda }}\right).\hfill \end{array}$$

(5)

The above equations can be derived from the standard FLRW equations either by time-dependent units rescaling and/(or only) rescaling of the expansion scale factor $a\to \lambda \left(t\right)a$. The rescaling ${\Lambda }_E\to {\Lambda }_E{\lambda }^2$ is coming from the Weyl transformations of the metric $g_{\mu \nu }\to {\lambda }^2{g}_{\mu \nu }$ since ${\Lambda }_E$ enters the Einstein equations via ${\Lambda }_E{g}_{\mu \nu }$; that is, a Weyl transformations where rescaling ${\Lambda }_E$ by ${\lambda }^2$ comes from the time rescaling along with $a\to \lambda \left(t\right)a$ to induce space rescaling. The choice of $\lambda$ within the Scale Invariant Vacuum paradigm removes the Einstein cosmological constant ${\Lambda }_E$ that restores scale invariance of the vacuum by making the last two terms in (4) and the last tree terms in () cancel each-other. That is, it is often considered $({\kappa }^2/{\lambda }^2={\Lambda }_E/3$ with $\kappa =-\dot{\lambda }/\lambda )$. While the other key relation $\dot{\kappa }=-{\kappa }^2$ is essential in turning $\ddot{\lambda }$ terms into $\dot{\lambda }$ for the cancellation of the last last tree terms in (). This way the dark energy problem is resolved within the SIV paradigm. See the Appendix for more details.

For our purpose here one needs to undo the time evolving unit of time due to the gravitational potential. That is, to construct reparametrization invariant symmetric FLRW cosmology one can apply general conformal Weyl transformation1$g_{\mu \nu }\to \lambda {\left(t\right)}^{-2}{g}_{\mu \nu }$. This is practically a re-scaling of the space expansion factor $a\to {\lambda }^{-1}a$ along with change in the lapse function $N\to {\lambda }^{-1}N$, which is effectively a time parametrization change but with no change to the unit of time nor the coordinates.

Such transformation can be undone by time-dependent change of units due to modification of the unit of second, induced by the change in the gravitational potential. Since both are conformal transformations the speed of light is unaffected. However, the freedom of reparametrization of a process (RISS), just like the freedom of choosing a coordinate system, makes the meaning of $\lambda \left(t\right)$ clear and justifies its dependence only on time.

That is, the changes are ${\Lambda }_E\to {\lambda }^{-2}{\Lambda }_E$, along with:

$$\rho \left(a\right)\to \rho (a/\lambda ),p\left(a\right)\to p(a/\lambda ),H={\displaystyle \frac{1}{a}}{\displaystyle \frac{da}{d\tau }}\to H-\left({\displaystyle \frac{\dot{\lambda }}{\lambda }}\right).$$

Doing such calculations explicitly in the RISS/SIV framework is hard and requires the development of specialized codes. However, one can circumvent this problem by utilizing the time-dependent units rescaling to remove the subsequent changes induced previously using $\lambda$. Since units wise ${\Lambda }_E$, $G\rho \left(a\right),$ and $Gp\left(a\right)$ are effectively unchanged because everywhere a is usually accompanied by dimension-full extra factor. Thus, the only change that stays after these two transformations is $H\to (H-{\displaystyle \frac{1}{\lambda }}{\displaystyle \frac{d\lambda }{d\tau }})=H(1-a{\lambda }^{\prime }/\lambda )$. This is used later to define rigorously a-dependent rescaling of H as $H_{new}$ in () using (1). Next, it can be shown that such minimal change can point towards the resolution of the Hubble tension upon the proper construction of $\lambda \left(a\right)$ that at this point is to be identified with $\lambda =N/N_0=\sqrt{(1+2\Psi )}/N_0$. Unlike SIV where a specific expression for $\lambda =t_0/t$ is derived that guarantees constant ${\Lambda }_E:=\Lambda /{\lambda }^2$ and G, which may not be appropriate for the BBN epoch as discussed in [18], RISS does not have a definite functional form for $\lambda \left(t\right)$ but allows for alternatives depending on the process to be studied. In the rest of the paper this freedom is explored to choose $\lambda \left(a\right)$ using SIV guidance since the fraction of the time for the BBN and until the decoupling is negligible compared to the age of the Universe ${\tau }_0\approx 13.8$ Gyr.

4. Resolving the Hubble Tension

The CMB spectrum can be computed in the Newtonian gauge:

$$d{s}^2=a^2\left((1+2\Phi )d{x}^2-(1+2\Psi )d{\eta }^2\right),$$

(6)

where $\Phi$ and $\Psi$ are gravitational potentials, see [19,20,21]. Clearly the conformal time $\eta$ is different from the proper time $\tau$ and the lapse $N=\sqrt{1+2\Psi }$ enters the expression for the Hubble function.

$$\begin{array}{ccc}& & H={\displaystyle \frac{1}{a}}{\displaystyle \frac{da}{d\tau }}={\displaystyle \frac{1}{aN\left(t\right)}}{\displaystyle \frac{da}{dt}}\hfill \\ & & \Rightarrow {\displaystyle \frac{N\left({\tau }_{dc}\right)}{N\left(t\right)}}\left({\displaystyle \frac{1}{a}}{\displaystyle \frac{da}{d{\tau }_{dc}}}\right)={\displaystyle \frac{N\left({\tau }_{now}\right)}{N\left(t\right)}}\left({\displaystyle \frac{1}{a}}{\displaystyle \frac{da}{d{\tau }_{now}}}\right).\hfill \end{array}$$

Here, t is a coordinate time, while the proper time now and during decoupling are given by $d{\tau }_{now}=N\left({\tau }_{now}\right)dt$ with t near $t_0$ now, and similarly $d{\tau }_{dc}=N\left({\tau }_{dc}\right)dt$ near decoupling. Therefore, by identifying the LHS with what the decoupling processes are experiencing in the early universe with $H_{0Planck}$ and the RHS with what the current observation are seeing, one has:

$$H_{0Planck}={\displaystyle \frac{N_0}{N_{dec}}}{H}_{0Local}$$

(7)

The ratio, $N_{dec}/N_0$ as a function of k is shown in Figure 1 where the potentials $\Phi$ and $\Psi$ are during decoupling epoch $z\approx 1100$ and current epoch $z\approx 0$. The potentials $\Phi$ and $\Psi$ are generated using CLASSy run for standard LCDM [22] since this is the more popular CMB code and therefore experienced researchers could easily validated this results; however, one can reproduce them using the CMBquick [23] as well. For $k<0.05$ Mpc⁻¹ one can clearly see the ratio $N_{dec}/N_0\approx 1.1$, suggesting at decoupling ${\lambda }_{dec}\approx 1.1$.

The value for $N_{dec}/N_0\approx 1.1$ agrees with the CMB deduced value and the observed local Hubble constant. Since the local measurements of the Hubble constant suggest $H_{0Local}=72.55$ km/s/Mpc [2,3], while the CMB based Planck value is $H_{0Planck}=67.36$ [1,3]. The presence of lapse function N that increases as one goes back in time suggests utilization of an SIV like conformal factor $\lambda \left(a\right)$. The choice to be considered is inspired by the way SIV time t is matched to standard time $\tau$: $(t-t_{in})/(t_0-t_{in})=(\tau -{\tau }_{in})/({\tau }_0-{\tau }_{in})$ with $t_0=1$ and $t_{BB}=t_{in}$ for the SIV time and ${\tau }_0=13.8$ Gyr and ${\tau }_{BB}=0$ for the standard time [1,10,21]. For the current study one defines $t\left(a\right)$ by using $(t-t_{in})/(1-t_{in})=a$ which gives 2:

$$\begin{array}{c}\hfill t=(1-x)a+x,\lambda \left(a\right)=1/t,\end{array}$$

(8)

$$\begin{array}{c}\hfill \kappa =-\dot{\lambda }/\lambda ,\varkappa =a{\lambda }^{\prime }/\lambda ,\xi =1+\varkappa ,\end{array}$$

(9)

here, x will be chosen to produce $\lambda \approx 1.1$ at $z_{dec}\approx 1100$ that is needed to match $N\left(t_{dec}\right)$. Other choices of $t\left(a\right)$ have been explored as well, for example SIV inspired $t\left(a\right)$ for the matter dominated epoch (see the footnote), but the results were not sensitive enough to the specific form, as long as the corresponding $\lambda$ has the desired properties. This is to be expected since the key idea is that $\lambda \approx 1.1$ at $z_{dec}\approx 1100$ is the physically important property as long as $\lambda$ and $\xi$ stay near one. The current choice of $\lambda$ provides simple meaning for x since $\xi \left(\lambda \right)=x\lambda$, with $a_0=1$ and ${\lambda }_0=1$ now, this gives ${\xi }_0=x$ which is related to the ratio of the two values of the Hubble constant.

The main approach here is to apply ${\lambda }^{-1}$ to $(N,a)$ this why one will modify $H\left(a\right)\to \lambda H\left({\lambda }^{-1}a\right)$ and then to undo much of the changes by using units rescaling with $\lambda$. The reason to undo the change by units rescaling is to make use of the symmetry and to assess the validity of the idea using readily available public codes. Doing calculations using the RISS/SIV framework is hard and requires the development of specialized codes. However, one can circumvent this problem by utilizing the time-dependent units rescaling to remove the subsequent changes induced previously by ${\lambda }^{-1}$. In doing so, units wise ${\Lambda }_E$, $G\rho \left(a\right),$ and $Gp\left(a\right)$ are effectively unchanged since everywhere a is accompanied by dimension-full extra factor. For example, for ${\rho }_m\sim 1/(a^3{R}_0^3$), where $R_0$ is the current size of the Universe; this way one has $a\to {\lambda }^{-1}a\to {\lambda }^{-1}a\lambda =a$. This is also seen from $d{s}^2=a^{2}d{\overrightarrow{x}}^2-N^{2}d{t}^2\to {(a/\lambda )}^2{\left(\lambda d\overrightarrow{x}\right)}^2-{(N/\lambda )}^2{\left(\lambda dt\right)}^2$ where the effect of the Weyl transformation $g_{\mu \nu }\to \lambda {\left(t\right)}^{-2}{g}_{\mu \nu }$ via $(N,a)\to (N,a)/\lambda$ has been indicated explicitly along with the units change $(dt,dx)\to \lambda (dt,dx)$. Since as discussed in Section 2 most of the CMB expressions are unaffected by the time-dependent units rescaling and only time dependent quantities with derivatives as in H will be affected.

Next, it can be shown that such minimal change points further towards the resolution of the Hubble tension upon the proper construction of $\lambda \left(a\right)$ and value that is to be identified with $\lambda =N/N_0=\sqrt{(1+2\Psi )}/N_0$ during decoupling. For the next demonstration, first the publicly available Mathematic code CMBquick [23] was utilized and then the results were validated and further improved by using CLASSy [22]. The changes are to allow runs with modified Hubble function $H_{new}\left(a\right)$ ():

$$\begin{array}{ccc}\hfill H& =& {\displaystyle \frac{da}{ad\tau }}\to (H-{\displaystyle \frac{1}{\lambda }}{\displaystyle \frac{d\lambda }{d\tau }})=H(1-a{\lambda }^{\prime }/\lambda )={\displaystyle \frac{da}{ad\tau }}\xi ,\hfill \end{array}$$

(10)

$$\begin{array}{ccc}& \Rightarrow & {\displaystyle \frac{da}{ad\tau }}\xi \left(a\right)=\sqrt{{\displaystyle \frac{4\pi }{3}}G\sum {\rho }_i},\xi \left(a\right)=(1-a{\lambda }^{\prime }/\lambda ),\hfill \end{array}$$

(11)

$$\begin{array}{ccc}& \Rightarrow & H_{new}\left(a\right)=\xi {\left(a\right)}^{-1}\sqrt{{\displaystyle \frac{4\pi }{3}}G\rho }=\xi {\left(a\right)}^{-1}{H}_{std}\left(a\right).\hfill \end{array}$$

(12)

The above expressions define rigorously a-dependent rescaling of H as $H_{new}$ in (), while the density dependent expressions that do not involve time derivatives are unaffected as already pointed out in Section 3. This allows standard computer codes to be easily modified and utilized. Although the Newtonian gauge was initially used to understand the problem and to justify the need for $\lambda \left(a\right)$ and its value at decoupling, but once such choice is made as in (8), the subsequent considerations are gauge independent.

When the Hubble function is modified according to the discussion leading to (8) and (). The differences between $H_{std}$ and $H_{new}$ are very hard to be demonstrated graphically since $1/\xi$ is very small deviation from 1. As a result, having $1/\xi \approx 1$ in the early universe, at radiation-matter equilibrium epoch and during decoupling implies that most of the other CMB properties such as TE/EE spectra, CMB lensing, and BAO should be reproduced as well while broader late-time expansion history will be guaranteed by the correct local $H_0$. Upon using $H_{new}$ one observes shift of the CMB spectra to the right of the original un-modified standard LCDM spectra, then by adjusting only h from $0.67$ to $0.74$ the graph moves back to the left and almost overlaps with the original CMB spectra that traditionally predicts $H_{0Planck}=67.36$ km/s/Mpc but now the $H_{new}\left(a\right)$ results in $H_{0Local}=72.55$ km/s/Mpc instead. Starting with CMBQuick by using a consecutive search along each of the seven main cosmological parameters, ${\Omega }_m,{\Omega }_b,A_s,n_s,{\tau }_{rei}$, and SIV $t_{in}$ and h demonstrated U-shape landscape for each one; this way a set of adjusted cosmological parameters can be found and now the difference from the original LCDM CMB spectra is mostly less than a few percents. Finally, these parameters were used as a starting point for Pawell search method utilizing CASSy. As it can be seen in Figure 2 the difference from the LCDM $C_{\ell }^{TT}$ is less then a present for most of $\ell >40$. The corresponding CMB TT-spectra is shown in Figure 3 . The new model ${\chi }^2$/dof=16.9 when the value for the LCDM is $14.8$, while for the EE-spectra (not shown) the value is 6.64 for both, and for the TE-spectra (not shown) they are 9.06 and 9.25. Not shown here, but the results for the EE and TE spectra agree well with observation and MCMC type runs are now in progress too.

While there is a growing body of data, LCDM is still the simplest model that describes well the CMB. By finding alternative models that can do as well as LCDM by comparing $C_{\ell }^{TT}$-spectra guarantees such models to be well positioned to compete with LCDM. The calculation presents arrives at such cosmological parameters only by comparing $C_{\ell }^{TT}$ but then the result is compared to other data sets found in the [24]. The original data sets indicated in this study on Figure 3 are: Planck 2018 [25] ACT DR6 2025 [26] SPT-3GD1 2025 [27] SPTpol 2017 [28] BICEP2/Keck 2015 [29]. Modern data processing is elaborate in how to add various data sets and how to estimate systematic and foreground errors [30]. Here I use the data to visualize how the theory compares to observation, the model parameters are derived by minimizing the deviation from the LCDM $C_{\ell }^{TT}$ that results in a few percent differences Figure 2 . In particular, the specific datasets used for visualization here are hosted at the NASA LAMBDA archive as sources for the figures presented there. That is, the TT power spectrum plot data sources at [24]. For more context on the historical measurements and data sets see [31].

5. Discussion and Conclusions

The presented resolution of the Hubble tension does not require any new physics or fields, just a minor adjustment to the time parametrization due to the difference in gravitational potential that influences the length of the second over the evolution of the universe. As one would expect, higher energy density in the early universe results in $\lambda \left(t\right)=d\tau /dt$ increasing towards the past with respect to the current time t, as seen from the correspondence to $N\left(k\right)=\sqrt{1+2\Psi \left(k\right)}$ from Figure 1.

Thus, according to (6), $a\to \lambda a$ is well justified and the value of $\lambda$ during the recombination is supported by the ration computed from the output of the CLASSy as sown in Figure 1. All of these supports the idea that the difference in gravitational potential, as seen from their ratios, is causing the Hubble tension as seen in (7).

Furthermore, unlike SIV where one cannot explain the meaning of $\lambda$, RISS clearly points to the difference between proper time and conformal time and the violation of their usual relation, according to (6) and utilizes a process as described to Section for testing and implementing the idea. That is, unlike the SIV theory prescribes a specific form $\lambda \left(t\right)=t_0/t$, the RISS paradigm is more relaxed about the choice of $\lambda$ that can pave the way towards determining the value of the cosmological constant and the phenomenon of dark matter as outlined in [10,11,12]. As such it will requires the development of specialized codes if the research is to fully comply with the SIV framework. However, here an alternative RISS approach of using $\lambda \left(a\right)$ has been utilized, based on the discussion in Section 3 and the derivation of the a-dependent rescaling of $H_{std}$ into $H\left(a\right)\to \xi {\left(a\right)}^{-1}H\left(a\right)$ ().

The present formalism has been implemented and visualized in Figure 3 and has resulted in resolution of the Hubble tension as demonstrated and explained in the previous sections. That is, as previously discussed in Section 2 epoch-dependent units rescaling symmetry has been already demonstrated. The effect of Weyl type time-reparametrization followed by its inverse epoch-dependent units rescaling affects only the time derivatives of a: that is, only H as defined by $\dot{a}/a$ (10), but not its $\rho$ dependent relation (). Thus, the use of $H_{new}$ at the background level propagates to the corresponding background dynamics, perturbation treatment and observables remain consistent since the rescaling factor $\xi {\left(a\right)}^{-1}$ remains close to one near the epoch of decoupling.

In conclusion, the main idea presented here is that the discrepancy between the CMB-inferred and locally measured values of $H_0$ can be understood without introducing new particles or fields, but rather through a reinterpretation of the rate of change of the time variable entering the cosmological description as influenced by the differences in the overall gravitational potential during the decoupling and the current epoch.